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Inverse modeling for data-driven physical simulation

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Title:
Inverse modeling for data-driven physical simulation
Creator:
Transue, Shane Michael
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Computer Science and Engineering, CU Denver
Degree Disciplines:
Computer science and information systems
Committee Chair:
Alaghband, Gita
Committee Members:
Choi, Min-Hyung
Biswas, Ashis Kumer
Yu, Kai
Walter, Zhiping

Notes

Abstract:
Inverse modeling through data-driven methods introduces a new research direction for connecting analytical abstractions with real-world observations. In contrast to analytical modeling, the integration of data-driven and analytical methods allow us to characterize and reconstruct underlying physical phenomena of complex systems using discrete observations. This enables the creation of hybrid models that establish core behavioral characteristics driven by analytical methods that can be augmented with data-driven systems. As data-driven methods continue to enable alternative forms of computing efficient solutions to problems that are hard to explicitly solve through analytical methods, we adapt this paradigm shift to create elegant models representing various physical phenomena. In this dissertation, we present novel algorithms for the real-time simulation of data-driven elastic materials, behavior-driven procedural geometry for 3D printed elastic materials, and introduce new directions in detailed respiratory analysis through thermal imaging. Specifically, we introduce methods for creating real-time data-driven Finite Element (FE) simulations for elastic materials, the automated perforation of volumetric meshes for 3D printing heterogeneous elastic materials, and introduce new quantitative metrics for vision-based respiratory analysis. We formulate, implement, and evaluate each of the proposed contributions to demonstrate the empowering nature of inverse modeling for computer aided design tools, interactive applications, and clinically deployable systems. Each of the proposed contributions defines an illustrative concept of how challenges in modeling complex real-world behaviors can be addressed through inverse modeling.

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University of Colorado Denver
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Auraria Library
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Copyright Shane M. Transue. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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Full Text
INVERSE MODELING FOR DATA-DRIVEN
PHYSICAL SIMULATION by
SHANE MICHAEL TRANSUE B.S., Metropolitan State University of Denver, 2011 M.S., University of Colorado Denver, 2014
A dissertation submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Computer Science and Information Systems Program
2019


©2019
SHANE MICHAEL TRANSUE ALL RIGHTS RESERVED
11


This dissertation for the Doctor of Philosophy degree by Shane Michael Transue has been approved for the
Computer Science and Information Systems Program
by
Gita Alaghband, Chair Min-Hyung Choi, Advisor Ashis Kumer Biswas Kai Yu
Zhiping Walter
May 18th,
m
2019


Transue, Shane M. (Ph.D., Computer Science and Information Systems)
Inverse Modeling for Data-driven Physical Simulation Dissertation directed by Associate Professor Min-Hyung Choi
ABSTRACT
Inverse modeling through data-driven methods introduces a new research direction for connecting analytical abstractions with real-world observations. In contrast to analytical modeling, the integration of data-driven and analytical methods allow us to characterize and reconstruct underlying physical phenomena of complex systems using discrete observations. This enables the creation of hybrid models that establish core behavioral characteristics driven by analytical methods that can be augmented with data-driven systems. As data-driven methods continue to enable alternative forms of computing efficient solutions to problems that are hard to explicitly solve through analytical methods, we adapt this paradigm shift to create elegant models representing various physical phenomena. In this dissertation, we present novel algorithms for the real-time simulation of data-driven elastic materials, behavior-driven procedural geometry for 3D printed elastic materials, and introduce new directions in detailed respiratory analysis through thermal imaging. Specifically, we introduce methods for creating real-time data-driven Finite Element (FE) simulations for elastic materials, the automated perforation of volumetric meshes for 3D printing heterogeneous elastic materials, and introduce new quantitative metrics for vision-based respiratory analysis.
We formulate, implement, and evaluate each of the proposed contributions to demonstrate the empowering nature of inverse modeling for computer aided design tools, interactive applications, and clinically deployable systems. Each of the proposed contributions defines an illustrative concept of how challenges in modeling complex real-world behaviors can be addressed through inverse modeling.
The form and content of this abstract are approved. I recommend its publication.
Approved: Min-Hyung Choi
rv


TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION........................................................................ 1
1.1 Research Motivation............................................................... 1
1.2 Research Contributions............................................................ 2
1.2.1 Data-driven Elastic Material Simulation........................................ 2
1.2.2 Procedural Perforation for Controlling Elasticity.............................. 3
1.2.3 Turbulent Exhale Flow Modeling and Analysis.................................... 4
II. INVERSE MODELING AND SIMULATION................................................... 5
2.1 Functional Modeling and Simulation................................................ 5
2.1.1 Forward Simulation............................................................. 6
2.1.2 Constrained Dynamics and Optimal Control....................................... 6
2.1.3 Inverse Simulation............................................................. 7
2.1.4 Data-driven Modeling through Imaging.......................................... 10
2.1.5 Physical Plausibility......................................................... 11
2.2 Hybrid Models: Analytical and Data-driven Models................................ 11
III. NEURAL ELEMENTS: DATA-DRIVEN ELASTICITY............................................ 13
3.1 Related Work: Simulation of Elastic Solids....................................... 15
3.1.1 Mass-spring Systems........................................................... 16
3.1.2 Constraint-based Solvers...................................................... 18
3.1.3 Deep Learning Approaches...................................................... 19
3.2 Method Overview.................................................................. 19
3.2.1 Contributions................................................................. 19
3.2.2 Finite Element Formulation for Neural Elements................................ 21
3.2.3 Corotational Formulation...................................................... 23
3.2.4 Data-driven FEA............................................................... 28
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3.3 Data-driven Element Behavior......................................................... 29
3.3.1 Element-based Material Response................................................... 30
3.3.2 Reference Elements, Coordinates, and Geometric Ratios............................. 34
3.3.3 Computation Separability.......................................................... 39
3.3.4 Element Network Model............................................................. 42
3.3.5 Material Network Model............................................................ 48
3.3.6 Covering the Material Property Domain............................................. 50
3.3.7 Extensions to New Element Types................................................... 51
3.4 Dynamic Neural Elements.............................................................. 51
3.4.1 Material Force Responses.......................................................... 51
3.4.2 Numerical Integradon.............................................................. 52
3.4.3 Dynamics Formulation.............................................................. 54
3.5 Elastic Material Data and Training................................................... 59
3.5.1 Synthetic Elastic Data............................................................ 59
3.5.2 Material Property Integration..................................................... 61
3.5.3 Displacements and Force Responses................................................. 63
3.5.4 Training.......................................................................... 65
3.6 Parallel Elements.................................................................... 66
3.7 Experiments and Dynamic Simulations.................................................. 67
3.7.1 Deformation: Compression.......................................................... 68
3.7.2 Deformation: Stretch.............................................................. 68
3.7.3 Deformation: Rotational Twist..................................................... 69
3.7.4 Interactive Deformations.......................................................... 70
3.7.5 Free-form Deformation and Collision............................................... 72
3.7.6 Quantitative Evaluation........................................................... 73
3.8 Natural Perturbations................................................................ 73
3.8.1 Network Numerical Variance........................................................ 75
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3.8.2 NULL Displacement Responses
75
IV. GENERATIVE DEFORMATION: 3D PRINTING BEHAVIOR.......................................... 77
4.1 Related Work....................................................................... 79
4.1.1 Internal Meshing............................................................... 79
4.2 Method Overview.................................................................... 81
4.2.1 Challenges in Printing Elastic Materials....................................... 83
4.2.2 Contributions.................................................................. 84
4.3 Geometric Perforation.............................................................. 85
4.3.1 Geometric Perforation.......................................................... 85
4.3.2 Mesh Perforation Algorithm..................................................... 88
4.3.3 Homogeneous Perforation........................................................ 89
4.3.4 Heterogeneous Perforation...................................................... 89
4.3.5 Element Constraints and Design................................................. 91
4.4 Generative Deformation............................................................ 92
4.5 3D Printing Process and Technologies.............................................. 94
4.5.1 Stereo-lithography (SLA)....................................................... 95
4.5.2 Fused Deposition Modeling (FDM)................................................ 96
4.6 Experimental Deformations......................................................... 97
4.7 Discussion........................................................................ 99
V. TURBULENT EXHALE FLOW: RESPIRATORY ANALYSIS..........................................101
5.1 Related Work......................................................................102
5.1.1 Challenges in Respiratory Analysis.............................................103
5.1.2 Surface Deformation Reconstruction.............................................104
5.2 Visualizing Exhale Flow...........................................................106
5.2.1 CO2 Visualization for Respiratory Analysis.....................................107
5.3 Modeling Exhale Flow..............................................................108
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5.3.1 Dense Exhale Modeling....................................................109
5.3.2 Dense Flow Reconstruction................................................Ill
5.3.3 Volumetric Approximation.................................................112
5.4 Experimental Setup and Application..........................................113
5.4.1 Obstructed Breathing.....................................................114
5.5 Discussion and Future Work..................................................115
5.5.1 Correlative 3D Modeling..................................................115
5.5.2 Clinical Deployment......................................................116
VI. CONCLUSION....................................................................118
REFERENCES............................................................................118
APPENDIX
A. Neural Element Memory Maps..................................................124
B. Linear Elasticity Finite Element Stiffness Matrices.........................125
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LIST OF FIGURES
FIGURE
3.0. 1 Neural Elements (NE) is a data-driven elastic material simulation method that closely ac-
curately replicates Finite Element simulations of solid elastic materials............. 14
3.0. 2 Illustration of two deformations applied to the Stanford dragon volumetric model. The de-
formations are imposed using a force-based manipulation tool integrated into an interactive application. The core simulation is built on the propose data-driven Neural Element (NE) elastic material model................................................................ 15
3.2.3 Illustration of two deformations applied to the Stanford dragon volumetric model. The de-
formations are imposed using a force-based manipulation tool integrated into an interactive application. The core simulation is built on the propose data-driven Neural Element (NE) elastic material model................................................................ 20
3.2.4 Dynamic behavior of a 2D triangle between the rest and current configurations (left). The element nodal displacements (do, di • d2) due to the deformation are defined by rotating the element to the optimal pre-deformation state with both elements coinciding at their center
of mass (right)........................................................................ 25
3.2.5 Dynamic behavior of a 3D tetrahedra between the rest and current configurations (left). The
element nodal displacements (do, d\, cfe, do) due to the deformation are defined by rotating the element to the optimal pre-deformation state with both elements coinciding at their center of mass (right)................................................................. 26
3.3.6 Illustration of the material response forces of a two-dimensional triangle element provid-
ing three observations that establish the foundation for the proposed data-driven material model. This includes multiple force responses for each displacement (left), response forces are not rotation invariant, but for rotated elements we can ensure response force components are non-zero (right)................................................................... 32
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3.3.7 Example of a reference triangle element R used for two-dimensional simulations. The cho-
sen reference geometry (left) is rotated by an arbitrary angle 6 (center) to ensure non-zero force component responses to an applied deformation for all nodes (right)........... 35
3.3.8 Example of a reference tetrahedral element It used for three-dimensional simulations. The
chosen reference geometry (left) is rotated arbitrarily (center) to ensure non-zero force component responses to an applied deformation for all nodes (right).................... 36
3.3.9 Parameterization and transformation of an arbitrary instance element I to the reference co-
ordinate system containing the reference element R. The node ordering generates matching pairs between the two elements In —>• Rn,Iu —> Ru, Iv —> Rv for all independent executions of each node between the elements........................................ 37
3.3.10 Geometric ratios of the response forces computed between the instance I and reference tetra-
hedra R. The ratio vector r is computed by I/R for all force components (right). This shows one of four possible node orderings where (n = a,w = b,u = c,v = d). Ratios are ordering dependent.................................................................... 39
3.3.11 Illustration of the reference coordinate system for the ratio and force response model based
on computing the material response based on each node independently. Since the element has 3 nodes (a, b, c), the computation is performed three times. The complete element behavior is then generated as the linear combination of each partial response......... 41
3.3.12 Illustration of the displacements imposed on a 3D reference tetrahedral element. Each dis-
placement and its associated material response is used to train an aggregate of material networks. Displacement components dx (left), dy (center), and dz (right) will be recombined into a complete element response.................................................... 43
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3.3.13 Interchangeable network designs that are provided to the core of the neural network material architecture (left). These networks are then assembled into the design required for the selected element type (right). The triangular element has six response forces which requires six network instances that are embedded in the element network. The value of di is defined as the displacement value expressed as dx or dy for 2D elements, each of which requires
its own network instance................................................................ 45
3.3.14 Illustration of the 3D tetrahedral element system that generates the partial material response
for a given displacement di where i e {x, y, z ). This generates the partial forces with respect to one displacement of the primary node in the reference element, thus there are 3 instances of this network for dx, dy, and dz.............................................. 46
3.3.15 Neural element material response network. This network represents the complete response
of the material based on the trained internal networks for each DOF and force response component due to the displacement imposed on any component of the 4 element nodes. These are the output forces that are used to drive the dynamics of the simulation............. 48
3.3.16 Illustration of the reference coordinate system for the ratio and force response model based
on computing the material response based on each node independently. Since the element has 3 nodes (a, b, c), the computation is performed three times (once for each node) for each degree of freedom. The complete element behavior is then generated as the linear combination of each partial response......................................................... 49
3.4.17 Illustration of the independent response forces for adjacent elements due to the coefficient
matrices and /i2 (left). The net forces (center) and displacements (right) of the nodes are averaged due to the differences in the responses provided from each element............. 57
3.5.18 Integration functions used to encode the material properties (E, v) and the displacement
applied to the reference element di to generate training datasets. These functions represent just two selected periodic functions that can be used to encoded these network inputs... 62
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3.5.19 Illustration of the sin(x) integration function applied to the displacement (d) as the input
to the training model (left). The material response along the provided DOF of the reference element is shown as the expected force (/) value (right). Note this is only a subset of the samples....................................................................................... 64
3.5.20 Illustration of the dGauss(x) integration function applied to the displacement (d) as the input to the training model (left). The material response along the provided DOF of the reference element is shown as the expected force (/) value (right). Note this is only a subset
of the samples.............................................................................. 64
3.5.21 Plots of the training error (loss) over the course of 5000 epochs. The training converges quite rapidly due to the lack of noise within the synthetic data and the simplicity of the waveforms. 65
3.7.22 Illustration of neural element meshes simulated in real-time. This includes interactive de-
formations (left), real-time stress analysis of elements (center), and dynamic boundary conditions for generating complex animations (right)............................................. 67
3.7.23 Illustration of neural element bar mesh being compressed through two fixed boundary con-
ditions. The left face nodes are fixed and the right face nodes are being directly moved (displaced) to reduce the distance between the two ends of the bar. This results in a compression of the material between these two boundary conditions.................................. 68
3.7.24 Illustration of neural element bar mesh being stretched. The left side face nodes are fixed
and the right face nodes are incrementally moved to increase the distance between the ends of the bar. This results in an elongation of the bar and a narrowing of the center due to the internal material displacements............................................................... 69
3.7.25 Illustration of a dynamic boundary condition that can be imposed on a mesh to create a twist deformation. The right side face nodes are rotated about the twist axis (x) and the rest of
the material behavior is generated through the neural element responses..................... 70
3.7.26 Illustration of neural element meshes simulated in real-time. This includes interactive de-
formations (left), real-time stress analysis of elements (center), and dynamic boundary conditions for generating complex animations (right)............................................. 71
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3.7.27 Illustration of a neural element model undergoing rigid-body translation (as it falls due to
applied gravity), resulting in a collision with the ground. Due to the mass and size of the ears, the model rolls after making contact with the ground............................... 72
3.7.28 Illustration of a subset of material responses generated by the neural element model for one
element during a simulation of 800 time-steps (frames). This illustrates the behavioral error associated with the neural element predicted value (pr) and the exact value (ex) provided by an FEA-based solution. For the selected 3D tetrahedral element, this is shown by the 12 component forces of the element. The waveform is sporadic due to the large deformation of the element through user interactions.......................................... 74
3.8.29 The initial randomized weights of the network contribute to residual non-zero response forces
provided for the input displacement of zero. This results in a constant non-zero shift in the response forces that contribute to a net positive or negative internal force within the element. This is corrected by removing the shift to obtain accurate force response estimations.... 76
4.0.1 Generative Deformation: The automated process of perforating volumetric simulation meshes that are 3D printed using elastic materials to obtain specific deformation behaviors.............. 78
4.1.2 Illustration comparing the difference between typical internal meshing techniques and CSG
modeling or automated perforation. Invalid geometry (center) of the hollowed region cannot be validly simulated due to geometric discontinuities. The valid generation of the geometric elements (right) must be followed for generating meshes that can be simulated through FEA methods.............................................................................. 81
4.2.3 Automated volumetric perforation pipeline. The input is a simple surface mesh and the main process consists of a four stages: (1) volumetric meshing and element-wise user constraints,
(2) FEA simulation of the solid mesh, (3) the automated perforation, and (4) the simulation of the perforated mesh. The result produced by the perforation algorithm provides a valid 3D printable surface mesh that can be printed using consumer-level 3D printers
XIII
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4.3.4 Automated volumetric perforation pipeline. The input is a simple surface mesh and the main
process consists of a four stages: (1) volumetric meshing and element-wise user constraints, (2) FEA simulation of the solid mesh, (3) the automated perforation, and (4) the simulation of the perforated mesh. The result produced by the perforation algorithm provides a valid 3D printable surface mesh....................................................... 86
4.3.5 Combinatoric representation of the micro-tetrahedra that are generated for each perforated
element. These enumerations define all possible micro-tetrahedra edge/corner elements within lookup tables.......................................................................... 87
4.3.6 Uniform inset of the Stanford bunny mesh. The inset values progress from 25% (left) to 50%
(center) to 75% (right) each specified by one value (0.25, 0.5, 0.75). At 100% elements become solid............................................................................. 89
4.3.7 Eieterogeneous geometric structure due to per-element specified inset values. This method allows for smooth gradients between elements and allows specific regions of the mesh to
be controlled through an FEA simulation result or user design......................... 90
4.3.8 Generative deformation: integration of design constraints, deformation behavior, and stress
analysis to generate heterogeneous geometric structures consisting of a single elastic material model. The design shape (top row) has specific design constraints that are optimized to generate different deformation and stress patterns (bend and stretch) that alter inset values.................................................................................... 92
4.4.9 Perforation design studio. This interactive design studio includes the ability to import, per-
forate, simulate, and export 3D printable surface meshes. The screenshots show two different meshes within the four-viewport design view..................................... 94
4.5.10 3D print setup and result for the Stanford dragon model printed on the FormLabs Form 1+
using an elastic resin. The sheer model (left) defined external support structures that allowed for a successful print (center, right)............................................ 96
4.5.11 Volumetric perforation printing process (SLA) illustrating the resin print (left), alcohol wash
(center), and our custom built acrylic 405[nm]-UV curing box (right).................. 96
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4.5.12 Final result of the perforated Stanford Bunny and Dragon models printed using an SLA-
based printer with a single flexible resin. Each region has variability in the elastic behavior (hard, soft) due to the gradient-based transitions of the perforated structure. Support materials have been carefully removed.................................................... 97
4.6.13 3D printed perforation meshes for testing the generative deformation pipeline. Each is a 2.0x2.0x4.5[cm]
single material perforated print....................................................... 98
4.6.14 Result of deforming five beams generated using the generative deformation algorithm. The
top row illustrates each beam subject to a deformation imposed by a horizontal load, inducing various deflection behaviors. The twist deformation (bottom row) is imposed by an applied torque to demonstrate nonlinear rotational deformations. Each exhibits unique deflections and localized deformations due to the changes in the heterogeneous perforations.. 99
4.7.15 Demonstration of the successful print of a complex perforated structure using an FDM 3D
printer. The TPU provides a relatively rigid result when printed into solid objects, however through the perforated geometry, the elastic material properties of the object can be drastically changed.........................................................................100
5.0.1 Resulting ( '02 density distribution images illustrating unique respiratory patterns between
individuals (top vs bottom rows). For each image sequence, one exhale period has been recorded and visualized, showing the clear separation between the nose-mouth distribution and density flow behaviors. These flow behaviors unique to each individual are based on their own physiological traits....................................................................102
5.1.2 Illustrations of the various models generated by depth imaging methods for respiratory analysis. Depth imaging (left), simple extruded depth (center), and region based methods (right), do not provide the model fidelity and accuracy that the iso-surface deformation model can provide. This model provides a methods for measuring both breathing rate, tidal volume with > 90% accuracy..............................................................................105
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5.1.3 Illustrations of the various models generated by depth imaging methods for respiratory anal-
ysis. Depth imaging (left), simple extruded depth (center), and region based methods (right), do not provide the model fidelity and accuracy that the iso-surface deformation model can provide. This model provides a methods for measuring both breathing rate and tidal volume....................................................................................106
5.3.4 Volumetric and density modeling of the CO2 exhale region defined by the view frustum of
the imaging device. The exhale flow region contains a non-linear CO2 concentration distribution function density over distance x. Each individual pixel /// y represents a continuous volume through which the exhale flows. From the value at each element ey, the final pixel pij is the projection of all element densities within v..........................110
5.3.5 Recorded turbulent exhale flow from the mouth (left), nose (center), and both nose and mouth
simultaneously (right). Through our imaging process, we obtain an accurate illustration of the CO2 density distribution and flow behavior with minimal background interference....Ill
5.3.6 Exhale flow reconstruction process. We approximate the reconstruction of the projected den-
sity volume by estimating the function density(x) using heuristic approximations. Methods derived on this design can be used to create numerous forms of different 3D exhale models......................................................................................112
5.3.7 Turbulent exhale optical flow vectors. The generated vector field illustrates the apparent
flow computed through a standard dense optical flow algorithm. The (top) row illustrates the original CO2 density images, and the (bottom) row illustrates the resulting vector norm-color-mapped flow........................................................................112
5.3.8 Exhale flow reconstruction process. We approximate the reconstruction of the projected den-
sity volume by estimating the function density(x) using heuristic approximations. Methods based on this design can be used to create numerous forms of different 3D exhale models......................................................................................113
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5.4.9 Overview of the respiratory monitoring setup that uses a CO2 camera to provide remote
exhale analysis. This includes the CO2 camera, software, and cross-section of the patients breathing area.............................................................................114
5.4.10 Real-time respiratory monitoring application designed to work with our CO2 imaging camera. This includes both a high-contrast visualization of the turbulent exhale flow with an applied heat map and the segmented exhale region that is used to generate the waveform
of the quantitative tidal volume estimation. This application provides the ability for clinicians to effectively communicate information about patient breathing behaviors in an intuitive way................................................................................114
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CHAPTER I
INTRODUCTION
The synergistic integration of analytical and data-driven methods inherently form a class of hybrid modeling techniques that enable a wide variety solutions to challenging interdisciplinary engineering problems. Formalizing the use of data-driven analytical methods, we introduce inverse modeling as a primary concept towards creating accurate models and simulations of real-world physical phenomena. This process combines domain knowledge from mechanical and aerospace engineering, graphics, and computer vision with data-driven methods from machine learning to create dynamic hybrid methods for modeling complex systems. Using inverse modeling as a core theme, we analyze how challenging problems in computational mechanics and medical imaging can be addressed through the use of hybrid models.
1.1 Research Motivation
The core idea behind inverse modeling is how we can integrate complex analytical systems with data-driven methods that allow us to capture real-world observations as problem constraints. In contrast to direct modeling, the objective is not only to generate an abstraction of the observed behavior, but to gain insight into the underlying physical phenomena and inversely reconstruct or describe this behavior. This can allow us to provide more accurate, efficient, or completely new techniques for modeling complex real-world systems. To do this, we look at existing analytical models and closely integrate data-driven components to form new hybrid modeling techniques that balance the benefits of these approaches. There is a wealth of different problems that can be primarily addressed using analytical methods or data-driven methods, but the mixture of these modeling techniques typically requires extensive domain knowledge and there are various forms of engineering problems where these two approaches are fundamentally incompatible. For example, most physical simulation methods are purely analytical with real-world experimental values providing accurate parameters for simulations, resulting in highly accurate analysis models. For data-driven methods, there are areas such as computer vision where trained models provide highly accurate results for object recognition and 3D scene modeling which can be difficult or less efficient for analytical models. The motivation of this work is to use inverse modeling to address prominent problems within
1


various domains including computer graphics, physics simulation, 3D printing, and medical imaging.
1.2 Research Contributions
The contributions of this dissertation represent solutions to three prominent problems in the held of computational mechanics, 3D printing elastic materials, and vision-based respiratory analysis. For each contribution, we outline the core problem, the general approach to solving the problem and how our solution methods introduce novel solution directions. The overview of each contribution is provided below, followed by a detailed outline of the solution direction.
• Neural Elements: Data-driven Elasticity - The neural element method introduces a new model derived from computational mechanics for creating physically simulating 3D elastic objects for realtime and interactive computer aided design tools and applications.
• Generative Deformation: 3D Printing Behavior - Generative deformation is the concept of modifying structural geometry to alter elastic or deformation behaviors of models. This method provides an automated perforation algorithm that can generate varied internal elastic structures to create design-oriented, 3D printable behaviors.
• Turbulent Exhale Flow: Respiratory Analysis - Analysis of breathing behaviors is well studied in computer vision, but techniques in thermal CO2 visualization provide the foundation for extracting new and meaningful quantitative metrics for evaluating pulmonary functions.
1.2.1 Data-driven Elastic Material Simulation
The simulation of deformable elastic materials represents one of the most fundamental and mature research topics in aerospace and mechanical engineering. The process for modeling flexible and deformable objects resides within many of the core topics in engineering, design, and more recently, computer graphics. Analytical models that define the constitutive equations that define how solid materials behave are derived from the core principles of continuum mechanics. These provide the ability to accurately evaluate how solid materials behave including displacements of model nodes and stress distributions within the material due the structural dynamics of the model. These methods provide highly accurate simulations of
2


complete structures and are used extensively throughout many computer aided design applications. From the computer graphics side, some of these core techniques have been integrated into animation and game design based on a wide variety of different methods each with varying levels of computational complexity and accuracy. Although the held of simulating elastic materials is popular, there are still limited options for real-time and interactive solid material simulations. In this work, we introduce a new hybrid formulation of elastic material dynamics that mixes analytical and data-driven methods to create interactive and real-time simulations of 2D and 3D models called Neural Elements. This is due to the core internal representation of material properties being encoded within neural networks. We provide an in-depth analysis of this approach and illustrate the accurate simulation of several different complex deformations using an interactive, real-time application for deforming elastic material models.
1.2.2 Procedural Perforation for Controlling Elasticity
Procedural generation of elastic structures provides the fundamental basis for controlling and designing 3D printed deformable object behaviors. The automation through generative algorithms provides flexibility in how design and functionality can be seamlessly integrated into a cohesive process that generates 3D prints with variable elasticity. Generative deformation introduces an automated method for perforating existing volumetric structures, promoting simulated deformations, and integrating stress analysis into a cohesive pipeline model that can be used with existing consumer-level 3D printers with elastic material capabilities. In this work, we present a consolidated implementation of the design, simulate, refine, and 3D print procedure based on the automated generation of heterogeneous lattice structures. We utilize Finite Element Analysis (FEA) metrics to generate perforated deformation models that adhere to deformation behaviors created within our design environment. We present the core algorithms, automated pipeline, and 3D print deformations of various objects. Quantitative results illustrate how the heterogeneous geometric structure can influence elastic material behaviors towards design objectives. Our method provides an automated open-source tool for quickly prototyping elastic 3D prints.
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1.2.3 Turbulent Exhale Flow Modeling and Analysis
Dense exhale flow through CO2 spectral imaging introduces a pivotal trajectory within non-contact respiratory analysis that consolidates several pulmonary evaluations into a single coherent monitoring process. Due to technical limitations and the limited exploration of respiratory analysis through this non-contact technique, this method has not been fully utilized to extract high-level respiratory behaviors through turbulent exhale analysis. In this work, we present a structural foundation for respiratory analysis of turbulent exhale flows through the visualization of dense CO2 density distributions using precisely refined thermal imaging device to target high-resolution respiratory modeling. We achieve spatial and temporal high-resolution flow reconstructions through the cooperative development of a thermal camera dedicated to respiratory analysis to drastically improve the precision of current exhale imaging methods. We then model turbulent exhale behaviors using a heuristic volumetric flow reconstruction process to generate sparse flow exhale models that are provided with real-world metrics through depth imaging. Together these contributions allow us to target the acquisition of numerous respiratory behaviors including, breathing rate, exhale strength and capacity, towards insights into lung functionality and tidal volume estimation.
4


CHAPTER II
INVERSE MODELING AND SIMULATION
This research covers inverse modeling and the behavioral reconstruction of continuous physical phenomena through data-driven modeling. The problem is defined as follows: given discrete instances forming a set of real-world physical states or observations recorded through various sensing modalities (depthimaging, thermal-imaging, synthetic data, etc.), how can we reconstruct models from partial datasets to replicate the observed behaviors defined as set of sparse physical constraints. The core of this problem integrates recorded behaviors as discrete objective states that define the set of observed physical behaviors to be replicated within a simulation or model. Since we can only obtain partial snapshots of the physical phenomena at discrete time intervals, mathematical or analytical models that could describe these behaviors are inherently underdetermined and require both data-driven and optimization methods to identify physically plausible solutions. A solution for this type of problem is defined as a physical simulation containing an idealized model that conforms to a set of recorded discrete states or data samples while adhering to the governing physical equations to replicate an observed behavior.
2.1 Functional Modeling and Simulation
Physical simulation is inherently interdisciplinary, incorporating components from computer graphics, applied mathematics, and mechanical engineering. The mathematical basis of physical simulations are derived from fundamental models within continuum and fluid dynamics and implemented through a variety of techniques explored within computer graphics using numerical methods from applied mathematics. In addition to these core simulation components, elements from electrical engineering such as signal processing are required for preprocessing recorded sensory data, filtering, and signal reconstruction. The research presented within this work is derived from numerous techniques in these domains to introduce new contributions of inverse modeling. Each contribution in this work represents incremental steps towards an overall inverse modeling framework that includes: sensor recording methods, data analysis and modeling procedures, optimization and control methods, and links between traditional problems and inversely simulated solutions.
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Due to the integration of several theoretical domains and the application of numerous techniques within each, this we provide a brief overview of the various methods, techniques, and terms that are defined within the context of inverse modeling. This includes a brief overview of the different forms of physical simulations in computer graphics, imaging devices and signal processing methods, constitutive models from theory of elasticity, thermal modeling, constrained dynamics, and how these are integrated into solutions based of inverse modeling problems.
2.1.1 Forward Simulation
In most forms of physical modeling, the process of forward simulation is mathematically defined as an Ordinary Differential Equation (ODE) that is formed using a governing equation, a set of initial conditions modifying the ODE as an Initial Value Problem (IVP), and solved at the equilibrium static state or through numerical integration as a dynamic system. This defines (traditional) dynamic simulations as the process of forming initial conditions, defining how discrete states are updated (constitutive model), and performing system updates. This process takes a simulated model from an initial statt forward in time to a new current state. Within the domain of elastic material theory these states are refereed to as the reference and current configurations of a body undergoing deformation. To solve or progress the simulation forward in time, numerical integration is used to compute the resulting displacements (or deformations) of the object over time due to external forces (loads) or user generated interactions. This is the most common form of simulation within computer graphics and animation, however due to the lack of controllability (generating objective-based behaviors) from a set of initial values, this form of direct simulation requires additional tools to improve its applicability in control-oriented design and animation.
2.1.2 Constrained Dynamics and Optimal Control
The backbone of analytical inverse modeling or simulation is based on significant contributions in constrained dynamics and optimal control from mechanical engineering. Traditional constraint-based dynamics introduces the ability to manipulate how rigid bodies move or interact based on prescribed paths, joints, or objectives. This low level form of controlling dynamic behavior of objects introduces an opti-
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mization process that allows for exact obtainable behaviors or least-squares approximations of nearest (non-reachable) objectives. Common examples of constrained dynamics include Boundary Value Problems (BVH), inverse kinematics in computer animation and end-effector movement in robotics. Optimal control introduces a similar concept as constraint-based dynamics, however provides a generalizable framework (and multiple techniques) for simulating complex systems while adhering to the governing physical model and arbitrary constraints. From a conceptual level, there are numerous techniques to facilitate optimal control within dynamic systems; however, most are derived from the same generalized components including: the state variables that represent the system, a set of control variables that can be modified to influence the system, and the cost function that defines how controls are manipulated to obtain the objective state. Unlike constraint-based modeling, optimal control also accounts for active modification of the system through control variables. The optimal control solution is a set of differential equations that minimize the cost function while generating the control states required for the system to reach the intended goal. An example of a solution method is the Linear-Quadratic Regulator (LQR) which reduces cost functions based on feedback controls. A common example of how this method is used is in simple rocket flight models where target velocity, fuel consumption, and landing are related to the control inputs (nozzle direction, burn rate) or the motor controls in an inverted pendulum.
2.1.3 Inverse Simulation
In the context of computational modeling, inverse simulation is defined as the process of observing an existing physical phenomena, formulating an appropriate idealized model, discretizing continuous quantities, and solving the underdetermined problem while adhering to a set of governing physical constraints. In general, an inverse simulation can be defined as a continuous set of boundary value problems where each of the discrete objective states define the boundary values (either Dirichlet or Neumann boundary conditions). The remaining intermediate states are the product of computing a simulation that adheres to the governing physical laws of motion while obtaining the set of boundary conditions defined at specific instances in time. The seminal paper that introduced an automated extension of this form of optimal control for animation idealized the concept as spacetime constraints [1], From a high-level perspective, space-
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time constraints provide the ability to define the state of a complex system at arbitrary (unique) instances in time and generate the control response needed to obtain the states while maintaining secondary interaction forces handled by the governing physical model. This process allows for complete control, while maintaining a constitutive model for generating realistic behaviors between discrete control states. This and similar methods are now heavily used within computer animation [2],
The premise of inverse modeling or simulation originates within aerospace engineering as a method of optimal control that attempts to identify the input controls needed to obtain a specific guided system response. This was particularly important within aerospace design due to requiring accurate simulations of how control systems would effect aerodynamic controllability and stability of engineered control systems. Common forms of engineering problems (IVP, BVP, etc.) and prominent solution methods such as Finite Element Analysis (FEA), Boundary Element Method (BEM), and numerous other techniques for solving Partial Differential Equations (PDEs) that reside within the core of applied mathematics and engineering analysis. These techniques all revolve around the principle of reconstructing and modeling physical phenomena through the solution of PDEs or time progression models based on systems of ODEs, BVPs, or optimal control.
In a broad sense, we define the process of obtaining an inverse solution to a given problem as the resulting set of control forces required to impose the behavior or through initial conditions and internal properties of the system. This leads to the formulation of two primary forms of inverse simulations: (1) an active inverse simulation which closely resembles optimal control in that, during the course of the simulation, input controls can be introduced to alter the current state behavior and (2) an passive form where the solution takes the form of the initial conditions and material properties required to replicate the observed behavior. For example, neural elements represents an actively controlled solution, whereas models generated through generative deformations are passive.
Active Control. The distinction between passive and active is that the passive simulation does not obtain control forces to modify the behavior. Active inverse simulations closely mirror the results from optimal control and can include both corrective dynamics (such as a robot trying to stay in one place, or balance) or formulated as an optimization problem where an approximate solution is accepted. Primary examples
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of active control systems include robotic control for stability, drone flight control, and 4D printing with heated or electrical activation methods.
Passive Control. Examples of passive inverse simulations are those that do not have any means of injecting external control forces; they simply rely on the optimization of the system properties. This primarily includes elastic material coefficient property optimization for Young’s Modulus (elastic Modulus E) and Poisson’s Ratio (v) for simulations of deformable bodies and controlled rigid-body simulations. For the application within deformable body simulation, we demonstrated the impact of the internal geometric structure [3] contributes to the result of the FEA simulation. For passive rigid-body inverse simulations, methods that optimize falling blocks to land in a pre-defined pattern [4] have become popular in commercial animation packages.
Though these concepts were utilized extensively through aerospace and mechanical engineering over several decades, the adoption of inverse modeling in computer graphics (spacetime constraints) occurred much later [1], This technique proved to work well for modeling typical constraint-based dynamic systems where the Degrees-of-Freedom (DOF) were manageable enough to formulate the system (even with the aid of automated tools), for example a series of linked joints where angles and lengths are well controlled (such as the jumping Pixar lamp). The underlying assumption is that each static state can obtained with minimal effort, allowing the intermediate states (physically plausible behavior) to be computed by a numerical optimization process. The problem with this approach is that as the complexity of the simulated object increases, these static states become increasingly difficult to define. Therefore, if we cannot generate realistic static states, the resulting behavior will also diverge from expected behavior. Therefore we need additional tools to assist in the process of reconstructing real-world physical behaviors to assist the existing set of mathematical optimization tools. Towards this objective, three notable contributions [5, 6, 7] have been made that attempt to encode high-level objectives as constraints. This includes the notion of imposing bending or twisting motions on simulated objects. We proposed a similar method using intuitive control metaphors within an interactive editing application [8],
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2.1.4 Data-driven Modeling through Imaging
Simulations that integrate idealized models with real-world behaviors require both accurate constitutive models and the ability to combine direct measurements with simulated systems. The process of integrating discrete constraints from recorded data within a dynamic system is inherently complex. In this work we define two primary methods for creating data-driven simulations based on how the data is used. The first variant leverages spacetime controls to partially define a discrete deformation or behavioral state of an object as an objective state that will be reached during the course of the simulation or model training procedure. This means that each recorded state defines a constraint that must be maintained during the course of the simulation, similar to how dynamic key-frames are used to define exact animation states of an object over time [9], The second variant assumes that we can collect and extract behavioral characteristics from the collected data to parameterize the simulation and obtain an objective result. In this instance, behavioral characteristics can be extracted directly from image data, or can be used to modify parameters of the physical simulation or material properties. Examples of this technique include using depth imaging for surface reconstruction, the estimation of elastic material properties, or the behavioral analysis of fluid flows.
Surface Modeling through 3D Scanning. Scanning and reconstructing 3D models from real-world objects is well studied and has made a large impact on how static models can be automatically generated from either visible light (RGB), stereo, or depth imaging. Both the reconstruction from visible light, Shape from Shading (SFS) and depth estimation methods have been developed to the point where commercial solutions provide high quality results. But there are still assumptions that these current solutions make, such as the object is static and we have a high overlap between scan pairs. Under these assumptions, techniques in feature estimation such as Fast Point Feature Histograms (FPFH) and variants of the Iterative Closest Point (ICP) can provide both rough and refined alignments between point-clouds generated from depth images. Under these constraints, alignments can be performed and the object can be reconstructed. However, in the instance where we minimize the total number of scans required to record the surface of the object, this process becomes challenging. To alleviate this problem in the most severe cases, SxStudio was developed to provide painting-based selections of overlap regions common to each scan to improve
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alignment success. This method introduces an intuitive interface to resolve alignment problems that cannot be addressed with current algorithms. This work resulted in a robust alignment compilation process that can be used to ensure scan alignments are successful. Once this process is complete, conventional loop closure algorithms can be used to minimize the global error distribution between all scan pairs. We have also developed a studio design application for painting-based scan alignment [10] and 3D object reconstruction. The development of this application provides a base framework upon which scanned object surfaces can be used to generate structural models from real-world objects. These methods lead to key features introduced in our more recent work [11], This prior research is relevant to the development of the proposed chest surface deformation modeling techniques we introduce for respiratory analysis.
2.1.5 Physical Plausibility
Derived from traditional computer graphics terminology, the notion of physical plausibility refers a perceptual and visual fidelity of the physical behavior within a system. If the behavior does not seem to violate the natural laws of physics, it is deemed physically plausible. From a technical perspective, this is defined as the set of probable physical states that could be obtained using a constitutive model that adheres to the natural laws of physics, given natural flaws within the system and measurement method. The most prominent example of physical plausibility is used within fluid simulations in computer graphics such as Smoothed Particle Flydrodynamics (SPFT) which varies drastically from traditional Computational Fluid Dynamics (CFD) techniques by sacrificing accuracy for real-time simulations. The justification in this reduction is based on the notion that the visual perception of the result is indistinguishable from the actual or accurate result. In the context of inverse modeling, physical plausibility is defined as one of the probable outcomes that satisfies the given observational constraints without violating the underlying physical model.
2.2 Hybrid Models: Analytical and Data-driven Models
The underlying theme of inverse modeling is the ability to seamlessly integrate analytical methods with data-driven methods from machine learning to generate new types of hybrid solutions. Through the rapid
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progression of machine and deep learning techniques, there has been a divergence between what kinds of problems can be solved effectively through analytical solutions and those that are better suited for solutions derived from data-driven or learning-oriented methods. The concise objective we propose is that there are numerous problem formulations that can benefit from the careful integration of these two methods. This is specifically true for the core analytical methods of physical simulation where even recent techniques concede that a purely data-driven physical simulation based on representing the force to displacement relationship of elastic materials is nearly impossible [12], While this means that there is a direct separation between solution methods that require purely analytical models and data-driven techniques, we illustrate that there are still numerous ways to integrate these abstractions into hybrid models.
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CHAPTER III
NEURAL ELEMENTS: DATA-DRIVEN ELASTICITY
The field of dynamics and physical simulation of elastic materials is populated with numerous formulations and simulation methods for modeling deformable objects. In a broad sense, the integration of computational mechanics, numerical analysis, and computer graphics incorporates a diverse set of techniques that must seamlessly mesh into a coherent model to simulate complex deformation behaviors. As the number of refined techniques for improving simulation accuracy, quality, and performance have grown, there are incremental jumps in how we simulate deformable objects. Yet, even with the highly refined models used in real-world engineering and graphics applications, there are new approaches that reformulate existing problems to create new and unique solutions. One of the main thrusts in computational mechanics is the open problem of how to integrate data-driven modeling techniques with established methods in computational mechanics and computer graphics. The difficulty in combining these two domains is that most proven methods from computational mechanics are completely analytical and rely on accurate elastic material behaviors driven by the theory of elasticity. These methods also rely on numerical techniques that do not seamlessly integrate with the mechanics of data-driven solutions. Therefore, the prospect of augmenting current analytical techniques with the expressiveness and flexibility of data-driven models can form a solid foundation for new methodologies in elastic material simulation. Through the introduction of the neural element framework, we aim to provide a new tool for real-time applications in Virtual and Augmented Reality (AR/VR), 3D printing of elastic material designs, game design, and computer graphics. This contribution can also enable a substantial leap in how interactive solid simulations are used within other fields such as aerospace, mechanical engineering, and interactive design, where design and modeling tools can benefit from use accurate interactive models and simulations.
Neural Elements (NE) introduces a hybrid formulation that integrates an analytical Finite Element (FE) elastic material model with data-driven machine learning techniques to inversely model material behaviors. The core objective of the neural element formulation is to encapsulate the relationship between potential nodal deformations and the material responses generated by solid elements to achieve a lightweight, computationally efficient simulation framework. The objective is to provide a flexible and accu-
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Figure 3.0.1: Neural Elements (NE) is a data-driven elastic material simulation method that closely accurately replicates Finite Element simulations of solid elastic materials.
rate simulation model for interactive and real-time applications.
Conceptually, the notion of combining the constitutive elastic material models with data-driven methods has been well established since the introduction of basic multilayer perceptron networks or neural networks. Even basic neural networks have been used to learn the behaviors of material samples, a task that is well suited to machine learning with experientially-driven analysis. Variants of this fundamental idea have been well explored in both mechanical and civil engineering for numerous application domain-oriented tasks and more recently in computer graphics and animation. Linking the stress-to-strain relationship exhibited by an arbitrary elastic material to a trained network through machine learning has been employed in numerous studies that operate on real-world experiments of elastic materials. From obtaining precise material responses to determining material parameters for analytical models, some forms of data-driven elastic modeling has been well established. However, within a significant portion of the engineering research directions related to data-driven elastic material modeling, many of these research directions are very objective-oriented. That is, they typically want to obtain highly accurate behaviors of exact real-world materials for very directed purposes based on highly controlled experiments. Within computer graphics, the opposite is true. From the computer animation and interactive simulation perspective, highly accurate data-driven models lack flexibility and are typically not well-suited for general simulations. Therefore, the objective is oriented towards creating flexible and computationally inexpensive elastic material models at the expense of accuracy.
Within computational mechanics, several graphics oriented methods have been proposed for the real-
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Figure 3.0.2: Illustration of two deformations applied to the Stanford dragon volumetric model. The deformations are imposed using a force-based manipulation tool integrated into an interactive application. The core simulation is built on the propose data-driven Neural Element (NE) elastic material model, time simulation of elastic solids. This includes Mass-Spring Systems (MSS), FEA-based elastic solids,
constraint-based systems, and complex simulation methods that mix particle and solid dynamics such as the Material Point Method (MPM). While these existing methods provide a wide range of capabilities for simulating numerous types of materials and physical phenomena, there are still relatively few alternatives for solid mechanics within real-time applications such as Computer Aided Design (CAD) tools, games, and modeling tools for 3D printing that require close approximations of the analytical solutions provided by finite element solutions. Within the current landscape of existing elastic material simulation methods, neural elements is aimed at providing an accurate method that can bridge the gap between high performance methods such as MSS and complete finite element models. The objective is to provide a close approximation to well-established constitutive models from continuum mechanics but supplement the fundamental operations through data-driven approaches. This provides different forms of optimizations and affords new features that can only be expressed through data-driven methods. Additionally, through the formulation of the hybrid method, we can also provide flexibility in the simulation framework to facilitate the deployment of the core method in various applications in computer graphics.
3.1 Related Work: Simulation of Elastic Solids
The simulation of solid materials incorporates contributions from numerous fields including continuum mechanics, numerical analysis, and computer graphics. From the contributions provided by the vast array of core algorithms that makes the simulation of solid elastic materials possible, there have been an assortment of different models proposed. These vary from highly accurate finite element models used in
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mechanical engineering to numerous hybrid models that aim at achieving real-time performance for applications in computer graphics and game design. Within this populated held, we provide an outline of where the neural element formulation for simulating elastic solids falls and how it is distinct from existing methods.
3.1.1 Mass-spring Systems
The most prominent method of simulating deformable objects within interactive computer graphics and game design is mass-spring systems. This is due to the simplicity and performance that can be obtained through the combination of multiple point masses with arbitrary springs. The flexibility and performance of this method for cloth simulation has established a widespread adoption of this method within most realtime graphics applications. For cloth simulation, the ability to formalize shear, bend, and structural components of cloth leads to physically plausible results that have been incrementally improved to provide realistic simulations within real-time time constraints. However, this method is ill-suited for the simulation of volumetric structures. In fact, there have been numerous contributions to the formulation of massspring meshes that propose different spring configurations, constraints, and even secondary algorithms that operate on mesh topology to improve the simulation of solid materials using this method. This is due to the inadequate representation of the material forces that are exhibited in real-world materials and how they are approximated through a collection of springs. Furthermore, in a series of contributions, [13, 14] illustrated that the MSS model and the FE formulation of elastic materials are incompatible. That is, there is no way to properly represent Hookean elastic material behaviors using the MSS formulation. For the two-dimensional case, there has been some progress in how spring configurations can be defined to approximate similar behaviors. Yet, in the three-dimensional case, there is no exact analog for reproducing the behaviors of volumetric objects through a set collection of springs. Additionally, the fundamental material properties of this method differ greatly in the semantic meaning of material behavior.
For mass-spring systems, the core material properties are generally defined as ks, the Hookean spring coefficient (normally identified as k in the f = kx relation) and kd which represents the damping coefficient the reduces kinetic energy of the system based on velocity. This is distinctly different than the coef-
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ficients used in continuum mechanics. For solid elastic Flookean materials based on the isotropic model proposed in continuum mechanics, the primary material coefficients can be defined as Young’s modulus (the modulus of elasticity) E and the Poisson ratio v of the material. The specification of these two coefficients provides a relatively consistent behaviors based on solid mechanics, even between different implementations. The problem is that there is no direct mapping between these quantities and the spring coefficients in mass-spring systems. To consolidate the behavioral differences between finite element meshes and mass-spring systems, there has been numerous attempts to formalize the material behavior relationship. In general, the objective of these approaches has been to specify geometric and elastic material properties and analytically derive the spring types and coefficients that are required to approximate the same behavior as an FEA-based model. This has lead to the numerous approaches presented by [15] below:
Geometric Approximation L. ^4l+^2 rv — 72 l0
Wilhelms et al. [16] ka = cEi 4"
Geometric Approximation on. u _ ZU . Iic- ||o|2
Van Gelder et al. [13] on . u _ EVeV 6L> . Iic- ||o|2
Geometric Approximation Zerbato etal. [17] k = ElEE2 + geneticalgorithm
Geometric Model + Elasticity ke = E if
Arnab etal. [18] kg — E Eiej 2l2(l-v)
Adjustment of Van Gelder et al. Macieletal. [19] k k0 coso + ki cosi h CQSi
Reference FEA Model Lloyd et al. [20] ktri = Ee Eh^E he V 1/iF red Z^e hd — J-hT? "'red ktetra = Ee ktetra2 = Ee kv = #3^§
Parameterization (tensile shear) , E(j2(3v+2)-i2 ~ 4l0h0(l+v) i El0{4v+l) Ked ~ 8(l+i>)
Baudetetal. [14] kf — G — 2(\+v) /„ _ 3El0 Kd ~ 8(l+i>)
FEA Approximation Natsupakpong et al. [21] h — A 1 E he — 3 T- 2 */ = (i + Is)
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The shear number of different illustrates the difficulty in specifying the elastic material coefficients to obtain similar behaviors from mass-spring models. The primary objective here is to retain the significant performance increase provided by MSS models, but provide the coefficients and accurate behaviors of solid materials as would typically be obtained using an FEA-based simulation. The complexity and inaccuracies illustrated by these approaches provides an optimal opportunity for an inverse modeling solution. Instead of attempting to replicate the inconsistency of the behaviors exhibited by MSS and FEA based models with analytical methods that only provide partial approximations of the desired result, a data-driven method can be employed.
3.1.2 Constraint-based Solvers
Constraint-based systems operate on particles as a fundamental building block for simulating every type of physical phenomenon. This includes the integration of rigid-body, deformable or elastic materials, gas, and fluid dynamics. The appeal of this approach is that it provides a unified simulation pipeline. That is, the constraint formulation that dictates the interaction between particles is consolidated into a single set of operations, namely, the generation and solution of constrained systems. The systems can then be efficiently computed for each simulation time-step, handling the interactions between all particle and constraint types. The appeal of these methods is that they can be aptly adapted to parallel architectures for simulating large quantities of particles. Since the premise of the method is derived from particle interactions integrated into larger continuous volumes, the number of particles required for most objects is still extensive. However, with the scalability of the approach, it makes graphics processing units a prime target for the deployment of these systems due to the high throughput that can be achieved. This provides a viable framework for creating highly adaptable deformation, granular, fluid, and rigid-body behaviors that can be used in a variety of real-time applications. This method has become a predominant simulation method for interactive and real-time graphics applications.
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3.1.3 Deep Learning Approaches
The prospect for introducing deep learning approaches for the simulation of solid elastic materials has been well described with respect to reconstructing the force-displacement relation for general deformable simulations as nearly impossible [12], Due to the inherent nature of the problem, the constitutive models and numerical methods are fundamentally ill-suited for learning-based adaptations. Based on the constraints of these formulations, [12] concede the formulation of a deep learning approach for completely replacing the constitutive model and integration methodologies of elastic material simulation. Rather, they provide a warping formulation that uses deep learning to modify elastic material behaviors to match that of complex non-linear formulations such as St. Venant-Kirchhoff and Neo-Hookean material models.
3.2 Method Overview
Neural elements proposes a method of simulating isotropic elastic materials for the simulation of solid deformable objects. This method is derived from a decomposition of the constitutive material model provided by the finite element formulation of isotropic elastic materials. Through the functional analysis of this model and inverse modeling, we transform this purely analytical method into a data-driven method based on a completely new compute model. This allows us to introduce several new contributions towards the simulation of real-time deformable objects that closely mimic the accuracy provided by traditional FEA-based solutions. These include: (1) introducing a physical model that resides between mass-spring and FEA-based systems, (2) trivial integration of heterogeneous materials, (3) native handling of dynamic boundary conditions, (4) dynamic topology due to the removal of matrix systems, and (5) a parallel friendly compute model.
3.2.1 Contributions
The neural element based simulation of elastic materials is aimed at providing a data-driven method for generating complex deformations of solid objects within interactive and real-time applications. The objective of this approach is to provide the accuracy and stability from Finite Element Analysis (FEA) methods of simulating deformable models to animation, game design, interactive prototyping, and 3D print design.
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y
y
N0
Ni
Nn
-V -V
Discrete Volumetric Elements Instance/Reference Formulation Element Behavior Network Training Real-time Elastic Deformations
Figure 3.2.3: Illustration of two deformations applied to the Stanford dragon volumetric model. The deformations are imposed using a force-based manipulation tool integrated into an interactive application. The core simulation is built on the propose data-driven Neural Element (NE) elastic material model.
Current challenges for deformable object simulation within many interactive applications is the computational complexity and inflexibility of existing solution methods general use. Through the introduction of the neural element design, we can provide a new alternative solution for simulating solid elastic materials that is more accurate than mass-spring systems and a more efficient FEA-based solution. Below is a brief overview of the primary contributions provided by the neural element formulation:
• Solid Elastic Materials: Introduce a method derived from FEA formulations of isotropic elastic materials to accurately simulate solid meshes. This provides a new method that lies between the performance but inaccuracy of mass-spring meshes and a complete finite element simulation of elastic materials which are computationally expensive.
• Interchangeable Solution Methods: Neural elements are composed of several building blocks that include the machine learning model used to predict material responses based on stress-to-strain relationships, numerical integration methods, and training-data. Each of these components are interchangeable, allowing for a flexible design interface of the core method.
• Heterogeneous Material Models: Due to the independent evaluation of all elements within a simulation, every element can be assigned a unique material. Since the core method evaluates this material property through a neural network, there is no additional cost or system reconstruction associated with changing element materials. This means that element materials can be changed dynamically at any time during a simulation.
• Dynamic Boundary Conditions: To define how parts of a mesh are fixed or move during a simulation is defined by a set of boundary conditions. Unlike finite element-based simulations where a
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complete system reconstruction is required to update boundary conditions, the neural element formulation natively handles both fixed and dynamic boundary conditions. Flexibility in defining these conditions provides a natural extension to interactive simulations.
• Dynamic Topology: Independent evaluation of discrete elements allows for instant changes in topology. Since there is no system matrices defined based on the topology of the simulated mesh, splits along element borders and the introduction of new elements can quickly be integrated to an existing mesh. This allows for behaviors such as ripping or tearing and cutting of elastic materials.
• Parallel Compute Model: The evaluation of individual elements is performed independently through parallel neural networks. Derived from the data-driven approach, we define a set of neural networks that can be dynamically instanced to provide element responses for material behaviors from parallel compute models. Therefore the method is extended to multi-core parallel computation with the option of extending this to Graphics Processing Units (GPU)
3.2.2 Finite Element Formulation for Neural Elements
In finite element simulations for elastic materials, the evaluation of an objects deformation behavior is directly evaluated through a set of governing equations and formalizations that define structural rigidity as an integrated system of elements. The objective quantities obtained through the solution of the system depend on the form of this analysis and the post-processing performed on the outcome of the solution. In the most common form used for elastic materials, the objective is to determine the nodal displacements, stress, and response forces incurred from an applied load and predefined boundary conditions. These boundary conditions and the applied load define the constraints that characterize the deformation of the object. From the computed result, a number of valuable quantities can be evaluated to determine the performance or behavior of the material and the geometric structure of the model. In most cases, the applied load induces external forces on the object from which the resulting deformation can be measured as nodal displacements or as stress to evaluate critical yield criteria related to plastic deformation, fracture, and other physical characteristics used within numerous engineering applications.
Interactive simulations that incorporate deformable objects based on finite element models have been
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highly optimized to model several different elastic material formulations. Through the rigorous study for improving numerical integration techniques, modal analysis for behavior optimization, and the various alternative material models that have been introduced, the adoption of deformable objects derived from solid mechanics has been limited. The performance improvements obtained by reducing the accuracy or limiting the deformable characteristics of solid objects has not pushed deformable models into the forefront of real-time simulations simply because they require a disproportionate portion of the computational load required for interactive graphics applications. Existing solutions introduced various techniques for approximating the behaviors of deformable solid materials, but many still rely on the premise of physically plausible behavior. Therefore, the visualized deformation may be adequate for most interactive applications where the absolute accuracy is difficult to evaluate; however, there are still other applications that benefit from the accurate deformation and stress evaluations provided by an FEA-driven approach. This is primary motivation towards using the isotropic elastic material model from the FEA formulation to derive an accurate and efficient data-driven dynamics model.
Neural elements at its core relies on the evaluation of the isotropic elastic material model for discrete element behaviors to generate synthetic material responses. The reason behind this is due to the dual objectives of the neural element derivation: (1) to generate accurate material responses through a computationally efficient decomposition of the primary linear elastic model and (2) provide a fast and flexible system architecture that enables both accurate simulations but also provides a variety of useful features for interactive applications. These include the ability to define heterogeneous mesh materials, easy-to-define boundary conditions, efficient handling of topology changes, motion-driven deformations, and parallel friendly compute model.
To provide these features, we decompose the standard elements of the isotropic elastic material model to exploit how elastic solids are formulated to obtain an efficient breakdown of material behaviors that can be replicated through a data-driven approach. This requires analysis of the core shape functions and the evaluation of the stress-to-strain relationship of basic elements commonly used in FEA formulations. Additionally, to evaluate this relationship for generating the constitutive model used within the neural element design, we also have to generate the FEA-based material responses of elements.
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FEA Formulation Overview. The premise of the Galerkin FEA formulation is to decompose the continuous geometry of an object into discrete elements that exhibit the behaviors of an isotropic elastic material. The generation of this discrete set of elements is known as meshing. This process divides the continuous geometry of a solid object or manifold surface into discrete elements based on a pre-selected primitive shape. For two-dimensional meshes this is typically triangles where each node has two Degrees of Freedom (DOFs) and for three-dimensional geometry, tetrahedral elements are typically selected where each node has three DOFs per node. These discrete elements are then characterized by a set of shape functions that describe how the governing equation (such as Flooke’s Law) is applied to represent the relationship between stress and strain. This process incorporates the formulation of the governing equations that dictate material behavior and provide the matrix equations required for solving for various quantities such as nodal displacements, material responses, and element stress. Neural element behavior is formed of the discrete observations of the behaviors generated by a pre-defmed FEA solution. This provides an exact model of the material behaviors that each neural element aims to replicate. To provide this material response and the evaluation of the stress-to-strain of an idealized material, we define the element stiffness matrix k of a given element type (triangle, tetrahedra, etc.) based on the shape functions that characterize its behavior. For each element in a mesh, this matrix is generated and then integrated into a larger global stiffness matrix K that defines the material and structural mechanics of a given mesh. If we are given a set of external forces fext, and the global stiffness matrix K, we can solve for the displacements U of the nodes within the mesh. To solve this we can simply obtain the solution to KU = fext, solving for U. Conversely, if we are given a set of displacements, we can solve for the resulting internal forces of an isolated element fint = KU.
3.2.3 Corotational Formulation
Linear stress-to-strain relations derived from elastic material properties have introduced an accurate finite element solution to analyze small deformations of continuous solid materials. The infinitesimal strain model can be used replicate deformations where the displacement of material points is infinitesimally small with respect to the scale of the objects size. However, due to the linear nature of the model and the
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assumption that displacements are small, this formulation introduces large artifacts in material behavior due to the large rotations that highly flexible materials allow. Therefore, to solve this problem and facilitate large rotational deformations, the corotational formulation was introduced to: (1) assume that each element is still accurately represented by the infinitesimal strain model and (2) account for the rotation of the material as the object deforms independently. By computing and applying the inverse of the rotation for each element at each time-step, the linear model can still be used to compute accurate deformation behaviors of the continuous material. This introduces the concept of geometrically non-linear material simulations through the corotational formulation of FEA simulations.
The corotational correction for elastic finite elements is required due to the computation of element displacements within the linear formulation based on the Cauchy stress tensor for elements undergoing rigid-body rotations. The problem with this formulation is that this formulation and the Cauchy stress tensor are not rotation invariant. That is, the rotation of the element is not correctly accounted for in the computation of the elastic stress-to-strain relationship. This results in large and physically incorrect artifacts for large rotational deformations, leading to increases in volume as elements within the material rotate away from their rest positions. To correct these artifacts, Stiffness Warping was introduced [22] to remove the artifacts that the linear elastic forces introduce on the deformable body. Initially, this was applied to the nodes of elements leading to two problems: (1) the rotation Ri of vertex i has to be computed by its adjacent vertices which is an ambiguous problem and (2) the elastic forces are not guaranteed to sum to zero, resulting in the introduction of ghost forces. This technique was improved through the transition to an element-based corotational formulation [23], The objective is to extract the deformation and rotational components of an element between its reference (rest) and current (deformed) state. The result of this process will provide an orthogonal rotation matrix It that represents the optimal rotation between the rest and deformed states of the element. This rotation can then be used to return the deformed element back to its rest coordinate space to compute nodal displacements and its inverse can be used to translate response forces back to the global coordinate system.
2D Triangle Element Rotation. The triangle element T in the reference configuration contains three nodes: Tr = {/(,. Iff and the element in its current deformed state is defined by the changes in
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node positions to Tc = {Qo, Qi, Q-i}- Any point p e Pi2, simply defined as: p = {px,py), has a set of Barycentric coordinates {pi, fh,lh) that uniquely define this point with respect to the element nodes. This Barycentric coordinate can then be computed in both the reference and current configurations.
Figure 3.2.4: Dynamic behavior of a 2D triangle between the rest and current configurations (left). The element nodal displacements {do, di,d-2) due to the deformation are defined by rotating the element to the optimal pre-deformation state with both elements coinciding at their center of mass (right).
The relationship between this point and the element can be integrated into a homogeneous form to satisfy the conditions for both the triangles rest state {Tr) and the triangles current state (Tc):
Po, Pi, P'2; Po ~Px Qo, Qi, Q2, Po ~qx
Tr = P0y Ply P2y Pi = py Tc = QOy Qly Q2y Pi = %
_ 1 1 1 _ A. _1_ l l 1 _ A. _1_
The relationship above can be used to solve for the Barycentric coordinate of the point, however this is not the objective of the formulation. Rather, the condition using this coordinate is used to create a relationship between the elements states. The Barycentric coordinate represents the same point in both the reference and current configurations of the element to formalize the linear transformation between the two provided states. Then the same condition holds for all points in both the rest and deformed elements, where p and q are different Cartesian coordinates but have the same Barycentric coordinate. Thus we can express the matrices in 3.2.1 using two equations: (1) Pp = p and (2) Qp = q. These can then be combined to form a 2D representation of the general equation presented in [23],
q = Qp = QP 1p = Ap which can be simplified to A = QP 1 (3.2.2)
Where A = QP-1 represents the transformation of the element between the rest and current configurations. The structure of A is defined by three components: (1) the element deformation, (2) rotation of
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the element, and (3) the elements translation. The 2D translation of the transformation can be directly extracted from this matrix using the values (. ty), however both the rotation and deformation components between the two element states are defined as the sub-matrix B that resides within A as shown in Equation 3.2.3.
-Boo -Boi t'X
-Bio Bn ty where B = Deformation and Rotation t = Translation (3.2.3)
0 0 1
Since the matrix B is a combination of both the deformation and rotation of the element, the matrix has to be decomposed into the rotation and deformation components. To solve this problem Polar Decomposition is used to extract the rotation (R) and deformation (U) matrices from B. Where we obtain the deformation U as the positive definite matrix of the decomposition and the rotation matrix It as the orthogonal matrix of the deformation gradient F. For the 2D formulation of this problem, there is a simple method proposed in [24] to obtain the decomposition of an arbitrary (2 x 2) matrix using eigen-vectors and eigenvalues to compute the decomposition of the orthogonal and positive definite matrices.
3D Tetrahedral Element Rotation. The tetrahedral element T in the reference configuration (r) contains four nodes: Tr = {/(,. /). /T /'-■,} and the element in its current (c) deformed state is defined by the changes in these node positions to: Tc = {Qo, Q i • Qi■ Qo}. In the 3D element case, any point p e R3, simply defined as: p = {px,Py,Pz}, has a set of Barycentric coordinates (/ii, fh, Ih, IU) that uniquely define this point with respect to the element. The relationship between this point and the element in the rest state can be integrated into a homogeneous form to satisfy:
Qi
Qo Q2
Figure 3.2.5: Dynamic behavior of a 3D tetrahedra between the rest and current configurations (left). The element nodal displacements (do, d\,d-2, do) due to the deformation are defined by rotating the element to the optimal pre-deformation state with both elements coinciding at their center of mass (right).
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(3.2.4)
P0* Plx P2X p3x A> Px Qox Q lx Q2x Q3x /3o qx
P0y Ply P2y P3y Pi Py T, — QOy Qly Q 2y Q 3y Pi %
p0z Pi, P2z P3, 02 Pz ± c Qo* Qlz Q2z Q3z P2 Qz
1111 P 3 1 1111 At 1
Using the same formulation as the 2D case, these matrices form two equations: (1) Pfi = p and (2) (JrS = q. These can then be combined to form the 3D representation of the equation presented in [23],
q = Q(3 = QP lp = Ap which can be simplified to A = QP 1 (3.2.5)
As with the two-dimensional case, A = QP-1 represents the transformation of the element between the rest and current configurations. The structure of A is defined by three components: (1) the element deformation, (2) rotation of the element, and (3) the elements translation. The translation of the transformation can be directly extracted from this matrix using the values I ,r tz). The structure of the matrix A and the sub-matrix B that resides within A is shown in Equation 3.2.6.
A
-Boo -Boi B>02 tX
-Bio Bn B>12 ty
B>20 B>21 B>22 tz
0 0 0 1
where B = Deformation and Rotation
Translation (3.2.6)
To perform the decomposition of B for an arbitrary (3 x 3) matrix, there are several different approaches. These range from using Singular Value Decomposition (SVD) methods that are typically expensive to iterative solutions that can sacrifice accuracy for performance [25], As many of these techniques stem from the abstract mathematical process of decomposing any arbitrary matrix A, they can be accurate but impose large bottlenecks in real-time simulations due to their computational complexity. This is due to the generality of A.
Within the fixed domain of a continuous simulation there are other assumptions that can significantly reduce the problems solution space. Since the deformations in the simulation are considered continuous, the current decomposition will only shift slightly from the previous decomposition result assuming a small simulation time-step. Based on this assumption, an efficient method has been introduced to incrementally extracting the rotation matrix from the decomposition using an iterative solution [26], This method operates off the notion that we can significantly reduce the generality of A to effectively converge on the
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optimal decomposition. This method uses an initial guess that is the result of the decomposition from the previous time-step (or the from the initial rest position) to iterate towards the optimal decomposition of the current time-step.
3.2.4 Data-driven FEA
The concept of data-driven finite element simulations has been well established for introducing a variety of different methods to augment and improve existing analytical FEA formulations. Finite element simulations are also inherently partially data-driven to some degree due to the evaluation or measurement of the material properties used to parameterize the shape functions that describe element behavior. In fact, one of the common forms of a data-driven FEA method is the evaluation of element responses to deformations performed on real-world materials measured in precise experimental setups. From these initial contributions, research into data-driven FEA rapidly expanded from the evaluation of material responses from lab-driven experiments, to improving the accuracy of simulated materials, creating energy-based sampling for improving dynamic element behaviors, and using Deep Learning to augment linear elastic models to incorporate non-linear material behaviors. Many of these data-driven methods have aimed at providing more accurate simulation results for various engineering applications or have attempted to resolve disparities between real-world material behaviors and simulation results. While there has been tremendous progress in the number of methods that augment, improve, or extend the functionality of the finite element method for elastic materials, many existing methods line the periphery of the core computational FEA formulation. This is because most of the core formulations within the FEA solution method are incompatible with or cannot be expressed as data-driven computations. This is because the core solutions for solving systems of equations or integrating over time are based on discrete numerical algorithms that approximate analytical methods. Since these forms of problems are not well suited to the application of a predefined training domain to generate solutions, most data-driven methods augment the formulation of these core solutions.
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3.3 Data-driven Element Behavior
Inverse modeling for data-driven elastic materials have defined a prominent research area that attempts to integrate solid mechanics with new methods in machine learning to improve the accuracy of simulated materials. Various data-driven methods have been employed within continuum mechanics and elastic material property modeling for improving simulation accuracy and providing solutions for inconsistencies observed between simulated and real-world materials. The premise of modeling elastic element behaviors on the measurements of real-world materials provides a basis for improving the utility and accuracy of FEA simulations for numerous engineering applications and advanced 3D printing techniques [27], Based on these contributions, there are three primary classes of methods that use data-driven techniques to augment or improve elastic material simulations: (1) the use of real-world measurements to both formulate and predict material properties, (2) the use of data-driven methods as numerical algorithms to supplement or replace the constitutive model of elastic materials, and (3) the use of data-driven techniques to augment elastic material behaviors (Ex. augmenting simulations of linear materials to exhibit nonlinear behaviors). Neural elements presents a method that falls within the second class of data-driven methods to enable a constitutive model derived from FEA-driven elastic material behaviors. Due to this formulation replacing the core numerical algorithms used in most FEA elastic material simulations, we are able to exploit this new computational model to provide an accurate and flexible simulation of solid materials that can be used within interactive or real-time applications.
One of the prominent challenges associated with deriving a data-driven physical simulation is how the data can be used to augment or improve simulation accuracy, utility, or performance. Depending on the objective of the data-driven model, there are numerous constraints imposed by the formulations of the governing equations and integration techniques that limit how the data can be integrated into existing systems. In the general form of dynamics, the process of updating element stiffness matrices and numerical integration are not well suited for being directly replaced by methods from machine learning. In most instances, these constraints direct how most machine learning methods are applied to FEA simulation methods. For solid mechanics, the primary target is the constitutive model of the simulated elastic materials.
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This is because the relationship between stress and strain [28] or material force responses is well established between a given behavior and sample responses that can be obtained through repeatable experiments. The most common form this process is related to the experiential derivation of complex material properties obtained from performing real-world deformation analysis. While the foundation of the neural elements derivation is built on a similar premise, the key difference is that the objective is to completely replace the constitutive elastic material model with a data-driven synthetic model. The introduction of a synthetic model is characterized by the virtual generation of synthetic elastic material data that is used to define the accurate behaviors of FEA elements, but with a completely new computational model based on elemental force responses. The objective of this model is to introduce light-weight material mechanics that are highly adaptable to real-time applications, but maintain the accuracy and behavior of fully formulated linear elastic model behavior obtained using an FEA simulation.
3.3.1 Element-based Material Response
Elastic materials undergo strain e = d/dX(x — X) as a function of the reference state X of the element and the current configuration x based on the displacements that have altered the state of the elements geometry. Due to this strain, the material exhibits a response that is a function of the applied deformation and the properties of the material. If we currently know the displacements of the deformation, then the solution to determining the material response is to multiply the global stiffness matrix K with the current global displacements U to obtain the internal response forces /„,/ = KU. These internal forces represent the material response to some given displacement of each element node for each degree of freedom. As the primary distinction between the solution provided by the formulating the solid as an elastic continuum versus other simpler simulation methods such as mass-spring meshes, the result of a single nodal displacement along one degree of freedom incurs multiple response forces within the element. The number of response force components is the same as the FEA system size that depends on the number of nodes in the element type and the total degrees of freedom. In the instance of a two-dimensional triangle element, the displacement within one degree of freedom of a single node will result in six response forces due to the three nodes, each having two DOFs. For a three-dimensional tetrahedra, the element will generate a
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response force vector for each of the four nodes, resulting in 12 response force components. In the neural element formulation, the objective is to encompass these material responses within data-driven model that can generate accurate deformations through inversely defining the desired behavioral traits.
Based on the behavior of the material response forces that are generated by providing a set of nodal displacements, we observe three primary factors that will characterize the formulation of the proposed data-driven approach: (01) the nodal displacement in one degree of freedom results in the generation of multiple forces acting upon all nodes within the element. Observation (02) verifies that the response force components of the material are not rotation invariant, as noted in the corotational formulation. Meaning that for every geometric instance of the element, the response forces are different depending on the formulation of the element stiffness matrix. For observation (03), we identify that there are different element configurations that result in special response force conditions. One unique condition is when the displacement of a node is applied to an element symmetrical about a primary axis and the displacement acts along the same axis. Due to the symmetry in this case, the perpendicular force (in the y direction) is zero as shown in Figure 3.3.6 (left). Flowever, if the elements orientation is changed, then the condition imposing the zero force component will change, resulting in six non-zero response forces as shown in Figure
3.3.6 (right).
From observation one, we formulate that for each DOF, there will be a set of n â–  d response force components where n is the number of nodes in the element and d is the DOF dimensionality. This means that for every node and DOF, there arc v â–  d response force components that must be estimated by the constitutive model to provide an element force response. Stemming from observation two, this new model will also have to account for how forces change when an elements geometry or orientation is changed. Since the base FEA formulation of the elastic material behavior is not rotation invariant, we cannot use rotations to translate forces between coordinate systems.
Since simulated meshes are generally composed of a large number of elements, all of which vary in size, orientation, or geometric structure, the prediction of numerous element response forces based on these factors would require arbitrarily complex networks. Additionally, the trained network would also have to account for the potential material properties of the element (E, v). This leads to an infeasible
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â–  X
X
(01) Element Response Components
(02) No Rotation Invariance
Figure 3.3.6: Illustration of the material response forces of a two-dimensional triangle element providing three observations that establish the foundation for the proposed data-driven material model. This includes multiple force responses for each displacement (left), response forces are not rotation invariant, but for rotated elements we can ensure response force components are non-zero (right).
prospect for a computationally inexpensive solution due to all of the possible response forces that would have to be predicted based on this large number of inputs. Conservatively, counting an input for each factor (a large underestimate), we would need at least five inputs (scale, rotation, structure, elastic modulus, and Poisson Ratio) to account for this variance. Based on the expressible entropy of a neural network with this many inputs and 6 outputs for a 2D element or 12 outputs for a 3D element, this solution quickly becomes unreasonable. This is due to the complexity of the network structure required to establish these relationships accurately. Furthermore, what is the prospect of this solution, given that current elastic constitutive models already provide exact response forces in concise closed-form solutions?
These observations lead to the development of several challenging problems that must be addressed within the new data-driven model. The solutions to these problems presented in this work establish the framework of the neural element derivation and also provide context of the data-driven method with respect to existing techniques. We enumerate the core problems as geometric variance, material variance, rotation invariant transformations, and computational complexity.
Geometric Variance. How can the wide geometric variance of elements be accounted for? Every element has different material responses based on a number of different identified factors. Therefore, the representation of the training domain and network structure required to provide accurate predictions makes this problem ill-suited for many machine learning techniques. Additionally, any solution to this problem also
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has to compete with the computational complexity of the closed-form equations provided by continuum mechanics used in the governing matrix equations. These equations specifically formulate the relationship between shape functions and the geometry of the element based on interpolation as described in the FEA formulation for elastic elements.
Material Variance. Flow can complex material behaviors be modeled through computationally inexpensive models to reconstruct an entire simulation that approximates the accuracy of an FEA formulation? This is where the proposed method differs considerably from sample-driven techniques that use real-world measurements to provide more accurate simulation results of physical materials. Rather than providing estimated values that loosely connect real-world materials to simulated elements, our method proposes to completely replace the constitutive model. This means that all material responses are all generated from an aggregate of synchronized neural networks.
Rotation Invariant Transformations. Flow can we obtain transformations of response forces without rotations? From the premise of a data-driven method, we have identified that representing all material responses given from any element geometry for any orientation is ill-suited for a data-driven solution. If we assume that we could obtain a complete set of material force responses, how can they be translated to all elements within a simulated mesh? Even if we can generate the material responses from a given element, the responses would have to be transformed to the global instance of every element. Therefore, we require a solution that circumvents the rotation invariance of the FEA formulation to transform element responses between coordinate systems.
Computational Complexity. What is the computational complexity of the proposed method compared to the matrix form used in a standard FEA simulations of elastic materials? This problem is complex due to the number of factors that contribute to the overall algorithmic structure of an FEA simulation. This includes the creation or update of element stiffness matrices for all elements, the construction of the global stiffness matrix, the polar decomposition of all elements, and the integration of the system (explicit or implicit). Based on the formulation of the neural element network architecture, the computational complexity of both methods will be analyzed.
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To solve these problems, we decompose the closed-form equations of the isotropic linear elastic material model into a fundamental set of behaviors that can be trivially captured through training a synchronized aggregate of light-weight neural networks. To do this we combine observations two and three to eliminate the variational complexity of element material responses through the separation of the geometric and material components of the underlying elastic material model. We then introduce the concept of geometric ratios to handle the variance of element shapes and force transformations, and create a new type of element called a reference element that precisely defines elastic material behaviors based on data-driven inverse modeling.
3.3.2 Reference Elements, Coordinates, and Geometric Ratios
Reference Elements. In a simulation of an elastic material composed of discrete elements, we can evaluate a material response from a set of displacements that define the stress to strain relationship within a single element. If it is assumed that the mesh is composed of an idealized, homogeneous, isotropic material, the material responses of the this element represents the same behavior as all elements minus the geometric variance of each element instance. This is a key observation towards constructing the discrete training domain that can be used to represent the material response due to an displacement or strain imposed on this element. Based on this objective, the idea is to numerically limit this domain as much as possible while still providing a complete representation of the materials behavior. We combine both the displacement-to-material response of a single element with the minimization of the training domain needed to represent the behavior through the creation of a reference element. This element provides a stress-to-strain stand-in that provides a material response reference for all elements within the simulation. This approach provides two important contributions: (1) the element can be defined to have a single constant geometric configuration, isolating the material response from the geometry and (2) this configuration can be decomposed to nodal contributions to further minimize the training domain. In the NE formulation, this represents a recurring theme of decomposition and minimization to build a data-compatible model by consistently minimizing the training domain based on the FEA formulation.
A reference element is simply defined as a constant instance of the element-type used within the sim-
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ulation. This element can be defined by any single element as long as it adheres to two critical constraints: (1) the geometry is not degenerate and (2) the material response of the element is non-zero for any applied deformation. From observation three, this means that the element can be of regular shape, but may require changes to meet the second constraint. The simplest solution address second constraint is to change the orientation of the element by an arbitrary rotation. This ensures that for any displacement applied to any node of the element will result in some non-zero response. For example, in the two-dimensional case, the reference element can be defined as an equilateral triangle with an arbitrary rotation (9), resulting in the generation of non-zero responses by the element in both the x and y directions for all nodes as shown in Figure 3.3.7.
y
Reference Element (invalid)
Reference Arbitrary Rotation
Non-zero Force Components
Figure 3.3.7: Example of a reference triangle element R used for two-dimensional simulations. The chosen reference geometry (left) is rotated by an arbitrary angle 9 (center) to ensure non-zero force component responses to an applied deformation for all nodes (right).
This formulation can be generalized to any element-type with nodes that have any number of DOFs. This means that the basic premise of the NE method can be applied to 1, 2, or 3 dimensional simulations for any element type. This is because a reference element can be defined for each simulation instance, as long as the material response of the element is non-zero for all deformations. Similar to the 2D case, the three-dimensional reference element can be defined as tetrahedra with an arbitrary rotation. The difference here is that to ensure the constraint of non-zero response forces, the element must be rotated using an arbitrary quaternion, or three Euler rotations (9X, 9y, 9Z). This creates response force components in the x, y, and z directions for all nodes as shown in Figure 3.3.8.
The reference element provides a reduction in the element parameter variance that allows us to model
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y
v
y
-x
X
-y
Reference Element (3D)
-y
Reference Arbitrary Rotation
-y
Non-zero Force Components
Figure 3.3.8: Example of a reference tetrahedral element R used for three-dimensional simulations. The chosen reference geometry (left) is rotated arbitrarily (center) to ensure non-zero force component responses to an applied deformation for all nodes (right).
material behaviors independent of the elements geometry (since the geometry is constant). Given the reference, we can now quantitatively evaluate the stress-to-strain relationship of the element and express this relationship within a discrete training domain. Based on the isotropic elastic material formulation, we can manipulate the reference geometry using different displacements to elicit behavioral responses from the element. From the reference element, we can then establish this behavior as a relationship between Young’s Modulus (E), Poisson ratio (v), the applied displacement (d), and the generated material response.
Reference Coordinates. Generating data-driven material responses from a reference element only provides a partial solution towards building a complete response model. Since the response forces are only valid for the reference element, we must establish a relationship between arbitrary element configurations and the response forces that can be generated from the reference. To do this we provide a parameterization that transforms an arbitrary element into the reference coordinate system and an inverse transformation that allows us to extract the correct element forces from the data-driven reference response. This process includes three stages: (1) parameterization of arbitrary elements in the reference coordinate system, (2) the creation of a compute model based on evaluating the response forces of each element node independently, and (3) performing the inverse transformation of the material response from the reference to the corotated coordinate system.
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The parametrization of an instance element I, is defined by the iterative evaluation of each node in the element. For a given element, we pick a predefined node ordering. From this ordering, we will evaluate the element response with respect to this selected node within the reference coordinate system. The sequence in Figure 3.3.9 illustrates this process for a simple 2D triangle element. The instance element is parameterized with respect to the currently selected node n. This represents the current node that will be displaced to generate a set of internal response forces from the other nodes u and v within the element. The result is two elements, the instance I and the reference R, both located within the reference coordinate system as shown in Figure 3.3.9 (right).
Figure 3.3.9: Parameterization and transformation of an arbitrary instance element I to the reference coordinate system containing the reference element R. The node ordering generates matching pairs between the two elements In —> II,,. I„ —> It,,. lr —> Rv for all independent executions of each node between the elements.
Since the elements have different geometric shapes but the same number of nodes, they share matching pairs of node labels. This plays a pivotal role in the inverse operation required to relate the instance and reference elements. The insight is that the geometry only plays an intermediate role within how the two element responses differ. If we relate the response values from these two elements, per component, then we can generate a map between the instance and reference elements. The solution to this relationship is defined as the ratio between nodal force response components. This introduces a new concept based on geometric ratios for representing the inverse operation that allows us to transform reference material responses to element instances. This ratio transformation will allow us to generalize material responses to all instance elements within a simulated mesh.
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Geometric Ratios. Given two isotropic elements with the same material properties, we can evaluate the response forces of the material as internal forces within each element due to an imposed displacement. If we displace the same node in each element and the elements are different, then we will generate unique response forces within each element. If the geometry of the two elements is the same, we would expect the result will be consistent, that is, the internal responses are the same. From our assumption that internal force components are always non-zero by construction, we can relate the responses as a ratio between internal forces between these elements. Given a instance element I with nodes /(). /1.I2- .... /„ and reference element It with nodes lit,. H\, It2..... It,, we can establish a ratio between that relates these responses as shown in Equation 3.3.1.
r{I, R) = [Io/Ro, h/Ri,h/R2,In/Rn]T (3.3.1)
Since a displacement incurs multiple response forces for all nodes, we have to consider the relationship between all force components for all nodes. This will result in a ratio vector that has n â–  d ratios, one ratio for each component (DOF) of each node. However, this vector of ratios is only valid for one node ordering between the instance and reference elements. For each node, the parametrization of the instance element will change. Therefore, if we consider the number of nodes as n, we will have n ratio vectors of length n â–  d that represent the full relationship between the two response behaviors provided by the two elements. For example, if we select the first node of a tetrahedral element and transform it to the reference coordinate system we obtain the two elements shown in Figure 3.3.10.
The 12 response force components from each element that are paired with each other based on the node and component labeling to generate the complete set of ratios for the selected node ordering. Expanding the ratio vector based on this example, we can construct the 12 component ratio vector r if we label the internal forces according to their node and force component: Nodes n, w, u, and v have response force components x, y, and z.
= Inx/R-nx = Iw x/Rwx = lux/Rux = Ivx/Rvx
r[AY] = Iny/Rny r[BY] = Iwy/Rwy r[CY] = Iuy/Ruy r[DY] = Ivy/Rvy
r[AZ\ = Inz/Rnz r[BZ] = Iwz/Rwz r[CZ] = Iuz/Ruz r[DZ] = Ivz/Rvz
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Figure 3.3.10: Geometric ratios of the response forces computed between the instance I and reference tetrahedra R. The ratio vector r is computed by I/R for all force components (right). This shows one of four possible node orderings where (n = a, w = b, u = c,v = d). Ratios are ordering dependent.
The reason that each node ordering requires a ratio vector is due to the computational separation of each node. That is, we use the same network structures to evaluate the material response with respect to each node of an element and reconstruct the total contributions from each node as a linear combination. This formulation is possible due to the isotropic behavior of the material formulation. Using the same network structures to evaluate all nodes of all elements both reduces the number of light-weight networks required to reconstruct material response and also considerably reduces the internal complexity of each network. For the instance elements included within a simulation, we pre-compute these ratios which remain constant during the simulation. Since the Poisson ratio (v) of an elements material directly relates to how forces are directed, changing this material property will invoke an update of these ratio vectors. If the Poisson ratio of the elements material is changed during the simulation, then the ratios stored within the element are automatically updated.
3.3.3 Computation Separability
Material element responses are composed of a large number of interdependent behaviors that vary drastically given different material and geometric properties. Attempting to replicate these behaviors based on the parametrization of an entire element will result in networks that require a high level of complexity to accurately reproduce the desired behavior. If the network structure required to model the material, geometric variance, and the resulting behavior becomes too complex, any contributions or advantages of this
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design will quickly diminish. The objective of the NE formulation is to provide a computationally efficient and accurate method for predicting the complete behavior of an elastic material element. To obtain this, the formulation is carefully designed to provide the highest level of accuracy while maintaining a reasonable computational complexity. The design specifically looks at how far the FEA formulation of an isotropic material can be deconstructed to provide a light-weight and flexible adaptation into a data-driven model. We then use the decomposition to separate the model into a large number of synchronized operations.
The decomposition of the FEA formulation for an isotropic elastic material are based on determining how to model equivalent element behavior while minimizing network complexity. This initial step builds towards a new direction in the application of neural networks for elastic material modeling. Instead of requiring an extensive number of internal layers as commonly employed within Deep Learning techniques or attempting to encode physical behaviors within a Convolutional Neural Network (CNN), we look at how we can match the deconstruction of an elements primitive behaviors into a large set of minute, easy to replicate behaviors. To do this, we introduce the two forms of decomposing an element into its most fundamental components by analyzing the structure of the FEA formulation. The first is an observation about the general structure of the matrix equations derived for an isotropic material in FEA. The global stiffness matrix is composed of several individual rows and columns that represent the component-wise contributions of each element stiffness matrix assembled into an integrated system. Since these rows and columns are partitioned into the components of the discrete nodes represented within the system, linear isotropic materials are composed of simple linear combinations of component-wise integration of multiple elements. Element contributions are obtained during the construction of the global stiffness matrix. Based on this decompose elements in a two stage process: (1) node separation and (2) axis or DOF separation. These operations provide the foundation for the aggregate network system that defines the core of the NE formulation.
The premise of separating an element into trivial components that can be computed independently aligns with the prospect of using an aggregate system of light-weight networks to improve accuracy and computational complexity. Based on node separation, we break the element into its n nodes and compute
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the response for each independently. We then decompose each node into its primary degrees of freedom to compute the elements behavior on each primary axis. This significantly reduces the complexity of the behaviors that must be replicated by each network, but increases the total number of networks. With respect to the reference element and the geometric ratios, we perform these computations and store them for each node within the element.
y y y
Figure 3.3.11: Illustration of the reference coordinate system for the ratio and force response model based on computing the material response based on each node independently. Since the element has 3 nodes (a, b, c), the computation is performed three times. The complete element behavior is then generated as the linear combination of each partial response.
Partial Material Responses. Splitting the computation of the elements material response results in the generation of partial element force responses. The material response independently computed for each node will result in n â–  d force responses for each node stored in the vector p, where p is the partial force vector and i corresponds to the index of the node within the element. Partial responses also have an important relationship with the ratios that are computed for each node. To fully account for the geometric and material response of the element, we use the precomputed ratios and the response forces computed per node to reconstruct the elements material response. This means that for every element, there are n partial forces responses and n ratio vectors of length n â–  d. The linear combination of these components then represents the full response of the element due to the applied deformation. The n partial computations are performed at each simulation time-step for all elements.
Internal Force Reconstitution. Separation of the nodes and each of their degrees of freedom requires a reconstitution process that reassembles the partial internal forces generated from the network aggregate.
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This is because each network only computes a partial result of the neural elements response. To extract the final response of the element subject to the applied deformation, all of the partial forces must be reconstituted by performing a linear combination of the partial forces generated by each node. This result is defined as the sum of the partial force vector />, multiplied by the ratio vector r, computed for element node as shown in Equation 3.3.2.
n
fiat — E ri ■ Pi where r is the ratio vector and p is partial force vector for node i (3.3.2)
i=0
This generates the complete response of the element. To provide a consistent reconstruction, a constant node ordering is provided to create memory maps that can be efficiently executed to combine the partial internal forces. This is because as the node order changes with respect to the reference element as nodes are computed independently. Therefore, we must keep track of which partial forces correspond to each node as the element is shifted and the partial forces are generated within the reference coordinate system.
3.3.4 Element Network Model
The proposed neural element model relating ratio and partial force calculations allows for the separation of the geometry and material response of simulated elements. This separation is critical to the parametriza-tion of the data-driven model used to generate material responses of individual elements. Since the ratio transformation allows us to map how forces from the reference element to an arbitrary instance element, we can limit the computation of the material response to the geometry of the reference element. Using this constant geometry, we focus on how to parametrize the material properties of the element and the stress-to-strain relationship exhibited by applying variable displacements. From deconstructing the model into a set of force components along the DOFs for each node, we define a set of networks that map directly to these behaviors. This means that for each degree of freedom in the material force responses, there is an associated network that will be responsible for replicating the desired material behavior. In the general case, for elements with n nodes and d degrees of freedom, we have n â–  d total networks that replicate the response imposed by an individual one-dimensional displacement (dx, dy, dz). This generates an instance
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of the network aggregate that can handle the partial forces generated from the displacements along one degree of freedom. Therefore, to generate the complete response we use multiple instances of the same network aggregate to generate the complete response of the element from the displacements imposed on all nodes. This formulates a total of 2 • n, • d networks for two-dimensional simulations and 3 ■ n ■ d for three-dimensional simulations. This formulation allows us to obtain a set of constant light-weight networks that can trivially capture a smaller portion of more complex material behaviors. Since the separability of the core material response provides a reusable architecture, we can easily instantiate a small number of networks to compute material responses for all elements. This also means that we can aggressively simplify the internal structure of each network due to the designed simplicity in the material response signal. These networks are then aggregated into an element material model that combines the contributions from all networks. This model is then capable of providing complete material responses of an individual element based on a set of n -dimensional displacements di. An illustration of the displacements (dx, dy, and dz) that are used as part of the input to the material network that are applied to the 3D tetrahedral reference element is shown in Figure 3.3.12.
y y
Element Displacement (dx) Element Displacement (dy)
y
Element Displacement (dz)
Figure 3.3.12: Illustration of the displacements imposed on a 3D reference tetrahedral element. Each displacement and its associated material response is used to train an aggregate of material networks. Displacement components dx (left), dy (center), and dz (right) will be recombined into a complete element response.
To represent the material properties of the isotropic elastic material, we define elasticity through Young’s Modulus (E) and the Poisson ratio (v). Since the stress-to-strain relationship is trained based on the elements response to imposed deformations, we express this as a variable displacement d. From the
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nodal separation provided by model, we only have to consider the displacement of the node n within the reference element, the other nodes within the element are considered fixed boundary conditions. This is possible because of the ratio transformation that allows us to apply the deformation of the selected node and DOF to the response generated from node n in the reference element. The number of displacements applied to node n of the reference element matches the number of DOFs, but these are done through separate networks. Thus the resulting element model for each DOF has three inputs: (E, v, d) and one material response output (/).
Evaluating displacements of an element along its DOFs to evaluate the material response is effective for elastic material models that can be recombined after each component is individually calculated. This is compatible with the linear elastic model due to the contribution of response forces that can be predicted from a trained network. The implementation of this network is flexible and only requires the replication of a one-dimensional waveform. Since there are numerous methods varying in network and computational complexity, we briefly analyze prominent candidates.
Network Design. The premise of the neural element approach is to provide a functional decomposition of internal material responses that can be trivially replicated through any number of numerical learning approaches. Candidates for this include multilayer perceptron networks (neural networks), Convolutional Neural Networks (CNNs), and Long Short Term Memory Networks (LSTMs). The design and integration of material properties as periodic input functions indicate that LSTM-based networks would provide an ideal network architecture for the material integration method proposed in Section 3.5. Flowever, the overhead associated with internal state and the complexity of the network for such simple functions incurs a higher overhead than what can be provided through simple multilayer neural networks. This does not eliminate the potential for using these networks with the proposed method as they are required for elastic material responses that are much more complicated than linear elasticity. Depending on the minimization that could be obtained using a CNN architecture for reconstructing the simplified one-dimensional response, this method could also be employed. The design and implementation of the neural element approach allows this component to be interchangeable depending on the required material response com-
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plexity and desired accuracy. The network type and architecture are completely interchangeable.
The design objective of the partial force responses provides a simple one-dimensional waveform that can easily be replicated through the use of a multilayer perceptron. For the initial design, this form of network is used due to the minimal number of operations required to execute the model at runtime. By reducing the internal structure in both hidden layers and layer node count, we can ensure an efficient balance between accuracy and performance that provides a stable simulation.
Triangular Element Network. In the instance of a two-dimensional triangular element, there are three nodes, and two DOFs. Using the system size of the triangular element, there will be six networks that represent the component responses of this element. If we assume that the element is composed of the nodes a, b, and c, the corresponding force components are defined in a force response vector F of size 6. For each response component, we create the network system that will represent the material responses of the material using the reference element. For simplicity, we illustrate how the inputs of two minimal multilayer perceptron networks correspond to the proposed neural element architecture and network system for
the triangular element in Figure 3.3.13.
Figure 3.3.13: Interchangeable network designs that are provided to the core of the neural network material architecture (left). These networks are then assembled into the design required for the selected element type (right). The triangular element has six response forces which requires six network instances that are embedded in the element network. The value of di is defined as the displacement value expressed as dx or dy for 2D elements, each of which requires its own network instance.
The neural element architecture defines a hierarchy of systems that define how interchangeable networks communicate to reproduce the material response of an individual element. Using the network architecture
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as a black-box, we abstract the connections of the inputs and partial force outputs into a consolidated element network. The presented sequence of diagrams illustrates how each component is abstracted into a higher level of the neural element structural hierarchy. This is due to the organization of the networks used to replicate trivial responses at the lowest level and combine these partial forces into the complete element response at the highest level. For the partial material response forces, the networks also have to be evaluated multiple times depending on the displacement DOFs. Thus for two-dimensional simulations there are two instances of the partial material networks corresponding to dx, dy and for three-dimensional simulations there are three instances of the partial material networks corresponding to dx, dy, and dz.
Tetrahedral Element Network. The tetrahedral network is derived from the same network aggregate formulation as the triangular element. The increase in complexity of this network is derived from the element including four nodes and an additional degree of freedom. Since the element has 4 nodes (a, b, c, d), the corresponding force component vector F has components: fci, and /,/, where i G {x,y,z}. The
network system for the tetrahedral element is shown in Figure 3.3.15.
Figure 3.3.14: Illustration of the 3D tetrahedral element system that generates the partial material response for a given displacement ck where i G {x,y, z}. This generates the partial forces with respect to one displacement of the primary node in the reference element, thus there are 3 instances of this network for dx,
dy, and dz.
Partial element forces only represent the material response due to the displacement of one node along one degree of freedom. Therefore, for each node we have to generate 4 sets of 12 response forces for each DOF in the tetrahedral element. This results in the combination of 4 • 3 • 12 = 36 total partial force com-
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ponents for each node. Thus, for each DOF of the simulated system, there must be multiple instances of the partial material response network, but all instances are always generated using the same 36 unique partial force behaviors. While this provides a complex network structure, these operations can be optimized based on real-time performance objectives. This is not a significant memory overhead due to the lightweight nature of the networks that are in general, composed of a small set of weights and the definition of the networks architecture. During execution, these structures decompose to a set of multiplications and additions as the internal operations of the networks are evaluated. Additionally, the partial response vectors can be preallocated and reused for each response force calculation and the node memory map for the element is constant.
Once the partial material response force network is defined for each DOF, they are combined to form the complete element material response. This represents the complete set of combined partial forces that creates the behavioral response of the elastic material for the given element. That is, for each node and its displacements in the dx, dy, and dz directions, we predict three partial force vectors. We then transform these partial force predictions to the instance geometry of the element using the elements nodal ratios. Finally, the forces are accumulated based on their individual components and then mapped to the correct nodes of the instance element using the nodal memory map presented in Section 3.3.3.
To illustrate the final form of the element material response network, we use the entire diagram from Figure 3.3.14 and insert this network into the complete element material response network 12 times to generate the final network presented in Figure 3.3.15. This complete network takes as input the material properties of the element and the 12 displacements of the elements nodes (3 displacements per node). This allows a material response to be predicted as a function of the displacement of all nodes in all directions for any instance element in the simulation.
This model can now be used to predict the behavior of all neural elements within a simulated model based on the elements assigned material and the current deformation state of the element in the simulation. This provides the general framework from which any parametrization of a neural element can be defined. However, depending on the type of simulation and the trained network type, there are imposed limitations on which materials can be expressed using this method. Specifically, the domain of the material proper-
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Figure 3.3.15: Neural element material response network. This network represents the complete response of the material based on the trained internal networks for each DOF and force response component due to the displacement imposed on any component of the 4 element nodes. These are the output forces that are used to drive the dynamics of the simulation.
ties that can be simulated is exceedingly large for what small and efficient networks can represent. From the limitation of the entropy within a network as a function of its architecture, we have to provide an additional solution to representing a wide variety of different materials. Many approaches present a collection of named material classes that can be reasonable represented within a data-driven method. However, while we provide this solution as a subset of possible material properties that we can simulate, we also introduce a continuous material network model that allows arbitrary elastic material properties to be assigned to an element.
3.3.5 Material Network Model
The fundamental problem of representing an unbounded range of material properties is that they cannot be expressed in terms of a data-driven method for one network. While exceedingly complex material behaviors based on several properties could be expressed through a large or deep network, we present an alternative based on decomposing the elastic material domain. For the Poisson ratio (v) of the material,
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this is not required due to the fact that v e [0,0.5] for a vast majority of materials. However, as this ratio approaches 0.5, the response forces become much larger in magnitude due to the material behavior and can present a challenge in training for a data-driven approach. The primary problem is in representing Young’s modulus. This is because the domain of this material coefficient exists in the range [0, oo]. While there is no infinitely rigid real-world material, we must provide a reasonable coefficient range that can be represented through the neural element derivation. Our solution to this problem is to define a reasonable domain of Young’s modulus and discretize the interval into a collection of network instances based on the design presented in Figure 3.3.15. Each instance can then be trained on a different set of material responses based on the intervals range. Here we present a simple example of how we can define four networks that are each trained on a different range of Young’s modulus, allowing each to converge on a reasonable training set. This concept is illustrated in Figure 3.4.17 for a range of E = 1000 to E = 10000.
E = 1000 E = 5500 E = 10000
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Element Model [0] Element Model [1] Element Model [2] Element Model [3]
Material Response Material Response Material Response Material Response
Figure 3.3.16: Illustration of the reference coordinate system for the ratio and force response model based on computing the material response based on each node independently. Since the element has 3 nodes (a, b, c), the computation is performed three times (once for each node) for each degree of freedom. The complete element behavior is then generated as the linear combination of each partial response.
This method defines a larger set of networks that can be used to represent material properties of the elements that are commonly used within real-time simulations. Although this technique is transparent to the use of the neural element method, there are some considerations regarding the accuracy of this method. This is due to the edge cases between the network coverage areas. The discrete boundary between the training sets can introduce slight variances in material behavior depending on the characteristics of the training set used to define the networks. Although, if each element models domain is sufficiently covered in the training data, then the loss in accuracy and stability should be minimal.
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3.3.6 Covering the Material Property Domain
The elastic material coefficient domain can have a considerable range to represent through a collection of discrete networks. Therefore, we look at the constraints of a real-time simulation and impose reasonable constraints on the material properties of the elements that may be simulated. This can provide a limit on the number of intervals generated which can reduce the training required to replicate an arbitrary number of elastic materials.
Real-time Simulation Constraints on the Material Domain. Elastic material properties can vary drastically depending on the types of materials being simulated, each having their own theoretical domain of potential values. Due to this, the coverage of some the elastic material properties can lead to an arbitrarily large numerical domain. Since this is incompatible with the limited domain that can be represented within the training set, we introduce constraints on the domains In the case of Poisson’s Ratio, the numerical domain is (mostly) limited lo v e [0,0.5] (with the exception of more complex materials or structures that exhibit negative Poisson Ratios). This domain can be covered with some discretization of the domain split into different training sets based on the desired accuracy of each network. In the case of the elastic modulus, the discretization of the numerical domain of the variable is much more extensive, E e [0, oo] leading to a limitation in representing all possible values. There are practical limits that ground this value to reality much quicker than anything approaching infinity (for example the elastic modulus of a metal-like materials). Considering the limitations of what is represented within interactive simulations that assume discrete time-steps, floating-point representations, and rasterization, we can further limit the higher bound of this domain. The premise of this reduction is based on two components: (1) the potential perception of infinitesimally small deformations observed in real-time simulations and (2) a reasonable bound on external forces. If the induced deformation of an object is negligible based on the rigidity of the material, then the object approximates a rigid-body and should be replaced by this simpler formulation. If the elastic properties of the material require external force magnitudes that would rarely if ever, be present within the simulation the object should also be replaced with a rigid-body. We also take into consideration that for minute displacements of the element material, the resulting numerical change can be effectively removed
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through the rasterization of the object.
3.3.7 Extensions to New Element Types
The proposed element material network architecture can be extended and generalized to other elements such as two-dimensional quadrilaterals and hexahedral solids. The ability to generalize this method to each form of element is based in how nodal response forces are accumulated through independent computations. Since these properties are common to all discrete elements used within FEA simulations, it can be extended to incorporate new element types. The complexity of the network structure depends on the number of nodes within the element and the DOFs of the system, therefore most simple elements such as tetrahedral elements can be efficiently approximated using this approach. Additionally, since most realtime FEA simulations for continuous materials use three-dimensional objects, the DOF of the system will typically be constrained to three.
3.4 Dynamic Neural Elements
Neural elements introduce a new discrete method for approximating the behaviors of solid materials for real-time and interactive simulations. This requires the development of a dynamic system formulation that is compatible with the proposed neural element design. Building on several established methods, we build a dynamic simulation compatible with the neural element design that is computationally efficient, allows for complex features such as real-time topology changes, and stability. To achieve this, we look at how we can convert element material responses into complex mesh structures and the form of numerical integration required to drive the core dynamics of the system while providing stability and performance.
3.4.1 Material Force Responses
The neural element formulation provides a method for predicting a complete set of internal material responses as a set of forces based on a deformation applied to an element. This process can be used to efficiently determine the material response of an instance element within a simulation. But this only considers the response of an isolated individual element. Since most simulated volumetric meshes are composed of
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numerous connected elements that form a topology, the element contributions to the continuum requires an integration of these element forces. Due to the data-driven nature of the neural element formulation, we are required to operate on the material response of each element independently. However, since we are establishing the response forces of each element, we can compute the net forces acting upon the discrete nodes within the system. Therefore, for each element in the system we predict the material response unique to each elements state and then combine these forces at each node. This process assumes a fairly straight-forward simulation setup based on the evaluation of displacements and the accumulation of response forces. These response forces can then be integrated into a standard numerical integration process to generate the material states such as velocity and position.
3.4.2 Numerical Integration
Numerical integration of dynamic systems is well studied within numerical analysis for methods in mechanics and computer graphics and represents a relatively mature state with regards to accuracy and stability. Due to this there are numerous numerical integration methods that have been introduced, each with their own discrete formulation, performance, and stability characteristics. The two primary classes of numerical integrators are defined as explicit and implicit methods. These two classes are characterized by the underlying mathematical formulations that define how quantities are evaluated with respect to discretized functions of time for solving Ordinary and Partial Differential Equations (ODEs and PDEs).
Explicit integration. This form of numerical integration represents some of the most computationally efficient methods due to the simplicity in the premise of the prediction model which is defined by the current system state to predict the next state. This essentially characterizes the state of the dynamics in the next simulation time-step based on the projection of the current state. Numerically this is straight-forward and computationally inexpensive. Methods in this class include: explicit Euler integration (forward Euler), the explicit Runge-Kutta family (which includes Euler, RK2, RK4, RK5), central differences, and Verlet, amongst several variants of these approaches. While these methods provide efficient and reasonably accurate state updates for dynamic systems, they also incur numerical instability, a highly studied domain within numerical analysis. In most instances, explicit methods are only conditionally stable, that is, in
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the projection of the current simulation state, the time-step (dl ) must be arbitrary small to ensure stability. This behavior is a relational function between the simulation time-step size and the modal behavior of the dynamic object. In both rigid-body and elastic material simulations, avoiding this instability requires reducing the time-step size. This incurs three primary problems: (1) as the time-step is reduced, the object appears to slow down due to the higher number of iterations required to perform state updates for an interactive or real-time simulation, (2) this higher number of system solutions drastically increases computational load, and (3) there are instances where the reduction of the time-step to an infinitesimal value will still not provide numerical stability of the system. These constraints typically eliminate the use of explicit methods for simulations more complex than those based on rigid-body dynamics.
Implicit integration. Numerical integration based on implicit methods are derived from the same premise of obtaining the system state to predict the next state. However the basis of the implicit method is a slight variation within the numerical formulation that incurs the solution of a non-linear system of equations. This automatically incurs two two constraints for dynamic system formulations: (1) since the solution to a non-linear system of equations is required, this system has to be formulated and updated as required by the constitutive model and (2) solving non-linear equations, even with iterative solvers is significantly more computationally expensive then explicit methods. The catch is that for this performance hit, most methods provide unconditional stability. This is a critical component to providing stable simulations with reasonable time-steps, especially for models composed of elastic materials. Numerous methods including: implcit Euler (backward Euler), Houbolt, Wilson-61, and Newmark integration have been proposed, all of which provide reasonable computation times with unconditional stability or parameters that enable stability. For real-time dynamics that incorporate elastic materials, many formulations use these well-established methods due to the unconditional stability provided. Additionally, the formulations of the models based on the elastic material models from FEA are well suited for using matrix form ODE representations for integrating system states.
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3.4.3 Dynamics Formulation
Neural elements introduces a data-driven method for generating elastic material responses based on the imposed deformations of individual elements. From the structural dynamics governing equation presented in the FEA overview, we make a slight adjustment based on the integration of the data-driven element responses that are generated from the neural element formulation. The modified governing equation simply interchanges the global stiffness matrix K and the displacements x, with the response force vector representation r provided by the neural element model. This provides a slightly different formulation of the dynamics equation:
+ Cvt+dt + rt+dt — ft+dt (3.4.1)
where M is represented by a lumped mass matrix, C represents the damping matrix, and / represents the external forces acting upon the system. With this formulation we also store the system state vectors related to the discrete nodes for the displacement dt, velocity vt, and acceleration at. In the FEA formulation, element stiffness matrices ki are generated for each element independently and then combined into the global stiffness matrix K that composes the structure and topology of the simulated mesh. This means that the construction of the global stiffness matrix is defined by the system size of the mesh nodes and DOFs. This results in 2n and 3n matrix sizes for two-dimensional and three-dimensional simulations respectively. This is the formulation currently used within most FEA simulations due to how system states and relational behaviors are encoded from the formalization and discretization of the governing equations. By construction this direction has a couple of drawbacks for the neural element formulation: (1) matrix sizes can become excessively large for meshes with high element counts, leading to sparse matrices that take a considerable amount of memory to store. This problem is directly solved by dense matrix representations that account for the sparsity of a matrix by introducing indexed lists of non-zero values that can be stored within a significantly smaller memory footprint. This is an acceptable solution to this problem, however there remains a small amount of overhead for managing dense representations. (2) The driving force of the neural element formulation is to replace the precisely defined element stiffness matrix with a data-driven model. This means that instead of integrating the element stiffness matrices into the global stiffness ma-
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trix, we must evaluate each elements response to imposed deformations and integrate their response into the dynamics equation.
Development of the core structures that implement the dynamics equation can take two potential directions based on the formulation: (1) the entire system can be formed as a matrix equation, based on the formalization of the governing equation and (2) the evaluation of individual elements and their reactions to the strain within the material. The key difference in these mechanical formulations is how the integration of system elements is handled. For the matrix equation form of the governing equation, the elastic material model is defined to map the relationship between external forces and displacements. This means that over time, the external force vector can be integrated to obtain accurate calculations of the displacements exhibited by the nodes of the system. When the inverse is proposed, as in the case where displacements of an element generate response forces, the integration of the element contributions tot he systems is handled through force consolidation.
Due to the neural element formulation, the response forces of the material is handled per-element. Therefore, the we consider the internal material response r as the primary driver of the elastic material behavior. Because of this, there are implications for the selection and design of the numerical integration algorithms that can be used to drive the system. Purely explicit techniques are easily adapted to this form because they can take as input the mesh node displacements, velocity, and acceleration and provide updated displacements and velocities for the next simulation time-step with direct equations, for example: explicit Euler, central differences, and the Runge-Kutta family of methods. Elowever, many of these do not provide the stability characteristics required to ensure large-deformations will not numerically diverge. Therefore, the challenge is combining the per-element formulation provided by the neural element design with a stable integration method.
For simulations driven by data-driven networks, there have been attempts at implementing implicit integration methods that approximate the Jacobian of the material response to solve the non-linear system of equations and ensure stability, but this requires both the numerical approximation of the Jacobian and solving the non-linear system. Alternative approaches have also been directed at obtaining stable explicit integration techniques by slightly modifying the formulation to impose behavioral constraints on the
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modes of the system. For performance, stability, and the ability to adapt the NE formulation to a parallel model, we employ numerically stable explicit integration.
Stable Explicit Integration. The premise of the improved stability method is to bound system frequencies by providing coefficient matrices that will provide stability within the numerical integration update.
To use these coefficient matrices, we initialize several parameters of the simulation including the time-step dt, the mass matrix M, the rest stiffness matrix of each element K0 (linear FEA), and the damping matrix C. With these initial values, we can compute the coefficient matrices /31 and r>2 that are used in the integration performed for each time-step. These are defined as:
/?i = [I + \-dt- M-1C + \-dt2 â–  M_1K0]_1 x [I + \-dt- M-1C] p2 = \[l + \-dt-M~1K0}-1
The integration also requires the initialization of the displacement and velocity vectors using the newly computed coefficient matrices. We pre-compute and store these values per element as element stiffness matrices and state vectors. It is assumed that the simulation represents a simple initial-value problem where these quantities are provided by the initialization of the simulation.
ddt = do + I3\ ■ dt • v0 + /?2 • dt2 • a0 vdt = vo + — • dt ■ (ao + a dt)
This is where we alter the formulation presented in [29] and change the representation of the mass matrix M, stiffness matrix K, and damping matrix C to represent the structural and dynamic properties of a single element. This means that there are a total of n mass, stiffness, and damping matrices for a mesh with n elements. Immediately, this causes a problem within the stable explicit method. This is because the coefficient matrices d\ and d2 are now dependent on the element stiffness matrix instead of the global stiffness matrix. This means that for neighboring elements, there will be different coefficient matrices that pull the elements in different directions.
Deformation and Integration Update. The simulation update contains several steps related to generating displacements within the corotational coordinate system, evaluating the element material responses, ensuring the net force of each element is zero, and the actual integration step. For the evaluation of the
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y
y
Element Coefficient Matrices Element Response Deformations
y
Updated Deformation State
Figure 3.4.17: Illustration of the independent response forces for adjacent elements due to the coefficient matrices f3\ and r>2 (left). The net forces (center) and displacements (right) of the nodes are averaged due to the differences in the responses provided from each element.
material responses for each element, we define that the response vector r is returned by the neural element function NE(e) that takes a deformed element and provides the response of the element as internal forces. The function EqualizeNetForce(r) ensures that the net internal force of the element is zero. This is accomplished by performing a mean-shift on each force component. This update is performed per-element within the main simulation loop.
For each element within the simulated model:
dt+dt = dt + pi ■ dt • vt + ■ dt2 ■ a
r t+dt = NE(dt)
rt+dt = Fquali/.eNetForcefr,,,//.)
vt+dt = [M + ^ • dt ■ C]-1 • [M[vt + ^ • dt ■ at] + ^ • dt ■ (ft+dt - rt+dt)] &t+dt = [ft+dt Cvf+(if rf+(if]
Where ft+dt represents the external forces acting upon the current element. Due to the integration constraints based on the coefficient matrices, the force responses from each element are consolidated between each time-step. This resolves the problem of discontinuities that would be introduced within the mesh topology between elements. Although this allows this form of integration to be used with the distributed set of elements, this also introduces error coefficients that result from the deformation of independent ele-
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ments. The introduction of this error is justified by the performance improvements and flexibility afforded to the framework of the simulation. This is one of the benefits of evaluating each element independently and quickly resolving material response discrepancies between elements. While this is a side-effect of this integration method, the core method can be adopted to alternative integration methods.
The evaluation of the neural element function that generates the material response is transparent with respect to the integration method. This allows for the formulation of the integration technique to be interchangeable. As with most deformable simulations, the selection of the integration method depends on the accuracy and computational complexity of the method and the target application domain. The stable explicit method is employed by default due to the simplicity of the update, in that it does not require solving a non-linear system of equations. This method also enables a trivial parallel implementation of the primary computational load of the technique which is the evaluation of the response forces from the neural element network.
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3.5 Elastic Material Data and Training
Exploration of data-driven methods that improve elastic material behaviors or introduce real-world approximations of material properties is challenging due to the acquisition of sufficient training data that is required to define complex elastic material behaviors. Numerous methods that utilize real-world data can obtain accurate stress-to-strain relationships through rigorous experimentation and integrate these relationships into finite element based simulations to improve the accuracy of the simulated materials. Many of these methods that primarily aim to replicate the behaviors of real-world elastic materials are dominated by models that are more accurate for non-linear behaviors such as the St. Venant-Kirchhoff, Neo-Hookean, and Hyperelastic formulations. Additionally, most of these models are computed off-line (not real-time or interactive) using accurate solution methods such as allowing Newtons method to converge to very small error tolerances. The data acquisition process of these methods is challenging to extend past the parametrization of nonlinear material models due to the complexity in obtaining accurate experimental results by manipulating elements reliably to formulate a complete model based off of data observations alone. While the neural element formulation does not completely eliminate the use of the elastic material formulation, it defines balanced contributions from existing material response formulations and data-driven observations that could be augmented with real-world data similar to existing methods. The objective of the approach is to provide an efficient method for generating high-performance solid elastic material simulations used in real-time and interactive applications based on observations from a variety of sources that are compatible with the approach. Therefore, the premise the neural element approach can also rely on other forms of data sources including real-world experimental observations, existing FEA models, and even slight modifications of existing model behaviors through data augmentation and nonlinear manipulations.
3.5.1 Synthetic Elastic Data
The premise of the neural element method is to integrate the behaviors of elastic solids formalized by a data-driven source to replace the constitutive model of elastic materials. Through the decomposition of the
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isotropic material model, we have introduced a system of neural networks that can be trained to replicate the DOF specific behaviors of an element. Naturally, this method requires an extensive amount of data to replicate the material behaviors of an element based on every node and DOF. The required data and experimental procedure could be minimized to provide an accurate representation of the material, but this is secondary of the objective of the NE formulation. Since we aim to provide a flexible method for interactive simulation for elastic solids, we are more interested in the ability to formalize the model to maximize performance and how the model can be adapted to graphics applications. Therefore, we build the core concept on the presented network architecture and utilize synthetic data generated from exact FEA simulations in an attempt to replicate the accuracy of the purely analytical solution. The primary motivation for this direction is not to limit the source of this data-driven method, but to establish the formalization of the core architectures that allow us to mix the analytical and machine learning components into a cohesive model.
The synthetic training data required to establish the stress-to-strain relationship of the isotropic elastic material is derived from the solution of the analytical FEA model evaluated on the reference element. The core data generation algorithm for this process is as follows: (1) generate a variant of the elastic material properties used to represent the element (E, v), (2) induce and record a displacement d applied to the origin node of the reference element to generate response forces, (3) for each of the element response forces, record all n â–  d force responses as inputs to the network aggregate. Each networks inputs (E,v, di) and output / is stored independently. To represent a variance of the material properties as defined by the material network model, samples are generated based on changes in E, v, and the displacement di described in Section 3.5.2.
Synthetic data also benefits from a complete absence of noise. While the sampling of the material properties used as input is still discrete, the synthesized waveforms contain very little variance. This allows us to train on these datasets to obtain very low error rates as the training converges. This also factors into the ability to minimize the internal structure of the employed network architecture, thus improving the performance of the method overall.
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3.5.2 Material Property Integration
Every network within the material network model requires the generation of the input values E, v, and di along with the expected value / to create training datasets. To maintain generalized training sets, the samples are generated through the use of integration functions. This allows the material properties and responses to be integrated as continuous periodic functions that can be used with a variety of different network architectures including LSTM networks. Since the integration functions are continuous and periodic, we can ensure that generated expected values follow smooth response curves. The generation of these samples are based on interchangeable functions that can be varied to shift the density of training samples based on specific objectives. We introduce these integration functions for each input variable including the material properties E and v, the displacement value di. Due to the responses of the elastic material, the output response force within each DOF can also be continuous and periodic, but this also depends on how the three waveforms are synchronized as discrete samples provided to the input of each network.
Having each input value oscillate through a continuous periodic function allows us to specify several characteristics of these integration functions, the sample combinations they generate, and the behaviors of the response forces. This includes: (1) the definition of the integration function itself (such as sin, cos, or various others), (2) the number of samples per period, and (3) the frequency of this function. The selection of these properties determines the characteristics of the material response forces that are generated from the applied displacements. This also introduces the concept of relative frequencies. Due to the interplay between the material properties and the applied displacement, the response force magnitude can fluctuate wildly. This is an undesirable characteristic of the training data due to the low rate of convergence on these high magnitude spikes or anomalies within the dataset. The objective in specifying the integration function, resolution, and frequency of the dataset is driven by which types of responses should occupy more of the dataset samples and which types of signals can promote high convergence rates.
Integration Functions. The integration functions that characterize the generation of the material response datasets for the reference element can be defined as any continuous periodic function. For trivial functions, the periodic nature can be used directly in the dataset generation process. This means that any well
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behaved functions can be used to vary the material properties and displacement within a training set. For the implemented training set, we provide two primary integration functions: I\(x) = sin(x) function and a variation of the first derivative of the Gaussian identified as h{x) = dGauss(x). This function is defined in Equation 3.5.1. The plots of the two integration functions are shown in Figure 3.5.18.
h(x) = sin(x)
h(x) = d,Ga,uss(x)
dg(x, a) dx
x _^ rr\ 2,7( "
(3.5.1)
Integration Function: sin(x)
Integration Function: dGaussMod(x)
Figure 3.5.18: Integration functions used to encode the material properties (E, v) and the displacement applied to the reference element di to generate training datasets. These functions represent just two selected periodic functions that can be used to encoded these network inputs.
The primary purpose of the integration functions is to provide a continuous function that defines the network input values based on the combination of possible inputs. This is due to the number of possible combinations that can be expressed for each variable. For example if we set Young’s modulus E = 1000, the Poisson ratio v e [0, 0.5]. Therefore, these variables are completely independent, leading to an enumerated two-dimensional sample space. To ensure that there is sufficient coverage of these domains, we have to provide variations of these two variables that generate an adequate representation of the possible material property combinations. This is not only critical to the accuracy of the proposed method. If there is insufficient coverage of the training domain, then the resulting forces will introduce higher vibration modes within the system, leading to numerical instability.
The concept behind providing different integration functions is based on the density of the samples used to represent this two-dimensional material domain. For instance, if the sin(x) function is used to represent the displacement imposed on the reference element, there is a reasonably uniform sample dis-
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tribution. Therefore, for any potential displacement applied to the element, the trained network should provide a reasonable force response. However, in the case of the dGauss(x) function, the samples will represent a higher concentration of smaller displacements. This means that the network will provide a more accurate approximation of response forces for small displacements and fewer samples for large displacements. Since the network will perform a form of interpolation between training set samples, there are different integration functions that can work, but depending on the required accuracy and simulation behavior, there may be benefits to selecting one over another.
3.5.3 Displacements and Force Responses
The generation of the training set based on expressing the material properties and displacement applied to the reference element is handled through the solution of the equation F = KU. Since we can formulate an isotropic FEA system that has one element (the reference element) characterized by its element stiffness matrix K, we can apply the deformation defined by the training set as U to create a set of response forces F. For a given training configuration, we express the range of Young’s modulus between Emin and Emax- Similarly, for the Poisson ratio, we define the range as and These quantities characterize the integration functions (their amplitudes) effectively defining a distribution of samples between these minimum and maximum values. A similar range is used to describe the displacement applied to the reference element where d~ and d 1 represent the negative and positive bounds on the displacement. For the value of this displacement range to be effective, we assume that the scale of the reference element has been normalized such that its nodes he on the unit circle (2D) or unit sphere (3D). Then reasonable values of the displacement can be defined as: d~ e [0, —1] and d1 e [0,1], This essentially trains the network to handle instances up to where the element will invert. From the solution of the finite element method for the isotropic elastic material, this displacement range can be set higher, and the correct response forces will still be provided, even in the case of element inversion.
Following the standard process of normalizing the network inputs, we present the plots of the input datasets used for training the network models. Due to the large number of networks, we select an individual example. This includes the distribution of the normalized samples for both the sin(x) function shown
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Training Data: lntegrationFunc=sin(x) Training Data: lntegrationFunc=sin{x)
Samples [n] Samples [n]
Figure 3.5.19: Illustration of the sin(x) integration function applied to the displacement (d) as the input to the training model (left). The material response along the provided DOF of the reference element is shown as the expected force (/) value (right). Note this is only a subset of the samples.
in Figure 3.5.19 and the dGauss(x) function shown in Figure 3.5.20.
In these instances, the material properties are encoded using the sin(x) function while the displacement is encoded using the dGauss(x) function. The characteristics of the output waveform are defined by the integration functions and the relative frequency of the inputs. This means that the complexity of the expected output depends on the relative behaviors of the three input functions and can be modified by altering the parameters of the training set. The objective here is to maximize the coverage of the material property domain while providing the highest convergence with the smallest network structure possible. To provide the complete training model for the system of networks, we generate the training sets provided in
Training Data: lntegrationFunc=dGauss(x) Training Data: lntegrationFunc=dGauss(x)
Samples [n] Samples [n]
Figure 3.5.20: Illustration of the dGa.uss(x) integration function applied to the displacement (d) as the input to the training model (left). The material response along the provided DOF of the reference element is shown as the expected force (/) value (right). Note this is only a subset of the samples.
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Figures 3.5.19 and 3.5.20 for n • d networks. This means that for a two-dimensional simulation we will generate 6-2 = 12 networks, each containing their own E, v, and d inputs and / output and for a three-dimensional simulation we will generate 12 • 3 = 36 networks. In our implementation we utilize the Fast Artificial Neural Network Library (FANN) that provides a simplified C interface to optimized network implementations for basic multilayer perception networks. Since the networks are composed of a minimal number of layers represented by simple weights, we store 36 individual 4KB networks.
3.5.4 Training
The training process defined by the NE formulation is designed to rapidly converge to the simplified waveforms that contain no noise due to the use of synthetic data. Therefore, the error behavior observed over the course of a large number of epochs converges quickly. This error provides a validation of the simplicity of the network models that are required to replicate the expected force values for the deformations applied to the reference element. This allows each network to be defined by the simplest internal architecture possible to replicate material responses efficiently. Since the original finite element formulation performs numerous matrix operations that can be effectively optimized, this task is not inherently trivial. Therefore, there is a delicate balance between the target accuracy obtained by each network and the performance of the networks internal architecture. The training error for networks containing 4 and 6 hidden nodes are shown in Figure 3.7.22.
0 10 20 30 40 0 10 20 30 40
Training Epoch [n] Training Epoch [n]
Figure 3.5.21: Plots of the training error (loss) over the course of 5000 epochs. The training converges quite rapidly due to the lack of noise within the synthetic data and the simplicity of the waveforms.
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3.6 Parallel Elements
The individual material responses of elements provides the foundational outline for performing parallel executions of the networks used to generate material behaviors. The premise of the parallelization that can be performed using this method is based on the execution of these networks as separate instances.
As an advantage provided by the formulation of the material network that uses nodal separation, all of the independent networks are the same for all elements, even for different material properties. This means that once we obtain the trained networks, we can use multiple instances of the same networks for all elements. For the parallel computation of elements, instancing removes the dependency constraints between elements. Therefore, the execution of the material network can be made trivial for all elements by providing each computational unit with its own local copy of the networks required to compute the elements response. Additionally, since the networks are constant, they can be preloaded and store as constant instances. Furthermore, they are represented by a small number of weights so the memory overhead is remains minimal.
The primary challenge associated with the parallel form of the NE formulation is the synchronization and data dependency imposed on the resolution of the net forces acting upon each node. Since each element may have numerous adjacent neighbors, the resolution of these net force additions are the bottleneck of the implementation. This is because the current solution incorporates the use of atomic operations to ensure that the contention over the nodal forces is resolved.
The current implementation utilizes OpenMP to enable parallel execution on multi-core CPUs. Flow-ever, based on the design of the parallel version of the NE design, this method can also be extended to a Graphics Processing Unit (GPU) where each CUDA core or stream processor can store local copies of the trained networks to allow each compute unit to process an individual element. Flowever, based on the same bottleneck that is imposed within the core design, the use of atomic floating point additions can be used to resolve race conditions, but will still cause a significant reduction in the overall performance.
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3.7 Experiments and Dynamic Simulations
The main contribution of the neural element method is a scalable and flexible simulation framework for deformable solid objects. Since the method is heavily derived from the FEA-based formulation, the accuracy of this method resides within the realm of physically plausible behavior but defines new real-time alternative to exact solutions. Unlike prior methods such as mass-spring systems that use arbitrary internal connections and material (spring) coefficients, this method provides standard coefficients to define elastic material behavior. To illustrate the power of the inverse modeling approach used to derive the NE model, we can look at the prediction accuracies within the model and the resulting simulation behaviors. This includes several different standard deformations, the interactive deformation of arbitrary meshes, and data-driven modifications of material properties.
Interactive Deformation Real-time Stress Analysis Dynamic Boundary Conditions
Figure 3.7.22: Illustration of neural element meshes simulated in real-time. This includes interactive deformations (left), real-time stress analysis of elements (center), and dynamic boundary conditions for generating complex animations (right).
The neural element formulation provides a new method of simulating elastic materials based on the large collection of trained networks, but is hard to visualize without demonstrating the simulation of dynamic behaviors the model provides. To illustrate the capabilities of this core material model, we present several standard deformation behaviors. This includes various forms of deformations that include: (1) stretching and compression, (2) cantilever beams, (3) twist deformations, and (4) interactive deformations through pulling localized regions. The following deformation examples have been generated with Young’s modulus of E = 10000 and Poisson ratio of v = 0.3.
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Frame [100] Frame [120] Frame [140] Frame [160]
Figure 3.7.23: Illustration of neural element bar mesh being compressed through two fixed boundary conditions. The left face nodes are fixed and the right face nodes are being directly moved (displaced) to reduce the distance between the two ends of the bar. This results in a compression of the material between these two boundary conditions.
3.7.1 Deformation: Compression
Compression of a neural element mesh is characterized by fixing two boundary conditions applied to each end of the mesh and then reducing the distance between these fixed nodes. As the distance between the fixed ends is reduced, the material is subject to the behavior induced by the selected Poisson ratio (v = 0.3). Initially, this defines that the middle of the deformed bar will begin to bulge out in the center due to the displacement of the internal material. The secondary behavior exhibited by this deformation is the buckling induced within the material as the slight numerical variances cause the material to displace perpendicular to the compression, as shown in Figure 3.7.23.
The von Mises stress magnitudes are shown using the color map and indicate a mixed representation of the normal and shear stresses computed within each element. The stress patterns illustrate the differences in how the internal forces begin to introduce buckling behaviors that are indicated by the high stress regions that are incurred during the compression deformation.
3.7.2 Deformation: Stretch
Stretch deformations are characterized by boundary conditions that are the same as those used within the compression deformation. That is, the left face nodes are fixed and the right face nodes are incrementally moved to increase the distance between the two ends of the bar. As this deformation occurs, the Poisson ratio of the material is clearly shown due to the contraction of the material within the center of the
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Frame [20] Frame [40] Frame [60]
Figure 3.7.24: Illustration of neural element bar mesh being stretched. The left side face nodes are fixed and the right face nodes are incrementally moved to increase the distance between the ends of the bar. This results in an elongation of the bar and a narrowing of the center due to the internal material displacements.
bar. This is because of the material displacements within the center becoming elongated as the material is stretched as shown in Figure 3.7.24.
3.7.3 Deformation: Rotational Twist
The introduction of this deformation provides an illustration of a dynamic boundary condition that can be defined through arbitrary node manipulations. Specifically, we fix the left side face nodes of the bar mesh and the rotate the right side face nodes about the twist axis. This boundary condition is imposed by simply altering the right face node positions directly by rotating them about this axis. The neural element material model will then automatically account for the changes in these node positions and propagate the imposed deformation.
For highly elastic materials, the twist deformation results in substantial displacements of the material elements. The correct behavior of this example is only possible through the polar decomposition of the elements deformation and rotation components. If this form of large deformation is applied to FEA simulations that do not incorporate this process, then as the material is twisted, the resulting volume of the mesh will grow considerably. This results in extremely large errors within the simulation as a consequence of the underlying material model not being rotation invariant.
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2.92e+3
fl
I
O.Oe+O
Frame [20]
Frame [40]
Frame [120]
Frame [80]
Frame [100]
Frame [140]
Frame [160]
Frame [180]
Frame [60]
Frame [200] Frame [220] Frame [240]
Figure 3.7.25: Illustration of a dynamic boundary condition that can be imposed on a mesh to create a twist deformation. The right side face nodes are rotated about the twist axis (x) and the rest of the material behavior is generated through the neural element responses.
An important note about this is that the boundary condition can be changed at any time and does not require a system reconstruction as with FEA-based simulations. This is partially due to the formulation of the stable integration method used with the system. To provide the flexibility commonly required for interactive applications generally provided by simpler systems such as mass-spring models, dynamic boundary conditions allow the user to modify constraints at any point. For neural elements, this does impose any performance penalty due to a system reconstruction.
3.7.4 Interactive Deformations
The benefit of introducing real-time simulation models is that they can provide interactive deformations for illustrating real-time stress evaluations. This means that we can derive an interaction method for controlling localized regions of the material by applying arbitrary external forces or directly specifying exact node positions and view how the imposed constraint effects model behavior. In the application that allows real-time interaction with neural element models, we implement a simple picking scheme where specific
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faces or nodes of a model can be pulled based on manipulations based on movement. This provides a control metaphor for pulling on specific regions of the model to impose external forces that will lead to deformations. As the model deforms, new regions of the model can also be deformed. Examples of interactions with the Stanford bunny model are shown in Figure 3.7.26. In this example, there are multiple different interactions imposed on the model throughout the simulation. These external interactions are illustrated through the force vector imposed on the region of the model indicated. In total there are three individual deformations imposed on the bunny face, ear, and chest. Each interaction is shown over the course of several frames and the length of the applied force vector indicates the magnitude of the interaction.
2.34e+3
0.0e+0
Frame [100]
Frame [180]
Frame [40]
Frame [120]
Frame [200]
Frame [60]
Frame [80]
Frame [160]
Frame [240]
Frame [140]
Frame [220]
Figure 3.7.26: Illustration of neural element meshes simulated in real-time. This includes interactive deformations (left), real-time stress analysis of elements (center), and dynamic boundary conditions for generating complex animations (right).
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3.7.5 Free-form Deformation and Collision
The neural element formulation natively provides the ability to handle arbitrary rigid-body or free-form motion in addition to the deformations within the model geometry. Illustrating a simple example of this, we provide a simple collision between a dropped model and an infinite plane. For this example, we apply gravity and allow the model to drop from a predefined height. This allows the model to make contact with the ground and illustrate the element stresses within the model. Due to the weight shift of the model (due to the ears), the model rolls onto the ground where it rests as shown in the image sequence of Figure
3.7.27.
2.34e+3
if)
~~ if) 0)
___-M
LO
if)
~~ 0)
if)
c
-- o >
I
0.0e+0
Frame [20]
Frame [100]
Frame [180]
Frame [60]
Frame [40]
Frame [120]
Frame [140]
Frame [200]
Frame [220]
Frame [80]
Frame [160]
Frame [240]
Figure 3.7.27: Illustration of a neural element model undergoing rigid-body translation (as it falls due to applied gravity), resulting in a collision with the ground. Due to the mass and size of the ears, the model rolls after making contact with the ground.
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3.7.6 Quantitative Evaluation
The neural element model provides a valid method for the simulation of isotropic elastic materials within real-time applications. However, we can also quantitatively evaluate the response forces that are predicted by the data-driven model. Through the evaluation of the response forces compared to the exact FEA-based result, we can evaluate the accuracy of this method. Due to the high number of predicted forces that compose the data-driven component of the model, we have to select a representative subsection of the forces to evaluate. To do this, we have selected the deformation of an element from the beam stretch simulation. Under gravity, the material response is rather tame, only generating waveforms similar to a sine function with some abnormalities, as the beam oscillates between a compress and stretched state. Therefore, in the following response force prediction, we add significant interaction forces to generate more complex waveforms as shown in Figure 3.7.28. This illustrates, for each component in the element (3D tetrahedra), the exact and predicted force responses of the element as large-scale deformations are applied over the course of 800 simulation time-steps (frames). The predicted value (pr) is from the proposed NE formulation and the exact value (ex) is provided by an exact FEA solution of the material response.
3.8 Natural Perturbations
Analytical closed form solutions to the governing equations of motion and elasticity generate numerically precise values that result in pseudo implausible physical behaviors related to balance and symmetry. This is due to the numerical representation of the equations as they are computed. Neural elements do not natively exhibit this behavior due to the randomness introduced by both initialization and the specific characteristics of the training data. This leads to a more natural or expected result for simulating imperfect elements that compose a continuous material.
If the numerical variance within the network is minimal, this can provide natural behaviors to simulated objects based on the perturbations introduce in the elastic material responses. However, if these small fluctuations become too large, they can impact the accuracy of the simulation or in the worst case cause numerical instability. To alleviate this problem, we look at how natural variance in the network can
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Response Force (Exact)
0 100 200 300 400 500 600 700 800
Response Force (Exact)
0 100 200 300 400 500 600 700 800
Response Force (Exact)
, ft. i * ; —a_ex_fdy a_pr fdy

*
0 100 200 300 400 500 600 700 800
Response Force (Exact)
0 100 200 300 400 500 600 700 800
Response Force (Exact)
—a_ex_fdz —a_pr_fdz
VJ ——
0 100 200 300 400 500 600 700 800
Figure 3.7.28: Illustration of a subset of material responses generated by the neural element model for one element during a simulation of 800 time-steps (frames). This illustrates the behavioral error associated with the neural element predicted value (pr) and the exact value (ex) provided by an FEA-based solution. For the selected 3D tetrahedral element, this is shown by the 12 component forces of the element. The waveform is sporadic due to the large deformation of the element through user interactions.
be minimized and how we can ensure zero net internal force within simulated elements for numerical sta-
bility. This is enforced after the networks are used to compute the predicted material response. That is, for
each element, the internal response forces are mean shifted to ensure a zero net internal force.
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3.8.1 Network Numerical Variance
Precise system formulations in physical simulations have the ability to generate perfectly symmetrical or unique system responses due to how accurately they are defined. Behavioral symmetry and abnormal numerically perfect conditions (such as a perfectly inverted triangle) are the product of floating-point arithmetic and algebraic equations that do not introduce chaotic variance into the system. This is due to the precision in which all of the simulation geometry and initial state can be defined. For instance, if a simple line is balanced on a point vertically, the chaotic interactions in the real-world (vibration, airflow, friction, etc.) would result in a slight acceleration in the x or z directions, resulting in the rigid-body rotation of the object (it will simply fall over). However, for the same instance within a simulation we can define this perfectly vertical line, and specify that there is no x or z direction acceleration by the oversimplification of the real-world presented within the simulation. This will result in the line existing in a perpetual vertical state.
3.8.2 NULL Displacement Responses
The randomized weights used to initialize the neural network and the samples within the training set may introduce a non-zero bias within the response forces r(d) computed when the provided displacement d is equal to zero (dx, dy, dz = NULL). Since the training will not result in a zero magnitude response force provided by the network when the displacement equals zero, this non-zero network computation will introduce a constant shift of the predicted response force p(d). This does not effect the behavioral result of the response for non-zero displacements, but introduces a constant shift similar to an integration constant which effects the net force on the element due to the displacement. This shift does not invalidate the behavior of the material, but can lead to non-zero internal forces within the element. To provide an accurate representation of the network accuracy and resolve the non-zero internal net force, this offset can be resolved. The solution to this problem is to establish the NULL displacement offset eo required to generate a net zero material response force for the network. This value is obtained by collecting the prediction value for a displacement of zero and then subtracting it from all future prediction values: rid) = p(d) — eo.
Removing this shift results in an accurate replication of the elastic material response. To illustrate
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how this correction improves the accuracy of the response force prediction, we present the force responses of an element within a cantilever beam under the influence of gravity. The response force generated by a dx network and its associated offset correction is shown in Figure 3.8.29.
Figure 3.8.29: The initial randomized weights of the network contribute to residual non-zero response forces provided for the input displacement of zero. This results in a constant non-zero shift in the response forces that contribute to a net positive or negative internal force within the element. This is corrected by removing the shift to obtain accurate force response estimations.
In this instance, the constant shift embedded in the network evaluates to 7.728[iV], that is, for an input displacement of zero, the response force generated is 7.73[iV] in the x direction, contributing to the constant offset. This offset can be corrected for each network to reduce the prediction error of the response forces, but it does not completely eliminate the non-zero internal force of the element. It is important to note that this slight shift does not dramatically effect the behavioral result of the simulation, only the quantitative evaluation of the response forces with respect to the exact values. Additionally, the selection of the integration function also influences the error in the prediction.
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CHAPTER IV
GENERATIVE DEFORMATION: 3D PRINTING BEHAVIOR
3D printing technology is developing at a blistering pace and has expanded to wide reaching implications for both academic and industrial research in product design, engineering, and manufacturing [30], Yet, the control of elastic deformations of 3D printed objects remains challenging due to the complex interrelationship between the properties of available elastic materials, the density of internal lattice structures, and the connectivity of procedurally generated print geometry. To this end, several recent contributions have significantly improved the iterative design process to incorporate Finite Element Analysis (FEA)
[31] through various software packages to facilitate print optimizations for stress reduction [32], print material minimization [33], internal lattice generation [34], vibration reduction [35], print structures [36], implications of layering orientation, and numerous other metrics. While the design, simulate, refine, print pipeline has been well established for modifying characteristics of 3D prints, there still remains a limited set of tools that provide an end-to-end solution for providing precise control of deformation behaviors in 3d printed objects based on design constraints.
Several variations of simulation-based optimization pipelines have been introduced [37] within the domain of 3D printing to address a numerous challenges with printing controllable, structurally robust objects and parts. Most pipelines attempt to introduce an overarching bridge between the initial design of a part and the resulting print subject to several additional constraints that can be defined to optimize the print with respect to a given set of objectives. Significant contributions have been introduced through the precise control of elastic structures using micro-structures [38] and the behavioral optimization of multimaterial micro-structures [39], However, these advanced techniques assume that 3D printers with an extremely high print resolution or multiple mixable materials are available. The problem is that the cost and limited set of tools that are specifically designed for these expensive 3D printers are challenging to use with consumer-level printers that have limited elastic material capabilities.
In this work, we introduce a consolidated pipeline that integrates both automated perforation, deformation behavior, and stress analysis to provide an automated process that allows for the development of elastic 3D prints on consumer-level printers. Specifically we focus on combining generative modeling
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Automated Volumetric Perforation Output: Generative deformation
Figure 4.0.1: Generative Deformation: The automated process of perforating volumetric simulation meshes that are 3D printed using elastic materials to obtain specific deformation behaviors.
with FEA dynamics, but integrate a 1-to-l pipeline between the simulation and print geometry. That is, the procedural geometry generated by the generative algorithm is consistent between both the simulation model and the 3D surface model provided to the printer. This process is completely automated and integrated into a single design tool. This allows the application to be used for quickly generated perforated geometric structures for quickly changing the material properties of 3D printed elastic materials.
These contributions incorporate a design application that integrates existing volumetric mesh generation algorithms, FEA simulation, and perforated internal structure generation to quickly obtain 3D prints specifically targeted for consumer-level 3D printers. Building on this pipeline, shown in Figure 2, we facilitate a method for generative deformation which allows for a heterogeneous lattice structure to be generated with respect to deformation objectives. We define this process as the procedural generation of internal geometry that is derived from two primary metrics obtained through FEA simulation: (1) measures of tetrahedral element deformation using dihedral angles and (2) internal stress distributions derived from nonlinear FEA simulations [40,41] using VegaFEM [42], This objective does not only impact passive deformations of 3D prints, but can also be used to modify activated behaviors of 4D prints based on how heterogeneous structures impact actuated movements [43],
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4.1 Related Work
In general, generative design is loosely defined by the process of altering the geometric structure or material properties of a part or model to achieve a specific behavioral characteristic. Typically this is related to the stress analysis of the part for load tolerances, plastic deformation or fracture. Sine the objective is to maximize these tolerances or load capacity, the design can be optimized by making slight structural adjustments to the underlying 3D model. The process is defined a simple iterative design pipeline: (1) design a 3D part with the required specifications, (2) load the model into an FEA-based simulation to establish loads, boundary conditions related to the use case, and evaluate the stress distributions, follow by (3) running optimization algorithms that modify the design of the part to minimize stress, material use, or adjust the high-stress regions of the part to provide the tolerance required for the load condition.
There are numerous methods that are currently being explored by high-end labs for complex microstructure property and behavior estimation for high-end 3D printers [27], For high-end printers, there are two primary methods that can be used to generate precisely controlled micro-structures: (1) high resolution printers that can print dual materials, one as the primary material and a secondary that provides support for the primary that is then dissolved or (2) multi-material printers that can dynamically adjust the composition of the print material to adjust the elastic material properties of the print. Both of these solutions are typically removed from consumer-level printers that are commonly limited to very few print materials. This is due to the complexity of the 3D printer design required to facilitate dynamically mixed materials depending on a models material properties. Naturally, this approach to 3D printing complex models will inevitably become more common, but the ability to dynamic alter the underlying geometry of 3D prints is still a viable tool for changing a models deformation behaviors.
4.1.1 Internal Meshing
Consumer-level 3D printers are typically combined with design software provided by the printers manufacturer, or open-source alternatives. These applications are referred to as slicing programs because they provide the pre-processing required to transform 3D models into solid print instruction that the printer
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can parse to generate the model. These applications are called slicers due to the nature of slicing a 3D model into discrete layers that can be replicated by a 3D printer. Within most of these applications, there are additional functions that can be integrated into this basic slicing process. This includes important features such as internal meshing which allows for solid internal regions of a mesh to be hollowed out and replaced with a sparse mesh structure. Typically this greatly reduces the print material used to create the model and can significantly reduce print time. While this similar geometric approach to creating meshlike structures within a 3D print is common, it is distinctly different than the proposed perforation method. This is because most implementations that generate internal meshing essentially superimpose the mesh structure within hollow regions of a mesh without addressing the integrity of the resulting geometry. That is, all of new geometric shapes are simply placed into these regions without resolving the topology between the mesh structure and the original print surfaces. The consequence of performing these forms of naive internal meshing algorithms on meshes, is that the mesh can no longer be physically simulated. This is because to physically simulate a 3D mesh, all of the internal geometry must form one continuous topology that defines a solid material. For example, if there are two faces that intersect each other, but are not connected by unique nodes, then the model cannot be simulated. From computer graphics, this problem can be resolved for general techniques in meshing such as Constructive Solid Geometry (CSG), however this technique does not automatically generate FEA-based internal solid elements for elastic material simulations. CSG techniques could be paired with existing finite element meshing algorithms, however the resulting geometry will be extremely complex and limit the ability to use the simulation mesh within interactive applications. One of the objectives of the perforated structure is to maintain some level of interactive simulation based on complex structures. This means that the geometric complexity and number of the solid simulation elements (tetrahedra) has to be minimized. For a given volumetric tetrahedral mesh, the topology consistent perforation of the structure provides the minimal number of elements that can be used to simulate an element-wise sparsely hollow mesh.
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V
V
y
Solid 3D Model (input) Invalid Internal Mesh Topology Valid Perforation Topology
Figure 4.1.2: Illustration comparing the difference between typical internal meshing techniques and CSG modeling or automated perforation. Invalid geometry (center) of the hollowed region cannot be validly simulated due to geometric discontinuities. The valid generation of the geometric elements (right) must be followed for generating meshes that can be simulated through FEA methods.
4.2 Method Overview
Generative deformation is a process through which the objective is to generate specific deformation behaviors of 3D printed elastic materials. Through perforating a volumetric mesh, we define how voids or holes of different sizes can be introduced within a solid elastic mesh to alter its deformation behaviors. This contribution relates to the development of new methods that allow us to 3D print elastic material behaviors in addition to matching the static geometry of an object. The motivation behind this method is the simplicity of the core concept: inset elements within a mesh by different values to obtain different internal structures that will modify deformation behavior. The translation of this concept to an iterative design process is also fairly simple: for flexible regions make insets larger and for stiff regions, make inset values smaller. By completely automating this core algorithm, there are numerous design and data-driven tools that can be used to vary the complex distribution of element element perforation inset values. This includes painting-based modifications of inset values, FEA-driven optimizations of inset values, and user design constraints.
The generative deformation design pipeline proposes an iterative process through which specific deformation behaviors can be introduced to a solid elastic mesh. The process is has also been generalized to start with any valid source mesh that can be defined as any valid surface mesh. The core process involves four primary steps: (1) the volumetric meshing of the surface model, typically provided by meshing algo-
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rithins such as Tetgen [44], (2) the simulation of the solid volumetric mesh to determine how behaviors will effect the perforation inset values of the object, (3) the generation of the perforated mesh, and (4) simulation of the perforated mesh to validate mesh behavior. The output of this process is an automatically generated surface model that corresponds to the perforated mesh. The objective of this method is to integrate this process into a coherent easy-to-use design application that also includes the ability to perform finite element analysis on the design models to improve prototype accuracy. The implementation can also be used for rapid prototyping of perforated structures based on the simplicity of generating and 3D print-
ing perforated meshes. â–¡ Surface Mesh
Input: Source Mesh
(3) Automated Perforation
I User Constraints
Rigid
(1) Volumetric Meshing
(4) Perforated Simulation
Applied
Deformation
(2) Solid FEA Simulation
Output: 3D Printable Mesh
Figure 4.2.3: Automated volumetric perforation pipeline. The input is a simple surface mesh and the main process consists of a four stages: (1) volumetric meshing and element-wise user constraints, (2) FEA simulation of the solid mesh, (3) the automated perforation, and (4) the simulation of the perforated mesh. The result produced by the perforation algorithm provides a valid 3D printable surface mesh that can be printed using consumer-level 3D printers.
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4.2.1 Challenges in Printing Elastic Materials
Printing elastic materials represents a relatively challenging aspect of 3D printing deformable objects.
This is because for consumer-level printers, there is a limited selection of elastic materials that can be used and the few that can be are typically proprietary mixtures for resin-based Stereo-lithography (SLA) systems or loosely controlled and highly variable elastic filaments for Fused Deposition Modeling (FDM) printers. In addition to limited materials, this process is also challenging due to the parameterization of the printing process which can include large changes in behavior due to the number of layers, layer thickness, orientation, and adhesion. Additionally, for SLA prints, the post-print curing process can have a significant impact on how the model will deform. If the model is under-cured it will maintain flexibility but contain residue of the resin. If the model curred for an extended period and over-cured, the resin-based model will become brittle.
For most consumer-level 3D printers, the problem is the limited selection of elastic materials. SLA-based systems typically introduce a limited selection (ex. 1 proprietary) elastic resin. For FDM printers, the selection is larger due to the ability of other manufacturers to create flexible materials that are compatible with the specifications (filament diameter, melt temperatures, etc.) of most printers. These are types of accessible printers that cannot dynamically mix multiple elastic materials to obtain specific deformation characteristics. For example, even if we have a variety of multiple elastic materials (filament, resins), picking a single material may not provide an adequate behavior. Since these types of materials cannot be readily mixed in arbitrary proportions to generate different behaviors, we are limited to use a predefined selection. Typically, the use of the materials within the limited selection will not result in ideal or optimized behaviors.
The printing process itself naturally has a large impact on the deformation behaviors of an elastic print. These parameters can be modified through trial and error to produce desirable or consistent results, but they do not allow for control over the large changes in how the printed object will deform once printed. This is where we can introduce minute changes to the core geometry of the model to provide controllable modifications of the deformation behavior. Even slight modifications of local regions can dras-
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INVERSEMODELINGFORDATA-DRIVEN PHYSICALSIMULATION by SHANEMICHAELTRANSUE B.S.,MetropolitanStateUniversityofDenver,2011 M.S.,UniversityofColoradoDenver,2014 Adissertationsubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy ComputerScienceandInformationSystemsProgram 2019

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2019 SHANEMICHAELTRANSUE ALLRIGHTSRESERVED ii

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ThisdissertationfortheDoctorofPhilosophydegreeby ShaneMichaelTransue hasbeenapprovedforthe ComputerScienceandInformationSystemsProgram by GitaAlaghband,Chair Min-HyungChoi,Advisor AshisKumerBiswas KaiYu ZhipingWalter May18 th ,2019 iii

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Transue,ShaneM.Ph.D.,ComputerScienceandInformationSystems InverseModelingforData-drivenPhysicalSimulation DissertationdirectedbyAssociateProfessorMin-HyungChoi ABSTRACT Inversemodelingthroughdata-drivenmethodsintroducesanewresearchdirectionforconnecting analyticalabstractionswithreal-worldobservations.Incontrasttoanalyticalmodeling,theintegrationof data-drivenandanalyticalmethodsallowustocharacterizeandreconstructunderlyingphysicalphenomenaofcomplexsystemsusingdiscreteobservations.Thisenablesthecreationofhybridmodelsthatestablishcorebehavioralcharacteristicsdrivenbyanalyticalmethodsthatcanbeaugmentedwithdata-driven systems.Asdata-drivenmethodscontinuetoenablealternativeformsofcomputingefcientsolutions toproblemsthatarehardtoexplicitlysolvethroughanalyticalmethods,weadaptthisparadigmshiftto createelegantmodelsrepresentingvariousphysicalphenomena.Inthisdissertation,wepresentnovelalgorithmsforthereal-timesimulationofdata-drivenelasticmaterials,behavior-drivenproceduralgeometry for3Dprintedelasticmaterials,andintroducenewdirectionsindetailedrespiratoryanalysisthroughthermalimaging.Specically,weintroducemethodsforcreatingreal-timedata-drivenFiniteElementFE simulationsforelasticmaterials,theautomatedperforationofvolumetricmeshesfor3Dprintingheterogeneouselasticmaterials,andintroducenewquantitativemetricsforvision-basedrespiratoryanalysis. Weformulate,implement,andevaluateeachoftheproposedcontributionstodemonstratetheempoweringnatureofinversemodelingforcomputeraideddesigntools,interactiveapplications,andclinically deployablesystems.Eachoftheproposedcontributionsdenesanillustrativeconceptofhowchallenges inmodelingcomplexreal-worldbehaviorscanbeaddressedthroughinversemodeling. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:Min-HyungChoi iv

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TABLEOFCONTENTS CHAPTER I.INTRODUCTION........................................................................................................................1 1.1ResearchMotivation................................................................................................................1 1.2ResearchContributions............................................................................................................2 1.2.1Data-drivenElasticMaterialSimulation............................................................................2 1.2.2ProceduralPerforationforControllingElasticity..............................................................3 1.2.3TurbulentExhaleFlowModelingandAnalysis.................................................................4 II.INVERSEMODELINGANDSIMULATION............................................................................5 2.1FunctionalModelingandSimulation......................................................................................5 2.1.1ForwardSimulation............................................................................................................6 2.1.2ConstrainedDynamicsandOptimalControl.....................................................................6 2.1.3InverseSimulation..............................................................................................................7 2.1.4Data-drivenModelingthroughImaging.............................................................................10 2.1.5PhysicalPlausibility...........................................................................................................11 2.2HybridModels:AnalyticalandData-drivenModels..............................................................11 III.NEURALELEMENTS:DATA-DRIVENELASTICITY...........................................................13 3.1RelatedWork:SimulationofElasticSolids............................................................................15 3.1.1Mass-springSystems..........................................................................................................16 3.1.2Constraint-basedSolvers....................................................................................................18 3.1.3DeepLearningApproaches................................................................................................19 3.2MethodOverview....................................................................................................................19 3.2.1Contributions......................................................................................................................19 3.2.2FiniteElementFormulationforNeuralElements..............................................................21 3.2.3CorotationalFormulation...................................................................................................23 3.2.4Data-drivenFEA................................................................................................................28 v

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3.3Data-drivenElementBehavior................................................................................................29 3.3.1Element-basedMaterialResponse.....................................................................................30 3.3.2ReferenceElements,Coordinates,andGeometricRatios..................................................34 3.3.3ComputationSeparability...................................................................................................39 3.3.4ElementNetworkModel....................................................................................................42 3.3.5MaterialNetworkModel....................................................................................................48 3.3.6CoveringtheMaterialPropertyDomain............................................................................50 3.3.7ExtensionstoNewElementTypes.....................................................................................51 3.4DynamicNeuralElements.......................................................................................................51 3.4.1MaterialForceResponses..................................................................................................51 3.4.2NumericalIntegration........................................................................................................52 3.4.3DynamicsFormulation.......................................................................................................54 3.5ElasticMaterialDataandTraining..........................................................................................59 3.5.1SyntheticElasticData........................................................................................................59 3.5.2MaterialPropertyIntegration.............................................................................................61 3.5.3DisplacementsandForceResponses..................................................................................63 3.5.4Training..............................................................................................................................65 3.6ParallelElements.....................................................................................................................66 3.7ExperimentsandDynamicSimulations..................................................................................67 3.7.1Deformation:Compression................................................................................................68 3.7.2Deformation:Stretch..........................................................................................................68 3.7.3Deformation:RotationalTwist...........................................................................................69 3.7.4InteractiveDeformations....................................................................................................70 3.7.5Free-formDeformationandCollision................................................................................72 3.7.6QuantitativeEvaluation......................................................................................................73 3.8NaturalPerturbations...............................................................................................................73 3.8.1NetworkNumericalVariance.............................................................................................75 vi

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3.8.2NULLDisplacementResponses........................................................................................75 IV.GENERATIVEDEFORMATION:3DPRINTINGBEHAVIOR................................................77 4.1RelatedWork...........................................................................................................................79 4.1.1InternalMeshing................................................................................................................79 4.2MethodOverview....................................................................................................................81 4.2.1ChallengesinPrintingElasticMaterials............................................................................83 4.2.2Contributions......................................................................................................................84 4.3GeometricPerforation.............................................................................................................85 4.3.1GeometricPerforation........................................................................................................85 4.3.2MeshPerforationAlgorithm..............................................................................................88 4.3.3HomogeneousPerforation..................................................................................................89 4.3.4HeterogeneousPerforation.................................................................................................89 4.3.5ElementConstraintsandDesign........................................................................................91 4.4GenerativeDeformation..........................................................................................................92 4.53DPrintingProcessandTechnologies....................................................................................94 4.5.1Stereo-lithographySLA...................................................................................................95 4.5.2FusedDepositionModelingFDM...................................................................................96 4.6ExperimentalDeformations.....................................................................................................97 4.7Discussion................................................................................................................................99 V.TURBULENTEXHALEFLOW:RESPIRATORYANALYSIS.................................................101 5.1RelatedWork...........................................................................................................................102 5.1.1ChallengesinRespiratoryAnalysis...................................................................................103 5.1.2SurfaceDeformationReconstruction.................................................................................104 5.2VisualizingExhaleFlow..........................................................................................................106 5.2.1 CO 2 VisualizationforRespiratoryAnalysis......................................................................107 5.3ModelingExhaleFlow............................................................................................................108 vii

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5.3.1DenseExhaleModeling.....................................................................................................109 5.3.2DenseFlowReconstruction................................................................................................111 5.3.3VolumetricApproximation.................................................................................................112 5.4ExperimentalSetupandApplication.......................................................................................113 5.4.1ObstructedBreathing..........................................................................................................114 5.5DiscussionandFutureWork....................................................................................................115 5.5.1Correlative3DModeling....................................................................................................115 5.5.2ClinicalDeployment..........................................................................................................116 VI.CONCLUSION.............................................................................................................................118 REFERENCES..........................................................................................................................................118 APPENDIX A.NeuralElementMemoryMaps...............................................................................................124 B.LinearElasticityFiniteElementStiffnessMatrices................................................................125 viii

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LISTOFFIGURES FIGURE 3.0.1NeuralElementsNEisadata-drivenelasticmaterialsimulationmethodthatcloselyaccuratelyreplicatesFiniteElementsimulationsofsolidelasticmaterials..............................14 3.0.2IllustrationoftwodeformationsappliedtotheStanforddragonvolumetricmodel.Thedeformationsareimposedusingaforce-basedmanipulationtoolintegratedintoaninteractiveapplication.Thecoresimulationisbuiltontheproposedata-drivenNeuralElementNE elasticmaterialmodel............................................................................................................15 3.2.3IllustrationoftwodeformationsappliedtotheStanforddragonvolumetricmodel.Thedeformationsareimposedusingaforce-basedmanipulationtoolintegratedintoaninteractiveapplication.Thecoresimulationisbuiltontheproposedata-drivenNeuralElementNE elasticmaterialmodel............................................................................................................20 3.2.4Dynamicbehaviorofa2Dtrianglebetweentherestandcurrentcongurationsleft.The elementnodaldisplacements d 0 ;d 1 ;d 2 duetothedeformationaredenedbyrotatingthe elementtotheoptimalpre-deformationstatewithbothelementscoincidingattheircenter ofmassright........................................................................................................................25 3.2.5Dynamicbehaviorofa3Dtetrahedrabetweentherestandcurrentcongurationsleft.The elementnodaldisplacements d 0 ;d 1 ;d 2 ;d 3 duetothedeformationaredenedbyrotatingtheelementtotheoptimalpre-deformationstatewithbothelementscoincidingattheir centerofmassright.............................................................................................................26 3.3.6Illustrationofthematerialresponseforcesofatwo-dimensionaltriangleelementprovidingthreeobservationsthatestablishthefoundationfortheproposeddata-drivenmaterial model.Thisincludesmultipleforceresponsesforeachdisplacementleft,responseforces arenotrotationinvariant,butforrotatedelementswecanensureresponseforcecomponents arenon-zeroright................................................................................................................32 ix

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3.3.7Exampleofareferencetriangleelement R usedfortwo-dimensionalsimulations.Thechosenreferencegeometryleftisrotatedbyanarbitraryangle centertoensurenon-zero forcecomponentresponsestoanapplieddeformationforallnodesright..........................35 3.3.8Exampleofareferencetetrahedralelement R usedforthree-dimensionalsimulations.The chosenreferencegeometryleftisrotatedarbitrarilycentertoensurenon-zeroforcecomponentresponsestoanapplieddeformationforallnodesright..........................................36 3.3.9Parameterizationandtransformationofanarbitraryinstanceelement I tothereferencecoordinatesystemcontainingthereferenceelement R .Thenodeorderinggeneratesmatchingpairsbetweenthetwoelements I n ! R n ;I u ! R u ;I v ! R v forallindependent executionsofeachnodebetweentheelements......................................................................37 3.3.10Geometricratiosoftheresponseforcescomputedbetweentheinstance I andreferencetetrahedra R .Theratiovector r iscomputedby I=R forallforcecomponentsright.Thisshows oneoffourpossiblenodeorderingswhere n = a;w = b;u = c;v = d .Ratiosareorderingdependent....................................................................................................................39 3.3.11Illustrationofthereferencecoordinatesystemfortheratioandforceresponsemodelbased oncomputingthematerialresponsebasedoneachnodeindependently.Sincetheelement has 3 nodes a , b , c ,thecomputationisperformedthreetimes.Thecompleteelementbehavioristhengeneratedasthelinearcombinationofeachpartialresponse.........................41 3.3.12Illustrationofthedisplacementsimposedona3Dreferencetetrahedralelement.Eachdisplacementanditsassociatedmaterialresponseisusedtotrainanaggregateofmaterialnetworks.Displacementcomponents dx left, dy center,and dz rightwillberecombined intoacompleteelementresponse..........................................................................................43 x

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3.3.13Interchangeablenetworkdesignsthatareprovidedtothecoreoftheneuralnetworkmaterialarchitectureleft.Thesenetworksarethenassembledintothedesignrequiredforthe selectedelementtyperight.Thetriangularelementhassixresponseforceswhichrequires sixnetworkinstancesthatareembeddedintheelementnetwork.Thevalueof d i isdened asthedisplacementvalueexpressedas dx or dy for2Delements,eachofwhichrequires itsownnetworkinstance........................................................................................................45 3.3.14Illustrationofthe3Dtetrahedralelementsystemthatgeneratesthepartialmaterialresponse foragivendisplacement d i where i 2f x;y;z g .Thisgeneratesthepartialforceswithrespecttoonedisplacementoftheprimarynodeinthereferenceelement,thusthereare 3 instancesofthisnetworkfor dx , dy ,and dz .............................................................................46 3.3.15Neuralelementmaterialresponsenetwork.Thisnetworkrepresentsthecompleteresponse ofthematerialbasedonthetrainedinternalnetworksforeachDOFandforceresponsecomponentduetothedisplacementimposedonanycomponentofthe 4 elementnodes.These aretheoutputforcesthatareusedtodrivethedynamicsofthesimulation..........................48 3.3.16Illustrationofthereferencecoordinatesystemfortheratioandforceresponsemodelbased oncomputingthematerialresponsebasedoneachnodeindependently.Sincetheelement has 3 nodes a , b , c ,thecomputationisperformedthreetimesonceforeachnodeforeach degreeoffreedom.Thecompleteelementbehavioristhengeneratedasthelinearcombinationofeachpartialresponse...............................................................................................49 3.4.17Illustrationoftheindependentresponseforcesforadjacentelementsduetothecoefcient matrices 1 and 2 left.Thenetforcescenteranddisplacementsrightofthenodesare averagedduetothedifferencesintheresponsesprovidedfromeachelement.....................57 3.5.18Integrationfunctionsusedtoencodethematerialproperties E , v andthedisplacement appliedtothereferenceelement d i togeneratetrainingdatasets.Thesefunctionsrepresent justtwoselectedperiodicfunctionsthatcanbeusedtoencodedthesenetworkinputs........62 xi

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3.5.19Illustrationofthe sin x integrationfunctionappliedtothedisplacement d astheinput tothetrainingmodelleft.ThematerialresponsealongtheprovidedDOFofthereference elementisshownastheexpectedforce f valueright. Notethisisonlyasubsetofthe samples. ..................................................................................................................................64 3.5.20Illustrationofthe dGauss x integrationfunctionappliedtothedisplacement d asthe inputtothetrainingmodelleft.ThematerialresponsealongtheprovidedDOFofthereferenceelementisshownastheexpectedforce f valueright. Notethisisonlyasubset ofthesamples. ........................................................................................................................64 3.5.21Plotsofthetrainingerrorlossoverthecourseof 5000 epochs.Thetrainingconvergesquite rapidlyduetothelackofnoisewithinthesyntheticdataandthesimplicityofthewaveforms.65 3.7.22Illustrationofneuralelementmeshessimulatedinreal-time.Thisincludesinteractivedeformationsleft,real-timestressanalysisofelementscenter,anddynamicboundaryconditionsforgeneratingcomplexanimationsright.................................................................67 3.7.23Illustrationofneuralelementbarmeshbeingcompressedthroughtwoxedboundaryconditions.Theleftfacenodesarexedandtherightfacenodesarebeingdirectlymoveddisplacedtoreducethedistancebetweenthetwoendsofthebar.Thisresultsinacompressionofthematerialbetweenthesetwoboundaryconditions................................................68 3.7.24Illustrationofneuralelementbarmeshbeingstretched.Theleftsidefacenodesarexed andtherightfacenodesareincrementallymovedtoincreasethedistancebetweentheends ofthebar.Thisresultsinanelongationofthebarandanarrowingofthecenterduetothe internalmaterialdisplacements..............................................................................................69 3.7.25Illustrationofadynamicboundaryconditionthatcanbeimposedonameshtocreateatwist deformation.Therightsidefacenodesarerotatedaboutthetwistaxis ^ x andtherestof thematerialbehaviorisgeneratedthroughtheneuralelementresponses.............................70 3.7.26Illustrationofneuralelementmeshessimulatedinreal-time.Thisincludesinteractivedeformationsleft,real-timestressanalysisofelementscenter,anddynamicboundaryconditionsforgeneratingcomplexanimationsright.................................................................71 xii

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3.7.27Illustrationofaneuralelementmodelundergoingrigid-bodytranslationasitfallsdueto appliedgravity,resultinginacollisionwiththeground.Duetothemassandsizeofthe ears,themodelrollsaftermakingcontactwiththeground...................................................72 3.7.28Illustrationofasubsetofmaterialresponsesgeneratedbytheneuralelementmodelforone elementduringasimulationof 800 time-stepsframes.Thisillustratesthebehavioralerrorassociatedwiththeneuralelementpredictedvalue pr andtheexactvalue ex providedbyanFEA-basedsolution.Fortheselected3Dtetrahedralelement,thisisshownby the 12 componentforcesoftheelement.Thewaveformissporadicduetothelargedeformationoftheelementthroughuserinteractions....................................................................74 3.8.29Theinitialrandomizedweightsofthenetworkcontributetoresidualnon-zeroresponseforces providedfortheinputdisplacementofzero.Thisresultsinaconstantnon-zeroshiftinthe responseforcesthatcontributetoanetpositiveornegativeinternalforcewithintheelement. Thisiscorrectedbyremovingtheshifttoobtainaccurateforceresponseestimations.........76 4.0.1GenerativeDeformation:Theautomatedprocessofperforatingvolumetricsimulationmeshes thatare3Dprintedusingelasticmaterialstoobtainspecicdeformationbehaviors............78 4.1.2IllustrationcomparingthedifferencebetweentypicalinternalmeshingtechniquesandCSG modelingorautomatedperforation.Invalidgeometrycenterofthehollowedregioncannotbevalidlysimulatedduetogeometricdiscontinuities.Thevalidgenerationofthegeometricelementsrightmustbefollowedforgeneratingmeshesthatcanbesimulatedthrough FEAmethods.........................................................................................................................81 4.2.3Automatedvolumetricperforationpipeline.Theinputisasimplesurfacemeshandthemain processconsistsofafourstages:volumetricmeshingandelement-wiseuserconstraints, FEAsimulationofthesolidmesh,theautomatedperforation,andthesimulationoftheperforatedmesh.Theresultproducedbytheperforationalgorithmprovidesa valid3Dprintablesurfacemeshthatcanbeprintedusingconsumer-level3Dprinters........82 xiii

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4.3.4Automatedvolumetricperforationpipeline.Theinputisasimplesurfacemeshandthemain processconsistsofafourstages:volumetricmeshingandelement-wiseuserconstraints, FEAsimulationofthesolidmesh,theautomatedperforation,andthesimulationoftheperforatedmesh.Theresultproducedbytheperforationalgorithmprovidesa valid3Dprintablesurfacemesh.............................................................................................86 4.3.5Combinatoricrepresentationofthemicro-tetrahedrathataregeneratedforeachperforated element.Theseenumerationsdeneallpossiblemicro-tetrahedraedge/cornerelementswithin lookuptables..........................................................................................................................87 4.3.6UniforminsetoftheStanfordbunnymesh.Theinsetvaluesprogressfrom 25% leftto 50% centerto 75% righteachspeciedbyonevalue 0 : 25 , 0 : 5 , 0 : 75 .At 100% elementsbecomesolid..............................................................................................................................89 4.3.7Heterogeneousgeometricstructureduetoper-elementspeciedinsetvalues.Thismethod allowsforsmoothgradientsbetweenelementsandallowsspecicregionsofthemeshto becontrolledthroughanFEAsimulationresultoruserdesign.............................................90 4.3.8Generativedeformation:integrationofdesignconstraints,deformationbehavior,andstress analysistogenerateheterogeneousgeometricstructuresconsistingofasingleelasticmaterialmodel.Thedesignshapetoprowhasspecicdesignconstraintsthatareoptimized togeneratedifferentdeformationandstresspatternsbendandstretchthatalterinsetvalues..........................................................................................................................................92 4.4.9Perforationdesignstudio.Thisinteractivedesignstudioincludestheabilitytoimport,perforate,simulate,andexport3Dprintablesurfacemeshes.Thescreenshotsshowtwodifferentmesheswithinthefour-viewportdesignview.............................................................94 4.5.103DprintsetupandresultfortheStanforddragonmodelprintedontheFormLabsForm1+ usinganelasticresin.Theslicermodelleftdenedexternalsupportstructuresthatallowed forasuccessfulprintcenter,right.......................................................................................96 4.5.11VolumetricperforationprintingprocessSLAillustratingtheresinprintleft,alcoholwash center,andourcustombuiltacrylic 405[ nm ] -UVcuringboxright.................................96 xiv

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4.5.12FinalresultoftheperforatedStanfordBunnyandDragonmodelsprintedusinganSLAbasedprinterwithasingleexibleresin.Eachregionhasvariabilityintheelasticbehaviorhard,softduetothegradient-basedtransitionsoftheperforatedstructure.Supportmaterialshavebeencarefullyremoved.......................................................................................97 4.6.133Dprintedperforationmeshesfortestingthegenerativedeformationpipeline.Eachisa 2 : 0 x 2 : 0 x 4 : 5 [cm] singlematerialperforatedprint..............................................................................................98 4.6.14Resultofdeformingvebeamsgeneratedusingthegenerativedeformationalgorithm.The toprowillustrateseachbeamsubjecttoadeformationimposedbyahorizontalload,inducingvariousdeectionbehaviors.Thetwistdeformationbottomrowisimposedbyanappliedtorquetodemonstratenonlinearrotationaldeformations.Eachexhibitsuniquedeectionsandlocalizeddeformationsduetothechangesintheheterogeneousperforations......99 4.7.15DemonstrationofthesuccessfulprintofacomplexperforatedstructureusinganFDM3D printer.TheTPUprovidesarelativelyrigidresultwhenprintedintosolidobjects,however throughtheperforatedgeometry,theelasticmaterialpropertiesoftheobjectcanbedrasticallychanged.......................................................................................................................100 5.0.1Resulting CO 2 densitydistributionimagesillustratinguniquerespiratorypatternsbetween individualstopvsbottomrows.Foreachimagesequence,oneexhaleperiodhasbeenrecorded andvisualized,showingtheclearseparationbetweenthenose-mouthdistributionanddensityowbehaviors.Theseowbehaviorsuniquetoeachindividualarebasedontheirown physiologicaltraits.................................................................................................................102 5.1.2Illustrationsofthevariousmodelsgeneratedbydepthimagingmethodsforrespiratoryanalysis.Depthimagingleft,simpleextrudeddepthcenter,andregionbasedmethodsright, donotprovidethemodeldelityandaccuracythattheiso-surfacedeformationmodelcan provide.Thismodelprovidesamethodsformeasuringbothbreathingrate,tidalvolume with > 90% accuracy.............................................................................................................105 xv

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5.1.3Illustrationsofthevariousmodelsgeneratedbydepthimagingmethodsforrespiratoryanalysis.Depthimagingleft,simpleextrudeddepthcenter,andregionbasedmethodsright, donotprovidethemodeldelityandaccuracythattheiso-surfacedeformationmodelcan provide.Thismodelprovidesamethodsformeasuringbothbreathingrateandtidalvolume.........................................................................................................................................106 5.3.4Volumetricanddensitymodelingofthe CO 2 exhaleregiondenedbytheviewfrustumof theimagingdevice.Theexhaleowregioncontainsanon-linear CO 2 concentrationdistributionfunction density overdistance x .Eachindividualpixel p i;j representsacontinuousvolumethroughwhichtheexhaleows.Fromthevalueateachelement e i ,thenal pixel p i;j istheprojectionofallelementdensitieswithin v ..................................................110 5.3.5Recordedturbulentexhaleowfromthemouthleft,nosecenter,andbothnoseandmouth simultaneouslyright.Throughourimagingprocess,weobtainanaccurateillustrationof the CO 2 densitydistributionandowbehaviorwithminimalbackgroundinterference......111 5.3.6Exhaleowreconstructionprocess.Weapproximatethereconstructionoftheprojecteddensityvolumebyestimatingthefunction density x usingheuristicapproximations.Methodsderivedonthisdesigncanbeusedtocreatenumerousformsofdifferent3Dexhalemodels...........................................................................................................................................112 5.3.7Turbulentexhaleopticalowvectors.Thegeneratedvectoreldillustratesthe apparent owcomputedthroughastandarddenseopticalowalgorithm.Thetoprowillustrates theoriginal CO 2 densityimages,andthebottomrowillustratestheresultingvectornormcolor-mappedow..................................................................................................................112 5.3.8Exhaleowreconstructionprocess.Weapproximatethereconstructionoftheprojecteddensityvolumebyestimatingthefunction density x usingheuristicapproximations.Methodsbasedonthisdesigncanbeusedtocreatenumerousformsofdifferent3Dexhalemodels...........................................................................................................................................113 xvi

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5.4.9Overviewoftherespiratorymonitoringsetupthatusesa CO 2 cameratoprovideremote exhaleanalysis.Thisincludesthe CO 2 camera,software,andcross-sectionofthepatients breathingarea.........................................................................................................................114 5.4.10Real-timerespiratorymonitoringapplicationdesignedtoworkwithour CO 2 imagingcamera.Thisincludesbothahigh-contrastvisualizationoftheturbulentexhaleowwithan appliedheatmapandthesegmentedexhaleregionthatisusedtogeneratethewaveform ofthequantitativetidalvolumeestimation.Thisapplicationprovidestheabilityforclinicianstoeffectivelycommunicateinformationaboutpatientbreathingbehaviorsinanintuitiveway..............................................................................................................................114 xvii

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CHAPTERI INTRODUCTION Thesynergisticintegrationofanalyticalanddata-drivenmethodsinherentlyformaclassofhybridmodelingtechniquesthatenableawidevarietysolutionstochallenginginterdisciplinaryengineeringproblems. Formalizingtheuseofdata-drivenanalyticalmethods,weintroduce inversemodeling asaprimaryconcepttowardscreatingaccuratemodelsandsimulationsofreal-worldphysicalphenomena.Thisprocess combinesdomainknowledgefrommechanicalandaerospaceengineering,graphics,andcomputervision withdata-drivenmethodsfrommachinelearningtocreatedynamichybridmethodsformodelingcomplex systems.Usinginversemodelingasacoretheme,weanalyzehowchallengingproblemsincomputational mechanicsandmedicalimagingcanbeaddressedthroughtheuseofhybridmodels. 1.1ResearchMotivation Thecoreideabehindinversemodelingishowwecanintegratecomplexanalyticalsystemswithdatadrivenmethodsthatallowustocapturereal-worldobservationsasproblemconstraints.Incontrasttodirectmodeling,theobjectiveisnotonlytogenerateanabstractionoftheobservedbehavior,buttogain insightintotheunderlyingphysicalphenomenaandinverselyreconstructordescribethisbehavior.This canallowustoprovidemoreaccurate,efcient,orcompletelynewtechniquesformodelingcomplexrealworldsystems.Todothis,welookatexistinganalyticalmodelsandcloselyintegratedata-drivencomponentstoformnewhybridmodelingtechniquesthatbalancethebenetsoftheseapproaches.Thereis awealthofdifferentproblemsthatcanbeprimarilyaddressedusinganalyticalmethodsordata-driven methods,butthemixtureofthesemodelingtechniquestypicallyrequiresextensivedomainknowledgeand therearevariousformsofengineeringproblemswherethesetwoapproachesarefundamentallyincompatible.Forexample,mostphysicalsimulationmethodsarepurelyanalyticalwithreal-worldexperimental valuesprovidingaccurateparametersforsimulations,resultinginhighlyaccurateanalysismodels.For data-drivenmethods,thereareareassuchascomputervisionwheretrainedmodelsprovidehighlyaccurateresultsforobjectrecognitionand3Dscenemodelingwhichcanbedifcultorlessefcientforanalyticalmodels.Themotivationofthisworkistouseinversemodelingtoaddressprominentproblemswithin 1

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variousdomainsincludingcomputergraphics,physicssimulation,3Dprinting,andmedicalimaging. 1.2ResearchContributions Thecontributionsofthisdissertationrepresentsolutionstothreeprominentproblemsintheeldofcomputationalmechanics,3Dprintingelasticmaterials,andvision-basedrespiratoryanalysis.Foreachcontribution,weoutlinethecoreproblem,thegeneralapproachtosolvingtheproblemandhowoursolution methodsintroducenovelsolutiondirections.Theoverviewofeachcontributionisprovidedbelow,followedbyadetailedoutlineofthesolutiondirection. • NeuralElements:Data-drivenElasticity -The neuralelementmethod introducesanewmodel derivedfromcomputationalmechanicsforcreatingphysicallysimulating3Delasticobjectsforrealtimeandinteractivecomputeraideddesigntoolsandapplications. • GenerativeDeformation:3DPrintingBehavior -Generativedeformationistheconceptofmodifyingstructuralgeometrytoalterelasticordeformationbehaviorsofmodels.Thismethodprovidesanautomated perforation algorithmthatcangeneratevariedinternalelasticstructurestocreate design-oriented,3Dprintablebehaviors. • TurbulentExhaleFlow:RespiratoryAnalysis -Analysisofbreathingbehaviorsiswellstudiedin computervision,buttechniquesinthermal CO 2 visualizationprovidethefoundationforextracting newandmeaningfulquantitativemetricsforevaluatingpulmonaryfunctions. 1.2.1Data-drivenElasticMaterialSimulation Thesimulationofdeformableelasticmaterialsrepresentsoneofthemostfundamentalandmatureresearchtopicsinaerospaceandmechanicalengineering.Theprocessformodelingexibleanddeformable objectsresideswithinmanyofthecoretopicsinengineering,design,andmorerecently,computergraphics.Analyticalmodelsthatdenetheconstitutiveequationsthatdenehowsolidmaterialsbehaveare derivedfromthecoreprinciplesofcontinuummechanics.Theseprovidetheabilitytoaccuratelyevaluatehowsolidmaterialsbehaveincludingdisplacementsofmodelnodesandstressdistributionswithinthe materialduethestructuraldynamicsofthemodel.Thesemethodsprovidehighlyaccuratesimulationsof 2

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completestructuresandareusedextensivelythroughoutmanycomputeraideddesignapplications.From thecomputergraphicsside,someofthesecoretechniqueshavebeenintegratedintoanimationandgame designbasedonawidevarietyofdifferentmethodseachwithvaryinglevelsofcomputationalcomplexity andaccuracy.Althoughtheeldofsimulatingelasticmaterialsispopular,therearestilllimitedoptions forreal-timeandinteractivesolidmaterialsimulations.Inthiswork,weintroduceanewhybridformulationofelasticmaterialdynamicsthatmixesanalyticalanddata-drivenmethodstocreateinteractiveand real-timesimulationsof2Dand3Dmodelscalled NeuralElements .Thisisduetothecoreinternalrepresentationofmaterialpropertiesbeingencodedwithinneuralnetworks.Weprovideanin-depthanalysis ofthisapproachandillustratetheaccuratesimulationofseveraldifferentcomplexdeformationsusingan interactive,real-timeapplicationfordeformingelasticmaterialmodels. 1.2.2ProceduralPerforationforControllingElasticity Proceduralgenerationofelasticstructuresprovidesthefundamentalbasisforcontrollinganddesigning 3Dprinteddeformableobjectbehaviors.Theautomationthroughgenerativealgorithmsprovidesexibilityinhowdesignandfunctionalitycanbeseamlesslyintegratedintoacohesiveprocessthatgenerates 3Dprintswithvariableelasticity.Generativedeformationintroducesanautomatedmethodforperforatingexistingvolumetricstructures,promotingsimulateddeformations,andintegratingstressanalysisinto acohesivepipelinemodelthatcanbeusedwithexistingconsumer-level3Dprinterswithelasticmaterialcapabilities.Inthiswork,wepresentaconsolidatedimplementationofthedesign,simulate,rene, and3Dprintprocedurebasedontheautomatedgenerationofheterogeneouslatticestructures.Weutilize FiniteElementAnalysisFEAmetricstogenerateperforateddeformationmodelsthatadheretodeformationbehaviorscreatedwithinourdesignenvironment.Wepresentthecorealgorithms,automatedpipeline, and3Dprintdeformationsofvariousobjects.Quantitativeresultsillustratehowtheheterogeneousgeometricstructurecaninuenceelasticmaterialbehaviorstowardsdesignobjectives.Ourmethodprovides anautomatedopen-sourcetoolforquicklyprototypingelastic3Dprints. 3

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1.2.3TurbulentExhaleFlowModelingandAnalysis Denseexhaleowthrough CO 2 spectralimagingintroducesapivotaltrajectorywithinnon-contactrespiratoryanalysisthatconsolidatesseveralpulmonaryevaluationsintoasinglecoherentmonitoringprocess. Duetotechnicallimitationsandthelimitedexplorationofrespiratoryanalysisthroughthisnon-contact technique,thismethodhasnotbeenfullyutilizedtoextracthigh-levelrespiratorybehaviorsthroughturbulentexhaleanalysis.Inthiswork,wepresentastructuralfoundationforrespiratoryanalysisofturbulent exhaleowsthroughthevisualizationofdense CO 2 densitydistributionsusingpreciselyrenedthermalimagingdevicetotargethigh-resolutionrespiratorymodeling.Weachievespatialandtemporalhighresolutionowreconstructionsthroughthecooperativedevelopmentofathermalcameradedicatedtorespiratoryanalysistodrasticallyimprovetheprecisionofcurrentexhaleimagingmethods.Wethenmodel turbulentexhalebehaviorsusingaheuristicvolumetricowreconstructionprocesstogeneratesparseow exhalemodelsthatareprovidedwithreal-worldmetricsthroughdepthimaging.Togetherthesecontributionsallowustotargettheacquisitionofnumerousrespiratorybehaviorsincluding,breathingrate,exhale strengthandcapacity,towardsinsightsintolungfunctionalityandtidalvolumeestimation. 4

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CHAPTERII INVERSEMODELINGANDSIMULATION Thisresearchcoversinversemodelingandthebehavioralreconstructionofcontinuousphysicalphenomenathroughdata-drivenmodeling.Theproblemisdenedasfollows:givendiscreteinstancesforming asetofreal-worldphysicalstatesorobservationsrecordedthroughvarioussensingmodalitiesdepthimaging,thermal-imaging,syntheticdata,etc.,howcanwereconstructmodelsfrompartialdatasetsto replicatetheobservedbehaviorsdenedassetofsparsephysicalconstraints.Thecoreofthisproblemintegratesrecordedbehaviorsasdiscreteobjectivestatesthatdenethesetofobservedphysicalbehaviorsto bereplicatedwithinasimulationormodel.Sincewecanonlyobtainpartialsnapshotsofthephysicalphenomenaatdiscretetimeintervals,mathematicaloranalyticalmodelsthatcoulddescribethesebehaviors areinherentlyunderdeterminedandrequirebothdata-drivenandoptimizationmethodstoidentify physicallyplausible solutions.Asolutionforthistypeofproblemisdenedasaphysicalsimulationcontaining anidealizedmodelthatconformstoasetofrecordeddiscretestatesordatasampleswhileadheringtothe governingphysicalequationstoreplicateanobservedbehavior. 2.1FunctionalModelingandSimulation Physicalsimulationisinherentlyinterdisciplinary,incorporatingcomponentsfromcomputergraphics, appliedmathematics,andmechanicalengineering.Themathematicalbasisofphysicalsimulationsarederivedfromfundamentalmodelswithincontinuumanduiddynamicsandimplementedthroughavariety oftechniquesexploredwithincomputergraphicsusingnumericalmethodsfromappliedmathematics.In additiontothesecoresimulationcomponents,elementsfromelectricalengineeringsuchassignalprocessingarerequiredforpreprocessingrecordedsensorydata,ltering,andsignalreconstruction.Theresearch presentedwithinthisworkisderivedfromnumeroustechniquesinthesedomainstointroducenewcontributionsofinversemodeling.Eachcontributioninthisworkrepresentsincrementalstepstowardsan overallinversemodelingframeworkthatincludes:sensorrecordingmethods,dataanalysisandmodelingprocedures,optimizationandcontrolmethods,andlinksbetweentraditionalproblemsandinversely simulatedsolutions. 5

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Duetotheintegrationofseveraltheoreticaldomainsandtheapplicationofnumeroustechniques withineach,thisweprovideabriefoverviewofthevariousmethods,techniques,andtermsthataredenedwithinthecontextofinversemodeling.Thisincludesabriefoverviewofthedifferentformsofphysicalsimulationsincomputergraphics,imagingdevicesandsignalprocessingmethods,constitutivemodelsfromtheoryofelasticity,thermalmodeling,constraineddynamics,andhowtheseareintegratedinto solutionsbasedofinversemodelingproblems. 2.1.1ForwardSimulation Inmostformsofphysicalmodeling,theprocessofforwardsimulationismathematicallydenedasanOrdinaryDifferentialEquationODEthatisformedusingagoverningequation,asetofinitialconditions modifyingtheODEasanInitialValueProblemIVP,andsolvedattheequilibrium static stateorthrough numericalintegrationasa dynamic system.Thisdenestraditionaldynamicsimulationsastheprocess offorminginitialconditions,deninghowdiscretestatesareupdatedconstitutivemodel,andperformingsystemupdates.Thisprocesstakesasimulatedmodelfromaninitialstate forward intimetoanew currentstate.Withinthedomainofelasticmaterialtheorythesestatesarerefereedtoasthe reference and current congurationsofabodyundergoingdeformation.Tosolveorprogressthesimulationforwardin time,numericalintegrationisusedtocomputetheresultingdisplacementsordeformationsoftheobject overtimeduetoexternalforcesloadsorusergeneratedinteractions.Thisisthemostcommonformof simulationwithincomputergraphicsandanimation,howeverduetothelackofcontrollabilitygenerating objective-basedbehaviorsfromasetofinitialvalues,thisformofdirectsimulationrequiresadditional toolstoimproveitsapplicabilityincontrol-orienteddesignandanimation. 2.1.2ConstrainedDynamicsandOptimalControl Thebackboneofanalyticalinversemodelingorsimulationisbasedonsignicantcontributionsinconstraineddynamicsandoptimalcontrolfrommechanicalengineering.Traditional constraint-based dynamicsintroducestheabilitytomanipulatehowrigidbodiesmoveorinteractbasedonprescribedpaths, joints,orobjectives.Thislowlevelformofcontrollingdynamicbehaviorofobjectsintroducesanopti6

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mizationprocessthatallowsforexactobtainablebehaviorsorleast-squaresapproximationsofnearest non-reachableobjectives.CommonexamplesofconstraineddynamicsincludeBoundaryValueProblemsBVH, inversekinematics incomputeranimationandend-effectormovementinrobotics. Optimalcontrol introducesasimilarconceptasconstraint-baseddynamics,howeverprovidesageneralizable frameworkandmultipletechniquesforsimulatingcomplexsystemswhileadheringtothegoverning physicalmodelandarbitraryconstraints.Fromaconceptuallevel,therearenumeroustechniquestofacilitateoptimalcontrolwithindynamicsystems;however,mostarederivedfromthesamegeneralized componentsincluding:thestatevariablesthatrepresentthesystem,asetofcontrolvariablesthatcanbe modiedtoinuencethesystem,andthecostfunctionthatdeneshowcontrolsaremanipulatedtoobtain theobjectivestate.Unlikeconstraint-basedmodeling,optimalcontrolalsoaccountsfor active modicationofthesystemthroughcontrolvariables.Theoptimalcontrolsolutionisasetofdifferentialequations thatminimizethecostfunctionwhilegeneratingthecontrolstatesrequiredforthesystemtoreachtheintendedgoal.AnexampleofasolutionmethodistheLinear-QuadraticRegulatorLQRwhichreduces costfunctionsbasedonfeedbackcontrols.Acommonexampleofhowthismethodisusedisinsimple rocketightmodelswheretargetvelocity,fuelconsumption,andlandingarerelatedtothecontrolinputs nozzledirection,burnrateorthemotorcontrolsinaninvertedpendulum. 2.1.3InverseSimulation Inthecontextofcomputationalmodeling, inversesimulation isdenedastheprocessofobservinganexistingphysicalphenomena,formulatinganappropriateidealizedmodel,discretizingcontinuousquantities, andsolvingtheunderdeterminedproblemwhileadheringtoasetofgoverningphysicalconstraints.In general,aninversesimulationcanbedenedasacontinuoussetofboundaryvalueproblemswhereeach ofthediscreteobjectivestatesdenetheboundaryvalueseitherDirichletorNeumannboundaryconditions.Theremainingintermediatestatesaretheproductofcomputingasimulationthatadherestothe governingphysicallawsofmotionwhileobtainingthesetofboundaryconditionsdenedatspecicinstancesintime.Theseminalpaperthatintroducedanautomatedextensionofthisformofoptimalcontrol foranimationidealizedtheconceptas spacetimeconstraints [1].Fromahigh-levelperspective,space7

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timeconstraintsprovidetheabilitytodenethestateofacomplexsystematarbitraryuniqueinstances intimeandgeneratethecontrolresponseneededtoobtainthestateswhilemaintainingsecondaryinteractionforceshandledbythegoverningphysicalmodel.Thisprocessallowsforcompletecontrol,while maintainingaconstitutivemodelforgeneratingrealisticbehaviorsbetweendiscretecontrolstates.This andsimilarmethodsarenowheavilyusedwithincomputeranimation[2]. Thepremiseofinversemodelingorsimulationoriginateswithinaerospaceengineeringasamethod ofoptimalcontrolthatattemptstoidentifytheinputcontrolsneededtoobtainaspecicguidedsystemresponse.Thiswasparticularlyimportantwithinaerospacedesignduetorequiringaccuratesimulationsof howcontrolsystemswouldeffectaerodynamiccontrollabilityandstabilityofengineeredcontrolsystems. CommonformsofengineeringproblemsIVP,BVP,etc.andprominentsolutionmethodssuchasFinite ElementAnalysisFEA,BoundaryElementMethodBEM,andnumerousothertechniquesforsolving PartialDifferentialEquationsPDEsthatresidewithinthecoreofappliedmathematicsandengineering analysis.ThesetechniquesallrevolvearoundtheprincipleofreconstructingandmodelingphysicalphenomenathroughthesolutionofPDEsortimeprogressionmodelsbasedonsystemsofODEs,BVPs,or optimalcontrol. Inabroadsense,wedenetheprocessofobtaininganinversesolutiontoagivenproblemasthe resultingsetofcontrolforcesrequiredtoimposethebehaviororthroughinitialconditionsandinternal propertiesofthesystem.Thisleadstotheformulationoftwoprimaryformsofinversesimulations:an active inversesimulationwhichcloselyresemblesoptimalcontrolinthat,duringthecourseofthesimulation,inputcontrolscanbeintroducedtoalterthecurrentstatebehaviorandan passive formwherethe solutiontakestheformoftheinitialconditionsandmaterialpropertiesrequiredtoreplicatetheobserved behavior.Forexample,neuralelementsrepresentsanactivelycontrolledsolution,whereasmodelsgeneratedthroughgenerativedeformationsarepassive. ActiveControl .Thedistinctionbetweenpassiveandactiveisthatthepassivesimulationdoesnotobtain controlforcestomodifythebehavior.Activeinversesimulationscloselymirrortheresultsfromoptimal controlandcanincludebothcorrectivedynamicssuchasarobottryingtostayinoneplace,orbalance orformulatedasanoptimizationproblemwhereanapproximatesolutionisaccepted.Primaryexamples 8

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ofactivecontrolsystemsincluderoboticcontrolforstability,droneightcontrol,and4Dprintingwith heatedorelectricalactivationmethods. PassiveControl .Examplesofpassiveinversesimulationsarethosethatdonothaveanymeansofinjectingexternalcontrolforces;theysimplyrelyontheoptimizationofthesystemproperties.Thisprimarily includeselasticmaterialcoefcientpropertyoptimizationforYoung'sModuluselasticModulus E and Poisson'sRatio v forsimulationsofdeformablebodiesandcontrolledrigid-bodysimulations.Forthe applicationwithindeformablebodysimulation,wedemonstratedtheimpactoftheinternalgeometric structure[3]contributestotheresultoftheFEAsimulation.Forpassiverigid-bodyinversesimulations, methodsthatoptimizefallingblockstolandinapre-denedpattern[4]havebecomepopularincommercialanimationpackages. Thoughtheseconceptswereutilizedextensivelythroughaerospaceandmechanicalengineeringover severaldecades,theadoptionofinversemodelingincomputergraphics spacetimeconstraints occurred muchlater[1].Thistechniqueprovedtoworkwellformodelingtypicalconstraint-baseddynamicsystemswheretheDegrees-of-FreedomDOFweremanageableenoughtoformulatethesystemevenwith theaidofautomatedtools,forexampleaseriesoflinkedjointswhereanglesandlengthsarewellcontrolledsuchasthejumpingPixarlamp.Theunderlyingassumptionisthateach static statecanobtained withminimaleffort,allowingtheintermediatestatesphysicallyplausiblebehaviortobecomputedby anumericaloptimizationprocess.Theproblemwiththisapproachisthatasthecomplexityofthesimulatedobjectincreases,thesestaticstatesbecomeincreasinglydifculttodene.Therefore,ifwecannot generaterealisticstaticstates,theresultingbehaviorwillalsodivergefromexpectedbehavior.Therefore weneedadditionaltoolstoassistintheprocessofreconstructingreal-worldphysicalbehaviorstoassist theexistingsetofmathematicaloptimizationtools.Towardsthisobjective,threenotablecontributions [5,6,7]havebeenmadethatattempttoencode high-level objectivesasconstraints.Thisincludesthenotionofimposing bending or twisting motionsonsimulatedobjects.Weproposedasimilarmethodusing intuitivecontrolmetaphorswithinaninteractiveeditingapplication[8]. 9

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2.1.4Data-drivenModelingthroughImaging Simulationsthatintegrateidealizedmodelswithreal-worldbehaviorsrequirebothaccurateconstitutive modelsandtheabilitytocombinedirectmeasurementswithsimulatedsystems.Theprocessofintegratingdiscreteconstraintsfromrecordeddatawithinadynamicsystemisinherentlycomplex.Inthiswork wedenetwoprimarymethodsforcreatingdata-drivensimulationsbasedonhowthedataisused.The rstvariantleveragesspacetimecontrolstopartiallydeneadiscretedeformationorbehavioralstateof anobjectasanobjectivestatethatwillbereachedduringthecourseofthesimulationormodeltraining procedure.Thismeansthateachrecordedstatedenesaconstraintthatmustbemaintainedduringthe courseofthesimulation,similartohowdynamickey-framesareusedtodeneexactanimationstatesof anobjectovertime[9].Thesecondvariantassumesthatwecancollectandextractbehavioralcharacteristicsfromthecollecteddatatoparameterizethesimulationandobtainanobjectiveresult.Inthisinstance, behavioralcharacteristicscanbeextracteddirectlyfromimagedata,orcanbeusedtomodifyparameters ofthephysicalsimulationormaterialproperties.Examplesofthistechniqueincludeusingdepthimaging forsurfacereconstruction,theestimationofelasticmaterialproperties,orthebehavioralanalysisofuid ows. SurfaceModelingthrough3DScanning .Scanningandreconstructing3Dmodelsfromreal-worldobjectsiswellstudiedandhasmadealargeimpactonhowstaticmodelscanbeautomaticallygenerated fromeithervisiblelightRGB,stereo,ordepthimaging.Boththereconstructionfromvisiblelight,Shape fromShadingSFSanddepthestimationmethodshavebeendevelopedtothepointwherecommercial solutionsprovidehighqualityresults.Buttherearestillassumptionsthatthesecurrentsolutionsmake, suchastheobjectisstaticandwehaveahigh overlap betweenscanpairs.Undertheseassumptions,techniquesinfeatureestimationsuchasFastPointFeatureHistogramsFPFHandvariantsoftheIterative ClosestPointICPcanprovidebothroughandrenedalignmentsbetweenpoint-cloudsgeneratedfrom depthimages.Undertheseconstraints,alignmentscanbeperformedandtheobjectcanbereconstructed. However,intheinstancewhereweminimizethetotalnumberofscansrequiredtorecordthesurfaceof theobject,thisprocessbecomeschallenging.Toalleviatethisprobleminthemostseverecases, SxStudio wasdevelopedtoprovide painting-based selectionsofoverlapregionscommontoeachscantoimprove 10

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alignmentsuccess.Thismethodintroducesanintuitiveinterfacetoresolvealignmentproblemsthatcannotbeaddressedwithcurrentalgorithms.Thisworkresultedinarobustalignmentcompilationprocess thatcanbeusedtoensurescanalignmentsaresuccessful.Oncethisprocessiscomplete,conventional loopclosure algorithmscanbeusedtominimizetheglobalerrordistributionbetweenallscanpairs.We havealsodevelopedastudiodesignapplicationforpainting-basedscanalignment[10]and3Dobjectreconstruction.Thedevelopmentofthisapplicationprovidesabaseframeworkuponwhichscannedobject surfacescanbeusedtogeneratestructuralmodelsfromreal-worldobjects.Thesemethodsleadtokeyfeaturesintroducedinourmorerecentwork[11].Thispriorresearchisrelevanttothedevelopmentofthe proposedchestsurfacedeformationmodelingtechniquesweintroduceforrespiratoryanalysis. 2.1.5PhysicalPlausibility Derivedfromtraditionalcomputergraphicsterminology,thenotionofphysicalplausibilityrefersaperceptualandvisualdelityofthephysicalbehaviorwithinasystem.Ifthebehaviordoesnotseemtoviolatethenaturallawsofphysics,itisdeemed physicallyplausible .Fromatechnicalperspective,thisisdenedasthesetof probable physicalstatesthatcouldbeobtainedusingaconstitutivemodelthatadheres tothenaturallawsofphysics,givennaturalawswithinthesystemandmeasurementmethod.Themost prominentexampleofphysicalplausibilityisusedwithinuidsimulationsincomputergraphicssuchas SmoothedParticleHydrodynamicsSPHwhichvariesdrasticallyfromtraditionalComputationalFluid DynamicsCFDtechniquesbysacricingaccuracyforreal-timesimulations.Thejusticationinthisreductionisbasedonthenotionthatthevisualperceptionoftheresultisindistinguishablefromtheactual oraccurateresult.Inthecontextofinversemodeling,physicalplausibilityisdenedasoneoftheprobableoutcomesthatsatisesthegivenobservationalconstraintswithoutviolatingtheunderlyingphysical model. 2.2HybridModels:AnalyticalandData-drivenModels Theunderlyingthemeofinversemodelingistheabilitytoseamlesslyintegrateanalyticalmethodswith data-drivenmethodsfrommachinelearningtogeneratenewtypesof hybrid solutions.Throughtherapid 11

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progressionofmachineanddeeplearningtechniques,therehasbeenadivergencebetweenwhatkinds ofproblemscanbesolvedeffectivelythroughanalyticalsolutionsandthosethatarebettersuitedforsolutionsderivedfromdata-drivenorlearning-orientedmethods.Theconciseobjectiveweproposeisthat therearenumerousproblemformulationsthatcanbenetfromthecarefulintegrationofthesetwomethods.Thisisspecicallytrueforthecoreanalyticalmethodsofphysicalsimulationwhereevenrecent techniquesconcedethatapurelydata-drivenphysicalsimulationbasedonrepresentingtheforcetodisplacementrelationshipofelasticmaterialsis nearlyimpossible [12].Whilethismeansthatthereisadirect separationbetweensolutionmethodsthatrequirepurelyanalyticalmodelsanddata-driventechniques,we illustratethattherearestillnumerouswaystointegratetheseabstractionsintohybridmodels. 12

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CHAPTERIII NEURALELEMENTS:DATA-DRIVENELASTICITY Theeldofdynamicsandphysicalsimulationofelasticmaterialsispopulatedwithnumerousformulationsandsimulationmethodsformodelingdeformableobjects.Inabroadsense,theintegrationofcomputationalmechanics,numericalanalysis,andcomputergraphicsincorporatesadiversesetoftechniques thatmustseamlesslymeshintoacoherentmodeltosimulatecomplexdeformationbehaviors.Asthenumberofrenedtechniquesforimprovingsimulationaccuracy,quality,andperformancehavegrown,there areincrementaljumpsinhowwesimulatedeformableobjects.Yet,evenwiththehighlyrenedmodels usedinreal-worldengineeringandgraphicsapplications,therearenewapproachesthatreformulateexistingproblemstocreatenewanduniquesolutions.Oneofthemainthrustsincomputationalmechanicsis theopenproblemofhowtointegrate data-driven modelingtechniqueswithestablishedmethodsincomputationalmechanicsandcomputergraphics.Thedifcultyincombiningthesetwodomainsisthatmost provenmethodsfromcomputationalmechanicsarecompletelyanalyticalandrelyonaccurateelasticmaterialbehaviorsdrivenbythe theoryofelasticity .Thesemethodsalsorelyonnumericaltechniquesthat donotseamlesslyintegratewiththemechanicsofdata-drivensolutions.Therefore,theprospectofaugmentingcurrentanalyticaltechniqueswiththeexpressivenessandexibilityofdata-drivenmodelscan formasolidfoundationfornewmethodologiesinelasticmaterialsimulation.Throughtheintroductionof theneuralelementframework,weaimtoprovideanewtoolforreal-timeapplicationsinVirtualandAugmentedRealityAR/VR,3Dprintingofelasticmaterialdesigns,gamedesign,andcomputergraphics. Thiscontributioncanalsoenableasubstantialleapinhowinteractivesolidsimulationsareusedwithin othereldssuchasaerospace,mechanicalengineering,andinteractivedesign,wheredesignandmodeling toolscanbenetfromuseaccurateinteractivemodelsandsimulations. NeuralElementsNEintroducesahybridformulationthatintegratesananalyticalFiniteElement FEelasticmaterialmodelwithdata-drivenmachinelearningtechniquestoinverselymodelmaterialbehaviors.Thecoreobjectiveoftheneuralelementformulationistoencapsulatetherelationshipbetween potentialnodaldeformationsandthematerialresponsesgeneratedbysolidelementstoachievealightweight,computationallyefcientsimulationframework.Theobjectiveistoprovideaexibleandaccu13

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Figure3.0.1:NeuralElementsNEisadata-drivenelasticmaterialsimulationmethodthatcloselyaccuratelyreplicatesFiniteElementsimulationsofsolidelasticmaterials. ratesimulationmodelforinteractiveandreal-timeapplications. Conceptually,thenotionofcombiningtheconstitutiveelasticmaterialmodelswithdata-driven methodshasbeenwellestablishedsincetheintroductionofbasicmultilayerperceptronnetworksor neuralnetworks .Evenbasicneuralnetworkshavebeenusedtolearnthebehaviorsofmaterialsamples,a taskthatiswellsuitedtomachinelearningwithexperientially-drivenanalysis.Variantsofthisfundamentalideahavebeenwellexploredinbothmechanicalandcivilengineeringfornumerousapplication domain-orientedtasksandmorerecentlyincomputergraphicsandanimation.Linkingthe stress-to-strain relationshipexhibitedbyanarbitraryelasticmaterialtoatrainednetworkthroughmachinelearninghas beenemployedinnumerousstudiesthatoperateonreal-worldexperimentsofelasticmaterials.Fromobtainingprecisematerialresponsestodeterminingmaterialparametersforanalyticalmodels,someforms ofdata-drivenelasticmodelinghasbeenwellestablished.However,withinasignicantportionofthe engineeringresearchdirectionsrelatedtodata-drivenelasticmaterialmodeling,manyoftheseresearch directionsareveryobjective-oriented.Thatis,theytypicallywanttoobtainhighlyaccuratebehaviors ofexactreal-worldmaterialsforverydirectedpurposesbasedonhighlycontrolledexperiments.Within computergraphics,theoppositeistrue.Fromthecomputeranimationandinteractivesimulationperspective,highlyaccuratedata-drivenmodelslackexibilityandaretypicallynotwell-suitedforgeneralsimulations.Therefore,theobjectiveisorientedtowardscreatingexibleandcomputationallyinexpensive elasticmaterialmodelsattheexpenseofaccuracy. Withincomputationalmechanics,severalgraphicsorientedmethodshavebeenproposedforthereal14

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Figure3.0.2:IllustrationoftwodeformationsappliedtotheStanforddragonvolumetricmodel.Thedeformationsareimposedusingaforce-basedmanipulationtoolintegratedintoaninteractiveapplication.The coresimulationisbuiltontheproposedata-drivenNeuralElementNEelasticmaterialmodel. timesimulationofelasticsolids.ThisincludesMass-SpringSystemsMSS,FEA-basedelasticsolids, constraint-basedsystems,andcomplexsimulationmethodsthatmixparticleandsoliddynamicssuchas theMaterialPointMethodMPM.Whiletheseexistingmethodsprovideawiderangeofcapabilitiesfor simulatingnumeroustypesofmaterialsandphysicalphenomena,therearestillrelativelyfewalternatives forsolidmechanicswithinreal-timeapplicationssuchasComputerAidedDesignCADtools,games, andmodelingtoolsfor3Dprintingthatrequirecloseapproximationsoftheanalyticalsolutionsprovided byniteelementsolutions.Withinthecurrentlandscapeofexistingelasticmaterialsimulationmethods, neuralelementsisaimedatprovidinganaccuratemethodthatcanbridgethegapbetweenhighperformancemethodssuchasMSSandcompleteniteelementmodels.Theobjectiveistoprovideacloseapproximationtowell-establishedconstitutivemodelsfromcontinuummechanicsbutsupplementthefundamentaloperationsthroughdata-drivenapproaches.Thisprovidesdifferentformsofoptimizationsand affordsnewfeaturesthatcanonlybeexpressedthroughdata-drivenmethods.Additionally,throughthe formulationofthehybridmethod,wecanalsoprovideexibilityinthesimulationframeworktofacilitate thedeploymentofthecoremethodinvariousapplicationsincomputergraphics. 3.1RelatedWork:SimulationofElasticSolids Thesimulationofsolidmaterialsincorporatescontributionsfromnumerouseldsincludingcontinuum mechanics,numericalanalysis,andcomputergraphics.Fromthecontributionsprovidedbythevastarrayofcorealgorithmsthatmakesthesimulationofsolidelasticmaterialspossible,therehavebeenan assortmentofdifferentmodelsproposed.Thesevaryfromhighlyaccurateniteelementmodelsusedin 15

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mechanicalengineeringtonumeroushybridmodelsthataimatachievingreal-timeperformanceforapplicationsincomputergraphicsandgamedesign.Withinthispopulatedeld,weprovideanoutlineof wheretheneuralelementformulationforsimulatingelasticsolidsfallsandhowitisdistinctfromexisting methods. 3.1.1Mass-springSystems Themostprominentmethodofsimulatingdeformableobjectswithininteractivecomputergraphicsand gamedesignismass-springsystems.Thisisduetothesimplicityandperformancethatcanbeobtained throughthecombinationofmultiplepointmasseswitharbitrarysprings.Theexibilityandperformance ofthismethodforclothsimulationhasestablishedawidespreadadoptionofthismethodwithinmostrealtimegraphicsapplications.Forclothsimulation,theabilitytoformalizeshear,bend,andstructuralcomponentsofclothleadstophysicallyplausibleresultsthathavebeenincrementallyimprovedtoprovide realisticsimulationswithinreal-timetimeconstraints.However,thismethodisill-suitedforthesimulationofvolumetricstructures.Infact,therehavebeennumerouscontributionstotheformulationofmassspringmeshesthatproposedifferentspringcongurations,constraints,andevensecondaryalgorithms thatoperateonmeshtopologytoimprovethesimulationofsolidmaterialsusingthismethod.Thisisdue totheinadequaterepresentationofthematerialforcesthatareexhibitedinreal-worldmaterialsandhow theyareapproximatedthroughacollectionofsprings.Furthermore,inaseriesofcontributions,[13,14] illustratedthattheMSSmodelandtheFEformulationofelasticmaterialsare incompatible .Thatis,there isnowaytoproperlyrepresentHookeanelasticmaterialbehaviorsusingtheMSSformulation.Forthe two-dimensionalcase,therehasbeensomeprogressinhowspringcongurationscanbedenedtoapproximatesimilarbehaviors.Yet,inthethree-dimensionalcase,thereisnoexactanalogforreproducing thebehaviorsofvolumetricobjectsthroughasetcollectionofsprings.Additionally,thefundamentalmaterialpropertiesofthismethoddiffergreatlyinthesemanticmeaningofmaterialbehavior. Formass-springsystems,thecorematerialpropertiesaregenerallydenedas ks ,theHookeanspring coefcientnormallyidentiedas k inthe f = kx relationand kd whichrepresentsthedampingcoefcientthereduceskineticenergyofthesystembasedonvelocity.Thisisdistinctlydifferentthanthecoef16

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cientsusedincontinuummechanics.ForsolidelasticHookeanmaterialsbasedontheisotropicmodel proposedincontinuummechanics,theprimarymaterialcoefcientscanbedenedasYoung'smodulus themodulusofelasticity E andthePoissonratio v ofthematerial.Thespecicationofthesetwocoefcientsprovidesarelativelyconsistentbehaviorsbasedonsolidmechanics,evenbetweendifferentimplementations.Theproblemisthatthereisno direct mappingbetweenthesequantitiesandthespringcoefcientsinmass-springsystems.Toconsolidatethebehavioraldifferencesbetweenniteelementmeshes andmass-springsystems,therehasbeennumerousattemptstoformalizethematerialbehaviorrelationship.Ingeneral,theobjectiveoftheseapproacheshasbeentospecifygeometricandelasticmaterialpropertiesandanalyticallyderivethespringtypesandcoefcientsthatarerequiredtoapproximatethesame behaviorasanFEA-basedmodel.Thishasleadtothenumerousapproachespresentedby[15]below: GeometricApproximation Wilhelmsetal.[16] k = A 1 + A 2 l 2 0 k a = c P i A i 3 GeometricApproximation VanGelderetal.[13] 2 D : k c = E 2 P e A e j l 0 j 2 3 D : k c = E P e V e j l 0 j 2 GeometricApproximation Zerbatoetal.[17] k = E 1 + E 2 2 P i V i l 2 0 + geneticalgorithm GeometricModel+Elasticity Arnabetal.[18] k e = E P i 2 j V j l 2 i k g = E P i 2 j V j 2 l 2 i )]TJ/F22 7.9701 Tf 6.587 0 Td [(v AdjustmentofVanGelderetal. Macieletal.[19] k = k 0 cos 0 + P i k i cos i l 0 + P i l i cos i ReferenceFEAModel Lloydetal.[20] k tri = P e Eh p 3 4 k e rect = P e 5 16 hE k d rect = 7 16 hE k tetra = P e 2 p 2 21 lE k tetra 2 = P e 2 p 2 21 lE 4 5 k V = p 2 84 l 3 E 2 5 Parameterizationtensileshear Baudetetal.[14] k e = E j 2 v +2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(i 2 4 l 0 h 0 + v k f = G = E 2+ v k ed = El 0 v +1 8+ v k d = 3 El 0 8+ v FEAApproximation Natsupakpongetal.[21] k e = 3 + 2 k f = )]TJ/F20 7.9701 Tf 6.219 -4.569 Td [(5 12 + 3 4 17

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Theshearnumberofdifferentillustratesthedifcultyinspecifyingtheelasticmaterialcoefcients toobtainsimilarbehaviorsfrommass-springmodels.TheprimaryobjectivehereistoretainthesignicantperformanceincreaseprovidedbyMSSmodels,butprovidethecoefcientsandaccuratebehaviors ofsolidmaterialsaswouldtypicallybeobtainedusinganFEA-basedsimulation.Thecomplexityand inaccuraciesillustratedbytheseapproachesprovidesanoptimalopportunityforaninversemodelingsolution.InsteadofattemptingtoreplicatetheinconsistencyofthebehaviorsexhibitedbyMSSandFEA basedmodelswithanalyticalmethodsthatonlyprovidepartialapproximationsofthedesiredresult,a data-drivenmethodcanbeemployed. 3.1.2Constraint-basedSolvers Constraint-basedsystemsoperateonparticlesasafundamentalbuildingblockforsimulatingeverytype ofphysicalphenomenon.Thisincludestheintegrationofrigid-body,deformableorelasticmaterials,gas, anduiddynamics.Theappealofthisapproachisthatitprovidesa unied simulationpipeline.Thatis, theconstraintformulationthatdictatestheinteractionbetweenparticlesisconsolidatedintoasingleset ofoperations,namely,thegenerationandsolutionofconstrainedsystems.Thesystemscanthenbeefcientlycomputedforeachsimulationtime-step,handlingtheinteractionsbetweenallparticleandconstrainttypes.Theappealofthesemethodsisthattheycanbeaptlyadaptedtoparallelarchitecturesfor simulatinglargequantitiesofparticles.Sincethepremiseofthemethodisderivedfromparticleinteractionsintegratedintolargercontinuousvolumes,thenumberofparticlesrequiredformostobjectsisstill extensive.However,withthescalabilityoftheapproach,itmakesgraphicsprocessingunitsaprimetargetforthedeploymentofthesesystemsduetothehighthroughputthatcanbeachieved.Thisprovides aviableframeworkforcreatinghighlyadaptabledeformation,granular,uid,andrigid-bodybehaviors thatcanbeusedinavarietyofreal-timeapplications.Thismethodhasbecomeapredominantsimulation methodforinteractiveandreal-timegraphicsapplications. 18

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3.1.3DeepLearningApproaches Theprospectforintroducingdeeplearningapproachesforthesimulationofsolidelasticmaterialshas beenwelldescribedwithrespecttoreconstructingtheforce-displacementrelationforgeneraldeformable simulationsas nearlyimpossible [12].Duetotheinherentnatureoftheproblem,theconstitutivemodels andnumericalmethodsarefundamentallyill-suitedforlearning-basedadaptations.Basedontheconstraintsoftheseformulations,[12]concedetheformulationofadeeplearningapproachforcompletely replacingtheconstitutivemodelandintegrationmethodologiesofelasticmaterialsimulation.Rather,they providea warping formulationthatusesdeeplearningtomodifyelasticmaterialbehaviorstomatchthat ofcomplexnon-linearformulationssuchasSt.Venant-KirchhoffandNeo-Hookeanmaterialmodels. 3.2MethodOverview Neuralelementsproposesamethodofsimulatingisotropicelasticmaterialsforthesimulationofsolid deformableobjects.Thismethodisderivedfromadecompositionoftheconstitutivematerialmodelprovidedbytheniteelementformulationofisotropicelasticmaterials.Throughthefunctionalanalysisof thismodelandinversemodeling,wetransformthispurelyanalyticalmethodintoadata-drivenmethod basedonacompletelynewcomputemodel.Thisallowsustointroduceseveralnewcontributionstowards thesimulationofreal-timedeformableobjectsthatcloselymimictheaccuracyprovidedbytraditional FEA-basedsolutions.Theseinclude:introducingaphysicalmodelthatresidesbetweenmass-spring andFEA-basedsystems,trivialintegrationofheterogeneousmaterials,nativehandlingofdynamicboundaryconditions,dynamictopologyduetotheremovalofmatrixsystems,andaparallel friendlycomputemodel. 3.2.1Contributions Theneuralelementbasedsimulationofelasticmaterialsisaimedatprovidingadata-drivenmethodfor generatingcomplexdeformationsofsolidobjectswithininteractiveandreal-timeapplications.TheobjectiveofthisapproachistoprovidetheaccuracyandstabilityfromFiniteElementAnalysisFEAmethods ofsimulatingdeformablemodelstoanimation,gamedesign,interactiveprototyping,and3Dprintdesign. 19

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Figure3.2.3:IllustrationoftwodeformationsappliedtotheStanforddragonvolumetricmodel.Thedeformationsareimposedusingaforce-basedmanipulationtoolintegratedintoaninteractiveapplication.The coresimulationisbuiltontheproposedata-drivenNeuralElementNEelasticmaterialmodel. Currentchallengesfordeformableobjectsimulationwithinmanyinteractiveapplicationsisthecomputationalcomplexityandinexibilityofexistingsolutionmethodsgeneraluse.Throughtheintroductionof theneuralelementdesign,wecanprovideanewalternativesolutionforsimulatingsolidelasticmaterials thatismoreaccuratethanmass-springsystemsandamoreefcientFEA-basedsolution.Belowisabrief overviewoftheprimarycontributionsprovidedbytheneuralelementformulation: • SolidElasticMaterials :IntroduceamethodderivedfromFEAformulationsofisotropicelastic materialstoaccuratelysimulatesolidmeshes.Thisprovidesanewmethodthatliesbetweentheperformancebutinaccuracyofmass-springmeshesandacompleteniteelementsimulationofelastic materialswhicharecomputationallyexpensive. • InterchangeableSolutionMethods :Neuralelementsarecomposedofseveralbuildingblocksthat includethemachinelearningmodelusedtopredictmaterialresponsesbasedonstress-to-strainrelationships,numericalintegrationmethods,andtraining-data.Eachofthesecomponentsareinterchangeable,allowingforaexibledesigninterfaceofthecoremethod. • HeterogeneousMaterialModels :Duetotheindependentevaluationofallelementswithinasimulation,everyelementcanbeassignedauniquematerial.Sincethecoremethodevaluatesthismaterialpropertythroughaneuralnetwork,thereisnoadditionalcostorsystemreconstructionassociated withchangingelementmaterials.Thismeansthatelementmaterialscanbechangeddynamicallyat anytimeduringasimulation. • DynamicBoundaryConditions :Todenehowpartsofamesharexedormoveduringasimulationisdenedbyasetofboundaryconditions.Unlikeniteelement-basedsimulationswherea 20

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completesystemreconstructionisrequiredtoupdateboundaryconditions,theneuralelementformulationnativelyhandlesbothxedanddynamicboundaryconditions.Flexibilityindeningthese conditionsprovidesanaturalextensiontointeractivesimulations. • DynamicTopology :Independentevaluationofdiscreteelementsallowsforinstantchangesintopology.Sincethereisnosystemmatricesdenedbasedonthetopologyofthesimulatedmesh,splits alongelementbordersandtheintroductionofnewelementscanquicklybeintegratedtoanexisting mesh.Thisallowsforbehaviorssuchasrippingortearingandcuttingofelasticmaterials. • ParallelComputeModel :Theevaluationofindividualelementsisperformedindependentlythrough parallelneuralnetworks.Derivedfromthedata-drivenapproach,wedeneasetofneuralnetworks thatcanbedynamicallyinstancedtoprovideelementresponsesformaterialbehaviorsfromparallelcomputemodels.Thereforethemethodisextendedtomulti-coreparallelcomputationwiththe optionofextendingthistoGraphicsProcessingUnitsGPU 3.2.2FiniteElementFormulationforNeuralElements Inniteelementsimulationsforelasticmaterials,theevaluationofanobjectsdeformationbehaviorisdirectlyevaluatedthroughasetofgoverningequationsandformalizationsthatdenestructuralrigidityas anintegratedsystemofelements.Theobjectivequantitiesobtainedthroughthesolutionofthesystemdependontheformofthisanalysisandthepost-processingperformedontheoutcomeofthesolution.In themostcommonformusedforelasticmaterials,theobjectiveistodeterminethenodaldisplacements, stress,andresponseforcesincurredfromanappliedloadandpredenedboundaryconditions.These boundaryconditionsandtheappliedloaddenetheconstraintsthatcharacterizethedeformationofthe object.Fromthecomputedresult,anumberofvaluablequantitiescanbeevaluatedtodeterminetheperformanceorbehaviorofthematerialandthegeometricstructureofthemodel.Inmostcases,theapplied loadinducesexternalforcesontheobjectfromwhichtheresultingdeformationcanbemeasuredasnodal displacementsorasstresstoevaluatecriticalyieldcriteriarelatedtoplasticdeformation,fracture,and otherphysicalcharacteristicsusedwithinnumerousengineeringapplications. Interactivesimulationsthatincorporatedeformableobjectsbasedonniteelementmodelshavebeen 21

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highlyoptimizedtomodelseveraldifferentelasticmaterialformulations.Throughtherigorousstudyfor improvingnumericalintegrationtechniques,modalanalysisforbehavioroptimization,andthevariousalternativematerialmodelsthathavebeenintroduced,theadoptionofdeformableobjectsderivedfromsolid mechanicshasbeenlimited.Theperformanceimprovementsobtainedbyreducingtheaccuracyorlimitingthedeformablecharacteristicsofsolidobjectshasnotpusheddeformablemodelsintotheforefront ofreal-timesimulationssimplybecausetheyrequireadisproportionateportionofthecomputationalload requiredforinteractivegraphicsapplications.Existingsolutionsintroducedvarioustechniquesforapproximatingthebehaviorsofdeformablesolidmaterials,butmanystillrelyonthepremiseofphysically plausiblebehavior.Therefore,thevisualizeddeformationmaybeadequateformostinteractiveapplicationswheretheabsoluteaccuracyisdifculttoevaluate;however,therearestillotherapplicationsthat benetfromtheaccuratedeformationandstressevaluationsprovidedbyanFEA-drivenapproach.Thisis primarymotivationtowardsusingtheisotropicelasticmaterialmodelfromtheFEAformulationtoderive anaccurateandefcientdata-drivendynamicsmodel. Neuralelementsatitscorereliesontheevaluationoftheisotropicelasticmaterialmodelfordiscreteelementbehaviorstogenerate synthetic materialresponses.Thereasonbehindthisisduetothedual objectivesoftheneuralelementderivation:togenerateaccuratematerialresponsesthroughacomputationallyefcientdecompositionoftheprimarylinearelasticmodelandprovideafastandexible systemarchitecturethatenablesbothaccuratesimulationsbutalsoprovidesavarietyofusefulfeaturesfor interactiveapplications.Theseincludetheabilitytodeneheterogeneousmeshmaterials,easy-to-dene boundaryconditions,efcienthandlingoftopologychanges,motion-drivendeformations,andparallel friendlycomputemodel. Toprovidethesefeatures,wedecomposethestandardelementsoftheisotropicelasticmaterialmodel toexploithowelasticsolidsareformulatedtoobtainanefcientbreakdownofmaterialbehaviorsthatcan bereplicatedthroughadata-drivenapproach.Thisrequiresanalysisofthecoreshapefunctionsandthe evaluationofthestress-to-strainrelationshipofbasicelementscommonlyusedinFEAformulations.Additionally,toevaluatethisrelationshipforgeneratingtheconstitutivemodelusedwithintheneuralelementdesign,wealsohavetogeneratetheFEA-basedmaterialresponsesofelements. 22

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FEAFormulationOverview .Thepremiseofthe Galerkin FEAformulationistodecomposethecontinuousgeometryofanobjectintodiscreteelementsthatexhibitthebehaviorsofanisotropicelasticmaterial. Thegenerationofthisdiscretesetofelementsisknownas meshing .Thisprocessdividesthecontinuous geometryofasolidobjectormanifoldsurfaceintodiscreteelementsbasedonapre-selectedprimitive shape.Fortwo-dimensionalmeshesthisistypicallytriangleswhereeachnodehastwoDegreesofFreedomDOFsandforthree-dimensionalgeometry,tetrahedralelementsaretypicallyselectedwhereeach nodehasthreeDOFspernode.Thesediscreteelementsarethencharacterizedbyasetof shapefunctions thatdescribehowthegoverningequationsuchasHooke'sLawisappliedtorepresenttherelationship betweenstressandstrain.Thisprocessincorporatestheformulationofthegoverningequationsthatdictatematerialbehaviorandprovidethematrixequationsrequiredforsolvingforvariousquantitiessuchas nodaldisplacements,materialresponses,andelementstress.Neuralelementbehaviorisformedofthediscreteobservationsofthebehaviorsgeneratedbyapre-denedFEAsolution.Thisprovidesanexactmodel ofthematerialbehaviorsthateachneuralelementaimstoreplicate.Toprovidethismaterialresponseand theevaluationofthestress-to-strainofanidealizedmaterial,wedenethe elementstiffnessmatrix k ofa givenelementtypetriangle,tetrahedra,etc.basedontheshapefunctionsthatcharacterizeitsbehavior. Foreachelementinamesh,thismatrixisgeneratedandthenintegratedintoalarger globalstiffnessmatrix K thatdenesthematerialandstructuralmechanicsofagivenmesh.Ifwearegivenasetofexternal forces f ext ,andtheglobalstiffnessmatrix K ,wecansolveforthedisplacements U ofthenodeswithin themesh.Tosolvethiswecansimplyobtainthesolutionto KU = f ext ,solvingfor U .Conversely,if wearegivenasetofdisplacements,wecansolvefortheresultinginternalforcesofanisolatedelement f int = KU . 3.2.3CorotationalFormulation Linearstress-to-strainrelationsderivedfromelasticmaterialpropertieshaveintroducedanaccurateniteelementsolutiontoanalyzesmalldeformationsofcontinuoussolidmaterials.The innitesimalstrain modelcanbeusedreplicatedeformationswherethedisplacementofmaterialpointsisinnitesimally smallwithrespecttothescaleoftheobjectssize.However,duetothelinearnatureofthemodelandthe 23

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assumptionthatdisplacementsaresmall,thisformulationintroduceslargeartifactsinmaterialbehavior duetothelargerotationsthathighlyexiblematerialsallow.Therefore,tosolvethisproblemandfacilitatelargerotationaldeformations,thecorotationalformulationwasintroducedto:assumethateach elementisstillaccuratelyrepresentedbytheinnitesimalstrainmodelandaccountfortherotation ofthematerialastheobjectdeformsindependently.Bycomputingandapplyingtheinverseoftherotationforeachelementateachtime-step,thelinearmodelcanstillbeusedtocomputeaccuratedeformation behaviorsofthecontinuousmaterial.Thisintroducestheconceptofgeometricallynon-linearmaterial simulationsthroughthecorotationalformulationofFEAsimulations. Thecorotationalcorrectionforelasticniteelementsisrequiredduetothecomputationofelement displacementswithinthelinearformulationbasedontheCauchystresstensorforelementsundergoing rigid-bodyrotations.TheproblemwiththisformulationisthatthisformulationandtheCauchystresstensorarenot rotationinvariant .Thatis,therotationoftheelementisnotcorrectlyaccountedforinthecomputationoftheelasticstress-to-strainrelationship.Thisresultsinlargeandphysicallyincorrectartifacts forlargerotationaldeformations,leadingtoincreasesinvolumeaselementswithinthematerialrotate awayfromtheirrestpositions.Tocorrecttheseartifacts, StiffnessWarping wasintroduced[22]toremove theartifactsthatthelinearelasticforcesintroduceonthedeformablebody.Initially,thiswasappliedto thenodesofelementsleadingtotwoproblems:therotation R i ofvertex i hastobecomputedbyits adjacentverticeswhichisanambiguousproblemandtheelasticforcesarenotguaranteedtosumto zero,resultingintheintroductionofghostforces.Thistechniquewasimprovedthroughthetransitionto an element-based corotationalformulation[23].Theobjectiveistoextractthedeformationandrotational componentsofanelementbetweenitsreferencerestandcurrentdeformedstate.Theresultofthisprocesswillprovideanorthogonalrotationmatrix R thatrepresentstheoptimalrotationbetweentherestand deformedstatesoftheelement.Thisrotationcanthenbeusedtoreturnthedeformedelementbacktoits restcoordinatespacetocomputenodaldisplacementsanditsinversecanbeusedtotranslateresponse forcesbacktotheglobalcoordinatesystem. 2DTriangleElementRotation .Thetriangleelement T inthereferencecongurationcontainsthree nodes: T r = f P 0 ;P 1 ;P 2 g andtheelementinitscurrentdeformedstateisdenedbythechangesin 24

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nodepositionsto T c = f Q 0 ;Q 1 ;Q 2 g .Anypoint p 2 R 2 ,simplydenedas: p = p x ;p y ,hasasetof Barycentriccoordinates 1 ; 2 ; 3 thatuniquelydenethispointwithrespecttotheelementnodes.This Barycentriccoordinatecanthenbecomputedinboththereferenceandcurrentcongurations. Figure3.2.4:Dynamicbehaviorofa2Dtrianglebetweentherestandcurrentcongurationsleft.The elementnodaldisplacements d 0 ;d 1 ;d 2 duetothedeformationaredenedbyrotatingtheelementtothe optimalpre-deformationstatewithbothelementscoincidingattheircenterofmassright. Therelationshipbetweenthispointandtheelementcanbeintegratedintoahomogeneousformtosatisfy theconditionsforboththetrianglesreststate T r andthetrianglescurrentstate T c : T r = 2 4 P 0 x P 1 x P 2 x P 0 y P 1 y P 2 y 111 3 5 2 4 0 1 2 3 5 = 2 4 p x p y 1 3 5 T c = 2 4 Q 0 x Q 1 x Q 2 x Q 0 y Q 1 y Q 2 y 111 3 5 2 4 0 1 2 3 5 = 2 4 q x q y 1 3 5 .2.1 TherelationshipabovecanbeusedtosolvefortheBarycentriccoordinateofthepoint,howeverthisisnot theobjectiveoftheformulation.Rather,theconditionusingthiscoordinateisusedtocreatearelationship betweentheelementsstates.TheBarycentriccoordinaterepresentsthesamepointinboththereference andcurrentcongurationsoftheelementtoformalizethelineartransformationbetweenthetwoprovided states.Thenthesameconditionholdsforallpointsinboththerestanddeformedelements,where p and q aredifferentCartesiancoordinatesbuthavethesameBarycentriccoordinate.Thuswecanexpressthe matricesin3.2.1usingtwoequations: P = p and Q = q .Thesecanthenbecombinedtoforma 2Drepresentationofthegeneralequationpresentedin[23]. q = Q = QP )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 p = Ap whichcanbesimpliedto A = QP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 .2.2 Where A = QP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 representsthetransformationoftheelementbetweentherestandcurrentcongurations.Thestructureof A isdenedbythreecomponents:theelementdeformation,rotationof 25

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theelement,andtheelementstranslation.The2Dtranslationofthetransformationcanbedirectlyextractedfromthismatrixusingthevalues t x ;t y ,howeverboththerotationanddeformationcomponents betweenthetwoelementstatesaredenedasthesub-matrix B thatresideswithin A asshowninEquation 3.2.3. A = 2 4 B 00 B 01 t x B 10 B 11 t y 001 3 5 where B = DeformationandRotation t = Translation.2.3 Sincethematrix B isacombinationofboththedeformationandrotationoftheelement,thematrixhasto bedecomposedintotherotationanddeformationcomponents.Tosolvethisproblem PolarDecomposition isusedtoextracttherotation R anddeformation U matricesfrom B .Whereweobtainthedeformation U asthepositivedenitematrixofthedecompositionandtherotationmatrix R astheorthogonal matrixofthedeformationgradient F .Forthe2Dformulationofthisproblem,thereisasimplemethod proposedin[24]toobtainthedecompositionofanarbitrary 2 x 2 matrixusingeigen-vectorsandeigenvaluestocomputethedecompositionoftheorthogonalandpositivedenitematrices. 3DTetrahedralElementRotation .Thetetrahedralelement T inthereferenceconguration r containsfournodes: T r = f P 0 ;P 1 ;P 2 ;P 3 g andtheelementinitscurrent c deformedstateisdenedbythe changesinthesenodepositionsto: T c = f Q 0 ;Q 1 ;Q 2 ;Q 3 g .Inthe3Delementcase,anypoint p 2 R 3 , simplydenedas: p = f p x ;p y ;p z g ,hasasetofBarycentriccoordinates 1 ; 2 ; 3 ; 4 thatuniquelydenethispointwithrespecttotheelement.Therelationshipbetweenthispointandtheelementintherest statecanbeintegratedintoahomogeneousformtosatisfy: Figure3.2.5:Dynamicbehaviorofa3Dtetrahedrabetweentherestandcurrentcongurationsleft.The elementnodaldisplacements d 0 ;d 1 ;d 2 ;d 3 duetothedeformationaredenedbyrotatingtheelementto theoptimalpre-deformationstatewithbothelementscoincidingattheircenterofmassright. 26

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T r = 2 6 6 6 4 P 0 x P 1 x P 2 x P 3 x P 0 y P 1 y P 2 y P 3 y P 0 z P 1 z P 2 z P 3 z 1111 3 7 7 7 5 2 6 6 6 4 0 1 2 3 3 7 7 7 5 = 2 6 6 6 4 p x p y p z 1 3 7 7 7 5 T c = 2 6 6 6 4 Q 0 x Q 1 x Q 2 x Q 3 x Q 0 y Q 1 y Q 2 y Q 3 y Q 0 z Q 1 z Q 2 z Q 3 z 1111 3 7 7 7 5 2 6 6 6 4 0 1 2 3 3 7 7 7 5 = 2 6 6 6 4 q x q y q z 1 3 7 7 7 5 .2.4 Usingthesameformulationasthe2Dcase,thesematricesformtwoequations: P = p and Q = q .Thesecanthenbecombinedtoformthe3Drepresentationoftheequationpresentedin[23]. q = Q = QP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 p = Ap whichcanbesimpliedto A = QP )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .2.5 Aswiththetwo-dimensionalcase, A = QP )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 representsthetransformationoftheelementbetweenthe restandcurrentcongurations.Thestructureof A isdenedbythreecomponents:theelementdeformation,rotationoftheelement,andtheelementstranslation.Thetranslationofthetransformation canbedirectlyextractedfromthismatrixusingthevalues t x ;t y ;t z .Thestructureofthematrix A and thesub-matrix B thatresideswithin A isshowninEquation3.2.6. A = 2 6 6 6 4 B 00 B 01 B 02 t x B 10 B 11 B 12 t y B 20 B 21 B 22 t z 0001 3 7 7 7 5 where B = DeformationandRotation t = Translation.2.6 Toperformthedecompositionof B foranarbitrary 3 x 3 matrix,thereareseveraldifferentapproaches. TheserangefromusingSingularValueDecompositionSVDmethodsthataretypicallyexpensivetoiterativesolutionsthatcansacriceaccuracyforperformance[25].Asmanyofthesetechniquesstemfrom theabstractmathematicalprocessofdecomposinganyarbitrarymatrix A ,theycanbeaccuratebutimposelargebottlenecksinreal-timesimulationsduetotheircomputationalcomplexity.Thisisduetothe generalityof A . Withinthexeddomainofacontinuoussimulationthereareotherassumptionsthatcansignicantly reducetheproblemssolutionspace.Sincethedeformationsinthesimulationareconsideredcontinuous, thecurrentdecompositionwillonlyshiftslightlyfromthepreviousdecompositionresultassumingasmall simulationtime-step.Basedonthisassumption,anefcientmethodhasbeenintroducedtoincrementallyextractingtherotationmatrixfromthedecompositionusinganiterativesolution[26].Thismethod operatesoffthenotionthatwecansignicantlyreducethegeneralityof A toeffectivelyconvergeonthe 27

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optimaldecomposition.Thismethodusesaninitialguessthatistheresultofthedecompositionfromthe previoustime-steporthefromtheinitialrestpositiontoiteratetowardstheoptimaldecompositionofthe currenttime-step. 3.2.4Data-drivenFEA Theconceptofdata-drivenniteelementsimulationshasbeenwellestablishedforintroducingavariety ofdifferentmethodstoaugmentandimproveexistinganalyticalFEAformulations.Finiteelementsimulationsarealsoinherentlypartiallydata-driventosomedegreeduetotheevaluationormeasurementof thematerialpropertiesusedtoparameterizetheshapefunctionsthatdescribeelementbehavior.Infact, oneofthecommonformsofadata-drivenFEAmethodistheevaluationofelementresponsestodeformationsperformedonreal-worldmaterialsmeasuredinpreciseexperimentalsetups.Fromtheseinitialcontributions,researchintodata-drivenFEArapidlyexpandedfromtheevaluationofmaterialresponsesfrom lab-drivenexperiments,toimprovingtheaccuracyofsimulatedmaterials,creatingenergy-basedsampling forimprovingdynamicelementbehaviors,andusingDeepLearningtoaugmentlinearelasticmodelsto incorporatenon-linearmaterialbehaviors.Manyofthesedata-drivenmethodshaveaimedatproviding moreaccuratesimulationresultsforvariousengineeringapplicationsorhaveattemptedtoresolvedisparitiesbetweenreal-worldmaterialbehaviorsandsimulationresults.Whiletherehasbeentremendous progressinthenumberofmethodsthataugment,improve,orextendthefunctionalityoftheniteelement methodforelasticmaterials,manyexistingmethodslinetheperipheryofthecorecomputationalFEAformulation.ThisisbecausemostofthecoreformulationswithintheFEAsolutionmethodare incompatible withorcannotbeexpressedasdata-drivencomputations.Thisisbecausethecoresolutionsforsolving systemsofequationsorintegratingovertimearebasedondiscretenumericalalgorithmsthatapproximate analyticalmethods.Sincetheseformsofproblemsarenotwellsuitedtotheapplicationofapredened trainingdomaintogeneratesolutions,mostdata-drivenmethodsaugmenttheformulationofthesecore solutions. 28

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3.3Data-drivenElementBehavior Inversemodelingfordata-drivenelasticmaterialshavedenedaprominentresearchareathatattempts tointegratesolidmechanicswithnewmethodsinmachinelearningtoimprovetheaccuracyofsimulated materials.Variousdata-drivenmethodshavebeenemployedwithincontinuummechanicsandelasticmaterialpropertymodelingforimprovingsimulationaccuracyandprovidingsolutionsforinconsistencies observedbetweensimulatedandreal-worldmaterials.Thepremiseofmodelingelasticelementbehaviors onthemeasurementsofreal-worldmaterialsprovidesabasisforimprovingtheutilityandaccuracyof FEAsimulationsfornumerousengineeringapplicationsandadvanced3Dprintingtechniques[27].Based onthesecontributions,therearethreeprimaryclassesofmethodsthatusedata-driventechniquestoaugmentorimproveelasticmaterialsimulations:theuseofreal-worldmeasurementstobothformulate andpredictmaterialproperties,theuseofdata-drivenmethodsasnumericalalgorithmstosupplement orreplacetheconstitutivemodelofelasticmaterials,andtheuseofdata-driventechniquestoaugment elasticmaterialbehaviorsEx.augmentingsimulationsoflinearmaterialstoexhibitnonlinearbehaviors. Neuralelementspresentsamethodthatfallswithinthesecondclassofdata-drivenmethodstoenablea constitutivemodelderivedfromFEA-drivenelasticmaterialbehaviors.DuetothisformulationreplacingthecorenumericalalgorithmsusedinmostFEAelasticmaterialsimulations,weareabletoexploit thisnewcomputationalmodeltoprovideanaccurateandexiblesimulationofsolidmaterialsthatcanbe usedwithininteractiveorreal-timeapplications. Oneoftheprominentchallengesassociatedwithderivingadata-drivenphysicalsimulationishow thedatacanbeusedtoaugmentorimprovesimulationaccuracy,utility,orperformance.Dependingon theobjectiveofthedata-drivenmodel,therearenumerousconstraintsimposedbytheformulationsofthe governingequationsandintegrationtechniquesthatlimithowthedatacanbeintegratedintoexistingsystems.Inthegeneralformofdynamics,theprocessofupdatingelementstiffnessmatricesandnumerical integrationarenotwellsuitedforbeingdirectlyreplacedbymethodsfrommachinelearning.Inmostinstances,theseconstraintsdirecthowmostmachinelearningmethodsareappliedtoFEAsimulationmethods.Forsolidmechanics,theprimarytargetistheconstitutivemodelofthesimulatedelasticmaterials. 29

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Thisisbecausetherelationshipbetweenstressandstrain[28]ormaterialforceresponsesiswellestablishedbetweenagivenbehaviorandsampleresponsesthatcanbeobtainedthroughrepeatableexperiments.Themostcommonformthisprocessisrelatedtotheexperientialderivationofcomplexmaterial propertiesobtainedfromperformingreal-worlddeformationanalysis.Whilethefoundationoftheneural elementsderivationisbuiltonasimilarpremise,thekeydifferenceisthattheobjectiveistocompletely replacetheconstitutiveelasticmaterialmodelwithadata-driven syntheticmodel .Theintroductionofa syntheticmodelischaracterizedbythevirtualgenerationof synthetic elasticmaterialdatathatisusedto denetheaccuratebehaviorsofFEAelements,butwithacompletelynewcomputationalmodelbased onelementalforceresponses.Theobjectiveofthismodelistointroducelight-weightmaterialmechanics thatarehighlyadaptabletoreal-timeapplications,butmaintaintheaccuracyandbehavioroffullyformulatedlinearelasticmodelbehaviorobtainedusinganFEAsimulation. 3.3.1Element-basedMaterialResponse Elasticmaterialsundergostrain " = @=@X x )]TJ/F21 10.9091 Tf 11.313 0 Td [(X asafunctionofthereferencestate X oftheelement andthecurrentconguration x basedonthedisplacementsthathavealteredthestateoftheelementsgeometry.Duetothisstrain,thematerialexhibitsaresponsethatisafunctionoftheapplieddeformation andthepropertiesofthematerial.Ifwecurrentlyknowthedisplacementsofthedeformation,thenthesolutiontodeterminingthematerialresponseistomultiplytheglobalstiffnessmatrix K withthecurrent globaldisplacements U toobtaintheinternalresponseforces f int = KU .Theseinternalforcesrepresentthematerialresponsetosomegivendisplacementofeachelementnodeforeachdegreeoffreedom. Astheprimarydistinctionbetweenthesolutionprovidedbytheformulatingthesolidasanelasticcontinuumversusothersimplersimulationmethodssuchasmass-springmeshes,theresultofasinglenodaldisplacementalongonedegreeoffreedomincursmultipleresponseforceswithintheelement.Thenumber ofresponseforcecomponentsisthesameastheFEAsystemsizethatdependsonthenumberofnodesin theelementtypeandthetotaldegreesoffreedom.Intheinstanceofatwo-dimensionaltriangleelement, thedisplacementwithinonedegreeoffreedomofasinglenodewillresultinsixresponseforcesdueto thethreenodes,eachhavingtwoDOFs.Forathree-dimensionaltetrahedra,theelementwillgeneratea 30

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responseforcevectorforeachofthefournodes,resultingin 12 responseforcecomponents.Intheneural elementformulation,theobjectiveistoencompassthesematerialresponseswithindata-drivenmodelthat cangenerateaccuratedeformationsthroughinverselydeningthedesiredbehavioraltraits. Basedonthebehaviorofthematerialresponseforcesthataregeneratedbyprovidingasetofnodal displacements,weobservethreeprimaryfactorsthatwillcharacterizetheformulationoftheproposed data-drivenapproach:O1thenodaldisplacementinonedegreeoffreedomresultsinthegenerationof multipleforcesactinguponallnodeswithintheelement.ObservationO2veriesthattheresponseforce componentsofthematerialarenot rotationinvariant ,asnotedinthecorotationalformulation.Meaning thatforeverygeometricinstanceoftheelement,theresponseforcesaredifferentdependingontheformulationoftheelementstiffnessmatrix.ForobservationO3,weidentifythattherearedifferentelement congurationsthatresultinspecialresponseforceconditions.Oneuniqueconditioniswhenthedisplacementofanodeisappliedtoanelementsymmetricalaboutaprimaryaxisandthedisplacementactsalong thesameaxis.Duetothesymmetryinthiscase,theperpendicularforceinthe ^ y directioniszeroas showninFigure3.3.6left.However,iftheelementsorientationischanged,thentheconditionimposingthezeroforcecomponentwillchange,resultinginsixnon-zeroresponseforcesasshowninFigure 3.3.6right. Fromobservationone,weformulatethatforeachDOF,therewillbeasetof n d responseforce componentswhere n isthenumberofnodesintheelementand d istheDOFdimensionality.Thismeans thatforeverynodeandDOF,thereare n d responseforcecomponentsthatmustbeestimatedbythe constitutivemodeltoprovideanelementforceresponse.Stemmingfromobservationtwo,thisnewmodel willalsohavetoaccountforhowforceschangewhenanelementsgeometryororientationischanged. SincethebaseFEAformulationoftheelasticmaterialbehaviorisnotrotationinvariant,wecannotuse rotationstotranslateforcesbetweencoordinatesystems. Sincesimulatedmeshesaregenerallycomposedofalargenumberofelements,allofwhichvary insize,orientation,orgeometricstructure,thepredictionofnumerouselementresponseforcesbasedon thesefactorswouldrequirearbitrarilycomplexnetworks.Additionally,thetrainednetworkwouldalso havetoaccountforthepotentialmaterialpropertiesoftheelement E , v .Thisleadstoaninfeasible 31

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Figure3.3.6:Illustrationofthematerialresponseforcesofatwo-dimensionaltriangleelementproviding threeobservationsthatestablishthefoundationfortheproposeddata-drivenmaterialmodel.Thisincludes multipleforceresponsesforeachdisplacementleft,responseforcesarenotrotationinvariant,butfor rotatedelementswecanensureresponseforcecomponentsarenon-zeroright. prospectforacomputationallyinexpensivesolutionduetoallofthepossibleresponseforcesthatwould havetobepredictedbasedonthislargenumberofinputs.Conservatively,countinganinputforeachfactoralargeunderestimate,wewouldneedatleastveinputsscale,rotation,structure,elasticmodulus, andPoissonRatiotoaccountforthisvariance.Basedontheexpressibleentropyofaneuralnetworkwith thismanyinputsand 6 outputsfora2Delementor 12 outputsfora3Delement,thissolutionquicklybecomesunreasonable.Thisisduetothecomplexityofthenetworkstructurerequiredtoestablishthese relationshipsaccurately.Furthermore,whatistheprospectofthissolution,giventhatcurrentelasticconstitutivemodelsalreadyprovideexactresponseforcesinconcise closed-form solutions? Theseobservationsleadtothedevelopmentofseveralchallengingproblemsthatmustbeaddressed withinthenewdata-drivenmodel.Thesolutionstotheseproblemspresentedinthisworkestablishthe frameworkoftheneuralelementderivationandalsoprovidecontextofthedata-drivenmethodwithrespecttoexistingtechniques.Weenumeratethecoreproblemsas geometricvariance , materialvariance , rotationinvarianttransformations ,and computationalcomplexity . GeometricVariance .Howcanthewidegeometricvarianceofelementsbeaccountedfor?Everyelement hasdifferentmaterialresponsesbasedonanumberofdifferentidentiedfactors.Therefore,therepresentationofthetrainingdomainandnetworkstructurerequiredtoprovideaccuratepredictionsmakesthis problemill-suitedformanymachinelearningtechniques.Additionally,anysolutiontothisproblemalso 32

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hastocompetewiththecomputationalcomplexityoftheclosed-formequationsprovidedbycontinuum mechanicsusedinthegoverningmatrixequations.Theseequationsspecicallyformulatetherelationship betweenshapefunctionsandthegeometryoftheelementbasedoninterpolationasdescribedintheFEA formulationforelasticelements. MaterialVariance .HowcancomplexmaterialbehaviorsbemodeledthroughcomputationallyinexpensivemodelstoreconstructanentiresimulationthatapproximatestheaccuracyofanFEAformulation? Thisiswheretheproposedmethoddiffersconsiderablyfromsample-driventechniquesthatusereal-world measurementstoprovidemoreaccuratesimulationresultsofphysicalmaterials.Ratherthanproviding estimatedvaluesthatlooselyconnectreal-worldmaterialstosimulatedelements,ourmethodproposesto completelyreplacetheconstitutivemodel.Thismeansthatallmaterialresponsesareallgeneratedfrom anaggregateofsynchronized neuralnetworks . RotationInvariantTransformations .Howcanweobtaintransformationsofresponseforceswithout rotations?Fromthepremiseofadata-drivenmethod,wehaveidentiedthatrepresentingallmaterialresponsesgivenfromanyelementgeometryforanyorientationisill-suitedforadata-drivensolution.Ifwe assumethatwecouldobtainacompletesetofmaterialforceresponses,howcantheybetranslatedtoall elementswithinasimulatedmesh?Evenifwecangeneratethematerialresponsesfromagivenelement, theresponseswouldhavetobetransformedtotheglobalinstanceofeveryelement.Therefore,werequire asolutionthatcircumventstherotationinvarianceoftheFEAformulationtotransformelementresponses betweencoordinatesystems. ComputationalComplexity .Whatisthecomputationalcomplexityoftheproposedmethodcompared tothematrixformusedinastandardFEAsimulationsofelasticmaterials?Thisproblemiscomplexdue tothenumberoffactorsthatcontributetotheoverallalgorithmicstructureofanFEAsimulation.Thisincludesthecreationorupdateofelementstiffnessmatricesforallelements,theconstructionoftheglobal stiffnessmatrix,thepolardecompositionofallelements,andtheintegrationofthesystemexplicitorimplicit.Basedontheformulationoftheneuralelementnetworkarchitecture,thecomputationalcomplexityofbothmethodswillbeanalyzed. 33

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Tosolvetheseproblems,wedecomposetheclosed-formequationsoftheisotropiclinearelasticmaterialmodelintoafundamentalsetofbehaviorsthatcanbe triviallycaptured throughtrainingasynchronizedaggregateoflight-weightneuralnetworks.Todothiswecombineobservationstwoandthreeto eliminatethevariationalcomplexityofelementmaterialresponsesthroughtheseparationofthegeometric andmaterialcomponentsoftheunderlyingelasticmaterialmodel.Wethenintroducetheconceptof geometricratios tohandlethevarianceofelementshapesandforcetransformations,andcreateanewtypeof elementcalleda referenceelement thatpreciselydeneselasticmaterialbehaviorsbasedondata-driven inversemodeling. 3.3.2ReferenceElements,Coordinates,andGeometricRatios ReferenceElements .Inasimulationofanelasticmaterialcomposedofdiscreteelements,wecanevaluateamaterialresponsefromasetofdisplacementsthatdenethestresstostrainrelationshipwithina singleelement.Ifitisassumedthatthemeshiscomposedofanidealized,homogeneous,isotropicmaterial,thematerialresponsesofthethiselementrepresentsthesamebehaviorasallelementsminusthe geometricvarianceofeachelementinstance.Thisisakeyobservationtowardsconstructingthediscrete trainingdomainthatcanbeusedtorepresentthematerialresponseduetoandisplacementorstrainimposedonthiselement.Basedonthisobjective,theideaistonumericallylimitthisdomainasmuchas possiblewhilestillprovidingacompleterepresentationofthematerialsbehavior.Wecombineboththe displacement-to-materialresponseofasingleelementwiththeminimizationofthetrainingdomainneeded torepresentthebehaviorthroughthecreationofa referenceelement .Thiselementprovidesastress-tostrainstand-inthatprovidesamaterialresponsereferenceforallelementswithinthesimulation.This approachprovidestwoimportantcontributions:theelementcanbedenedtohaveasingleconstant geometricconguration,isolatingthematerialresponsefromthegeometryandthiscongurationcan bedecomposedtonodalcontributionstofurtherminimizethetrainingdomain.IntheNEformulation, thisrepresentsarecurringthemeofdecompositionandminimizationtobuildadata-compatiblemodelby consistentlyminimizingthetrainingdomainbasedontheFEAformulation. Areferenceelementissimplydenedasaconstantinstanceoftheelement-typeusedwithinthesim34

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ulation.Thiselementcanbedenedbyanysingleelementaslongasitadherestotwocriticalconstraints: thegeometryisnotdegenerateandthematerialresponseoftheelementisnon-zeroforanyapplied deformation.Fromobservationthree,thismeansthattheelementcanbeofregularshape,butmayrequire changestomeetthesecondconstraint.Thesimplestsolutionaddresssecondconstraintistochangethe orientationoftheelementbyanarbitraryrotation.Thisensuresthatforanydisplacementappliedtoany nodeoftheelementwillresultinsomenon-zeroresponse.Forexample,inthetwo-dimensionalcase,the referenceelementcanbedenedasanequilateraltrianglewithanarbitraryrotation ,resultinginthe generationofnon-zeroresponsesbytheelementinboththe ^ x and ^ y directionsforallnodesasshownin Figure3.3.7. Figure3.3.7:Exampleofareferencetriangleelement R usedfortwo-dimensionalsimulations.Thechosenreferencegeometryleftisrotatedbyanarbitraryangle centertoensurenon-zeroforcecomponent responsestoanapplieddeformationforallnodesright. Thisformulationcanbegeneralizedtoanyelement-typewithnodesthathaveanynumberofDOFs. ThismeansthatthebasicpremiseoftheNEmethodcanbeappliedto 1 , 2 ,or 3 dimensionalsimulations foranyelementtype.Thisisbecauseareferenceelementcanbedenedforeachsimulationinstance,as longasthematerialresponseoftheelementisnon-zeroforalldeformations.Similartothe2Dcase,the three-dimensionalreferenceelementcanbedenedastetrahedrawithanarbitraryrotation.Thedifference hereisthattoensuretheconstraintofnon-zeroresponseforces,theelementmustberotatedusinganarbitraryquaternion,orthreeEulerrotations x ; y ; z .Thiscreatesresponseforcecomponentsinthe ^ x , ^ y , and ^ z directionsforallnodesasshowninFigure3.3.8. Thereferenceelementprovidesareductionintheelementparametervariancethatallowsustomodel 35

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Figure3.3.8:Exampleofareferencetetrahedralelement R usedforthree-dimensionalsimulations.The chosenreferencegeometryleftisrotatedarbitrarilycentertoensurenon-zeroforcecomponentresponsestoanapplieddeformationforallnodesright. materialbehaviorsindependentoftheelementsgeometrysincethegeometryisconstant.Giventhereference,wecannowquantitativelyevaluatethestress-to-strainrelationshipoftheelementandexpress thisrelationshipwithinadiscretetrainingdomain.Basedontheisotropicelasticmaterialformulation, wecanmanipulatethereferencegeometryusingdifferentdisplacementstoelicitbehavioralresponses fromtheelement.Fromthereferenceelement,wecanthenestablishthisbehaviorasarelationshipbetweenYoung'sModulus E ,Poissonratio v ,theapplieddisplacement d ,andthegeneratedmaterial response. ReferenceCoordinates .Generatingdata-drivenmaterialresponsesfromareferenceelementonlyprovidesapartialsolutiontowardsbuildingacompleteresponsemodel.Sincetheresponseforcesareonly validforthereferenceelement,wemustestablisharelationshipbetweenarbitraryelementcongurations andtheresponseforcesthatcanbegeneratedfromthereference.Todothisweprovideaparameterization thattransformsanarbitraryelementintothereferencecoordinatesystemandaninversetransformation thatallowsustoextractthecorrectelementforcesfromthedata-drivenreferenceresponse.Thisprocess includesthreestages:parameterizationofarbitraryelementsinthereferencecoordinatesystem,the creationofacomputemodelbasedonevaluatingtheresponseforcesofeachelementnodeindependently, andperformingtheinversetransformationofthematerialresponsefromthereferencetothecorotated coordinatesystem. 36

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Theparametrizationofan instance element I ,isdenedbytheiterativeevaluationofeachnodein theelement.Foragivenelement,wepickapredenednodeordering.Fromthisordering,wewillevaluatetheelementresponsewithrespecttothisselectednodewithinthereferencecoordinatesystem.The sequenceinFigure3.3.9illustratesthisprocessforasimple2Dtriangleelement.Theinstanceelement isparameterizedwithrespecttothecurrentlyselectednode n .Thisrepresentsthecurrentnodethatwill bedisplacedtogenerateasetofinternalresponseforcesfromtheothernodes u and v withintheelement. Theresultistwoelements,theinstance I andthereference R ,bothlocatedwithinthereferencecoordinatesystemasshowninFigure3.3.9right. Figure3.3.9:Parameterizationandtransformationofanarbitraryinstanceelement I tothereferencecoordinatesystemcontainingthereferenceelement R .Thenodeorderinggeneratesmatchingpairsbetween thetwoelements I n ! R n ;I u ! R u ;I v ! R v forallindependentexecutionsofeachnodebetweenthe elements. Sincetheelementshavedifferentgeometricshapesbutthesamenumberofnodes,theysharematchingpairsofnodelabels.Thisplaysapivotalroleintheinverseoperationrequiredtorelatetheinstance andreferenceelements.Theinsightisthatthegeometryonlyplaysanintermediaterolewithinhowthe twoelementresponsesdiffer.Ifwerelatetheresponsevaluesfromthesetwoelements,percomponent, thenwecangenerateamapbetweentheinstanceandreferenceelements.Thesolutiontothisrelationship isdenedastheratiobetweennodalforceresponsecomponents.Thisintroducesanewconceptbasedon geometricratios forrepresentingtheinverseoperationthatallowsustotransformreferencematerialresponsestoelementinstances.Thisratiotransformationwillallowustogeneralizematerialresponsesto allinstanceelementswithinasimulatedmesh. 37

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GeometricRatios .Giventwoisotropicelementswiththesamematerialproperties,wecanevaluatethe responseforcesofthematerialasinternalforceswithineachelementduetoanimposeddisplacement.If wedisplacethesamenodeineachelementandtheelementsaredifferent,thenwewillgenerateunique responseforceswithineachelement.Ifthegeometryofthetwoelementsisthesame,wewouldexpect theresultwillbeconsistent,thatis,theinternalresponsesarethesame.Fromourassumptionthatinternalforcecomponentsarealwaysnon-zerobyconstruction,wecanrelatetheresponsesasaratiobetween internalforcesbetweentheseelements.Givenainstanceelement I withnodes I 0 ;I 1 ;I 2 ;:::;I n andreferenceelement R withnodes R 0 ;R 1 ;R 2 ;:::;R n wecanestablisharatiobetweenthatrelatestheseresponsesasshowninEquation3.3.1. r I;R =[ I 0 =R 0 ;I 1 =R 1 ;I 2 =R 2 ;:::;I n =R n ] T .3.1 Sinceadisplacementincursmultipleresponseforcesforallnodes,wehavetoconsidertherelationship betweenallforcecomponentsforallnodes.Thiswillresultinaratiovectorthathas n d ratios,oneratio foreachcomponentDOFofeachnode.However,thisvectorofratiosisonlyvalidforonenodeordering betweentheinstanceandreferenceelements.Foreachnode,theparametrizationoftheinstanceelement willchange.Therefore,ifweconsiderthenumberofnodesas n ,wewillhave n ratiovectorsoflength n d thatrepresentthefullrelationshipbetweenthetworesponsebehaviorsprovidedbythetwoelements. Forexample,ifweselecttherstnodeofatetrahedralelementandtransformittothereferencecoordinatesystemweobtainthetwoelementsshowninFigure3.3.10. The 12 responseforcecomponentsfromeachelementthatarepairedwitheachotherbasedonthe nodeandcomponentlabelingtogeneratethecompletesetofratiosfortheselectednodeordering.Expandingtheratiovectorbasedonthisexample,wecanconstructthe 12 componentratiovector r ifwe labeltheinternalforcesaccordingtotheirnodeandforcecomponent:Nodes n , w , u ,and v haveresponse forcecomponents x , y ,and z . r [ AX ]= I nx =R nx r [ BX ]= I wx =R wx r [ CX ]= I ux =R ux r [ DX ]= I vx =R vx r [ AY ]= I ny =R ny r [ BY ]= I wy =R wy r [ CY ]= I uy =R uy r [ DY ]= I vy =R vy r [ AZ ]= I nz =R nz r [ BZ ]= I wz =R wz r [ CZ ]= I uz =R uz r [ DZ ]= I vz =R vz 38

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Figure3.3.10:Geometricratiosoftheresponseforcescomputedbetweentheinstance I andreference tetrahedra R .Theratiovector r iscomputedby I=R forallforcecomponentsright.Thisshowsoneof fourpossiblenodeorderingswhere n = a;w = b;u = c;v = d .Ratiosareorderingdependent. Thereasonthateachnodeorderingrequiresaratiovectorisduetothecomputationalseparationof eachnode.Thatis,weusethesamenetworkstructurestoevaluatethematerialresponsewithrespectto eachnodeofanelementandreconstructthetotalcontributionsfromeachnodeasalinearcombination. Thisformulationispossibleduetotheisotropicbehaviorofthematerialformulation.Usingthesamenetworkstructurestoevaluateallnodesofallelementsbothreducesthenumberoflight-weightnetworks requiredtoreconstructmaterialresponseandalsoconsiderablyreducestheinternalcomplexityofeach network.Fortheinstanceelementsincludedwithinasimulation,wepre-computetheseratioswhichremainconstantduringthesimulation.SincethePoissonratio v ofanelementsmaterialdirectlyrelatesto howforcesaredirected,changingthismaterialpropertywillinvokeanupdateoftheseratiovectors.Ifthe Poissonratiooftheelementsmaterialischangedduringthesimulation,thentheratiosstoredwithinthe elementareautomaticallyupdated. 3.3.3ComputationSeparability Materialelementresponsesarecomposedofalargenumberofinterdependentbehaviorsthatvarydrasticallygivendifferentmaterialandgeometricproperties.Attemptingtoreplicatethesebehaviorsbasedon theparametrizationofanentireelementwillresultinnetworksthatrequireahighlevelofcomplexityto accuratelyreproducethedesiredbehavior.Ifthenetworkstructurerequiredtomodelthematerial,geometricvariance,andtheresultingbehaviorbecomestoocomplex,anycontributionsoradvantagesofthis 39

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designwillquicklydiminish.TheobjectiveoftheNEformulationistoprovideacomputationallyefcientandaccuratemethodforpredictingthecompletebehaviorofanelasticmaterialelement.Toobtain this,theformulationiscarefullydesignedtoprovidethehighestlevelofaccuracywhilemaintainingareasonablecomputationalcomplexity.ThedesignspecicallylooksathowfartheFEAformulationofan isotropicmaterialcanbedeconstructedtoprovidealight-weightandexibleadaptationintoadata-driven model.Wethenusethedecompositiontoseparatethemodelintoalargenumberofsynchronizedoperations. ThedecompositionoftheFEAformulationforanisotropicelasticmaterialarebasedondetermining howtomodelequivalentelementbehaviorwhileminimizingnetworkcomplexity.Thisinitialstepbuilds towardsanewdirectionintheapplicationofneuralnetworksforelasticmaterialmodeling.Insteadof requiringanextensivenumberofinternallayersascommonlyemployedwithinDeepLearningtechniques orattemptingtoencodephysicalbehaviorswithinaConvolutionalNeuralNetworkCNN,welookat howwecanmatchthedeconstructionofanelementsprimitivebehaviorsintoalargesetofminute,easy toreplicatebehaviors.Todothis,weintroducethetwoformsofdecomposinganelementintoitsmost fundamentalcomponentsbyanalyzingthestructureoftheFEAformulation.Therstisanobservation aboutthegeneralstructureofthematrixequationsderivedforanisotropicmaterialinFEA.Theglobal stiffnessmatrixiscomposedofseveralindividualrowsandcolumnsthatrepresentthecomponent-wise contributionsofeachelementstiffnessmatrixassembledintoanintegratedsystem.Sincetheserowsand columnsarepartitionedintothecomponentsofthediscretenodesrepresentedwithinthesystem,linear isotropicmaterialsarecomposedofsimplelinearcombinationsofcomponent-wiseintegrationofmultiple elements.Elementcontributionsareobtainedduringtheconstructionoftheglobalstiffnessmatrix.Based onthisdecomposeelementsinatwostageprocess:nodeseparationandaxisorDOFseparation. TheseoperationsprovidethefoundationfortheaggregatenetworksystemthatdenesthecoreoftheNE formulation. Thepremiseofseparatinganelementintotrivialcomponentsthatcanbecomputedindependently alignswiththeprospectofusinganaggregatesystemoflight-weightnetworkstoimproveaccuracyand computationalcomplexity.Basedonnodeseparation,webreaktheelementintoits n nodesandcompute 40

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theresponseforeachindependently.Wethendecomposeeachnodeintoitsprimarydegreesoffreedomto computetheelementsbehavioroneachprimaryaxis.Thissignicantlyreducesthecomplexityofthebehaviorsthatmustbereplicatedbyeachnetwork,butincreasesthetotalnumberofnetworks.Withrespect tothereferenceelementandthegeometricratios,weperformthesecomputationsandstorethemforeach nodewithintheelement. Figure3.3.11:Illustrationofthereferencecoordinatesystemfortheratioandforceresponsemodelbased oncomputingthematerialresponsebasedoneachnodeindependently.Sincetheelementhas 3 nodes a , b , c ,thecomputationisperformedthreetimes.Thecompleteelementbehavioristhengeneratedasthe linearcombinationofeachpartialresponse. PartialMaterialResponses .Splittingthecomputationoftheelementsmaterialresponseresultsinthe generationof partial elementforceresponses.Thematerialresponseindependentlycomputedforeach nodewillresultin n d forceresponsesforeachnodestoredinthevector p i where p isthepartialforce vectorand i correspondstotheindexofthenodewithintheelement.Partialresponsesalsohaveanimportantrelationshipwiththeratiosthatarecomputedforeachnode.Tofullyaccountforthegeometric andmaterialresponseoftheelement,weusetheprecomputedratiosandtheresponseforcescomputed pernodetoreconstructtheelementsmaterialresponse.Thismeansthatforeveryelement,thereare n partialforcesresponsesand n ratiovectorsoflength n d .Thelinearcombinationofthesecomponentsthen representsthefullresponseoftheelementduetotheapplieddeformation.The n partialcomputationsare performedateachsimulationtime-stepforallelements. InternalForceReconstitution .Separationofthenodesandeachoftheirdegreesoffreedomrequiresa reconstitutionprocessthatreassemblesthepartialinternalforcesgeneratedfromthenetworkaggregate. 41

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Thisisbecauseeachnetworkonlycomputesapartialresultoftheneuralelementsresponse.Toextract thenalresponseoftheelementsubjecttotheapplieddeformation,allofthepartialforcesmustbereconstitutedbyperformingalinearcombinationofthepartialforcesgeneratedbyeachnode.Thisresult isdenedasthesumofthepartialforcevector p i multipliedbytheratiovector r i computedforelement nodeasshowninEquation3.3.2. f int = n X i =0 r i p i where r istheratiovectorand p ispartialforcevectorfornode i .3.2 Thisgeneratesthecompleteresponseoftheelement.Toprovideaconsistentreconstruction,aconstantnodeorderingisprovidedtocreatememorymapsthatcanbeefcientlyexecutedtocombinethe partialinternalforces.Thisisbecauseasthenodeorderchangeswithrespecttothereferenceelementas nodesarecomputedindependently.Therefore,wemustkeeptrackofwhichpartialforcescorrespondto eachnodeastheelementisshiftedandthepartialforcesaregeneratedwithinthereferencecoordinate system. 3.3.4ElementNetworkModel Theproposedneuralelementmodelrelatingratioandpartialforcecalculationsallowsfortheseparation ofthegeometryandmaterialresponseofsimulatedelements.Thisseparationiscriticaltotheparametrizationofthedata-drivenmodelusedtogeneratematerialresponsesofindividualelements.Sincetheratio transformationallowsustomaphowforcesfromthereferenceelementtoanarbitraryinstanceelement, wecanlimitthecomputationofthematerialresponsetothegeometryofthereferenceelement.Usingthis constantgeometry,wefocusonhowtoparametrizethematerialpropertiesoftheelementandthestressto-strainrelationshipexhibitedbyapplyingvariabledisplacements.Fromdeconstructingthemodelinto asetofforcecomponentsalongtheDOFsforeachnode,wedeneasetofnetworksthatmapdirectlyto thesebehaviors.Thismeansthatforeachdegreeoffreedominthematerialforceresponses,thereisan associatednetworkthatwillberesponsibleforreplicatingthedesiredmaterialbehavior.Inthegeneral case,forelementswith n nodesand d degreesoffreedom,wehave n d totalnetworksthatreplicatethe responseimposedbyanindividualone-dimensionaldisplacement dx , dy , dz .Thisgeneratesaninstance 42

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ofthenetworkaggregatethatcanhandlethepartialforcesgeneratedfromthedisplacementsalongone degreeoffreedom.Therefore,togeneratethecompleteresponseweusemultipleinstancesofthesame networkaggregatetogeneratethecompleteresponseoftheelementfromthedisplacementsimposedon allnodes.Thisformulatesatotalof 2 n d networksfortwo-dimensionalsimulationsand 3 n d for three-dimensionalsimulations.Thisformulationallowsustoobtainasetofconstantlight-weightnetworksthatcantriviallycaptureasmallerportionofmorecomplexmaterialbehaviors.Sincetheseparabilityofthecorematerialresponseprovidesareusablearchitecture,wecaneasilyinstantiateasmallnumber ofnetworkstocomputematerialresponsesforallelements.Thisalsomeansthatwecanaggressivelysimplifytheinternalstructureofeachnetworkduetothedesignedsimplicityinthematerialresponsesignal. Thesenetworksarethenaggregatedintoan elementmaterialmodel thatcombinesthecontributionsfrom allnetworks.Thismodelisthencapableofprovidingcompletematerialresponsesofanindividualelementbasedonasetof n -dimensionaldisplacements d i .Anillustrationofthedisplacements dx , dy ,and dz thatareusedaspartoftheinputtothematerialnetworkthatareappliedtothe3DtetrahedralreferenceelementisshowninFigure3.3.12. Figure3.3.12:Illustrationofthedisplacementsimposedona3Dreferencetetrahedralelement.Each displacementanditsassociatedmaterialresponseisusedtotrainanaggregateofmaterialnetworks.Displacementcomponents dx left, dy center,and dz rightwillberecombinedintoacompleteelement response. Torepresentthematerialpropertiesoftheisotropicelasticmaterial,wedeneelasticitythrough Young'sModulus E andthePoissonratio v .Sincethestress-to-strainrelationshipistrainedbasedon theelementsresponsetoimposeddeformations,weexpressthisasavariabledisplacement d .Fromthe 43

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nodalseparationprovidedbymodel,weonlyhavetoconsiderthedisplacementofthenode n withinthe referenceelement,theothernodeswithintheelementareconsideredxedboundaryconditions.Thisis possiblebecauseoftheratiotransformationthatallowsustoapplythedeformationoftheselectednode andDOFtotheresponsegeneratedfromnode n inthereferenceelement.Thenumberofdisplacements appliedtonode n ofthereferenceelementmatchesthenumberofDOFs,butthesearedonethroughseparatenetworks.ThustheresultingelementmodelforeachDOFhasthreeinputs: E , v , d andonematerial responseoutput f . EvaluatingdisplacementsofanelementalongitsDOFstoevaluatethematerialresponseiseffective forelasticmaterialmodelsthatcanberecombinedaftereachcomponentisindividuallycalculated.This iscompatiblewiththelinearelasticmodelduetothecontributionofresponseforcesthatcanbepredicted fromatrainednetwork.Theimplementationofthisnetworkisexibleandonlyrequiresthereplication ofaone-dimensionalwaveform.Sincetherearenumerousmethodsvaryinginnetworkandcomputational complexity,webrieyanalyzeprominentcandidates. NetworkDesign .Thepremiseoftheneuralelementapproachistoprovideafunctionaldecomposition ofinternalmaterialresponsesthatcanbetriviallyreplicatedthroughanynumberofnumericallearning approaches.Candidatesforthisincludemultilayerperceptronnetworksneuralnetworks,Convolutional NeuralNetworksCNNs,andLongShortTermMemoryNetworksLSTMs.ThedesignandintegrationofmaterialpropertiesasperiodicinputfunctionsindicatethatLSTM-basednetworkswouldprovide anidealnetworkarchitectureforthematerialintegrationmethodproposedinSection3.5.However,the overheadassociatedwithinternalstateandthecomplexityofthenetworkforsuchsimplefunctionsincurs ahigheroverheadthanwhatcanbeprovidedthroughsimplemultilayerneuralnetworks.Thisdoesnot eliminatethepotentialforusingthesenetworkswiththeproposedmethodastheyarerequiredforelastic materialresponsesthataremuchmorecomplicatedthanlinearelasticity.DependingontheminimizationthatcouldbeobtainedusingaCNNarchitectureforreconstructingthesimpliedone-dimensional response,thismethodcouldalsobeemployed.Thedesignandimplementationoftheneuralelementapproachallowsthiscomponenttobe interchangeable dependingontherequiredmaterialresponsecom44

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plexityanddesiredaccuracy.Thenetworktypeandarchitecturearecompletelyinterchangeable. Thedesignobjectiveofthepartialforceresponsesprovidesasimpleone-dimensionalwaveform thatcaneasilybereplicatedthroughtheuseofamultilayerperceptron.Fortheinitialdesign,thisformof networkisusedduetotheminimalnumberofoperationsrequiredtoexecutethemodelatruntime.Byreducingtheinternalstructureinbothhiddenlayersandlayernodecount,wecanensureanefcientbalance betweenaccuracyandperformancethatprovidesastablesimulation. TriangularElementNetwork .Intheinstanceofatwo-dimensionaltriangularelement,therearethree nodes,andtwoDOFs.Usingthesystemsizeofthetriangularelement,therewillbesixnetworksthatrepresentthecomponentresponsesofthiselement.Ifweassumethattheelementiscomposedofthenodes a , b ,and c ,thecorrespondingforcecomponentsaredenedinaforceresponsevector F ofsize 6 .For eachresponsecomponent,wecreatethenetworksystemthatwillrepresentthematerialresponsesofthe materialusingthereferenceelement.Forsimplicity,weillustratehowtheinputsoftwominimalmultilayerperceptronnetworkscorrespondtotheproposedneuralelementarchitectureandnetworksystemfor thetriangularelementinFigure3.3.13. Figure3.3.13:Interchangeablenetworkdesignsthatareprovidedtothecoreoftheneuralnetworkmaterialarchitectureleft.Thesenetworksarethenassembledintothedesignrequiredfortheselectedelement typeright.Thetriangularelementhassixresponseforceswhichrequiressixnetworkinstancesthatare embeddedintheelementnetwork.Thevalueof d i isdenedasthedisplacementvalueexpressedas dx or dy for2Delements,eachofwhichrequiresitsownnetworkinstance. Theneuralelementarchitecturedenesahierarchyofsystemsthatdenehowinterchangeablenetworks communicatetoreproducethematerialresponseofanindividualelement.Usingthenetworkarchitecture 45

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asablack-box,weabstracttheconnectionsoftheinputsandpartialforceoutputsintoaconsolidatedelementnetwork.Thepresentedsequenceofdiagramsillustrateshoweachcomponentisabstractedintoa higherleveloftheneuralelementstructuralhierarchy.Thisisduetotheorganizationofthenetworksused toreplicatetrivialresponsesatthelowestlevelandcombinethesepartialforcesintothecompleteelement responseatthehighestlevel.Forthepartialmaterialresponseforces,thenetworksalsohavetobeevaluatedmultipletimesdependingonthedisplacementDOFs.Thusfortwo-dimensionalsimulationsthereare twoinstancesofthepartialmaterialnetworkscorrespondingto dx , dy andforthree-dimensionalsimulationstherearethreeinstancesofthepartialmaterialnetworkscorrespondingto dx , dy ,and dz . TetrahedralElementNetwork .Thetetrahedralnetworkisderivedfromthesamenetworkaggregateformulationasthetriangularelement.Theincreaseincomplexityofthisnetworkisderivedfromtheelement includingfournodesandanadditionaldegreeoffreedom.Sincetheelementhas 4 nodes a , b , c , d ,the correspondingforcecomponentvector F hascomponents: f ai , f bi , f ci ,and f di where i 2f x;y;z g .The networksystemforthetetrahedralelementisshowninFigure3.3.15. Figure3.3.14:Illustrationofthe3Dtetrahedralelementsystemthatgeneratesthepartialmaterialresponse foragivendisplacement d i where i 2f x;y;z g .Thisgeneratesthepartialforceswithrespecttoonedisplacementoftheprimarynodeinthereferenceelement,thusthereare 3 instancesofthisnetworkfor dx , dy ,and dz . Partialelementforcesonlyrepresentthematerialresponseduetothedisplacementofonenodealongone degreeoffreedom.Therefore,foreachnodewehavetogenerate 4 setsof 12 responseforcesforeach DOFinthetetrahedralelement.Thisresultsinthecombinationof 4 3 12=36 totalpartialforcecom46

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ponentsforeachnode.Thus,foreachDOFofthesimulatedsystem,theremustbemultipleinstancesof thepartialmaterialresponsenetwork,butallinstancesarealwaysgeneratedusingthesame 36 uniquepartialforcebehaviors.Whilethisprovidesacomplexnetworkstructure,theseoperationscanbeoptimized basedonreal-timeperformanceobjectives.Thisisnotasignicantmemoryoverheadduetothelightweightnatureofthenetworksthatareingeneral,composedofasmallsetofweightsandthedenition ofthenetworksarchitecture.Duringexecution,thesestructuresdecomposetoasetofmultiplicationsand additionsastheinternaloperationsofthenetworksareevaluated.Additionally,thepartialresponsevectorscanbepreallocatedandreusedforeachresponseforcecalculationandthenodememorymapforthe elementisconstant. OncethepartialmaterialresponseforcenetworkisdenedforeachDOF,theyarecombinedtoform thecomplete elementmaterialresponse .Thisrepresentsthecompletesetofcombinedpartialforcesthat createsthebehavioralresponseoftheelasticmaterialforthegivenelement.Thatis,foreachnodeandits displacementsinthe dx , dy ,and dz directions,wepredictthreepartialforcevectors.Wethentransform thesepartialforcepredictionstotheinstancegeometryoftheelementusingtheelementsnodalratios. Finally,theforcesareaccumulatedbasedontheirindividualcomponentsandthenmappedtothecorrect nodesoftheinstanceelementusingthenodalmemorymappresentedinSection3.3.3. Toillustratethenalformoftheelementmaterialresponsenetwork,weusetheentirediagramfrom Figure3.3.14andinsertthisnetworkintothecompleteelementmaterialresponsenetwork 12 timesto generatethenalnetworkpresentedinFigure3.3.15.Thiscompletenetworktakesasinputthematerial propertiesoftheelementandthe 12 displacementsoftheelementsnodes 3 displacementspernode.This allowsamaterialresponsetobepredictedasafunctionofthedisplacementofallnodesinalldirections foranyinstanceelementinthesimulation. Thismodelcannowbeusedtopredictthebehaviorofall neuralelements withinasimulatedmodel basedontheelementsassignedmaterialandthecurrentdeformationstateoftheelementinthesimulation. Thisprovidesthegeneralframeworkfromwhichanyparametrizationofaneuralelementcanbedened. However,dependingonthetypeofsimulationandthetrainednetworktype,thereareimposedlimitations onwhichmaterialscanbeexpressedusingthismethod.Specically,thedomainofthematerialproper47

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Figure3.3.15:Neuralelementmaterialresponsenetwork.Thisnetworkrepresentsthecompleteresponse ofthematerialbasedonthetrainedinternalnetworksforeachDOFandforceresponsecomponentdueto thedisplacementimposedonanycomponentofthe 4 elementnodes.Thesearetheoutputforcesthatare usedtodrivethedynamicsofthesimulation. tiesthatcanbesimulatedisexceedinglylargeforwhatsmallandefcientnetworkscanrepresent.From thelimitationoftheentropywithinanetworkasafunctionofitsarchitecture,wehavetoprovideanadditionalsolutiontorepresentingawidevarietyofdifferentmaterials.Manyapproachespresentacollection ofnamedmaterialclassesthatcanbereasonablerepresentedwithinadata-drivenmethod.However,while weprovidethissolutionasasubsetofpossiblematerialpropertiesthatwecansimulate,wealsointroduce acontinuous materialnetworkmodel thatallowsarbitraryelasticmaterialpropertiestobeassignedtoan element. 3.3.5MaterialNetworkModel Thefundamentalproblemofrepresentinganunboundedrangeofmaterialpropertiesisthattheycannot beexpressedintermsofadata-drivenmethodforonenetwork.Whileexceedinglycomplexmaterialbehaviorsbasedonseveralpropertiescouldbeexpressedthroughalargeordeepnetwork,wepresentan alternativebasedondecomposingtheelasticmaterialdomain.ForthePoissonratio v ofthematerial, 48

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thisisnotrequiredduetothefactthat v 2 [0 ; 0 : 5] foravastmajorityofmaterials.However,asthisratioapproaches 0 : 5 ,theresponseforcesbecomemuchlargerinmagnitudeduetothematerialbehavior andcanpresentachallengeintrainingforadata-drivenapproach.Theprimaryproblemisinrepresenting Young'smodulus.Thisisbecausethedomainofthismaterialcoefcientexistsintherange [0 ; 1 ] .While thereisnoinnitelyrigidreal-worldmaterial,wemustprovideareasonablecoefcientrangethatcan berepresentedthroughtheneuralelementderivation.OursolutiontothisproblemistodeneareasonabledomainofYoung'smodulusanddiscretizetheintervalintoacollectionofnetworkinstancesbased onthedesignpresentedinFigure3.3.15.Eachinstancecanthenbetrainedonadifferentsetofmaterial responsesbasedontheintervalsrange.HerewepresentasimpleexampleofhowwecandenefournetworksthatareeachtrainedonadifferentrangeofYoung'smodulus,allowingeachtoconvergeonareasonabletrainingset.ThisconceptisillustratedinFigure3.4.17forarangeof E =1000 to E =10000 . Figure3.3.16:Illustrationofthereferencecoordinatesystemfortheratioandforceresponsemodelbased oncomputingthematerialresponsebasedoneachnodeindependently.Sincetheelementhas 3 nodes a , b , c ,thecomputationisperformedthreetimesonceforeachnodeforeachdegreeoffreedom.The completeelementbehavioristhengeneratedasthelinearcombinationofeachpartialresponse. Thismethoddenesalargersetofnetworksthatcanbeusedtorepresentmaterialpropertiesof theelementsthatarecommonlyusedwithinreal-timesimulations.Althoughthistechniqueistransparenttotheuseoftheneuralelementmethod,therearesomeconsiderationsregardingtheaccuracyofthis method.Thisisduetotheedgecasesbetweenthenetworkcoverageareas.Thediscreteboundarybetweenthetrainingsetscanintroduceslightvariancesinmaterialbehaviordependingonthecharacteristics ofthetrainingsetusedtodenethenetworks.Although,ifeachelementmodelsdomainissufciently coveredinthetrainingdata,thenthelossinaccuracyandstabilityshouldbeminimal. 49

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3.3.6CoveringtheMaterialPropertyDomain Theelasticmaterialcoefcientdomaincanhaveaconsiderablerangetorepresentthroughacollectionof discretenetworks.Therefore,welookattheconstraintsofareal-timesimulationandimposereasonable constraintsonthematerialpropertiesoftheelementsthatmaybesimulated.Thiscanprovidealimiton thenumberofintervalsgeneratedwhichcanreducethetrainingrequiredtoreplicateanarbitrarynumber ofelasticmaterials. Real-timeSimulationConstraintsontheMaterialDomain .Elasticmaterialpropertiescanvarydrasticallydependingonthetypesofmaterialsbeingsimulated,eachhavingtheirowntheoreticaldomainof potentialvalues.Duetothis,thecoverageofsometheelasticmaterialpropertiescanleadtoanarbitrarily largenumericaldomain.Sincethisisincompatiblewiththelimiteddomainthatcanberepresentedwithin thetrainingset,weintroduceconstraintsonthedomainsInthecaseofPoisson'sRatio,thenumericaldomainismostlylimitedto v 2 [0 ; 0 : 5] withtheexceptionofmorecomplexmaterialsorstructuresthat exhibitnegativePoissonRatios.Thisdomaincanbecoveredwithsomediscretizationofthedomainsplit intodifferenttrainingsetsbasedonthedesiredaccuracyofeachnetwork.Inthecaseoftheelasticmodulus,thediscretizationofthenumericaldomainofthevariableismuchmoreextensive, E 2 [0 ; 1 ] leading toalimitationinrepresentingallpossiblevalues.Therearepracticallimitsthatgroundthisvaluetorealitymuchquickerthananythingapproachinginnityforexampletheelasticmodulusofametal-like materials.Consideringthelimitationsofwhatisrepresentedwithininteractivesimulationsthatassume discretetime-steps,oating-pointrepresentations,andrasterization,wecanfurtherlimitthehigherbound ofthisdomain.Thepremiseofthisreductionisbasedontwocomponents:thepotentialperceptionof innitesimallysmalldeformationsobservedinreal-timesimulationsandareasonableboundonexternalforces.Iftheinduceddeformationofanobjectisnegligiblebasedontherigidityofthematerial,then theobjectapproximatesarigid-bodyandshouldbereplacedbythissimplerformulation.Iftheelastic propertiesofthematerialrequireexternalforcemagnitudesthatwouldrarelyifever,bepresentwithinthe simulationtheobjectshouldalsobereplacedwitharigid-body.Wealsotakeintoconsiderationthatfor minutedisplacementsoftheelementmaterial,theresultingnumericalchangecanbeeffectivelyremoved 50

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throughtherasterizationoftheobject. 3.3.7ExtensionstoNewElementTypes Theproposedelementmaterialnetworkarchitecturecanbeextendedandgeneralizedtootherelements suchastwo-dimensionalquadrilateralsandhexahedralsolids.Theabilitytogeneralizethismethodto eachformofelementisbasedinhownodalresponseforcesareaccumulatedthroughindependentcomputations.SincethesepropertiesarecommontoalldiscreteelementsusedwithinFEAsimulations,itcan beextendedtoincorporatenewelementtypes.Thecomplexityofthenetworkstructuredependsonthe numberofnodeswithintheelementandtheDOFsofthesystem,thereforemostsimpleelementssuchas tetrahedralelementscanbeefcientlyapproximatedusingthisapproach.Additionally,sincemostrealtimeFEAsimulationsforcontinuousmaterialsusethree-dimensionalobjects,theDOFofthesystemwill typicallybeconstrainedtothree. 3.4DynamicNeuralElements Neuralelementsintroduceanewdiscretemethodforapproximatingthebehaviorsofsolidmaterialsfor real-timeandinteractivesimulations.Thisrequiresthedevelopmentofadynamicsystemformulation thatiscompatiblewiththeproposedneuralelementdesign.Buildingonseveralestablishedmethods,we buildadynamicsimulationcompatiblewiththeneuralelementdesignthatiscomputationallyefcient, allowsforcomplexfeaturessuchasreal-timetopologychanges,andstability.Toachievethis,welookat howwecanconvertelementmaterialresponsesintocomplexmeshstructuresandtheformofnumerical integrationrequiredtodrivethecoredynamicsofthesystemwhileprovidingstabilityandperformance. 3.4.1MaterialForceResponses Theneuralelementformulationprovidesamethodforpredictingacompletesetofinternalmaterialresponsesasasetofforcesbasedonadeformationappliedtoanelement.Thisprocesscanbeusedtoefcientlydeterminethematerialresponseofaninstanceelementwithinasimulation.Butthisonlyconsiders theresponseofanisolatedindividualelement.Sincemostsimulatedvolumetricmeshesarecomposedof 51

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numerousconnectedelementsthatformatopology,theelementcontributionstothecontinuumrequires anintegrationoftheseelementforces.Duetothedata-drivennatureoftheneuralelementformulation, wearerequiredtooperateonthematerialresponseofeachelementindependently.However,sincewe areestablishingtheresponseforcesofeachelement,wecancomputethenetforcesactinguponthediscretenodeswithinthesystem.Therefore,foreachelementinthesystemwepredictthematerialresponse uniquetoeachelementsstateandthencombinetheseforcesateachnode.Thisprocessassumesafairly straight-forwardsimulationsetupbasedontheevaluationofdisplacementsandtheaccumulationofresponseforces.Theseresponseforcescanthenbeintegratedintoastandardnumericalintegrationprocess togeneratethematerialstatessuchasvelocityandposition. 3.4.2NumericalIntegration Numericalintegrationofdynamicsystemsiswellstudiedwithinnumericalanalysisformethodsinmechanicsandcomputergraphicsandrepresentsarelativelymaturestatewithregardstoaccuracyandstability.Duetothistherearenumerousnumericalintegrationmethodsthathavebeenintroduced,eachwith theirowndiscreteformulation,performance,andstabilitycharacteristics.Thetwoprimaryclassesofnumericalintegratorsaredenedas explicit and implicit methods.Thesetwoclassesarecharacterizedbythe underlyingmathematicalformulationsthatdenehowquantitiesareevaluatedwithrespecttodiscretized functionsoftimeforsolvingOrdinaryandPartialDifferentialEquationsODEsandPDEs. Explicitintegration .Thisformofnumericalintegrationrepresentssomeofthemostcomputationally efcientmethodsduetothesimplicityinthepremiseofthepredictionmodelwhichisdenedbythecurrentsystemstatetopredictthenextstate.Thisessentiallycharacterizesthestateofthedynamicsinthe nextsimulationtime-stepbasedontheprojectionofthecurrentstate.Numericallythisisstraight-forward andcomputationallyinexpensive.Methodsinthisclassinclude:explicitEulerintegrationforwardEuler, theexplicitRunge-KuttafamilywhichincludesEuler,RK2,RK4,RK5,centraldifferences,andVerlet, amongstseveralvariantsoftheseapproaches.Whilethesemethodsprovideefcientandreasonablyaccuratestateupdatesfordynamicsystems,theyalsoincur numericalinstability ,ahighlystudieddomain withinnumericalanalysis.Inmostinstances,explicitmethodsareonlyconditionallystable,thatis,in 52

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theprojectionofthecurrentsimulationstate,thetime-step dt mustbearbitrarysmalltoensurestability.Thisbehaviorisarelationalfunctionbetweenthesimulationtime-stepsizeandthemodalbehaviorof thedynamicobject.Inbothrigid-bodyandelasticmaterialsimulations,avoidingthisinstabilityrequires reducingthetime-stepsize.Thisincursthreeprimaryproblems:asthetime-stepisreduced,theobjectappearstoslowdownduetothehighernumberofiterationsrequiredtoperformstateupdatesforan interactiveorreal-timesimulation,thishighernumberofsystemsolutionsdrasticallyincreasescomputationalload,andthereareinstanceswherethereductionofthetime-steptoaninnitesimalvaluewill stillnotprovidenumericalstabilityofthesystem.Theseconstraintstypicallyeliminatetheuseofexplicit methodsforsimulationsmorecomplexthanthosebasedonrigid-bodydynamics. Implicitintegration .Numericalintegrationbasedonimplicitmethodsarederivedfromthesamepremise ofobtainingthesystemstatetopredictthenextstate.Howeverthebasisoftheimplicitmethodisaslight variationwithinthenumericalformulationthatincursthesolutionofanon-linearsystemofequations. Thisautomaticallyincurstwotwoconstraintsfordynamicsystemformulations:sincethesolution toanon-linearsystemofequationsisrequired,thissystemhastobeformulatedandupdatedasrequired bytheconstitutivemodelandsolvingnon-linearequations,evenwithiterativesolversissignicantly morecomputationallyexpensivethenexplicitmethods.Thecatchisthatforthisperformancehit,most methodsprovide unconditionalstability .Thisisacriticalcomponenttoprovidingstablesimulationswith reasonabletime-steps,especiallyformodelscomposedofelasticmaterials.Numerousmethodsincluding:implcitEulerbackwardEuler,Houbolt,Wilson,andNewmarkintegrationhavebeenproposed, allofwhichprovidereasonablecomputationtimeswithunconditionalstabilityorparametersthatenable stability.Forreal-timedynamicsthatincorporateelasticmaterials,manyformulationsusethesewellestablishedmethodsduetotheunconditionalstabilityprovided.Additionally,theformulationsofthe modelsbasedontheelasticmaterialmodelsfromFEAarewellsuitedforusingmatrixformODErepresentationsforintegratingsystemstates. 53

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3.4.3DynamicsFormulation Neuralelementsintroducesadata-drivenmethodforgeneratingelasticmaterialresponsesbasedonthe imposeddeformationsofindividualelements.Fromthestructuraldynamicsgoverningequationpresented intheFEAoverview,wemakeaslightadjustmentbasedontheintegrationofthedata-drivenelementresponsesthataregeneratedfromtheneuralelementformulation.Themodiedgoverningequationsimply interchangestheglobalstiffnessmatrix K andthedisplacements x ,withtheresponseforcevectorrepresentation r providedbytheneuralelementmodel.Thisprovidesaslightlydifferentformulationofthe dynamicsequation: Ma t + dt + Cv t + dt + r t + dt = f t + dt .4.1 where M isrepresentedbyalumpedmassmatrix, C representsthedampingmatrix,and f representsthe externalforcesactinguponthesystem.Withthisformulationwealsostorethesystemstatevectorsrelated tothediscretenodesforthedisplacement d t ,velocity v t ,andacceleration a t .IntheFEAformulation, elementstiffnessmatrices k i aregeneratedforeachelementindependentlyandthencombinedintothe globalstiffnessmatrix K thatcomposesthestructureandtopologyofthesimulatedmesh.Thismeansthat theconstructionoftheglobalstiffnessmatrixisdenedbythesystemsizeofthemeshnodesandDOFs. Thisresultsin 2 n and 3 n matrixsizesfortwo-dimensionalandthree-dimensionalsimulationsrespectively.ThisistheformulationcurrentlyusedwithinmostFEAsimulationsduetohowsystemstatesand relationalbehaviorsareencodedfromtheformalizationanddiscretizationofthegoverningequations.By constructionthisdirectionhasacoupleofdrawbacksfortheneuralelementformulation:matrixsizes canbecomeexcessivelylargeformesheswithhighelementcounts,leadingto sparsematrices thattakea considerableamountofmemorytostore.Thisproblemisdirectlysolvedbydensematrixrepresentations thataccountforthesparsityofamatrixbyintroducingindexedlistsofnon-zerovaluesthatcanbestored withinasignicantlysmallermemoryfootprint.Thisisanacceptablesolutiontothisproblem,however thereremainsasmallamountofoverheadformanagingdenserepresentations.Thedrivingforceofthe neuralelementformulationistoreplacethepreciselydenedelementstiffnessmatrixwithadata-driven model.Thismeansthatinsteadofintegratingtheelementstiffnessmatricesintotheglobalstiffnessma54

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trix,wemustevaluateeachelementsresponsetoimposeddeformationsandintegratetheirresponseinto thedynamicsequation. Developmentofthecorestructuresthatimplementthedynamicsequationcantaketwopotentialdirectionsbasedontheformulation:theentiresystemcanbeformedasamatrixequation,basedonthe formalizationofthegoverningequationandtheevaluationofindividualelementsandtheirreactionsto thestrainwithinthematerial.Thekeydifferenceinthesemechanicalformulationsishowtheintegration ofsystemelementsishandled.Forthematrixequationformofthegoverningequation,theelasticmaterialmodelisdenedtomaptherelationshipbetweenexternalforcesanddisplacements.Thismeansthat overtime,theexternalforcevectorcanbeintegratedtoobtainaccuratecalculationsofthedisplacements exhibitedbythenodesofthesystem.Whentheinverseisproposed,asinthecasewheredisplacementsof anelementgenerateresponseforces,theintegrationoftheelementcontributionstothesystemsishandled throughforceconsolidation. Duetotheneuralelementformulation,theresponseforcesofthematerialishandledper-element. Therefore,theweconsidertheinternalmaterialresponse r astheprimarydriveroftheelasticmaterial behavior.Becauseofthis,thereareimplicationsfortheselectionanddesignofthenumericalintegration algorithmsthatcanbeusedtodrivethesystem.Purelyexplicittechniquesareeasilyadaptedtothisform becausetheycantakeasinputthemeshnodedisplacements,velocity,andaccelerationandprovideupdateddisplacementsandvelocitiesforthenextsimulationtime-stepwithdirectequations,forexample: explicitEuler,centraldifferences,andtheRunge-Kuttafamilyofmethods.However,manyofthesedo notprovidethestabilitycharacteristicsrequiredtoensurelarge-deformationswillnotnumericallydiverge. Therefore,thechallengeiscombiningtheper-elementformulationprovidedbytheneuralelementdesign withastableintegrationmethod. Forsimulationsdrivenbydata-drivennetworks,therehavebeenattemptsatimplementingimplicit integrationmethodsthatapproximatetheJacobianofthematerialresponsetosolvethenon-linearsystemofequationsandensurestability,butthisrequiresboththenumericalapproximationoftheJacobian andsolvingthenon-linearsystem.Alternativeapproacheshavealsobeendirectedatobtainingstableexplicitintegrationtechniquesbyslightlymodifyingtheformulationtoimposebehavioralconstraintsonthe 55

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modesofthesystem.Forperformance,stability,andtheabilitytoadapttheNEformulationtoaparallel model,weemploynumericallystableexplicitintegration. StableExplicitIntegration .Thepremiseoftheimprovedstabilitymethodistoboundsystemfrequenciesbyprovidingcoefcientmatricesthatwillprovidestabilitywithinthenumericalintegrationupdate. Tousethesecoefcientmatrices,weinitializeseveralparametersofthesimulationincludingthetime-step dt ,themassmatrix M ,thereststiffnessmatrixofeachelement K 0 linearFEA,andthedampingmatrix C .Withtheseinitialvalues,wecancomputethecoefcientmatrices 1 and 2 thatareusedinthe integrationperformedforeachtime-step.Thesearedenedas: 1 =[ I + 1 2 dt M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C + 1 4 dt 2 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 0 ] )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 [ I + 1 2 dt M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C ] 2 = 1 2 [ I + 1 2 dt M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 0 ] )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 Theintegrationalsorequirestheinitializationofthedisplacementandvelocityvectorsusingthenewly computedcoefcientmatrices.Wepre-computeandstorethesevaluesperelementaselementstiffness matricesandstatevectors.Itisassumedthatthesimulationrepresentsasimpleinitial-valueproblem wherethesequantitiesareprovidedbytheinitializationofthesimulation. d dt = d 0 + 1 dt v 0 + 2 dt 2 a 0 v dt = v 0 + 1 2 dt a 0 + a dt Thisiswherewealtertheformulationpresentedin[29]andchangetherepresentationofthemassmatrix M ,stiffnessmatrix K ,anddampingmatrix C torepresentthestructuralanddynamicpropertiesof asingleelement .Thismeansthatthereareatotalof n mass,stiffness,anddampingmatricesforameshwith n elements.Immediately,thiscausesaproblemwithinthestableexplicitmethod.Thisisbecausethecoefcientmatrices 1 and 2 arenowdependentontheelementstiffnessmatrixinsteadoftheglobalstiffness matrix.Thismeansthatforneighboringelements,therewillbedifferentcoefcientmatricesthat pull the elementsindifferentdirections. DeformationandIntegrationUpdate .Thesimulationupdatecontainsseveralstepsrelatedtogeneratingdisplacementswithinthecorotationalcoordinatesystem,evaluatingtheelementmaterialresponses, ensuringthenetforceofeachelementiszero,andtheactualintegrationstep.Fortheevaluationofthe 56

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Figure3.4.17:Illustrationoftheindependentresponseforcesforadjacentelementsduetothecoefcient matrices 1 and 2 left.Thenetforcescenteranddisplacementsrightofthenodesareaverageddue tothedifferencesintheresponsesprovidedfromeachelement. materialresponsesforeachelement,wedenethattheresponsevector r isreturnedbytheneuralelement functionNE e thattakesadeformedelementandprovidestheresponseoftheelementasinternalforces. ThefunctionEqualizeNetForce r ensuresthatthenetinternalforceoftheelementiszero.Thisisaccomplishedbyperformingamean-shiftoneachforcecomponent.Thisupdateisperformedper-element withinthemainsimulationloop. Foreachelementwithinthesimulatedmodel: d t + dt = d t + 1 dt v t + 2 dt 2 a r t + dt = NE d t r t + dt = EqualizeNetForce r t + dt v t + dt =[ M + 1 2 dt C ] )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 M [ v t + 1 2 dt a t ]+ 1 2 dt f t + dt )]TJ/F21 10.9091 Tf 10.909 0 Td [(r t + dt a t + dt = M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 [ f t + dt )]TJ/F45 10.9091 Tf 10.909 0 Td [(Cv t + dt )]TJ/F45 10.9091 Tf 10.909 0 Td [(r t + dt ] Where f t + dt representstheexternalforcesactinguponthecurrentelement.Duetotheintegrationconstraintsbasedonthecoefcientmatrices,theforceresponsesfromeachelementareconsolidatedbetween eachtime-step.Thisresolvestheproblemofdiscontinuitiesthatwouldbeintroducedwithinthemesh topologybetweenelements.Althoughthisallowsthisformofintegrationtobeusedwiththedistributed setofelements,thisalsointroduceserrorcoefcientsthatresultfromthedeformationofindependentele57

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ments.Theintroductionofthiserrorisjustiedbytheperformanceimprovementsandexibilityafforded totheframeworkofthesimulation.Thisisoneofthebenetsofevaluatingeachelementindependently andquicklyresolvingmaterialresponsediscrepanciesbetweenelements.Whilethisisaside-effectofthis integrationmethod,thecoremethodcanbeadoptedtoalternativeintegrationmethods. Theevaluationoftheneuralelementfunctionthatgeneratesthematerialresponseistransparentwith respecttotheintegrationmethod.Thisallowsfortheformulationoftheintegrationtechniquetobeinterchangeable.Aswithmostdeformablesimulations,theselectionoftheintegrationmethoddependsonthe accuracyandcomputationalcomplexityofthemethodandthetargetapplicationdomain.Thestableexplicitmethodisemployedbydefaultduetothesimplicityoftheupdate,inthatitdoesnotrequiresolving anon-linearsystemofequations.Thismethodalsoenablesatrivialparallelimplementationoftheprimarycomputationalloadofthetechniquewhichistheevaluationoftheresponseforcesfromtheneural elementnetwork. 58

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3.5ElasticMaterialDataandTraining Explorationofdata-drivenmethodsthatimproveelasticmaterialbehaviorsorintroducereal-worldapproximationsofmaterialpropertiesischallengingduetotheacquisitionofsufcienttrainingdatathatis requiredtodenecomplexelasticmaterialbehaviors.Numerousmethodsthatutilizereal-worlddatacan obtainaccuratestress-to-strainrelationshipsthroughrigorousexperimentationandintegratetheserelationshipsintoniteelementbasedsimulationstoimprovetheaccuracyofthesimulatedmaterials.Many ofthesemethodsthatprimarilyaimtoreplicatethebehaviorsofreal-worldelasticmaterialsaredominatedbymodelsthataremoreaccuratefornon-linearbehaviorssuchastheSt.Venant-Kirchhoff,NeoHookean,andHyperelasticformulations.Additionally,mostofthesemodelsarecomputedoff-linenot real-timeorinteractiveusingaccuratesolutionmethodssuchasallowingNewtonsmethodtoconverge toverysmallerrortolerances.Thedataacquisitionprocessofthesemethodsischallengingtoextendpast theparametrizationofnonlinearmaterialmodelsduetothecomplexityinobtainingaccurateexperimentalresultsbymanipulatingelementsreliablytoformulateacompletemodelbasedoffofdataobservationsalone.Whiletheneuralelementformulationdoesnotcompletelyeliminatetheuseoftheelastic materialformulation,itdenesbalancedcontributionsfromexistingmaterialresponseformulationsand data-drivenobservationsthatcouldbeaugmentedwithreal-worlddatasimilartoexistingmethods.The objectiveoftheapproachistoprovideanefcientmethodforgeneratinghigh-performancesolidelastic materialsimulationsusedinreal-timeandinteractiveapplicationsbasedonobservationsfromavariety ofsourcesthatarecompatiblewiththeapproach.Therefore,thepremisetheneuralelementapproachcan alsorelyonotherformsofdatasourcesincludingreal-worldexperimentalobservations,existingFEA models,andevenslightmodicationsofexistingmodelbehaviorsthroughdataaugmentationandnonlinearmanipulations. 3.5.1SyntheticElasticData Thepremiseoftheneuralelementmethodistointegratethebehaviorsofelasticsolidsformalizedbya data-drivensourcetoreplacetheconstitutivemodelofelasticmaterials.Throughthedecompositionofthe 59

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isotropicmaterialmodel,wehaveintroducedasystemofneuralnetworksthatcanbetrainedtoreplicate theDOFspecicbehaviorsofanelement.Naturally,thismethodrequiresanextensiveamountofdatato replicatethematerialbehaviorsofanelementbasedoneverynodeandDOF.Therequireddataandexperimentalprocedurecouldbeminimizedtoprovideanaccuraterepresentationofthematerial,butthisis secondaryoftheobjectiveoftheNEformulation.Sinceweaimtoprovideaexiblemethodforinteractivesimulationforelasticsolids,wearemoreinterestedintheabilitytoformalizethemodeltomaximize performanceandhowthemodelcanbeadaptedtographicsapplications.Therefore,webuildthecore conceptonthepresentednetworkarchitectureandutilize synthetic datageneratedfromexactFEAsimulationsinanattempttoreplicatetheaccuracyofthepurelyanalyticalsolution.Theprimarymotivation forthisdirectionisnottolimitthesourceofthisdata-drivenmethod,buttoestablishtheformalizationof thecorearchitecturesthatallowusto mix theanalyticalandmachinelearningcomponentsintoacohesive model. Thesynthetictrainingdatarequiredtoestablishthestress-to-strainrelationshipoftheisotropicelasticmaterialisderivedfromthesolutionoftheanalyticalFEAmodelevaluatedonthereferenceelement. Thecoredatagenerationalgorithmforthisprocessisasfollows:generateavariantoftheelasticmaterialpropertiesusedtorepresenttheelement E , v ,induceandrecordadisplacement d appliedto theoriginnodeofthereferenceelementtogenerateresponseforces,foreachoftheelementresponse forces,recordall n d forceresponsesasinputstothenetworkaggregate.Eachnetworksinputs E , v , d i andoutput f isstoredindependently.Torepresentavarianceofthematerialpropertiesasdenedby thematerialnetworkmodel,samplesaregeneratedbasedonchangesin E , v ,andthedisplacement d i describedinSection3.5.2. Syntheticdataalsobenetsfromacompleteabsenceofnoise.Whilethesamplingofthematerial propertiesusedasinputisstilldiscrete,thesynthesizedwaveformscontainverylittlevariance.Thisallowsustotrainonthesedatasetstoobtainverylowerrorratesasthetrainingconverges.Thisalsofactors intotheabilitytominimizetheinternalstructureoftheemployednetworkarchitecture,thusimprovingthe performanceofthemethodoverall. 60

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3.5.2MaterialPropertyIntegration Everynetworkwithinthematerialnetworkmodelrequiresthegenerationoftheinputvalues E , v ,and d i alongwiththeexpectedvalue f tocreatetrainingdatasets.Tomaintaingeneralizedtrainingsets,the samplesaregeneratedthroughtheuseof integrationfunctions .Thisallowsthematerialpropertiesand responsestobeintegratedascontinuousperiodicfunctionsthatcanbeusedwithavarietyofdifferentnetworkarchitecturesincludingLSTMnetworks.Sincetheintegrationfunctionsarecontinuousandperiodic, wecanensurethatgeneratedexpectedvaluesfollowsmoothresponsecurves.Thegenerationofthese samplesarebasedoninterchangeablefunctionsthatcanbevariedtoshiftthedensityoftrainingsamples basedonspecicobjectives.Weintroducetheseintegrationfunctionsforeachinputvariableincludingthe materialproperties E and v ,thedisplacementvalue d i .Duetotheresponsesoftheelasticmaterial,the outputresponseforcewithineachDOFcanalsobecontinuousandperiodic,butthisalsodependsonhow thethreewaveformsaresynchronizedasdiscretesamplesprovidedtotheinputofeachnetwork. Havingeachinputvalueoscillatethroughacontinuousperiodicfunctionallowsustospecifyseveral characteristicsoftheseintegrationfunctions,thesamplecombinationstheygenerate,andthebehaviorsof theresponseforces.Thisincludes:thedenitionoftheintegrationfunctionitselfsuchassin,cos,or variousothers,thenumberofsamplesperperiod,andthefrequencyofthisfunction.Theselection ofthesepropertiesdeterminesthecharacteristicsofthematerialresponseforcesthataregeneratedfrom theapplieddisplacements.Thisalsointroducestheconceptofrelativefrequencies.Duetotheinterplay betweenthematerialpropertiesandtheapplieddisplacement,theresponseforcemagnitudecanuctuatewildly.Thisisanundesirablecharacteristicofthetrainingdataduetothelowrateofconvergenceon thesehighmagnitudespikesoranomalieswithinthedataset.Theobjectiveinspecifyingtheintegration function,resolution,andfrequencyofthedatasetisdrivenbywhichtypesofresponsesshouldoccupy moreofthedatasetsamplesandwhichtypesofsignalscanpromotehighconvergencerates. IntegrationFunctions .Theintegrationfunctionsthatcharacterizethegenerationofthematerialresponse datasetsforthereferenceelementcanbedenedasanycontinuousperiodicfunction.Fortrivialfunctions,theperiodicnaturecanbeuseddirectlyinthedatasetgenerationprocess.Thismeansthatanywell 61

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behavedfunctionscanbeusedtovarythematerialpropertiesanddisplacementwithinatrainingset.For theimplementedtrainingset,weprovidetwoprimaryintegrationfunctions: I 1 x = sin x functionanda variationoftherstderivativeoftheGaussianidentiedas I 2 x = dGauss x .Thisfunctionisdened inEquation3.5.1.TheplotsofthetwointegrationfunctionsareshowninFigure3.5.18. I 1 x = sin x I 2 x = dGauss x = @g x; @x = )]TJ/F21 10.9091 Tf 22.82 7.38 Td [(x 3 p 2 e )]TJ/F9 4.9813 Tf 9.624 3.258 Td [(x 2 2 2 .5.1 Figure3.5.18:Integrationfunctionsusedtoencodethematerialproperties E , v andthedisplacementappliedtothereferenceelement d i togeneratetrainingdatasets.Thesefunctionsrepresentjusttwoselected periodicfunctionsthatcanbeusedtoencodedthesenetworkinputs. Theprimarypurposeoftheintegrationfunctionsistoprovideacontinuousfunctionthatdenesthe networkinputvaluesbasedonthecombinationofpossibleinputs.Thisisduetothenumberofpossible combinationsthatcanbeexpressedforeachvariable.ForexampleifwesetYoung'smodulus E =1000 , thePoissonratio v 2 [0 ; 0 : 5] .Therefore,thesevariablesarecompletelyindependent,leadingtoanenumeratedtwo-dimensionalsamplespace.Toensurethatthereissufcientcoverageofthesedomains,we havetoprovidevariationsofthesetwovariablesthatgenerateanadequaterepresentationofthepossible materialpropertycombinations.Thisisnotonlycriticaltotheaccuracyoftheproposedmethod.Ifthere isinsufcientcoverageofthetrainingdomain,thentheresultingforceswillintroducehighervibration modeswithinthesystem,leadingtonumericalinstability. Theconceptbehindprovidingdifferentintegrationfunctionsisbasedonthedensityofthesamples usedtorepresentthistwo-dimensionalmaterialdomain.Forinstance,ifthe sin x functionisusedto representthedisplacementimposedonthereferenceelement,thereisareasonablyuniformsampledis62

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tribution.Therefore,foranypotentialdisplacementappliedtotheelement,thetrainednetworkshould provideareasonableforceresponse.However,inthecaseofthe dGauss x function,thesampleswill representahigherconcentrationofsmallerdisplacements.Thismeansthatthenetworkwillprovidea moreaccurateapproximationofresponseforcesforsmalldisplacementsandfewersamplesforlargedisplacements.Sincethenetworkwillperformaformofinterpolationbetweentrainingsetsamples,there aredifferentintegrationfunctionsthatcanwork,butdependingontherequiredaccuracyandsimulation behavior,theremaybebenetstoselectingoneoveranother. 3.5.3DisplacementsandForceResponses Thegenerationofthetrainingsetbasedonexpressingthematerialpropertiesanddisplacementappliedto thereferenceelementishandledthroughthesolutionoftheequation F = KU .Sincewecanformulate anisotropicFEAsystemthathasoneelementthereferenceelementcharacterizedbyitselementstiffnessmatrix K ,wecanapplythedeformationdenedbythetrainingsetas U tocreateasetofresponse forces F .Foragiventrainingconguration,weexpresstherangeofYoung'smodulusbetween E min and E max .Similarly,forthePoissonratio,wedenetherangeas v min and v max .Thesequantitiescharacterizetheintegrationfunctionstheiramplitudeseffectivelydeningadistributionofsamplesbetween theseminimumandmaximumvalues.Asimilarrangeisusedtodescribethedisplacementappliedtothe referenceelementwhere d )]TJ/F18 10.9589 Tf 9.824 -3.959 Td [(and d + representthenegativeandpositiveboundsonthedisplacement.For thevalueofthisdisplacementrangetobeeffective,weassumethatthescaleofthereferenceelementhas beennormalizedsuchthatitsnodeslieontheunitcircleDorunitsphereD.Thenreasonablevalues ofthedisplacementcanbedenedas: d )]TJ/F23 10.9091 Tf 11.527 -3.959 Td [(2 [0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1] and d + 2 [0 ; 1] .Thisessentiallytrainsthenetwork tohandleinstancesuptowheretheelementwillinvert.Fromthesolutionoftheniteelementmethodfor theisotropicelasticmaterial,thisdisplacementrangecanbesethigher,andthecorrectresponseforces willstillbeprovided,eveninthecaseofelementinversion. Followingthestandardprocessofnormalizingthenetworkinputs,wepresenttheplotsoftheinput datasetsusedfortrainingthenetworkmodels.Duetothelargenumberofnetworks,weselectanindividualexample.Thisincludesthedistributionofthenormalizedsamplesforboththe sin x functionshown 63

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Figure3.5.19:Illustrationofthe sin x integrationfunctionappliedtothedisplacement d astheinputto thetrainingmodelleft.ThematerialresponsealongtheprovidedDOFofthereferenceelementisshown astheexpectedforce f valueright. Notethisisonlyasubsetofthesamples. inFigure3.5.19andthe dGauss x functionshowninFigure3.5.20. Intheseinstances,thematerialpropertiesareencodedusingthe sin x functionwhilethedisplacementisencodedusingthe dGauss x function.Thecharacteristicsoftheoutputwaveformaredened bytheintegrationfunctionsandtherelativefrequencyoftheinputs.Thismeansthatthecomplexityof theexpectedoutputdependsontherelativebehaviorsofthethreeinputfunctionsandcanbemodiedby alteringtheparametersofthetrainingset.Theobjectivehereistomaximizethecoverageofthematerial propertydomainwhileprovidingthehighestconvergencewiththesmallestnetworkstructurepossible.To providethecompletetrainingmodelforthesystemofnetworks,wegeneratethetrainingsetsprovidedin Figure3.5.20:Illustrationofthe dGauss x integrationfunctionappliedtothedisplacement d asthe inputtothetrainingmodelleft.ThematerialresponsealongtheprovidedDOFofthereferenceelement isshownastheexpectedforce f valueright. Notethisisonlyasubsetofthesamples. 64

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Figures3.5.19and3.5.20for n d networks.Thismeansthatforatwo-dimensionalsimulationwewill generate 6 2=12 networks,eachcontainingtheirown E , v ,and d inputsand f outputandforathreedimensionalsimulationwewillgenerate 12 3=36 networks.InourimplementationweutilizetheFast ArticialNeuralNetworkLibraryFANNthatprovidesasimpliedCinterfacetooptimizednetworkimplementationsforbasicmultilayerperceptionnetworks.Sincethenetworksarecomposedofaminimal numberoflayersrepresentedbysimpleweights,westore 36 individual4KBnetworks. 3.5.4Training ThetrainingprocessdenedbytheNEformulationisdesignedtorapidlyconvergetothesimpliedwaveformsthatcontainnonoiseduetotheuseofsyntheticdata.Therefore,theerrorbehaviorobservedover thecourseofalargenumberofepochsconvergesquickly.Thiserrorprovidesavalidationofthesimplicityofthenetworkmodelsthatarerequiredtoreplicatetheexpectedforcevaluesforthedeformationsappliedtothereferenceelement.Thisallowseachnetworktobedenedbythesimplestinternalarchitecture possibletoreplicatematerialresponsesefciently.Sincetheoriginalniteelementformulationperforms numerousmatrixoperationsthatcanbeeffectivelyoptimized,thistaskisnotinherentlytrivial.Therefore, thereisadelicatebalancebetweenthetargetaccuracyobtainedbyeachnetworkandtheperformanceof thenetworksinternalarchitecture.Thetrainingerrorfornetworkscontaining 4 and 6 hiddennodesare showninFigure3.7.22. Figure3.5.21:Plotsofthetrainingerrorlossoverthecourseof 5000 epochs.Thetrainingconverges quiterapidlyduetothelackofnoisewithinthesyntheticdataandthesimplicityofthewaveforms. 65

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3.6ParallelElements Theindividualmaterialresponsesofelementsprovidesthefoundationaloutlineforperformingparallel executionsofthenetworksusedtogeneratematerialbehaviors.Thepremiseoftheparallelizationthat canbeperformedusingthismethodisbasedontheexecutionofthesenetworksas separateinstances . Asanadvantageprovidedbytheformulationofthematerialnetworkthatusesnodalseparation,allofthe independentnetworksarethesameforallelements,evenfordifferentmaterialproperties.Thismeans thatonceweobtainthetrainednetworks,wecanusemultipleinstancesofthesamenetworksforallelements.Fortheparallelcomputationofelements,instancingremovesthedependencyconstraintsbetween elements.Therefore,theexecutionofthematerialnetworkcanbemadetrivialforallelementsbyprovidingeachcomputationalunitwithitsownlocalcopyofthenetworksrequiredtocomputetheelements response.Additionally,sincethenetworksareconstant,theycanbepreloadedandstoreasconstantinstances.Furthermore,theyarerepresentedbyasmallnumberofweightssothememoryoverheadisremainsminimal. TheprimarychallengeassociatedwiththeparallelformoftheNEformulationisthesynchronization anddatadependencyimposedontheresolutionofthenetforcesactinguponeachnode.Sinceeachelementmayhavenumerousadjacentneighbors,theresolutionofthesenetforceadditionsarethebottleneck oftheimplementation.Thisisbecausethecurrentsolutionincorporatestheuseofatomicoperationsto ensurethatthecontentionoverthenodalforcesisresolved. ThecurrentimplementationutilizesOpenMPtoenableparallelexecutiononmulti-coreCPUs.However,basedonthedesignoftheparallelversionoftheNEdesign,thismethodcanalsobeextendedtoa GraphicsProcessingUnitGPUwhereeachCUDAcoreorstreamprocessorcanstorelocalcopiesof thetrainednetworkstoalloweachcomputeunittoprocessanindividualelement.However,basedonthe samebottleneckthatisimposedwithinthecoredesign,theuseofatomicoatingpointadditionscanbe usedtoresolveraceconditions,butwillstillcauseasignicantreductionintheoverallperformance. 66

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3.7ExperimentsandDynamicSimulations Themaincontributionoftheneuralelementmethodisascalableandexiblesimulationframeworkfor deformablesolidobjects.SincethemethodisheavilyderivedfromtheFEA-basedformulation,theaccuracyofthismethodresideswithintherealmofphysicallyplausiblebehaviorbutdenesnewreal-time alternativetoexactsolutions.Unlikepriormethodssuchasmass-springsystemsthatusearbitraryinternalconnectionsandmaterialspringcoefcients,thismethodprovidesstandardcoefcientstodene elasticmaterialbehavior.ToillustratethepoweroftheinversemodelingapproachusedtoderivetheNE model,wecanlookatthepredictionaccuracieswithinthemodelandtheresultingsimulationbehaviors. Thisincludesseveraldifferentstandarddeformations,theinteractivedeformationofarbitrarymeshes,and data-drivenmodicationsofmaterialproperties. Figure3.7.22:Illustrationofneuralelementmeshessimulatedinreal-time.Thisincludesinteractivedeformationsleft,real-timestressanalysisofelementscenter,anddynamicboundaryconditionsfor generatingcomplexanimationsright. Theneuralelementformulationprovidesanewmethodofsimulatingelasticmaterialsbasedonthe largecollectionoftrainednetworks,butishardtovisualizewithoutdemonstratingthesimulationofdynamicbehaviorsthemodelprovides.Toillustratethecapabilitiesofthiscorematerialmodel,wepresent severalstandarddeformationbehaviors.Thisincludesvariousformsofdeformationsthatinclude: stretchingandcompression,cantileverbeams,twistdeformations,andinteractivedeformations throughpullinglocalizedregions.ThefollowingdeformationexampleshavebeengeneratedwithYoung's modulusof E =10000 andPoissonratioof v =0 : 3 . 67

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Figure3.7.23:Illustrationofneuralelementbarmeshbeingcompressedthroughtwoxedboundary conditions.Theleftfacenodesarexedandtherightfacenodesarebeingdirectlymoveddisplacedto reducethedistancebetweenthetwoendsofthebar.Thisresultsinacompressionofthematerialbetween thesetwoboundaryconditions. 3.7.1Deformation:Compression Compressionofaneuralelementmeshischaracterizedbyxingtwoboundaryconditionsappliedto eachendofthemeshandthenreducingthedistancebetweenthesexednodes.Asthedistancebetween thexedendsisreduced,thematerialissubjecttothebehaviorinducedbytheselectedPoissonratio v =0 : 3 .Initially,thisdenesthatthemiddleofthedeformedbarwillbegintobulgeoutinthecenter duetothedisplacementoftheinternalmaterial.Thesecondarybehaviorexhibitedbythisdeformationis thebucklinginducedwithinthematerialastheslightnumericalvariancescausethematerialtodisplace perpendiculartothecompression,asshowninFigure3.7.23. The vonMises stressmagnitudesareshownusingthecolormapandindicateamixedrepresentation ofthenormalandshearstressescomputedwithineachelement.Thestresspatternsillustratethedifferencesinhowtheinternalforcesbegintointroducebucklingbehaviorsthatareindicatedbythehighstress regionsthatareincurredduringthecompressiondeformation. 3.7.2Deformation:Stretch Stretchdeformationsarecharacterizedbyboundaryconditionsthatarethesameasthoseusedwithinthe compressiondeformation.Thatis,theleftfacenodesarexedandtherightfacenodesareincrementally movedtoincreasethedistancebetweenthetwoendsofthebar.Asthisdeformationoccurs,thePoissonratioofthematerialisclearlyshownduetothecontractionofthematerialwithinthecenterofthe 68

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Figure3.7.24:Illustrationofneuralelementbarmeshbeingstretched.Theleftsidefacenodesarexed andtherightfacenodesareincrementallymovedtoincreasethedistancebetweentheendsofthebar.This resultsinanelongationofthebarandanarrowingofthecenterduetotheinternalmaterialdisplacements. bar.Thisisbecauseofthematerialdisplacementswithinthecenterbecomingelongatedasthematerialis stretchedasshowninFigure3.7.24. 3.7.3Deformation:RotationalTwist Theintroductionofthisdeformationprovidesanillustrationofadynamicboundaryconditionthatcanbe denedthrougharbitrarynodemanipulations.Specically,wextheleftsidefacenodesofthebarmesh andtherotatetherightsidefacenodesaboutthetwistaxis.Thisboundaryconditionisimposedbysimply alteringtherightfacenodepositionsdirectlybyrotatingthemaboutthisaxis.Theneuralelementmaterial modelwillthenautomaticallyaccountforthechangesinthesenodepositionsandpropagatetheimposed deformation. Forhighlyelasticmaterials,thetwistdeformationresultsinsubstantialdisplacementsofthematerialelements.Thecorrectbehaviorofthisexampleisonlypossiblethroughthepolardecompositionof theelementsdeformationandrotationcomponents.IfthisformoflargedeformationisappliedtoFEA simulationsthatdonotincorporatethisprocess,thenasthematerialistwisted,theresultingvolumeofthe meshwillgrowconsiderably.Thisresultsinextremelylargeerrorswithinthesimulationasaconsequence oftheunderlyingmaterialmodelnotbeingrotationinvariant. 69

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Figure3.7.25:Illustrationofadynamicboundaryconditionthatcanbeimposedonameshtocreatea twistdeformation.Therightsidefacenodesarerotatedaboutthetwistaxis ^ x andtherestofthematerial behaviorisgeneratedthroughtheneuralelementresponses. Animportantnoteaboutthisisthattheboundaryconditioncanbechangedatanytimeanddoesnot requireasystemreconstructionaswithFEA-basedsimulations.Thisispartiallyduetotheformulationof thestableintegrationmethodusedwiththesystem.Toprovidetheexibilitycommonlyrequiredforinteractiveapplicationsgenerallyprovidedbysimplersystemssuchasmass-springmodels,dynamicboundary conditionsallowtheusertomodifyconstraintsatanypoint.Forneuralelements,thisdoesimposeany performancepenaltyduetoasystemreconstruction. 3.7.4InteractiveDeformations Thebenetofintroducingreal-timesimulationmodelsisthattheycanprovideinteractivedeformations forillustratingreal-timestressevaluations.Thismeansthatwecanderiveaninteractionmethodforcontrollinglocalizedregionsofthematerialbyapplyingarbitraryexternalforcesordirectlyspecifyingexact nodepositionsandviewhowtheimposedconstrainteffectsmodelbehavior.Intheapplicationthatallows real-timeinteractionwithneuralelementmodels,weimplementasimplepickingschemewherespecic 70

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facesornodesofamodelcanbepulledbasedonmanipulationsbasedonmovement.Thisprovidesacontrolmetaphorforpullingonspecicregionsofthemodeltoimposeexternalforcesthatwillleadtodeformations.Asthemodeldeforms,newregionsofthemodelcanalsobedeformed.Examplesofinteractions withtheStanfordbunnymodelareshowninFigure3.7.26.Inthisexample,therearemultipledifferent interactionsimposedonthemodelthroughoutthesimulation.Theseexternalinteractionsareillustrated throughtheforcevectorimposedontheregionofthemodelindicated.Intotaltherearethreeindividual deformationsimposedonthebunnyface,ear,andchest.Eachinteractionisshownoverthecourseofseveralframesandthelengthoftheappliedforcevectorindicatesthemagnitudeoftheinteraction. Figure3.7.26:Illustrationofneuralelementmeshessimulatedinreal-time.Thisincludesinteractivedeformationsleft,real-timestressanalysisofelementscenter,anddynamicboundaryconditionsfor generatingcomplexanimationsright. 71

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3.7.5Free-formDeformationandCollision Theneuralelementformulationnativelyprovidestheabilitytohandlearbitraryrigid-bodyorfree-form motioninadditiontothedeformationswithinthemodelgeometry.Illustratingasimpleexampleofthis, weprovideasimplecollisionbetweenadroppedmodelandaninniteplane.Forthisexample,weapplygravityandallowthemodeltodropfromapredenedheight.Thisallowsthemodeltomakecontact withthegroundandillustratetheelementstresseswithinthemodel.Duetotheweightshiftofthemodel duetotheears,themodelrollsontothegroundwhereitrestsasshownintheimagesequenceofFigure 3.7.27. Figure3.7.27:Illustrationofaneuralelementmodelundergoingrigid-bodytranslationasitfallsdueto appliedgravity,resultinginacollisionwiththeground.Duetothemassandsizeoftheears,themodel rollsaftermakingcontactwiththeground. 72

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3.7.6QuantitativeEvaluation Theneuralelementmodelprovidesavalidmethodforthesimulationofisotropicelasticmaterialswithin real-timeapplications.However,wecanalsoquantitativelyevaluatetheresponseforcesthatarepredicted bythedata-drivenmodel.ThroughtheevaluationoftheresponseforcescomparedtotheexactFEA-based result,wecanevaluatetheaccuracyofthismethod.Duetothehighnumberofpredictedforcesthatcomposethedata-drivencomponentofthemodel,wehavetoselectarepresentativesubsectionoftheforces toevaluate.Todothis,wehaveselectedthedeformationofanelementfromthebeamstretchsimulation. Undergravity,thematerialresponseisrathertame,onlygeneratingwaveformssimilartoasinefunction withsomeabnormalities,asthebeamoscillatesbetweenacompressandstretchedstate.Therefore,inthe followingresponseforceprediction,weaddsignicantinteractionforcestogeneratemorecomplexwaveformsasshowninFigure3.7.28.Thisillustrates,foreachcomponentintheelementDtetrahedra,the exactandpredictedforceresponsesoftheelementaslarge-scaledeformationsareappliedoverthecourse of 800 simulationtime-stepsframes.Thepredictedvalue pr isfromtheproposedNEformulationand theexactvalue ex isprovidedbyanexactFEAsolutionofthematerialresponse. 3.8NaturalPerturbations Analyticalclosedformsolutionstothegoverningequationsofmotionandelasticitygeneratenumerically precisevaluesthatresultinpseudoimplausiblephysicalbehaviorsrelatedtobalanceandsymmetry.This isduetothenumericalrepresentationoftheequationsastheyarecomputed.Neuralelementsdonotnativelyexhibitthisbehaviorduetotherandomnessintroducedbybothinitializationandthespeciccharacteristicsofthetrainingdata.Thisleadstoamorenaturalorexpectedresultforsimulatingimperfect elementsthatcomposeacontinuousmaterial. Ifthenumericalvariancewithinthenetworkisminimal,thiscanprovidenaturalbehaviorstosimulatedobjectsbasedontheperturbationsintroduceintheelasticmaterialresponses.However,ifthese smalluctuationsbecometoolarge,theycanimpacttheaccuracyofthesimulationorintheworstcase causenumericalinstability.Toalleviatethisproblem,welookathownaturalvarianceinthenetworkcan 73

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Figure3.7.28:Illustrationofasubsetofmaterialresponsesgeneratedbytheneuralelementmodelforone elementduringasimulationof 800 time-stepsframes.Thisillustratesthebehavioralerrorassociated withtheneuralelementpredictedvalue pr andtheexactvalue ex providedbyanFEA-basedsolution. Fortheselected3Dtetrahedralelement,thisisshownbythe 12 componentforcesoftheelement.The waveformissporadicduetothelargedeformationoftheelementthroughuserinteractions. beminimizedandhowwecanensurezeronetinternalforcewithinsimulatedelementsfornumericalstability.Thisisenforcedafterthenetworksareusedtocomputethepredictedmaterialresponse.Thatis,for eachelement,theinternalresponseforcesaremeanshiftedtoensureazeronetinternalforce. 74

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3.8.1NetworkNumericalVariance Precisesystemformulationsinphysicalsimulationshavetheabilitytogenerateperfectlysymmetricalor uniquesystemresponsesduetohowaccuratelytheyaredened.Behavioralsymmetryandabnormalnumericallyperfectconditionssuchasaperfectlyinvertedtrianglearetheproductofoating-pointarithmeticandalgebraicequationsthatdonotintroducechaoticvarianceintothesystem.Thisisduetothe precisioninwhichallofthesimulationgeometryandinitialstatecanbedened.Forinstance,ifasimple lineisbalancedonapointvertically,thechaoticinteractionsinthereal-worldvibration,airow,friction, etc.wouldresultinaslightaccelerationinthexorzdirections,resultingintherigid-bodyrotationofthe objectitwillsimplyfallover.However,forthesameinstancewithinasimulationwecandenethisperfectlyverticalline,andspecifythatthereisnoxorzdirectionaccelerationbytheoversimplicationofthe real-worldpresentedwithinthesimulation.Thiswillresultinthelineexistinginaperpetualverticalstate. 3.8.2NULLDisplacementResponses Therandomizedweightsusedtoinitializetheneuralnetworkandthesampleswithinthetrainingsetmay introduceanon-zerobiaswithintheresponseforces r d computedwhentheprovideddisplacement d is equaltozero dx;dy;dz =NULL.Sincethetrainingwillnotresultinazeromagnituderesponseforce providedbythenetworkwhenthedisplacementequalszero,thisnon-zeronetworkcomputationwillintroduceaconstantshiftofthepredictedresponseforce p d .Thisdoesnoteffectthebehavioralresultof theresponsefornon-zerodisplacements,butintroducesaconstantshiftsimilartoanintegrationconstant whicheffectsthenetforceontheelementduetothedisplacement.Thisshiftdoesnotinvalidatethebehaviorofthematerial,butcanleadtonon-zerointernalforceswithintheelement.Toprovideanaccurate representationofthenetworkaccuracyandresolvethenon-zerointernalnetforce,thisoffsetcanberesolved.ThesolutiontothisproblemistoestablishtheNULLdisplacementoffset 0 requiredtogeneratea netzeromaterialresponseforceforthenetwork.Thisvalueisobtainedbycollectingthepredictionvalue foradisplacementofzeroandthensubtractingitfromallfuturepredictionvalues: r d = p d )]TJ/F21 10.9091 Tf 10.909 0 Td [( 0 . Removingthisshiftresultsinanaccuratereplicationoftheelasticmaterialresponse.Toillustrate 75

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howthiscorrectionimprovestheaccuracyoftheresponseforceprediction,wepresenttheforceresponses ofanelementwithinacantileverbeamundertheinuenceofgravity.Theresponseforcegeneratedbya dx networkanditsassociatedoffsetcorrectionisshowninFigure3.8.29. Figure3.8.29:Theinitialrandomizedweightsofthenetworkcontributetoresidualnon-zeroresponse forcesprovidedfortheinputdisplacementofzero.Thisresultsinaconstantnon-zeroshiftintheresponse forcesthatcontributetoanetpositiveornegativeinternalforcewithintheelement.Thisiscorrectedby removingtheshifttoobtainaccurateforceresponseestimations. Inthisinstance,theconstantshiftembeddedinthenetworkevaluatesto 7 : 728[ N ] ,thatis,foraninputdisplacementofzero,theresponseforcegeneratedis 7 : 73[ N ] inthe ^ x direction,contributingtothe constantoffset.Thisoffsetcanbecorrectedforeachnetworktoreducethepredictionerroroftheresponseforces,butitdoesnotcompletelyeliminatethenon-zerointernalforceoftheelement.Itisimportanttonotethatthisslightshiftdoesnotdramaticallyeffectthebehavioralresultofthesimulation, onlythequantitativeevaluationoftheresponseforceswithrespecttotheexactvalues.Additionally,the selectionoftheintegrationfunctionalsoinuencestheerrorintheprediction. 76

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CHAPTERIV GENERATIVEDEFORMATION:3DPRINTINGBEHAVIOR 3Dprintingtechnologyisdevelopingatablisteringpaceandhasexpandedtowidereachingimplications forbothacademicandindustrialresearchinproductdesign,engineering,andmanufacturing[30].Yet, thecontrolofelasticdeformationsof3Dprintedobjectsremainschallengingduetothecomplexinterrelationshipbetweenthepropertiesofavailableelasticmaterials,thedensityofinternallatticestructures, andtheconnectivityofprocedurallygeneratedprintgeometry.Tothisend,severalrecentcontributions havesignicantlyimprovedtheiterativedesignprocesstoincorporateFiniteElementAnalysisFEA [31]throughvarioussoftwarepackagestofacilitateprintoptimizationsforstressreduction[32],printmaterialminimization[33],internallatticegeneration[34],vibrationreduction[35],printstructures[36], implicationsoflayeringorientation,andnumerousothermetrics.Whilethedesign,simulate,rene,print pipelinehasbeenwellestablishedformodifyingcharacteristicsof3Dprints,therestillremainsalimited setoftoolsthatprovideanend-to-endsolutionforprovidingprecisecontrolofdeformationbehaviorsin 3dprintedobjectsbasedondesignconstraints. Severalvariationsofsimulation-basedoptimizationpipelineshavebeenintroduced[37]withinthe domainof3Dprintingtoaddressanumerouschallengeswithprintingcontrollable,structurallyrobustobjectsandparts.Mostpipelinesattempttointroduceanoverarchingbridgebetweentheinitialdesignof apartandtheresultingprintsubjecttoseveraladditionalconstraintsthatcanbedenedtooptimizethe printwithrespecttoagivensetofobjectives.Signicantcontributionshavebeenintroducedthroughthe precisecontrolofelasticstructuresusingmicro-structures[38]andthebehavioraloptimizationofmultimaterialmicro-structures[39].However,theseadvancedtechniquesassumethat3Dprinterswithanextremelyhighprintresolutionormultiplemixablematerialsareavailable.Theproblemisthatthecostand limitedsetoftoolsthatarespecicallydesignedfortheseexpensive3Dprintersarechallengingtouse withconsumer-levelprintersthathavelimitedelasticmaterialcapabilities. Inthiswork,weintroduceaconsolidatedpipelinethatintegratesboth automatedperforation ,deformationbehavior,andstressanalysistoprovideanautomatedprocessthatallowsforthedevelopment ofelastic3Dprintsonconsumer-levelprinters.Specicallywefocusoncombininggenerativemodeling 77

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Figure4.0.1:GenerativeDeformation:Theautomatedprocessofperforatingvolumetricsimulation meshesthatare3Dprintedusingelasticmaterialstoobtainspecicdeformationbehaviors. withFEAdynamics,butintegratea1-to-1pipelinebetweenthesimulationandprintgeometry.Thatis, theproceduralgeometrygeneratedbythegenerativealgorithmisconsistentbetweenboththesimulation modelandthe3Dsurfacemodelprovidedtotheprinter.Thisprocessiscompletelyautomatedandintegratedintoasingledesigntool.Thisallowstheapplicationtobeusedforquicklygenerated perforated geometricstructuresforquicklychangingthematerialpropertiesof3Dprintedelasticmaterials. Thesecontributionsincorporateadesignapplicationthatintegratesexistingvolumetricmeshgenerationalgorithms,FEAsimulation,andperforatedinternalstructuregenerationtoquicklyobtain3D printsspecicallytargetedforconsumer-level3Dprinters.Buildingonthispipeline,showninFigure2, wefacilitateamethodfor generativedeformation whichallowsforaheterogeneouslatticestructuretobe generatedwithrespecttodeformationobjectives.Wedenethisprocessastheproceduralgenerationof internalgeometrythatisderivedfromtwoprimarymetricsobtainedthroughFEAsimulation:measuresoftetrahedralelementdeformationusingdihedralanglesandinternalstressdistributionsderived fromnonlinearFEAsimulations[40,41]usingVegaFEM[42].Thisobjectivedoesnotonlyimpactpassivedeformationsof3Dprints,butcanalsobeusedtomodifyactivatedbehaviorsof4Dprintsbasedon howheterogeneousstructuresimpactactuatedmovements[43]. 78

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4.1RelatedWork Ingeneral,generativedesignislooselydenedbytheprocessofalteringthegeometricstructureormaterialpropertiesofapartormodeltoachieveaspecicbehavioralcharacteristic.Typicallythisisrelated tothestressanalysisofthepartforloadtolerances,plasticdeformationorfracture.Sinetheobjectiveis tomaximizethesetolerancesorloadcapacity,thedesigncanbeoptimizedbymakingslightstructuraladjustmentstotheunderlying3Dmodel.Theprocessisdenedasimpleiterativedesignpipeline:design a3Dpartwiththerequiredspecications,loadthemodelintoanFEA-basedsimulationtoestablish loads,boundaryconditionsrelatedtotheusecase,andevaluatethestressdistributions,followbyrunningoptimizationalgorithmsthatmodifythedesignoftheparttominimizestress,materialuse,oradjust thehigh-stressregionsoftheparttoprovidethetolerancerequiredfortheloadcondition. Therearenumerousmethodsthatarecurrentlybeingexploredbyhigh-endlabsforcomplexmicrostructurepropertyandbehaviorestimationforhigh-end3Dprinters[27].Forhigh-endprinters,thereare twoprimarymethodsthatcanbeusedtogeneratepreciselycontrolledmicro-structures:highresolutionprintersthatcanprintdualmaterials,oneastheprimarymaterialandasecondarythatprovidessupportfortheprimarythatisthendissolvedormulti-materialprintersthatcandynamicallyadjustthe compositionoftheprintmaterialtoadjusttheelasticmaterialpropertiesoftheprint.Bothofthesesolutionsaretypicallyremovedfromconsumer-levelprintersthatarecommonlylimitedtoveryfewprint materials.Thisisduetothecomplexityofthe3Dprinterdesignrequiredtofacilitatedynamicallymixed materialsdependingonamodelsmaterialproperties.Naturally,thisapproachto3Dprintingcomplex modelswillinevitablybecomemorecommon,buttheabilitytodynamicaltertheunderlyinggeometry of3Dprintsisstillaviabletoolforchangingamodelsdeformationbehaviors. 4.1.1InternalMeshing Consumer-level3Dprintersaretypicallycombinedwithdesignsoftwareprovidedbytheprintersmanufacturer,oropen-sourcealternatives.Theseapplicationsarereferredtoas slicing programsbecausethey providethepre-processingrequiredtotransform3Dmodelsintosolidprintinstructionthattheprinter 79

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canparsetogeneratethemodel.Theseapplicationsarecalledslicersduetothenatureofslicinga3D modelintodiscretelayersthatcanbereplicatedbya3Dprinter.Withinmostoftheseapplications,there areadditionalfunctionsthatcanbeintegratedintothisbasicslicingprocess.Thisincludesimportantfeaturessuchas internalmeshing whichallowsforsolidinternalregionsofameshtobehollowedoutand replacedwithasparsemeshstructure.Typicallythisgreatlyreducestheprintmaterialusedtocreatethe modelandcansignicantlyreduceprinttime.Whilethissimilargeometricapproachtocreatingmeshlikestructureswithina3Dprintiscommon,itisdistinctlydifferentthantheproposedperforationmethod. Thisisbecausemostimplementationsthatgenerateinternalmeshingessentiallysuperimposethemesh structurewithinhollowregionsofameshwithoutaddressingtheintegrityoftheresultinggeometry.That is,allofnewgeometricshapesaresimplyplacedintotheseregionswithoutresolvingthetopologybetweenthemeshstructureandtheoriginalprintsurfaces.Theconsequenceofperformingtheseformsof naiveinternalmeshingalgorithmsonmeshes,isthatthemeshcannolongerbephysicallysimulated.This isbecausetophysicallysimulatea3Dmesh,alloftheinternalgeometrymustformonecontinuoustopologythatdenesasolidmaterial.Forexample,iftherearetwofacesthatintersecteachother,butarenot connectedbyuniquenodes,thenthemodelcannotbesimulated.Fromcomputergraphics,thisproblem canberesolvedforgeneraltechniquesinmeshingsuchasConstructiveSolidGeometryCSG,however thistechniquedoesnotautomaticallygenerateFEA-basedinternalsolidelementsforelasticmaterialsimulations.CSGtechniquescouldbepairedwithexistingniteelementmeshingalgorithms,howeverthe resultinggeometrywillbeextremelycomplexandlimittheabilitytousethesimulationmeshwithininteractiveapplications.Oneoftheobjectivesoftheperforatedstructureistomaintainsomelevelofinteractivesimulationbasedoncomplexstructures.Thismeansthatthegeometriccomplexityandnumberofthe solidsimulationelementstetrahedrahastobeminimized.Foragivenvolumetrictetrahedralmesh,the topologyconsistentperforationofthestructureprovidestheminimalnumberofelementsthatcanbeused tosimulateanelement-wisesparselyhollowmesh. 80

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Figure4.1.2:IllustrationcomparingthedifferencebetweentypicalinternalmeshingtechniquesandCSG modelingorautomatedperforation.Invalidgeometrycenterofthehollowedregioncannotbevalidly simulatedduetogeometricdiscontinuities.Thevalidgenerationofthegeometricelementsrightmustbe followedforgeneratingmeshesthatcanbesimulatedthroughFEAmethods. 4.2MethodOverview Generativedeformationisaprocessthroughwhichtheobjectiveistogeneratespecicdeformationbehaviorsof3Dprintedelasticmaterials.Through perforating avolumetricmesh,wedenehowvoidsor holesofdifferentsizescanbeintroducedwithinasolidelasticmeshtoalteritsdeformationbehaviors. Thiscontributionrelatestothedevelopmentofnewmethodsthatallowusto3Dprintelasticmaterialbehaviorsinadditiontomatchingthestaticgeometryofanobject.Themotivationbehindthismethodisthe simplicityofthecoreconcept:insetelementswithinameshbydifferentvaluestoobtaindifferentinternalstructuresthatwillmodifydeformationbehavior.Thetranslationofthisconcepttoaniterativedesign processisalsofairlysimple:forexibleregionsmakeinsetslargerandforstiffregions,makeinsetvaluessmaller.Bycompletelyautomatingthiscorealgorithm,therearenumerousdesignanddata-driven toolsthatcanbeusedtovarythecomplexdistributionofelementelementperforationinsetvalues.This includespainting-basedmodicationsofinsetvalues,FEA-drivenoptimizationsofinsetvalues,anduser designconstraints. Thegenerativedeformationdesignpipelineproposesaniterativeprocessthroughwhichspecicdeformationbehaviorscanbeintroducedtoasolidelasticmesh.Theprocessishasalsobeengeneralizedto startwithanyvalid sourcemesh thatcanbedenedasanyvalidsurfacemesh.Thecoreprocessinvolves fourprimarysteps:thevolumetricmeshingofthesurfacemodel,typicallyprovidedbymeshingalgo81

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rithmssuchasTetgen[44],thesimulationofthesolidvolumetricmeshtodeterminehowbehaviors willeffecttheperforationinsetvaluesoftheobject,thegenerationoftheperforatedmesh,andsimulationoftheperforatedmeshtovalidatemeshbehavior.Theoutputofthisprocessisanautomatically generatedsurfacemodelthatcorrespondstotheperforatedmesh.Theobjectiveofthismethodistointegratethisprocessintoacoherenteasy-to-usedesignapplicationthatalsoincludestheabilitytoperform niteelementanalysisonthedesignmodelstoimproveprototypeaccuracy.Theimplementationcanalso beusedforrapidprototypingofperforatedstructuresbasedonthesimplicityofgeneratingand3Dprintingperforatedmeshes. Figure4.2.3:Automatedvolumetricperforationpipeline.Theinputisasimplesurfacemeshandthemain processconsistsofafourstages:volumetricmeshingandelement-wiseuserconstraints,FEAsimulationofthesolidmesh,theautomatedperforation,andthesimulationoftheperforatedmesh.The resultproducedbytheperforationalgorithmprovidesavalid3Dprintablesurfacemeshthatcanbeprinted usingconsumer-level3Dprinters. 82

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4.2.1ChallengesinPrintingElasticMaterials Printingelasticmaterialsrepresentsarelativelychallengingaspectof3Dprintingdeformableobjects. Thisisbecauseforconsumer-levelprinters,thereisalimitedselectionofelasticmaterialsthatcanbeused andthefewthatcanbearetypicallyproprietarymixturesforresin-basedStereo-lithographySLAsystemsorlooselycontrolledandhighlyvariableelasticlamentsforFusedDepositionModelingFDM printers.Inadditiontolimitedmaterials,thisprocessisalsochallengingduetotheparameterizationofthe printingprocesswhichcanincludelargechangesinbehaviorduetothenumberoflayers,layerthickness, orientation,andadhesion.Additionally,forSLAprints,thepost-printcuringprocesscanhaveasignicantimpactonhowthemodelwilldeform.Ifthemodelisunder-cureditwillmaintainexibilitybutcontainresidueoftheresin.Ifthemodelcurredforanextendedperiodandover-cured,theresin-basedmodel willbecomebrittle. Formostconsumer-level3Dprinters,theproblemisthelimitedselectionofelasticmaterials.SLAbasedsystemstypicallyintroducealimitedselectionex.1proprietaryelasticresin.ForFDMprinters, theselectionislargerduetotheabilityofothermanufacturerstocreateexiblematerialsthatarecompatiblewiththespecicationslamentdiameter,melttemperatures,etc.ofmostprinters.Thesearetypes ofaccessibleprintersthatcannotdynamicallymixmultipleelasticmaterialstoobtainspecicdeformationcharacteristics.Forexample,evenifwehaveavarietyofmultipleelasticmaterialslament,resins, pickingasinglematerialmaynotprovideanadequatebehavior.Sincethesetypesofmaterialscannotbe readilymixedinarbitraryproportionstogeneratedifferentbehaviors,wearelimitedtouseapredened selection.Typically,theuseofthematerialswithinthelimitedselectionwillnotresultinidealoroptimizedbehaviors. Theprintingprocessitselfnaturallyhasalargeimpactonthedeformationbehaviorsofanelastic print.Theseparameterscanbemodiedthroughtrialanderrortoproducedesirableorconsistentresults,buttheydonotallowforcontroloverthelargechangesinhowtheprintedobjectwilldeformonce printed.Thisiswherewecanintroduceminutechangestothecoregeometryofthemodeltoprovidecontrollablemodicationsofthedeformationbehavior.Evenslightmodicationsoflocalregionscandras83

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ticallyalterdeformationbehaviors.Weattempttoleveragethisobservationandcreatevarieddensitiesof thematerialdistributiontoalterdeformationbehaviors.Bydoingthis,weprovideanefcientmethodfor changingbehaviorsthatmaybetheresultofalimitedmaterialselectionorbytheprintprocessitself. 4.2.2Contributions Toprovidealight-weightandopen-sourcesolutionforgeneratingperforatedstructuresin3Dprinting,we provideasetofcorecontributionsprovidedwithinourimplementation.Thecontributionsinclude: • AutomatedPerforation -Givenanarbitrarysurfacemesh,wecangenerateavolumetricmeshthat canbeperforatedanddirectly3Dprintedwiththeuseofopen-sourceorprinter-specicslicingsoftware.Foraprovidedsetofappliedtoallelements,thisprocessiscompletelyautomatedandwill generatea3Dprintablesurfacemesh. • 1-to-1SimulationGeometry -Forperforatedstructures,wegenerateboththeinternaltetrahedral andexternalsurfacefacesforboththeexternalandinternalstructureoftheperforatedmesh.This meansthatwecreateacontinuousmanifoldmeshthatcanbesimulatedusingFEAmethodstodeterminehowtheperforationaffectsdeformationbehaviors. • GenerativeDeformations -Weprovideasimpledeformationrecordingframeworkforalteringperforationinsetdensitiestoaltermeshbehaviorbasedonniteelementsimulations. • Non-uniformDensities -Theperforationalgorithmiscapableofgeneratingbothhomogeneousinsetsallelementsareinsetbythesamevalueorheterogeneousinsetsthatcreatenon-uniformdistributionsperforatedelementswithintheentiremeshvolume.Thisformofinternalmeshingismuch morecomplexthanuniforminsets,butprovidesasignicantlyhigherlevelofcontroloverlocalized deformationbehaviors. • PrototypingPipeline -Usingtheperforation,design,andsimulationmethodweprovideasaniterativedesignpipeline,thismethodcanbeusedtoquicklymodifytheelasticmaterialbehaviorsof 3Dprintedobjects.Thisincludesaninteractiveapplicationthatcanbeusedtodesignandgenerate perforatedmeshes. 84

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4.3GeometricPerforation Printingelasticstructuresthatfollowaprescribeddeformationbehaviorisdifcultfornumerousreasons including:materialpropertiesarelimitedtothosethatarecompatiblewiththe3Dprinter,theinternalgeometricstructureplaysasignicantroleinhowtheobjectdeforms,theprintingprocessitself layers,method,etc.largelyinuencesbehavior,thecuringprocessisnotonlyuniquebetweendifferentmaterialsbutevenslightvariancesinapplyingthesamecuringmethodtothesamematerialcanresult indifferentbehaviors,andsimpleandcommonoperationssuchasscalingcandrasticallyalterhowa printbehaves.Thesevecontributingfactorscontributetosubstantialchallengesindevelopinggeneralizedalgorithmsthatcanprovidespecicdeformationbehaviorsof3Dprintedelasticmaterials. Toaddressthewidevariancesinhowthesefactorscontributetothedeformationbehaviorofa3D print,proceduralgeometryandsimulationhavebecomecriticaltothedesignprocess.Basedontheserequirements,itisbenecialtohaveamethodologythatcanquicklyadjustthegeometricdistributionsthat modifybehaviorbasedontheconditionsofthesimulatedobjectivebehavior.Toachievethiswitharealtimedesignelement,wehaveimplementedageometricallyreducedformof inset-based proceduralgeometrytoformulateanautomatedformelasticcontrol:automated perforation .Thisprovidesavariety ofdifferentdesigntoolsthatcanbeemployedtomodifythegeometricmaterialdistributionoftheprint basedonvariousfunctionsincluding:latticethickness,gradientfunctions,per-elementconstraints,and precisepainting-baseduseradjustments. 4.3.1GeometricPerforation Theprocessofperforatingavolumetricstructureisderivedfromaninsetoperationbasedontheshapeof theinterfacebetweenadjacentelements.Thisoperationdenesasinglescalarvalue i where i 2 [0 ; 1] denesthe inset oftheelement.Foratetrahedralmesh,theinterfaceelementisatrianglebetweenneighboringtetrahedralelementsandtheinsetoperationisdenedasfollows:givenaninterfacetriangle t and inset i ,computethelinesparalleltoeachedgeofthetriangle a;b;c thatcreatethreeintersectionpoints oftheinsetasshowninFigure4.3.4.Iftwoneighboringtetrahedrahavedifferentinsetvalues i , j ,then 85

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each2Dedgeelementisdividedintotworegions,oneforeachinsetvalue.Thisisthereasonthattheheterogeneousornon-uniformperforationbecomessignicantlymorecomplexthanexistinginternalmeshingalgorithms.Whentheinsetsofadjacentelementsdonotmatch,wehavetondawaytoresolvethe differencesbetweenthegeometrygeneratedbybothinsetvalues.Oncethisproblemisresolved,theoperationwillresultinthecreationofvoidswithinbothelementsAandBthataltertheeffectiveelasticityof eachelement. Thisoperationhasbeenintroducedin3Dprintingforbiomassstructures[45];however,thepresentedformulationis invalid withrespecttotherequirementsofsimulationgeometry.Thisisbecausethis na veformofthealgorithmintroducescoplanarandunconnectedfacesandnointernalelements.TosimulateanelasticobjectusingFEA,thevolumetricmeshmustprovideacohesiveandvalidsetofelements thatdiscretelyrepresentthecontinuumofthematerial.Therefore,wemustgeneratenon-overlappinginternalelementswithcoherentindicesthatformacontinuum. Figure4.3.4:Automatedvolumetricperforationpipeline.Theinputisasimplesurfacemeshandthemain processconsistsofafourstages:volumetricmeshingandelement-wiseuserconstraints,FEAsimulationofthesolidmesh,theautomatedperforation,andthesimulationoftheperforatedmesh.The resultproducedbytheperforationalgorithmprovidesavalid3Dprintablesurfacemesh. Todenethe3Dformoftheperforationalgorithmweassume:asurfacemeshisprovidedandanexistingmeshingalgorithmsuchas Delaunay -basedTetgen[44]isusedtogenerateavolumetrictetrahedral mesh.Basedonthismesh,wegenerateagraph G ofallinternaltrianglefaces F sharedbetweenneighboringtetrahedralelementsandperformthe2Dinsetoperationinparallelforallfaces.Fromthis,weconsolidateallnodesandgenerateelementindicestoformacollectionof micro-tetrahedra thatcomposethe edgeelementsoftheperforatedmesh.Thecomplicationcomesfromthreechallenges:elementindices canbearbitrarilyordered,thecongurationofthepairwiseinterfacebetweendifferentinsetsrequires anenumerationofpossibleelementcongurationsthatdependonhowtheinsetvalues i and j relate,and 86

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basedonthetopologyofthevolumetricmesh,eachedgemayhavenoadjacency,oneadjacentortwo adjacenttetrahedra.Thus,eachedgemayrequirethegenerationof 3 , 6 ,or 9 internalmicro-tetrahedra dependingonthelocalinterfacetopology.Additionally,withthegenerationofalledgeelementsfrom Figure4.3.4,thereremainvoidswithinthecornersofeachoriginaltetrahedrathatmustalsobelled. Forarbitrarilyorderedelementindiceswedeneasimplecorrelationmapforeachinterfacebetween adjacentelements.Foreachinsetcongurationcase, i = j , i 6 = j , ij ,weemploy anenumeratedgenerationofallpossibleedgecongurations.Thisresultsinanelementgeneratinglookup tableofedgeelementsasshowninFigure4.3.5leftsharedbetweenanytwotetrahedraas: 3 -Tetedge, 6 -Tetedge,or 9 -Tetedges.Similarly,theinsetvaluesalsoimpactthegenerationofthecornerelements, resultinginanotherelementlookuptablethatillustratesthemicro-tetrahedraofthecornersasshownin Figure4.3.5right. Figure4.3.5:Combinatoricrepresentationofthemicro-tetrahedrathataregeneratedforeachperforated element.Theseenumerationsdeneallpossiblemicro-tetrahedraedge/cornerelementswithinlookup tables. Intheinstancewherenodepositionsbecomearbitraryclose ,theyareweldedtoreducegeometric complexityandminimizethenumberoflowqualityslivertetrahedra.TheseformsoftetrahedrasignicantlyreducethequalityoftheFEAsimulationsduetotheinterpolationsgeneratedintheshapefunctions ofthesimulatedelements.Thegenerationofthemicro-tetrahedraforallelementsresultsinaperforated, continuousmeshthatcanbesimulatedforstressevaluation. 87

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4.3.2MeshPerforationAlgorithm Thealgorithmassumesasinputatetrahedralmesh M ,andareal-valuedinsetvalue i orsetofinsetvalues I where j I j isequaltothenumberofelementsinthemeshand i 2 [0 ; 1] 8 i 2 I .First,allelementinsetsarecomputedinparallelbasedoneachelement'sassociatedinsetvalue I [ i ] .Thenforeachinterface betweenadjacentelements,micro-tetrahedraaregeneratedaccordingtotheedgerelationships 3 , 6 or 9 TetvariantsbetweentheelementinsetgeometryusingthelookupcongurationsinFigure4.3.5.This formstheinitialpassofthealgorithmthatgeneratesallofthemicro-tetrahedrathatresolvethedifferences intheinterfaceinsetsbetweentheelementsofthesourcemesh.Finally,themicro-tetrahedraelements forallnon-interfaceedgesonthesurfaceoftheobjectandinternalcornersarecomputed.Thisresultsin thecompletesetof internal , surface ,and corner micro-tetrahedrathatcomposetheperforatedmesh.The outputisacollectionofallgeneratedmicro-tetrahedraelementsandfacesthatrepresenttheperforated tetrahedralmesh P .Thismeshcontainsthesetofnodes,elements,andfacesthatcanbeusedforboth simulationandasaprintablesurfacemesh. Algorithm1 P ERFORATION M , I Input: TetrahedralMesh: M nodesN , elementsT Real-valuedElementInsets: I j I j = j T j Output: TetrahedralMesh: P 1: P:nodes N Parallelfor: 2: for each t 2 T do 3: InsetTetrahedraElement t , I [ i ] 4: endfor //Computegraphofinternalfaces 5: G =GenerateInternalFaceGraph M , F //Computeinterfaces/types,6,9 6: E =GenerateElementInterfaces G //Generateallmicro-tetrahedraelements 7: P:elements Internal MicroTets E , G 8: P:elements Surface MicroTets E , G 9: P:elements Corner MicroTets E , G 10: return P Thisalgorithminherentlycoversbothhomogeneousandheterogeneouselementperforationsdependingontheinsetvalueswithin I .Ifthisinsetarraycontainsuniformvalues,thentheoutputwillbe ahomogeneousperforation.Ifeveryvaluewithinthisarrayisdifferent,thentheoutputwillbeanon88

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Figure4.3.6:UniforminsetoftheStanfordbunnymesh.Theinsetvaluesprogressfrom 25% leftto 50% centerto 75% righteachspeciedbyonevalue 0 : 25 , 0 : 5 , 0 : 75 .At 100% elementsbecomesolid. uniformheterogeneousstructure.ThefullversionoftheperforationalgorithmsareimplementedinC++ andavailableatourlabwebsite:http://graphics.ucdenver.edu/generativedeformation.html 4.3.3HomogeneousPerforation Auniformorhomogeneousvolumetricperforationcanbeobtainedbyprovidinga constant insetvalue. Thisspeciesthatforallelementswithinthetetrahedralmesh,eachisassignedthesameinsetpercentageorvalue,resultinginauniformstructurethroughoutthemesh.TheperforationoftheStanfordbunny modelisshowninFigure4.3.6withuniforminsetsbasedonvariousinsetpercentages. Therearetwoprimaryformsofspecifyingthisuniforminset:apercentage-basedinsetvalue where 5% indicatesaverysmallinset,leadingtoverythinedgeelementsand 95% indicatesanelement thatisalmostsolidandanabsoluteinsetvalue.Fortheabsoluteinsetvalue,alledgeswillbedened asthesamethicknessregardlessoftheelementssize.Incontrasttotheabsoluteedgethickness,thepercentageinsetmethodwillhavelargerinsetedgesthansmallerelementswhenthesamepercentagevalueis provided.Thisisbecausethegeometryoftheelementandtheinsetoperationwillbelargerasexpressed asapercentage. 4.3.4HeterogeneousPerforation Toprovidemeaningfulcontrolwithinthedeformationofanobjectprintedwithonematerial,themassdistribution,internalstructure,andmemberthicknessmustbevariedtomodifytheelasticbehaviorwithin 89

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theprint.Towardsthisobjective,weintroducetheabilitytospecify per-element insetvaluesasthebasis forcreatingaheterogeneousmeshstructure.Thisprovidestheabilitytospecifyperlocalregion,howthe materialshouldbehave.Sincethisisdoneattheelementlevel,theresolutionofthiscontrolonlydepends onthenumberofelementsgeneratedwithinthevolumetricmesh.Sincemostmeshingalgorithmsprovide directcontroloverthisparameter,itiseasytovary.However,astheresolutionisincreased,theinsetarraywillalsobecomemuchlarger.Duetothenumerouspotentialmethodsfordeningthearrayofinset values,andtheresolutionoftheprovidedmesh,thecompositionoftheobjectmayvarydrastically.Toillustratethepotentialdifferencesinmeshgenerations,Figure4.3.7showshowdifferentgradientswithin themodelresultindifferentgeometriccompositionswhichwillresultindifferentlocalizeddeformation behaviors. Figure4.3.7:Heterogeneousgeometricstructureduetoper-elementspeciedinsetvalues.Thismethod allowsforsmoothgradientsbetweenelementsandallowsspecicregionsofthemeshtobecontrolled throughanFEAsimulationresultoruserdesign. Providinganautomatedmethodforgeneratingheterogeneousinternalstructuresisonlyaspowerful asthemethodsbywhichtheinsetvaluescanbedened.Therefore,ourgenerativedeformationrelieson theperforationalgorithmasabasisforimplementingthedynamicbehaviorsobservedthroughthesimulationoftheoriginaltetrahedralmodel.Inadditiontothesimulatedmodel,theremustalsobeamethod throughwhichconstraintscanbeimposedonthesimulation.Toaddresstheseweintroduceastagewithin theautomatedpipelinefor elementconstraintdesign toincorporatehowbehaviorsareoptimizedwithina simulation. 90

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4.3.5ElementConstraintsandDesign Simulatedprototypingandoptimizationprovideanidealbasisforreningandadjustingcontroloverdeformableobjects;however,fromadesignperspective,objectivesaretypicallyachievedthroughspecifying boundaryconditionsandsparsesetofconstraints.Anyalgorithmicschemethatcanautomatetheprocess ofcontrollingdeformationsshouldalsoallowforthedenitionofboundaryandbehavioralcharacteristics criticaltothefunctionaldesignoftheprint.Toachievethiswithintheperforationpipeline,weintroduce anintermediatedesignstagewhereexplicitconstraintscanbedenedas design-specic insetvalues.Utilizingselectionorpainting-basedtechniques,theinsetvaluesofelementscanbedirectlycontrolled.For higher-levelmoreindirectcontroloftheseconstraintsweintroducea gradient-falloff algorithmforeditinginsetswithingeneralregions.Thisalgorithmusesinternalfaceadjacenciestoperformabreadth-rst searchthroughwhichthemathematicallydenedfallofffunctionwillapplysmoothinsettransitionsbased onadjacency.Thisprovidesamethodforintroducingcontrolledsmoothgradientswithintheperforation thatdonotdependontheFEAsimulationandallowforexplicitcontroloftheperforation.Toallowfora mixturebetweensetsofuserdenedinsetconstraintsorboundaryconditionsandsimulation-driveninset values,individualelementscanbeassignedtohavefreesimulationcontrolledorconstantvalueuserdenedinsetsasshowninFigure4.3.8.Toillustratethisconcept,weprovidethree bar meshes,eachwith theirowndesignconstraintA,B,C.Eachdesignisthensubjecttothedeformationillustratedontheleft bendandstretch. ThegenerativedesigncreatedinFigure4.3.8illustratesthesubtlechangestoelementgeometrythat canimprovethedeformationcharacteristicsofprintedmodels.Foreachdesign,thegenerativedesignhas inuencedthenon-uniforminsetvaluesofthebargeometrysubjecttothedesignconstraints.Thiscan isobservedbyeachdesignmaintainingitsconstraintswhileallowthefreeelementstobechangedbased onthetypeoftheapplieddeformation.ThelastrowinFigure4.3.8illustratesthedifferencesmadeto thestructuralgeometryofthemeshbasedonthedeformationandresultingchangesmadethroughthe generativeprocess. 91

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Figure4.3.8:Generativedeformation:integrationofdesignconstraints,deformationbehavior,andstress analysistogenerateheterogeneousgeometricstructuresconsistingofasingleelasticmaterialmodel.The designshapetoprowhasspecicdesignconstraintsthatareoptimizedtogeneratedifferentdeformation andstresspatternsbendandstretchthatalterinsetvalues. 4.4GenerativeDeformation Theprocessof generativedeformation incorporatestheproceduralgenerationofgeometricstructuresthat promotedesign-specicdeformationbehaviors.Throughthesimpleobservationthatexistingmethods explore[38],varyingthethicknessordesignofinternalmicro-structuresvariestheelasticcharacteristics oftheobjectanddoingthistocarefullyselectedregionscansignicantlychangedeformationbehaviors. Tobuildonthisobservationandprovideanautomatedpipelineforgeneratingperforatedstructures,we introduceamethodthatcombinesdesign,elementdeformationbehavior,andstressanalysisintoasingle consolidateddeformationmodel.Theobjectiveofthisdeformationandstressanalysisistoevaluatehow thegeometriccompositionofthemodelshouldbevariedtopromotedeformationorreduceelementstress. Thismodelproducedthroughthisprocessisthenconstructedusingvolumetricperforation,resultingin a3Dprintablesurfacemeshthatdenesthegeneratedstructuralelasticity.Wethenprintseveralmodels basedonthisprocessandevaluateprintanddeformationbehaviorperformance. Inthegenerativeprocess,weformaniterativedesigncyclethatincorporatesthedesign,renement, andrecordingofsimulationbehaviorstoprovidecontroloverhowmeshstructuresaregeneratedusingthe 92

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perforationoperation.Toprovidethiscontrol,wepresentaweightedthreecomponentformalizationof ourgenerativealgorithmbasedon:userdesignconstraints,elasticdeformationobjectives,and stress-basedreinforcement.Thesecontrolmechanismsallowforapplicationspecicconstraints,deformationobjectivesoftheprint,andstrengthoptimizationthroughsimulation-drivenstressanalysis.Eachof thesecomponentsareintegratedintothisweightedmodelbasedonarecordingofthemesh M underan appliedloadinthesimulation: R M = c e i + d e j + s e j ; 8 e 2 M .4.1 where R isarecordedFEAsimulationofthemeshincludingthesetofuserdenedinset-constrainedelements c e i andnormalizeddeformationandstressvaluesoftheelementsets d e j , s e j representthe deformationandstressofallfreeelements e j inmesh M .Thecoefcients , 2 [0 ; 1] representabias towardspromotinglargerdeformations orreducingstress =1 )]TJ/F21 10.9091 Tf 11.824 0 Td [( .Forthedeformationfunction d e ,weemployasimplemetricbasedonthedihedralangles[46]ofthetetrahedralelement e .This providesarotationinvariantmeasureofdeformationwithinanelementdenedasthemagnitudeofsix independentangles ij 2 [0 ; ] betweenadjacentfacesineachelement.Forthestressstates,westore theCauchystresstensorevaluatedpostpolardecompositionthatcanthenbeusedtocomputethe von Mises scalarrepresentationofthestressineachelement.Bothofthesevaluesareevaluatedwithrespectto Equation4.4.1togeneratethenalinsetvalueswithinthearray I thatdenesalloftheper-elementinsets forthemeshbasedonrecording R .ThisprocessisoutlinedinAlgorithm2. Algorithm2 G EN D EFORMATION M , R M , , Input: TetrahedralMesh: M FEARecording R of M with n elementstates Deformationandstresscoefcients ; Output: Real-valuedInsetArray: I scalarinsets 2 [0 ; 1] 1: for eachconstrainedelement e i 2 M do 2: I [ i ] = c e i 3: endfor 4: for eachfreeelement e j 2 M do 5: for eachframe f in R do 6: d [ j ] R:dihedralAngleMag e j 7: s [ j ] R:vonMisesStress e j 8: endfor 9: I [ j ]= mean d + mean s 10: endfor 11: return I 93

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Figure4.4.9:Perforationdesignstudio.Thisinteractivedesignstudioincludestheabilitytoimport,perforate,simulate,andexport3Dprintablesurfacemeshes.Thescreenshotsshowtwodifferentmesheswithin thefour-viewportdesignview. Tofacilitatethegenerativedeformationpipelinewithinamanageabledesignenvironment,weprovideadesignapplicationthatprovides:importingsurfacemeshes,volumetricmeshing,FEAsimulation, perforationtools,andtheabilitytoexportperforatedmeshesforprinting.Screenshotsofourperforation designstudioisshowninFigure4.4.9.Theimplementationcurrentlyimplementsrudimentaryformsof theentiregenerativedeformationprocess. 4.53DPrintingProcessandTechnologies Stemmingfromtheintroductionofthegenerativedeformationpipelineandtheperforationalgorithm,we experimentallyverifyvariousdeformationbehaviorsonvarious3Dprintedperforatedmeshes.Theexperimentalprintswereobtainedthroughthepresentedperforationalgorithmandtheuseoftwoconsumerlevel3Dprinters.ForourprintsweusedaFormlabsForm1+stereolithographySLA3Dprinter[47] withitssingle exible resinandaMonopriceVoxelFusedDepositionModelingFDMprinterusing aThermoplasticpolyurethaneTPUlament.Thesetwoprintersrepresentsomeofthemostcommon typesof3Dprintertechnologythathavebeenwidelyavailable.Theoutcomeoftheperforationprocessis dependentupontheabilitytocreatequalityprintsbasedonthegeometryproducedthroughthisapproach. Whiletheprintqualityprovidedbyeachofthesedifferenttypesofmachinescanbesignicant,ourprimaryobjectiveistosuccessfullyprintperforatedgeometryonbothtypesof3Dprinters. Alargeportionofconsumer-level3Dprintersrequiretheadditionof supportstructures toprintover94

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hangswithincomplexmodels.Supportstructuresaredenedasanyadditionalgeometryaddedtoa3D printedmeshthatallowstheprintprocesstocompletesuccessfully.Typically,supportsareaddedbyslicer applicationsthatcomputeoverhangsandautomaticallyaddressthembyaddingadditionalsupportsbelow theoverhangs.ThisisanimportantconsiderationforbothSLAandFDMprintingtechniques.Inourperforationapproach,theideawastoprovideareasonableenoughresolutionoftheelementstructuretoavoid thegenerationofsupportstructures within theactualperforatedvoids.ForbothSLAandFDMprinting techniquesbasedontheelasticprintmaterialsweused,wewereabletosuccessfullyprintanumberof perforatedstructureswithouthavingtoexplicitlyaddsupporttotheoverhangregionsinherentlyintroducedwithinperforatedstructures. 4.5.1Stereo-lithographySLA TheprintprocessusingtheFormlabsForm1+SLAprinterisbasedonlaser-basedhardeningofliquid resinmaterials.Sinceourobjectiveistoprintelasticmaterials,weusedtheelasticmaterialresinprovided byFormLabs.Thisresinrepresentsaproprietarymixturethatallowsprintedobjectstobeexiblepostcure.ToprintperforatedmodelsusingtheForm1+,weusethe Preform slicingapplicationprovidedby themanufacturer.Thisallowsustoposition,orient,andgeneratesupportstructuresfortheprintedmodel. Theperforatedmodelwasloadedandpre-processedfortheprintbasedonthisprocess.Withintheperforatedstructure,weremovedallsupportsthatweregeneratedwithintheinternalstructureofthemesh.The setupandprintsequenceoftheStanforddragonmodelusingtheForm1+isshowninFigure4.5.10. Theresin-basedprintingmethodresultsinasignicantresidualresidueoftheelasticresin.Since theperforatedstructurecontainsalargenumberofvoidswheretheresincanbecometrapped,wemust ensurethatthetypicalpost-processingoftheprintaccountsforthisresidualmaterial.Intheinstanceof ourprints,wewereabletosuccessfullyremovetheresidualresinfromwithintheperforatedstructure.To post-processandcuretheSLAprints,wefollowthestandardprocessrecommendedforresin-basedprintersasshowninFigure4.5.11.Wethendesignedandbuiltacustom 405[ nm ] UVcuringboxtoensurethat therewasadequatereectionsofthehardeningwavelengthtopenetratetheinternalstructuresoftheperforatedmeshatleasttosomedegree. 95

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Figure4.5.10:3DprintsetupandresultfortheStanforddragonmodelprintedontheFormLabsForm 1+usinganelasticresin.Theslicermodelleftdenedexternalsupportstructuresthatallowedfora successfulprintcenter,right. Qualitatively,theperforationalgorithmprovidespromisingresultsforprintingheterogeneousinternalstructures,evenwhensupportstructuresarerequired.Inourexperimentation,wewereabletoobtain successfulprintswithoutinsertinginternalsupportstructuresintheperforatedvoids.However,animplicationofthetetrahedralinsetsisthatstressregionsmaybeinducedatthecornersofinsetswhichare pronetotearing,butthisismaterialdependent.Our3Dprintedbunnyanddragonmodelsareshownin Figure4.5.12. 4.5.2FusedDepositionModelingFDM Representingthemostaccessibleformof3Dprinting,fuseddepositionmodelingisaprocessthatuses heatedextrudernozzlestomeltcontinuousplasticlamentsintothelayersthatcomposea3Dprinted Figure4.5.11:VolumetricperforationprintingprocessSLAillustratingtheresinprintleft,alcohol washcenter,andourcustombuiltacrylic 405[ nm ] -UVcuringboxright. 96

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Figure4.5.12:FinalresultoftheperforatedStanfordBunnyandDragonmodelsprintedusinganSLAbasedprinterwithasingleexibleresin.Eachregionhasvariabilityintheelasticbehaviorhard,soft duetothegradient-basedtransitionsoftheperforatedstructure.Supportmaterialshavebeencarefully removed. model.Thisformofconsumer-levelprinterhasintroducedavarietyofslightdifferencesinhowmodels canbe3Dprinted.Mostimportantly,thereisawiderselectionofalternativematerialsthatcanbeused foravarietyofinterestingapplications.Specically,thereareanincreasingnumberofelasticorexible materialsthatcannowbeusedwithmostFDMmachineswithsomeadditionalconsiderationswithinthe printprocess.ToexperimentwithprintingperforatedstructuresusinganFDMmachine,weemploythe MonopriceVoxel,usingathermoplasticpolyurethanelament.Toaccommodateforthechangetothis elasticmaterial,wearerequiredtoreducetheprintspeedofthisprinterto 10[ mm=s ] . 4.6ExperimentalDeformations Forourquantitativeexperimentation,veperforationsweregeneratedandprintedusinga 8 x 4 x 4 tetrahedralbarastheinitialinputmesh.ThesedesignswerecreatedthroughtheprocessillustratedinFigure 4.3.8toprovideuser-denedconstraintstheelementsoftheprint.Usingthisprocess, 5 uniquedesigns wereintroducedandthenmodiedusingthegenerativedeformationalgorithm.Thepremiseofthisexperimentistogenerateperforatedmeshesthatalterdeectionandrotationalexibilityofeachprintsubjectto designconstraintsandmeasuretheresultingdeformationbehaviorsunderincrementalloads.Thenalized 3DprintsofthesemodelsareshowninFigure4.6.13asDesign[0]-[4]. 97

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Figure4.6.13:3Dprintedperforationmeshesfortestingthegenerativedeformationpipeline.Eachisa 2 : 0 x 2 : 0 x 4 : 5 [cm]singlematerialperforatedprint. Forbothexperiments,thedeectiondisplacementsandrotationanglesfromthereferencecongurationhavebeenmeasured[48]toevaluatethedeformationofeachmodel.Theseloadswereappliedusing incrementalweightsat 100 gintervalsundergravitytoobtaintheforcesandtorquesrequiredtoevaluate thedifferencesinthedeformationbehaviorsofeachdesign.Thesemeasurementshavebeenrecordedto identifytheimpactofthegeometricvarianceintroducedbyeachgenerateddesign.Inbothexperiments, theheterogeneousdesignofeachmodelintroduceschangesinbothbehavioraldeformationsanddeectionsatequilibriumthatcanbeadjustedtoachievespecicdesign-orientedobjectivesusingasinglematerial.TheresultsinFigure4.6.14illustratetheresultingdeectiontoprowandtwistdeformationsbottomrowastheresultoftheappliedload.Theillustrateddeformationsaretheresultofapplyingthesame loadforeachdesigntodeterminethedeformationbehaviorandmaximumdeection.Throughtheseexperiments,itisclearthatgeometriccompositionofthematerialmodiesdeformationbehaviorandthe maximumdeectionofthesinglematerialprint. ThedeformationsoftheperforatedprintsshowninFigure4.6.14indicatethattheelasticbehavioroftheuniformmaterialcanbecontrolledbymodifyinginternalgeometricinsetsandthrough simulation-driveninsetoptimizationwecangenerateheterogeneousstructuresthatpromotespeciclocalizeddeformationcharacteristics.Thisindicatesthatfromdifferentgeometricstructureswecanenable commonobjectivessuchasmaterialvolumeminimization,stressreduction,anduser-denedconstraints whilemaintainingprescribeddeformationbehaviors.Thisprovidesanimportanttoolforrapidprototypingthroughmodelsimulationfordeformableprintswhenalimitednumberofelasticmaterialssuchas curableresinsandlamentsareavailable.Additionally,theresultofthegenerativedeformationillustratestheafnityofeachshapetothedesigneddeformation,providinganeffectivebehavioralcontrol 98

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Figure4.6.14:Resultofdeformingvebeamsgeneratedusingthegenerativedeformationalgorithm.The toprowillustrateseachbeamsubjecttoadeformationimposedbyahorizontalload,inducingvarious deectionbehaviors.Thetwistdeformationbottomrowisimposedbyanappliedtorquetodemonstrate nonlinearrotationaldeformations.Eachexhibitsuniquedeectionsandlocalizeddeformationsduetothe changesintheheterogeneousperforations. schemeforelastic3Dprints. 4.7Discussion Throughthisworkwehaveintroducedanautomatedmethodforgeneratingheterogeneousmaterialstructuresthroughtheuseofvolumetricperforation.Thepresentedmethodprovidesagenerativedeformation pipelinethatenablesuserconstraintsandstressanalysisusingrecordedFEAsimulationstocontrolthe elasticbehaviorsofelastic3Dprintedobjects.Thecontributionsofthisworkareconsolidatedwithinan open-sourcetoolforusewithvarioustypesofconsumer-level3Dprinterswithlimitedselectionsofexiblematerials. WehavealsodemonstratedthatthismethodiscompatiblewithbothSLAandFDM-based3Dprintingtechnologies.Thisillustratesthattheproposedmethodcanbeusedwithconsumer-level3Dprinters thatsufferfromlowerprintqualityduetotheprintmethod.WhileFDMprinterscanstrugglewithbridge structuresandstringingartifacts,theproposedmethodcanstillbeusedtoproduceelasticstructuresthat canbeoptimizedtowardsspecicdeformationbehaviors.TheimagesinFigure4.7.15illustratetheresult ofprintingaperforatedformoftheStanfordBunnymodelusinganinexpensiveTPUlament.Aswith theSLA-basedprinterresult,wewereabletoobtainprintsthatdidnotrequireinternalsupportstructures withintheperforation.Thisisimportantduetothepotentialinuenceoftheinternalsupportstructures 99

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thatmayalterbehaviorduetotheinabilityordifcultyofremovingthesupportswithintheperforatedelements.Basedonourexperimentation,layeradhesionproblemsaremorefrequentwiththeFDM-based printswhencomparedtotheSLA-basedprints. Figure4.7.15:DemonstrationofthesuccessfulprintofacomplexperforatedstructureusinganFDM3D printer.TheTPUprovidesarelativelyrigidresultwhenprintedintosolidobjects,howeverthroughthe perforatedgeometry,theelasticmaterialpropertiesoftheobjectcanbedrasticallychanged. 100

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CHAPTERV TURBULENTEXHALEFLOW:RESPIRATORYANALYSIS Breathinganalysisrepresentsoneofthemostprominentdiagnosistechniquesuniversallyusedbypulmonologistsinbothclinicalandsleepstudiestoidentifycharacteristicsofnumerousconditionsthatcanbe identiedbyirregularbreathingpatterns,inconsistenciesinairow,orabnormaltidalvolumemeasurements.Theresultsofthisanalysishavebroadimplicationsduetocommondiagnosisofpulmonaryand developmentrelatedconditionsinducedbyabnormalbreathingbehaviorswhichincludeChronicObstructivePulmonaryDiseaseCOPD,lungfunctionality,cognitivedevelopment,impactsinthefacialdevelopmentofchildren,SleepDisorderedBreathingSDB,andevenpotentialcorrelationswithSuddenInfant DeathSyndromeSIDS.Breathinganalysistoolsanddevicesprovidenumerousmeasurementsthatcontributeimportantunderlyingindicatorsofvariouspulmonaryconditionsthatcanimprovediagnosisresults withoutextensiveorinvasivetesting.Usingdenseexhaleowanalysisthrough CO 2 spectralimagingintroducesapivotaltrajectorywithinnon-contactrespiratoryanalysisthatconsolidatesseveralpulmonary evaluationsintoasinglecoherentmonitoringtechnique.Duetotechnicallimitationsandthelimitedexplorationofrespiratoryanalysisthroughthis non-contact technique,thismethodhasnotbeenfullyutilizedtoextracthigh-levelrespiratorybehaviorsthroughturbulentexhaleanalysis.Inthiswork,wepresent astructuralfoundationforrespiratoryanalysisofturbulentexhaleowsthroughthevisualizationofdense CO 2 densitydistributionsusingpreciselyrenedthermalimagingdevicetotargethigh-resolutionrespiratorymodeling.Weachievespatialandtemporalhigh-resolutionowreconstructionsthroughthecooperativedevelopmentofathermalcameradedicatedtorespiratoryanalysistodrasticallyimprovetheprecision ofcurrentexhaleimagingmethods.Wethenmodelturbulentexhalebehaviorsusingaheuristicvolumetric owreconstructionprocesstogeneratesparseowexhalemodels.Togetherthesecontributionsallowus totargettheacquisitionofnumerousrespiratorybehaviorsincluding,breathingrate,exhalestrengthand capacity,towardsinsightsintolungfunctionalityandtidalvolumeestimation.Basedonthis,theprimary objectiveofthisresearchistointroduceanewpivotaldirectioninvision-basedrespiratoryanalysisfor clinicaldeploymentstorealizethebroaderimpactsofthismethod. 101

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Figure5.0.1:Resulting CO 2 densitydistributionimagesillustratinguniquerespiratorypatternsbetween individualstopvsbottomrows.Foreachimagesequence,oneexhaleperiodhasbeenrecordedandvisualized,showingtheclearseparationbetweenthenose-mouthdistributionanddensityowbehaviors. Theseowbehaviorsuniquetoeachindividualarebasedontheirownphysiologicaltraits. 5.1RelatedWork Accuratenon-contactrespiratoryanalysishasrecentlygainedpopularitywithinthedomainsofwireless signalprocessing[49]andcomputervision[50]toautomateandsignicantlybroadentheclassofquantitativerespiratorymetricsthatnon-contactmethodscanreliablyaddress.Numeroustechniquesexist forbothcontactandnon-contactrespiratoryanalysis[51],howeverallofthesemethods indirectly infer breathingbehaviorsorutilizecorrelationfunctionsforrespiratoryanalysis.Techniqueswithincomputer visionhaveintroducedthermalinfraredcameraswithspectralltersfor CO 2 imagingforrespiratoryanalysis[52],howevertheapplicabilityofthesetechniquestocomprehensiverespiratoryanalysisisseverely underdevelopedandtheadoptionofthesemethodshasbeenverylimited.Thisisduetothreeprimaryfactorssharedbetweenmostpriorvision-basedtechniques:priorobjectivesonlyemphasizesimplequantitativemeasuressuchasrespiratoryrate[53]withinlimitedRegionsofInterestRoIandstrength[54], limitingpotentialhigh-levelbehavioralanalysis,priordeviceslackthesensitivityrequiredtomonitorsubtledensityvariancesandcomplexowsbehaviorsforidentifyingrespiratoryconditions,and frame-ratelimitationsinhibittheabilitytoaccuratelycapturerapidandturbulentrespiratorybehaviors. Whiletherearerelatedapproachesthatalsouseinfraredimaging[55,56],theclinicalimpactoftheprior researchdirectionshasbeenextremelylimited.Thisisdueinparttothechallengesassociatedwithextractingmeaningfulandquantitativemetricsfromturbulentexhaleowstoimprovemonitoringmethods 102

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andincreasediagnosisaccuracies.Duetothesubstantialopendirectionsofthisformofanalysis,thereisa widevarietyofpotentialapplicationsthatcandirectlybenetfromsolutionstothesechallenges. 5.1.1ChallengesinRespiratoryAnalysis Theprimarylimitationsofexistingmethodologies[51]anddevicesarederivedfromtwoprimaryconstraints:mostgold-standarddevicesarecontact-basedandcaninherentlyalterapatientsnaturalbreathingbehaviorbecausetheyaretypicallyuncomfortableorunsuitableforlong-termstudies.Thisisdueto therequireduseoftubesthatlimitairoworwearablesolutionsthatconstrictnaturalchestmovements andmeasurementsareobtainedthroughindirectmethods,thatis,mosttechniquesinferrespiratory characteristicsthroughrelatedobservationssuchaschestmovementsinsteadofdirectmeasurements. Currentstate-of-the-artmethodsthatinferbreathingbehaviorsusewearabledevicesthatlimitnatural movementandmethodsthatapproximatedirectmeasurementstypicallyrelyonairowrestrictingtubes ordevices.Thisincludesspirometers[57],wearableaccelerometers,andimpedance-basedchestbelts, anddepthimagingdevicesthateithermonitorrestrictedairoworinferrespiratorybehaviorsindirectly throughchestmovements.Theseconstraintsresultinseveralcomplicationsthatlimittheabilitytoimprovethequalityandtypeofinformationaboutrespiratorybehaviorsobtainedusingcontact-basedorindirectmethods.Furthermore,currentsolutionsarenotonlylimitedinaccuracybuttheyfundamentally awedinhowtheyrepresentbreathingbehaviorsduetothreeprimaryreasons:directcontactmethods restrictairow,makingnormalbreathingstrenuous,indirectmethodsinferbreathingbehaviorfrom chestmovementwhichisdifferentbetweeneverypatientandonlypartiallyrepresentsactualairow,and allofthesemethodseitherrequirecontinuousattentionfromthepatienttousethedevicecorrectlyor constantlyinterrupttheirattentionmakingthemconsciousabouttheircurrentbreathingbehaviorswhich candrasticallyskewanalysisresults.Thelastchallengeassociatedwiththesetechniquesisthattheyare limitedtorepresentingcomplexrespiratorybehaviorsthroughsimplequantitativevaluessuchasrespiratoryrate,continuouspressure,ortidalvolume.Whilethesemeasuresareimportant,theylacktheability toaddressprominentrespiratoryconditionssuchaschronicobstructedbreathing,severeobstructivesleep apnea,andqualitativeanalysisofrespiratorypatterns. 103

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Demonstratingtheimpactoftheconstraintsimposedbyexistingsolutions,evaluationofasimplied studycanbeusedtoexemplifyhowacaseofobstructedbreathingcouldbeevaluated.Ifapatientexerts excessiveefforttobreathe,asindicatedbychestmovementsmeasuredbyexistingsolutions,thereisno suitableexistingsolutionthatcananalyzethemovementofairwithoutfundamentallyalteringthepatients naturalbreathing,aswouldberequiredtoidentifyobstructedbreathing.Intheinstanceoftube-basedsolutions,thiscausedbyrestrictedairow,unnaturalposture,rebreathingexhaledairwithinthetube,maintainingproperconnectiontothetube,anddegradationofdataqualityduetothestrenuouseffortrequired tousethedeviceforanextendedperiodoftime.Forsolutionsthatmonitorchestmovements,theexertion willbeidentied,butcannotdirectlyrelatedtoobstructedbreathingwithoutacoordinatedairowanalysis,whichwouldrequireusingbothdevicesatthesametime.Toaddressthesechallenges,newsolutions thatcomplementexistingdevicesshouldprovidealternativeformsofdirectmeasurementsthatarenot strenuoustouseanddonotalterthenaturalbreathingofthepatient. 5.1.2SurfaceDeformationReconstruction Initialnon-contactmethods[58,59,60,61]usedforrespiratoryanalysishaverevolvedaroundtwoprimarymethodsofobtainingmetricssuchasbreathingrate[62]andtidalvolume[63].Botharebasedon inferring breathingcharacteristicsthroughchestmovementsbutdifferconsiderablybasedonthecore technologiesusedtoperformreal-timemeasurements.Therstmethodisbasedontheuseofwireless signalsthatcanbeusedtomeasurechestmovementsthroughphaseshiftsinasignalemittedbyatransmitterTxandreceivedbyareceiverRx.Thismethodhasthebenetofworkingthroughclothingand cancloselyinferchestmovementsovertimetoestimaterateandtidalvolume.Thesecondtypeofmethod relatestovision-basedtechniques,depthimaginghasbeenusedinnumerousstudiestodirectlymeasure chestmovementsasoscillationswithindepthimages.Thebenetofthissolutionisthatitprovidesa higherresolutionimageofthechestanditsbehavior,providingcloseranalysisofrespiratorypatternsthan wirelessmethodsthatcanonlymeasureonesmallregion 10[ cm ] .Whilethisprovidesadditionalmovementinformation,allvision-basedtechniquesaresubjecttobothclothingandbodymovementswhichrequiresadditionaltrackingalgorithms.Tosolvethisproblemweintroduceanewmethodforchestsurface 104

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reconstructionwithautomatedmovementtracking[64].Thesetupofour training 1 and real-timemonitoring congurationsareshownbelowinFigure5.1.2. Figure5.1.2:Illustrationsofthevariousmodelsgeneratedbydepthimagingmethodsforrespiratoryanalysis.Depthimagingleft,simpleextrudeddepthcenter,andregionbasedmethodsright,donotprovide themodeldelityandaccuracythattheiso-surfacedeformationmodelcanprovide.Thismodelprovidesa methodsformeasuringbothbreathingrate,tidalvolumewith > 90% accuracy. Tosubstantiallyimprovethequalityandmodelingtechniquesthatusedepth-basedimaging,our methodreconstructstheentirechestsurfacethroughtheuseofasingledepthimagingdevicewithskeletal tracking.Byencapsulatingthechestregionusingthedepthimageandreconstructingtheback,top,and bottomregions,wecandeneafullyenclosedvolume.Fromthedivergencetheorem,wecanextractan iso-surface ,formeasuringbreathingrateandtidalvolumeasrespiratorycharacteristicsbasedonthedeformationsofthechestduringthemonitoringperiod.Additionally,duetothedelityofouriso-surface modelcomparedtopriorresultsshowninFigure5.1.3,wecancloselymonitorchestdeformationsasa newformofbreathingpatternanalysistoidentifyabnormalitiesintherespiratorycycle[11].Thequalityanddynamicnatureofthismethodallowsustodetectsubtlechangesinthedeformationofthechest duringtherespiratorycycle.Whiletheiso-surfacereconstructioncanbevisualized,theprimarymethod withinthisapproachisbasedonmonitoringthesmalluctuationswithinthebehavioritself,reducingthe privacyexposurerisk.Thisworkalsocontributedtotheextendedstudyofthisapproachwithmultiple sensors[65]. 1 Thismethodrequiresaninitialtrainingphaseduetoindividualcharacteristicsinbodycompositionanddeformations.Machinelearning,implementedthroughanArticialNeuralNetworkANNisusedtocorrelatedeformationtotidalvolume. 105

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Figure5.1.3:Illustrationsofthevariousmodelsgeneratedbydepthimagingmethodsforrespiratoryanalysis.Depthimagingleft,simpleextrudeddepthcenter,andregionbasedmethodsright,donotprovide themodeldelityandaccuracythattheiso-surfacedeformationmodelcanprovide.Thismodelprovidesa methodsformeasuringbothbreathingrateandtidalvolume. 5.2VisualizingExhaleFlow Stemmingfromtheinitialintroductionofthermalcameraswithinindustrialapplicationsforidentifying thespectralsignatureofspecicgases,carbondioxide CO 2 imaginghasbeenprimarilyappliedtoexhalevisualizationandhasonlyrecentlybeenadaptedforrespiratoryanalysis[53,52].Still,whileincrementalprogresshasbeenmadetowardsgeneratingbehavioralmetricsforclinicalrespiratoryanalysis, theprimaryinformationobtainedusingexistingmethodsrevolvesaroundobservationssuchasbreathing rateand CO 2 concentrationlevels.However,forclinicalrespiratoryanalysisthesemetricsfailtoprovide meaningfulcontributionstodiagnosingconditionsandcanbeobtainedusinglesscomplexandcheaperalternativedevices.Duetothis,thermalimagingfordirect CO 2 analysishashadlimitedpenetrationwithin themedicalcommunity. Methodsforimaging CO 2 exhalebehaviorshavebeenintroducedasearlyas2005[53]inwhichthe rateandconcentrationofthe CO 2 distributioncanbemonitoreddirectlyusingasinglethermalcamera asshowninFigure1.Whilethisvisualizationmethodhasbeenwellexplored,ithasnotyetsignicantly penetratedthemedicalcommunityasafeasiblesolutionforrespiratoryanalysis.Oneoftheprimaryreasonsthatthisformofimaginghasnotbeenadoptedisthelackofaccuratequantitativemetricsbeyond breathingratethatcanbeusedtomonitoranddiagnosecommonpulmonaryconditions.Additionally,the constraintsimposedbythecomplexityinreliablyprocessingthegeneratedimagesdonotjustifytheinitial costoftheimagingdevice.Forthisclassof CO 2 imagingmethodstoprovideaviablerespiratoryanalysis 106

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solution,substantialprogressmustbemadeintermsofthequantitativeoutcomesthatcanbeusedforcontinuousmonitoringanddiagnosisobjectives.Throughthiscontribution,directly CO 2 imagingcanprovide thefoundationforprovidinganon-contactmethodforexhaleanalysisthatpromotesnaturalbreathingand canbeusedforlong-term,overnight,andpediatricpulmonarystudies. Oneoftheprimarychallengesforthisadaptationisthepotentialcostincurredfromusingacomplex thermalcameramustoutweighthebenetsandmetricsobtainedfromusingothersimpleandinexpensivedevices.Therefore,theintroductionofadditional,clinicallymeaningfulmetricsandbehavioranalysismethodsmustbeaddressedthroughtheuseofnew CO 2 imagingtechniquestojustifyboththecost andsystemcomplexityrequiredtodeploythismethodasaviablealternativetothesesmaller,effective, andcheapermonitoringdevices.Toadoptthesemethodstoourrespiratoryanalysisobject,wepresenta comprehensivesystemofalgorithmsandhardwaresolutionsthatdenetwosystemdesignsforcontinuouslymonitoringofexhaleowsusingCO2imagingforpulmonaryandrespiratoryanalysis.Basedon thiscontribution,wefocusontheadditionofseveralquantitativemetricsfornaturalbreathingthatcannotbeobtainedusingexistingsolutionsincluding:naturalexhalebehaviors,tidalvolume,nose-to-mouth distribution,velocity,strength,andpulmonaryconditionevaluation. 5.2.1 CO 2 VisualizationforRespiratoryAnalysis Aspartofourprincipleresearchwork[66],wepresentanovelimagingtechniquefornon-contact[67, 68]anddirectbreathinganalysisusingthecombinationofa CO 2 lteredthermalimagingcameraand softwaresuiteforanalyzingandextractingmeaningfulrespiratorymetricsfordiagnosingcommonpulmonaryconditions.Thepremiseofthisapproachisthatthroughthedirectvisualizationoftheexhaled CO 2 breathingpatterns,wecanprovideanon-contactmethodforevaluatingbothquantitativeandqualitativerespiratorybehaviorsthatissuitableforshort-term,long-term,andchildrenstudiesbypediatricians. Byremovingtheuseoftubesorwearabledevicesthatmayalterapatientsnormalbreathingbehavior,we arecapableofcapturingandrecordingapatientsnatural,orunconsciousrespiratorycharacteristics.That is,whentheuserofacontact-basedsolutionismonitored,theymustconsciouslyfocusontheirbreathing toeitherensuretheyareusingthedevicecorrectly,orbecomeawareoftheirbreathingduetotheuseof 107

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thedevice.Thiscaninherentlyalterapatientsrespiratorypatternswhichcanaffectpulmonaryrelateddiagnosisresults.Additionally,theabilitytomonitornuancedunconsciousbreathingbehaviorscanbeused withinlong-termsleepstudiesandmayprovideinsightintohowwecanexplorenewabnormalbreathing patternsthatarenotapparentinthedatacollectedusingexistingsolutions.ThisisanimportantcontributionforidentifyingcharacteristicsofSDBsuchasapneaandotherconditionsthatmustbecontinuouslymonitoredwhilelimitinginterferencefromthedevice.Therefore,unlikeexistingsolutionsthatare usedforlong-termmonitoringsuchasimpedancebeltsaspartofapolysomnographyPSGstudy,wecan monitorthepatientremotely.Thelabqualitysetupofourmonitoringsolutionincorporatesthreeprimary components:the CO 2 camerathatprovidesareal-timestreamoftheexhaledatathatcanbevisualized andrecorded,therespiratoryanalysissoftwarethatextractsmeaningfulmeasurementsoftheobserved exhalebehavior,andacoolmatenishedbackdropthatprovidesauniformbackgroundtemperatureto maximizetheaccuracyofourdataacquisitionprocessofensureaccuratequantitativemeasurements. Oursolutionalsobuildsonexistingtechniquestointroducesynergisticdiagnosistoolsthatcanleveragemeasurementsofbotheffortandphysicalexertionofthepatientandtheactual CO 2 airowtohelp identifycasesofobstructedbreathing.Throughtheuseofboth CO 2 anddepthimaging,wecanevaluatebothchestmovements[64],patientbreathingeffort[49],andhowtheyrelatetoairowmeasurements thatarenothamperedbyexternaltubingorwearabledevices.Wehavealsointroducedseveralnewforms ofobtainingexhalemetricsbasedon CO 2 imagingthatinclude:nose-to-mouthseparation,breathing strength,andowbehaviorreconstruction[66].Theintegrationofmultipleimagingtechniquesprovides aframeworkforextractingawealthofinformationaboutthesurroundingenvironment,thepatientsposture,andimportantinformationfortrackingthepatientsmovementsovertime[69].Thesecontributions provideindividualcomponentsofadeployablesystem,howeverareliabledeploymentofthesesystems requirescontinueddevelopment. 5.3ModelingExhaleFlow Theprimaryevaluationcriteriawithinrespiratoryanalysisrevolvesaroundthecollectionofalimitedset ofquantitativemetricssuchasbreathingrate,owanalysis,andtidalvolumeestimates.Extensivere108

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searchhasculminatednumerous contact [70]and non-contact [50,49]methodsthatobtainthesemetricswithpromisinglevelsofaccuracy.However,basedontheseexistingmethods,allcurrentrespiratory evaluationisperformedusing indirect methods,thatis,theyinfermeasurementsthroughsecondarysignalssuchasvisiblechestmovements,vibration,pressure,acceleration,orsound[70].Priormethodsusing spectralanalysisfor CO 2 visualization[53]haveintroducedadirectmeansofevaluatingrespiratorybehaviorsusing direct exhalemeasurementsforbreathingrate.Similartothisformofdirectanalysis,we measureandmodeltheexhaleowconsistingofthevisualizedthermalsignatureofthe CO 2 waveform. Whilepriormethodsonlyprovideabreathingrateevaluation,thisformofvisualizationisunderutilized duetoitsabilitytogeneratenumerousadditionalmetricssuchasnose/mouthdistribution,velocity,dissipation,behavioralcharacteristicsandeveninsightintolungefciencyincontrolledenvironments.Enable thesemetrics,theresultofourworksignicantlydivergesfromexisting CO 2 -basedimagingtechniques [53,52]inboththelevelofanalysisandtheresolutionofourmodeling.Todevelopadevicefordirectly analyzingturbulent CO 2 exhaleows[53],wehavecoordinatedthedevelopmentofa hyper -sensitive FLIRthermalcamerathatcontainsanembeddedspectrallterthatdirectlytargetsthe CO 2 spectralband -5[ m ].Fromourrequirementspecication 2 ,thedeviceprovidesraw CO 2 countimages thatcontain theinfraredwavelengthactivationcountswithinthe CO 2 absorptionband[71,72].Throughthedevelopmenttheseimagingmethodsandour direct measurementsofbreathingbehavior,weintroduceanewvectorinvision-basedclinicalrespiratoryanalysis.Thisincludesdirectowandthermalanalysisforsubtle alternationsinairowrelatedtoasthma,ChronicObstructivePulmonaryDiseaseCOPD,developmentalconditionsrelatedtonoseandmouthbreathingdistributions,cognitivefunction[73],sleepapnea,and SuddenInfantDeathSyndromeSIDS. 5.3.1DenseExhaleModeling Exhaleowbehaviormodelingprovidesabasisforevaluatinghigh-levelrespiratorycharacteristicsbased onasetofobservablephenomenathatisnotfacilitatedbycurrentmonitoringtechniques.Thisincludes momentaryuctuationswithinexhalestreamsthatinherentlycontributetosecondaryowbehaviors 2 FLIRA6788scInSbCCF, 640 x 512 resolution CO 2 countimages@ 30 120 [fps]withprogrammaticcameracontrolandraw dataacquisition. 109

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Figure5.3.4:Volumetricanddensitymodelingofthe CO 2 exhaleregiondenedbytheviewfrustumof theimagingdevice.Theexhaleowregioncontainsanon-linear CO 2 concentrationdistributionfunction density overdistance x .Eachindividualpixel p i;j representsacontinuousvolumethroughwhichthe exhaleows.Fromthevalueateachelement e i ,thenalpixel p i;j istheprojectionofallelementdensities within v . associatedwithobstructedbreathing,subtlechangesbetweennose-mouthbreathingdistributions,lung functionality,andtheabilitytoidentifyabnormalexhale CO 2 signatures.Toextendrespiratoryanalysistoincludethesemetrics,weintroduceadenseowreconstructionprocessincluding:owestimationthroughdenseopticalow,heuristic-basedowsliceextrapolation,andprovideavolumetric sparsescalareldrepresentationofrecordedexhalebehaviorsforextendedmonitoringperiods. Carbondioxidedensityimagesobtainedthroughourcameraarecharacterizedbytheprojectionof volumetricdensitiesoftheobservablegasowswithgeneralinfraredradiation,lteredtothespectral wavelengthintervalrequiredfor CO 2 imaging.Tomaximizeclarityinthismeasurement,weimprove thesensitivityofourrecordingmodelbyaddingathinmattesurfaceparalleltotheimagingplanethat containsauniformheatdistribution.ThroughtheviewfrustumofthecamerashowninFigure5.3.4,we modelthecontinuousvolume v ofpixel p i;j fromthissurfacetotheimageplane I asadiscretesetof n elementswithanunknowndensitydistributionfunction density asafunctionofdistance x .Thisperelementdensityfunction V e i isprojectedtopixel p i;j resultinginanirreversiblelossofthisdistribution. Ourmodelisbasedonheuristicapproximationsoftheinverse V )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 e i ofthisvolumetricprojectionto determinetheper-framescalardensityvalue v i;j;k withinasparsevoxelgrid: v i;j;k = V )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 p i;j = V )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 h n X i =1 density e i i 8 p 2 I .3.1 Sincetheconsolidationofthisdensityvolumetoanimagerepresentationis unrecoverable duetotheprojectionoftheper-elementdensities,thereconstructionisinherentlylimitedtoanapproximationofthe originalvolumebutpreservesoverallowbehavior.Figure5.3.5illustratestheresultingintensityimages 110

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Figure5.3.5:Recordedturbulentexhaleowfromthemouthleft,nosecenter,andbothnoseand mouthsimultaneouslyright.Throughourimagingprocess,weobtainanaccurateillustrationofthe CO 2 densitydistributionandowbehaviorwithminimalbackgroundinterference. oftheexhaleobtainedthroughthismethod,demonstratingmultipledenseturbulentows.Duetothisresult,wedonotlimitouranalysistoasub-imageRoI,ratherweconsidertheentireexhaleregiontomodel bothbehavioralcharacteristicsanddiffusionpropertiesofexhalesequencestobuildaper-patientrespiratoryprole. 5.3.2DenseFlowReconstruction Denseowreconstructionfromtwo-dimensionalimagingisinherentlyambiguousandcannotbedirectly recovered.Toapproximatearesultingdistributionoftheowwithinareconstructionweemployafour stepprocessforestimatingexhaledensityowbehaviorsbasedonconsecutive CO 2 imagepairsovertime andspace,outlinedinFigure5.3.8.Inthisprocesswecollectthesetof n densityimagesovertime t , computetheapparentowthroughdenseopticalow[74],emplacetheseowframesintoavolumetricvoxelgridasaseedsliceinthemiddleofthisvolume,extrapolatesliceowestimates,andconvertthesescalareldsintosparserepresentationsforeachframe.Thesparserepresentationisduetothe denseresolutionofthevolumetricgrid,whichencodesthedensityvalue,andowvectorofeachcell.We evaluateeachframeindependentlywithinthissinglecomputevolume.Thisresultsinan n framerecording,eachcomposedofasparse3Dscalareldthatapproximatesdisjointowbehaviorsrecordedineach frame.Intheinstancewherewerequirereal-worldmetrics,weemployafusionbetweendepthandthermalimagingtoprovidereal-worlddistancesbetweenpixelsinthethermalimage.Thisprovidesavalid basisforestablishingthevolumetricunitoftheexhalesequence. 111

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Figure5.3.6:Exhaleowreconstructionprocess.Weapproximatethereconstructionoftheprojecteddensityvolumebyestimatingthefunction density x usingheuristicapproximations.Methodsderivedon thisdesigncanbeusedtocreatenumerousformsofdifferent3Dexhalemodels. Figure5.3.7:Turbulentexhaleopticalowvectors.Thegeneratedvectoreldillustratesthe apparent owcomputedthroughastandarddenseopticalowalgorithm.Thetoprowillustratestheoriginal CO 2 densityimages,andthebottomrowillustratestheresultingvectornorm-color-mappedow. 5.3.3VolumetricApproximation Directexhaleanalysisthroughtheproposedvision-basedtechniquerequiresmultipleformsofsegmentationtocaptureandmodelexhalebehaviors.Sincethevisualizationof CO 2 representsacomplexsystem ofuiddynamicsdissipatingintothebackground,thiscanbeadifculttaskdependingonthetypeof analysisormodelconstructionbeingperformed.Foridealizedmodelsgeneratedfromlab-qualitydata, operationssuchasdenseopticalowcanbeappliedtorawintensityimagestogenerateoweldsas showninFigure5.3.7.Basedontheseparationbetweentheexhaleandthebackground,wecanenable severaldifferentmethodsforvolumereconstructionincludingsparsevoxelrepresentationsandimage112

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basedmethodsderivedfromShape-from-shadingtechniques.Withinourreal-timeapplication,the3D volumeofanexhaleisshowninFigure5.3.8. Figure5.3.8:Exhaleowreconstructionprocess.Weapproximatethereconstructionoftheprojecteddensityvolumebyestimatingthefunction density x usingheuristicapproximations.Methodsbasedonthis designcanbeusedtocreatenumerousformsofdifferent3Dexhalemodels. 5.4ExperimentalSetupandApplication Theexperimentalsetupoftheexhalemonitoringprocessrequiresthe CO 2 cameraorientedsoithasa clearviewoftheexhalecross-section.Thispreventsothersourcesofheatfrominterferingwiththethermalsignatureoftheexhale.Toapproximatehowthesystemcanbesetupwithinaclinicaldeployment,we havecreatedabasicsetupforshort-termmonitoringasshowninFigure5.4.9. Duringthemonitoringprocessweobtainreal-timeestimationsofvariousmetricsincludingtherespiratoryrate,relativeow,nose-to-mouthdistribution,respiratoryrate,andtidalvolume.Theannotated screenshotofourreal-timemonitoringapplicationisshowninFigure5.4.10. 113

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Figure5.4.9:Overviewoftherespiratorymonitoringsetupthatusesa CO 2 cameratoprovideremote exhaleanalysis.Thisincludesthe CO 2 camera,software,andcross-sectionofthepatientsbreathingarea. 5.4.1ObstructedBreathing Providingtwoformsofrespiratoryanalysismethodsopensthedoortonewtechniquesthatcanbeused toanalyzepulmonaryconditions.Fromthewealthofinformationprovidedbybothchestsurfacedeformationsandtheanalysisoftheactualexhaled CO 2 ,wecancomparativelyevaluatetheprospectof obstructedbreathing .Byevaluatingtherelationshipbetweentheexhibitedeffortandtheactualexhaledtidal volume,wecanmakeestimationsaboutthisrelationshiptodetermineifthereisanobstructionpotential makingitharderforthepatienttobreathe.Thisisofcriticalimportanceforpatientsthatsufferfromsleep apnea,especiallychildren,wherethesleepqualitycanhavealargeimpactonadolescentdevelopment. Figure5.4.10:Real-timerespiratorymonitoringapplicationdesignedtoworkwithour CO 2 imagingcamera.Thisincludesbothahigh-contrastvisualizationoftheturbulentexhaleowwithanappliedheatmap andthesegmentedexhaleregionthatisusedtogeneratethewaveformofthequantitativetidalvolumeestimation.Thisapplicationprovidestheabilityforclinicianstoeffectivelycommunicateinformationabout patientbreathingbehaviorsinanintuitiveway. 114

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5.5DiscussionandFutureWork Thepremiseofvision-based CO 2 exhaleanalysisrepresentsanimportantsteptowardsbroadeningthe numberofquantitativemetricsthatcanbeusedtoidentifyanddiagnosepotentialrespiratoryconditions. However,duetothedifcultyinestablishingaccurateandreliablemeasurementsatahigherdevicecost isstillanon-goingresearchproblem.Thedeviceusedtoobtainlab-qualityexhalemodelsprovideshighly accuratemetrics,butisonlysuitableforclinicaldeploymentsandlong-termstudies.Therearehowever alternativesthataresignicantlymorecosteffective,ataslightcosttotheaccuracyofthedevicedueto theimplementationofthe CO 2 lterdesign.Yet,evenwiththisslightdegradationinimagequality,the methodsweproposecandirectlyappliedtothesecost-effectivesolutions.Thiswillprovidetherststeps towardsimplementingmodularsystemsthatcanbeusedtoeffectivelyenablevision-basedrespiratory analysisasanintegralpartofnumerousmonitoringapplicationsincludingsportsmedicine,clinicaldeploymentsforconditiondiagnosis,andlong-termsleepstudies. 5.5.1Correlative3DModeling Modelingtechniquesbasedontheuseofanindividualcameraposechallengingproblemsforaccurately estimatingvolume.Thisisduetothelackofspatialcontextfortheacquiredmeasurements.Aspartof anon-goingsegmentofthisresearch,theestimationofvisualizedturbulentowsfromasinglecamera remainsanopenproblem.Wearecurrentlylookingathowtoaddressthisproblemandhavedesigneda numberofpotentialsolutionsincluding:thedevelopmentofamulti-modalsolutionthatusesdepth andthermalimagingtoprovideaccuratedepthestimationstoestablishthemetricsforthereconstructed owvolumes,thegenerationofcorrelatedimagesthatcontaintheexhalebehaviorfrommultipleviewingangles,andtheintroductionofmultiplethermalcamerastorecordexhalebehaviorssimultaneously.Eachofthesesolutionshavetheirownpotentialtoresolvethelimitedaccuracyinthe3Dvolumetricreconstructionsofexhaleows,buteachalsohasvariedchallenges.For,Depth-ThermalDTfusioncanbenon-trivialandhasevenreceivedsomeattentionasausefulcomputervisiontechnique.To generatethecorrelatedimagesfor,complexexperimentalsetupsarerequired.For,thetechnical 115

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challengesassociatedwiththismethodarerelatedtothesynchronizationofmultipledevicesanddetermininghowtointegratethesestreamsintoasinglevolumetricmodelandthecostofthissolutionisalso generallyhigher. 5.5.2ClinicalDeployment Aspartofaclinicaldeployment,ecosystemintegrationrepresentsanimportantsteptowardsbuildinga robustsystemthatcanbeusedbyclinicalstafftoperformstudiesandrecordreliabledatasetsforanalysis. Thisincludestheabilitytoquicklysetup,calibrate,andusethedeployedsystemwithminimaloperator intervention.Torealizethis,wemustintegratethecorecomponentsfromourlabsolutionintoamobile clinicalsetup.Toaccomplishthisthereareseveralchallengesrelatedtoreleasingoursystemintoaclinicalecosystemthatinclude:theeliminationofoutsidesourcesofinterferencesuchasthermalnoisefrom thepatientbodyheat,backgroundthermalsignaturesenvironment,otherequipment,bedding,etc., camerapositioneldofview,patientmovement,andtheintegrationofthismonitoringsolutionwith relateddevices.Addressingthesechallengesrequiresthedevelopmentofrobustalgorithmsthatcanaccountforavarietyofenvironmentalandpatientfactorsthatcandiminishthequalityoftherecordeddata. Therefore,toprovideasolutiontoeachofthesechallengeswedeneasetofobjectivesthatassistinthe developmentofareliablesystemdeployment: • BackgroundandNoiseMinimization .Backgroundremovalofthermalheatsourcesincludesremovingbedding,othermedicaldevices,andenvironmentalsignaturesfromeffectingtherecorded exhalebehavior.Whilethereisamattebackgroundsurfaceusedtovisualizetheexhale,thesurroundingpassivesourcesofheatallcontributetonoisepatternswithintheanalysis.Thisobjective relatestodevelopingalgorithmsthatcaneliminateorminimizeexternalsourcesofnoisetoprovide high-qualityexhalerecordingsforanalysis. • CameraDeployment .Thecoreofourtechniqueassumesthatwecancaptureaviablecross-section ofthepatientsbreathing.Toobtainthis,wemustprovideareliablesolutionforplacementofthe cameraanditsorientationforthisrequiredperspective.Thisobjectiverequiresthedevelopmentof asimplemanagementsystemthatcanbeusedfordeployingthecamerainanadequatepositionfor 116

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recordingexhalebehaviors.Duetopatientmovementorthestudyenvironmenttheoptimalposition maynotalwaysbeobtainable,howeverwecanimprovethenumberandqualityofthedesiredmeasurementsbyprovidingacamerapositionandorientationmanagementsystem. • DataProtectionandReliability .Recordingandstoringpatientdatarequiresspecicsystemsfor ensuringnotonlyreliabilityofthedataduringclinicalstudies,butalsorequirementsforprivacyand robustdatamanagementsystems.Thisobjectiverelatestothedevelopmentofreliablestoragemechanismstoensurethatrecordeddataisnotlostduringextendedsessionsandthattherecordeddata canbesecurelyhandledafterthedatacollectionprocess.Thesecanbeimplementedthroughinformationmanagementsystemsthatcommunicatewiththedeployedsolutionusedbylabtechniciansto ensurethatdatasecurityismaintained. • EcosystemIntegration .Clinicalecosystemsarecomplexandrequirecross-validationandredundancyforensuringaccuracybetweendevicesandmeasurements.Therefore,oursolutionshouldeffectivelymeshwellwithexistingsystemsallowingforsynchronizationandmeasurementvalidation. Thisincludesmeasurementvalidationwithexistingdevicesandtime-stampsfordirecteventcomparisonsinrecordeddatastreams.Thisobjectiveisrelatedtothedevelopmentofsoftwaresolutions forrecordingmeasurementsfrommultiplesourcestoenablecross-validationandtheintegrationdesignofmultiplehardwaredevicesforaclinicalsetup. • Privacy .Usingaimagingdeviceswithinaclinicalsetuprequiresadditionalscrutinywithrespect toprivacyconcerns.Thehighresolutiondatacollectedwiththethermalcamerarequiresspecialallowancesforbothviewingandstoringtherecordeddata. Duetothenoveltyofthe CO 2 imagingsystemforbroaderrespiratorystudies,thesechallengesrequireadditionalinclinicdevelopmenttoensurethatmeaningfulmeasurementscanbeobtainedthrougha robustsystemsetup.Throughaddressingthechallengesissuedabove,theproposedsystemoffersalarge impactbyintroducingseveralnewpulmonarymetrics.Butmoreimportantly,thedeploymentofthissystemallowsustosignicantlyprogressresearchrelatedtounconsciousbreathingbehaviorsduetothenoncontactnatureofoursolution. 117

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CHAPTERVI CONCLUSION Inthisworkwehaveintroducedafundamentalconceptofinversemodelingtobridgethegapbetweenanalyticalanddata-drivenmethodstointroducehybridmodelingtechniques.Throughthedemonstrationof thisconceptinthreeuniqueproblemdomains,weaimtopromotethedevelopmentandcross-pollination ofanalyticalanddata-drivenmethodsfrommachinelearning.Derivingtheunderlyingphysicalbehaviors ofcomplexreal-worldbehaviorsisstillachallengingtask.Theintegrationofgoverningequations,optimizationmethods,andobservationdrivenconstraintsrequiresabroadperspectiveofhowtodecompose andrebuildestablishedmodelsandsolutions.Throughourthreeprimarycontributions,wehavedemonstratedthatanalyticalmethodsthatcanbedifculttointegratewithdata-drivenmethodscanstillreach somelevelofsynergy.Forreal-timedata-drivenniteelementsweintroducetheconceptofneuralelementstoreplacematerialbehaviorsandaugmentthesystemwithneuralnetworks.Extendingthistothe simulationandproceduralgenerationofmeshgeometry,weintroducedanautomatedmethodforchanging thedeformationbehaviorsof3Dprintedobjects.Broadeningtheunderlyingconceptofinversemodeling, wealsoaddressedhowtoextractmeaningfulmetricsforrespiratoryanalysisfrom CO 2 lteredthermal images.Utilizingtherelationshipbetweenanalyticalanddata-drivenmethodscanprovideprofoundimplicationsinavarietyofeldsandtheyshouldbeintegratedtoprovidethemostaccurateandcomputationallyefcientsolutionstochallengingreal-worldproblems. 118

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APPENDIX A.NeuralElementMemoryMaps TriangleElementNodeMemoryMap .Thememoryorderforthe2Dtriangularelementisdenedfor thepoints A , B ,and C withcomponents x and y .Theorderingoftheelementsthatmapsthepartialforces per-nodetothereconstitutedforceresponseisprovidedbelow: Ordering = f A;B;C g ; f B;A;C g ; f C;A;B g A.1 Theresponseforcevector f representsthereconstitutionoftheforcesgeneratedbytheper-nodepartial forcesmultipliedwiththenoderatiosstoredinarray n .Forthe2Dtriangularelementthisdenes 3 nodes times 2 dimensionsresultingin 6 totalresponseforces. f [ AX ]= n [ A ][ AX ]+ n [ B ][ BX ]+ n [ C ][ BX ] A x f [ AY ]= n [ A ][ AY ]+ n [ B ][ BY ]+ n [ C ][ BY ] A y f [ BX ]= n [ A ][ BX ]+ n [ B ][ AX ]+ n [ C ][ CX ] B x f [ BY ]= n [ A ][ BY ]+ n [ B ][ AY ]+ n [ C ][ CY ] B y f [ CX ]= n [ A ][ CX ]+ n [ B ][ CX ]+ n [ C ][ AX ] C x f [ CY ]= n [ A ][ CY ]+ n [ B ][ CY ]+ n [ C ][ AY ] C y TetrahedralElementNodeMemoryMap .Thememoryorderforthe3Dtetrahedralelementisdened forthepoints A , B , C ,and D withcomponents xy ,and z .Theorderingoftheelementsthatmapsthe partialforcesper-nodetothereconstitutedforceresponseisprovidedbelow: Ordering = f A;B;C;D g ; f B;A;C;D g ; f C;A;B;D g ; f D;A;B;C g A.2 Theresponseforcevector f representsthereconstitutionoftheforcesgeneratedbytheper-nodepartialforcesmultipliedwiththenoderatiosstoredinarray n .Forthe3Dtetrahedralelementthisdenes 4 nodestimes 3 dimensionsresultingin 12 totalresponseforces. r [ AX ]= n [ A ][ AX ]+ n [ B ][ BX ]+ n [ C ][ BX ]+ n [ D ][ BX ] A x r [ AY ]= n [ A ][ AY ]+ n [ B ][ BY ]+ n [ C ][ BY ]+ n [ D ][ BY ] A y r [ AZ ]= n [ A ][ AZ ]+ n [ B ][ BZ ]+ n [ C ][ BZ ]+ n [ D ][ BZ ] A z r [ BX ]= n [ A ][ BX ]+ n [ B ][ AX ]+ n [ C ][ CX ]+ n [ D ][ CX ] B x r [ BY ]= n [ A ][ BY ]+ n [ B ][ AY ]+ n [ C ][ CY ]+ n [ D ][ CY ] B y r [ BZ ]= n [ A ][ BZ ]+ n [ B ][ AZ ]+ n [ C ][ CZ ]+ n [ D ][ CZ ] B z r [ CX ]= n [ A ][ CX ]+ n [ B ][ CX ]+ n [ C ][ AX ]+ n [ D ][ DX ] C x r [ CY ]= n [ A ][ CY ]+ n [ B ][ CY ]+ n [ C ][ AY ]+ n [ D ][ DY ] C y r [ CZ ]= n [ A ][ CZ ]+ n [ B ][ CZ ]+ n [ C ][ AZ ]+ n [ D ][ DZ ] C z r [ DX ]= n [ A ][ DX ]+ n [ B ][ DX ]+ n [ C ][ DX ]+ n [ D ][ AX ] D x r [ DY ]= n [ A ][ DY ]+ n [ B ][ DY ]+ n [ C ][ DY ]+ n [ D ][ AY ] D y r [ DZ ]= n [ A ][ DZ ]+ n [ B ][ DZ ]+ n [ C ][ DZ ]+ n [ D ][ AZ ] D z Thesememorymapsprovidetheinversionoftheorderingtoreconstitutetheresponseforcesoftheelementduetohowtheyarecomputedindependentlywithrespecttoeachnode. 124

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B.LinearElasticityFiniteElementStiffnessMatrices TriangularElementStiffnessMatrix .Two-dimensionalFEAsimulationscanbeformulatedusingseveraldifferentelementtypesincludingtrusses,beams,andquadrilaterals.However,forarbitrarygeometric boundaryconditionsthemostgeneralizableelementisasimpletriangle.Thiselementischaracterized bythegeometryoftheelementandaconstantthickness t .Theelementstiffnessmatrixisdenedbythis thickness,theelementshapefunctionmatrix B andthe2Dplanestressorplanestrainmatrix D whichis parametrizedbyYoung'smodulus E theModulusofElasticityandthematerialsPoissonRatio v . k = Z V B T DB dV = t B T DB B.1 Theshapefunctionsoftheelementaredenedbythegeometriccongurationofthenodes a , b ,and c , eachwiththeirown x and y componentsandthearea A oftheelementcomputedfromthenodecoordinates: 2 A = a x b y )]TJ/F21 10.9091 Tf 9.976 0 Td [(c y + b x c y )]TJ/F21 10.9091 Tf 9.976 0 Td [(a y + c x a y )]TJ/F21 10.9091 Tf 9.976 0 Td [(b y .Theshapefunctionmatrix B isdenedasafunction ofdifferencesinthecoordinatesdenedbycoefcients a , b , c , a , b ,and c . B = 1 2 A " a 0 b 0 c 0 0 a 0 b 0 c a a b b c c # B.2 Where a = b y )]TJ/F21 10.9091 Tf 11.509 0 Td [(c y , b = c y )]TJ/F21 10.9091 Tf 11.509 0 Td [(a y , c = a y )]TJ/F21 10.9091 Tf 11.509 0 Td [(b y , a = c x )]TJ/F21 10.9091 Tf 11.509 0 Td [(b x , b = a x )]TJ/F21 10.9091 Tf 11.509 0 Td [(c x ,and c = b x )]TJ/F21 10.9091 Tf 11.509 0 Td [(a x . Forthematerialmatrix,therearetwovariantsthatcanbeuseddependingontheformoftheanalysis,the planestressandstrainmatrices.Forthe2Delementtrainingweusetheplanestrainformulation. D = E 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(v 2 " 1 v 0 v 10 00 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(v 2 # D = E + v )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 v " 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(vv 0 v 1 )]TJ/F21 10.9091 Tf 10.91 0 Td [(v 0 00 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 v 2 # B.3 Thestressofthetriangularelementisdenedby = DB u where =[ x y xy ] T .Followingcommon convention, representsnormalstressand representsshearstress.Basedonasimilarformulation,we canalsodenetheelementstiffnessmatrixfor3Dtetrahedralelements. TetrahedralElementStiffnessMatrix .Themostcommonelementtypesforthree-dimensionalFEA simulationsareregularhexahedralortetrahedralsolids.WhilehexahedralelementsarecommonforFEA simulationsthatuseembeddedgeometrytolinklow-resolutionsimulationmesheswithhigh-resolution surfacemesh,theyuselargescaleelementmatricesthatcontainnumerouscomponents.Fromthepremise oftheneuralelementapproachitismoreadvantageoustobasedmaterialbehaviorsoffoftetrahedralelements.Theelementstiffnessmatrixofalineartetrahedralelementisbasedonthevolumeoftheelement V ,theshapefunctionmatrix,andtheisotropicmaterialmatrix D .Weassumethatthetetrahedralelement iscomposedofthenodes a , b , c ,and d . k = Z V B T DB dV = V B T DB B.4 Thevolumeofthetetrahedralelementcanbeevaluatedfromthepositionofthenodesthatdenetheelement.ThisisevaluatedbycomputingthedeterminantofthematrixshowninEquationB.5.Whenan elementisrepresentedbyasetofindices,forexamplewhenitisloadedfromageneratedmesh,theorderingmustproduceapositivevolume.Iftheprovidedindicesoftheelementdoesnotprovideapositive volume,thentheindicescanbereorderedtoproduceapositivevalue.Inourformulation,thisiscorrect whennewsimulationmeshesareinitialized. 125

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V = 1 6 1 a x a y a z 1 b x b y b z 1 c x c y c z 1 d x d y d z B.5 Theshapefunctionsofthetetrahedralelementarecomputedasthepartialderivativesofthenodaldisplacementfunctions.Weinitiallydenethesefunctionswithrespecttothenodesoftheelement.Thefour primaryshapefunctionsaredenedinas: N 1 = 1 6 V 1 + 1 x + 1 y + 1 z N 2 = 1 6 V 2 + 2 x + 2 y + 2 z N 3 = 1 6 V 3 + 3 x + 3 y + 3 z N 4 = 1 6 V 4 + 4 x + 4 y + 4 z B.6 Wherewecanevaluatethecoefcients , , ,and throughcomputingthedeterminantsofthefollowingsetofmatrices.Thiswillgenerateatotalof 16 coefcientsthatdenetheshapefunctions. 1 = b x b y b z c x c y c z d x d y d z 2 = )]TJ/F25 10.9589 Tf 10.303 15.878 Td [( a x a y a z c x c y c z d x d y d z 3 = a x a y a z b x b y b z d x d y d z 4 = )]TJ/F25 10.9589 Tf 10.303 15.878 Td [( a x a y a z b x b y b z c x c y c z 1 = )]TJ/F25 10.9589 Tf 10.303 19.166 Td [( 1 b y b z 1 c y c z 1 d y d z 2 = 1 a y a z 1 c y c z 1 d y d z 3 = )]TJ/F25 10.9589 Tf 10.303 19.166 Td [( 1 a y a z 1 b y b z 1 d y d z 4 = 1 a y a z 1 b y b z 1 c y c z 1 = 1 b y b z 1 c y c z 1 d y d z 2 = )]TJ/F25 10.9589 Tf 10.303 19.166 Td [( 1 a y a z 1 c y c z 1 d y d z 3 = 1 a y a z 1 b y b z 1 d y d z 4 = )]TJ/F25 10.9589 Tf 10.303 19.166 Td [( 1 a y a z 1 b y b z 1 c y c z 1 = )]TJ/F25 10.9589 Tf 10.303 19.166 Td [( 1 b y b z 1 c y c z 1 d y d z 2 = 1 a y a z 1 c y c z 1 d y d z 3 = )]TJ/F25 10.9589 Tf 10.303 19.166 Td [( 1 a y a z 1 b y b z 1 d y d z 4 = 1 a y a z 1 b y b z 1 c y c z Usingthecomputedcoefcients,thepartialderivativesoftheshapefunctions N 1 through N 4 canbecomputedtodirectlyformtheshapefunctionmatrix B .Thisportionofthederivationisomitted,butwithin theimplementationtheseareevaluatedwithinamatrix. B = 2 6 6 6 6 6 6 6 4 @N 1 @x 00 @N 2 @x 00 @N 3 @x 00 @N 4 @x 00 0 @N 1 @y 00 @N 2 @y 00 @N 3 @y 00 @N 4 @y 0 00 @N 1 @z 00 @N 2 @z 00 @N 3 @z 00 @N 4 @z @N 1 @y @N 1 @x 0 @N 2 @y @N 2 @x 0 @N 3 @y @N 3 @x 0 @N 4 @y @N 4 @x 0 @N 1 @z 0 @N 1 @x @N 2 @z 0 @N 2 @x @N 3 @z 0 @N 3 @x @N 4 @z 0 @N 4 @x 0 @N 1 @z @N 1 @y 0 @N 2 @z @N 2 @y 0 @N 3 @z @N 3 @t 0 @N 4 @z @N 4 @y 3 7 7 7 7 7 7 7 5 B.7 Thelastcomponentofthetetrahedralelementstiffnessmatrixistheisotropicmaterialtensor.Thedenitionofthismatrixisparametrizedbytherelationshipbetweentheelasticmodulus E andthePoisson ratio v .AswiththedenitionoftheelasticmaterialforanFEA-basedsimulation,weexpressthematerial withinaneuralelementbasedonthesesameproperties.Thisisoneoftheprimaryobjectivesandcontributionstowardsintroducingadiscretemodelforallowingreal-timesimulationmeshestobedenedwith respecttothecoefcientsdenedwithincontinuummechanics. 126

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D = E + v )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 v 2 6 6 6 6 4 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(vvv 000 v 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(vv 000 vv 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(v 000 000 1 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 v 2 00 0000 1 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 v 2 0 00000 1 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 v 2 3 7 7 7 7 5 B.8 Formechanicalproblemswithinstatics,thefollowingrelationdenesthepurposeoftheelementstiffnessmatrixforrelatingforces F appliedtoamodeledstructurecharacterizedbythecombinationofall elementstoformtheglobalelementstiffnessmatrix K withtheglobalnodedisplacements U . KU = F B.9 Sincetheneuralelementformulationforgoestheprocessofassemblingelementstiffnessmatricesintothe globalstiffnessmatrix,weomitthenotionofthelocalandglobalelementcoordinatesorelementindices. ThestressevaluationofdeformedelementsiscomputedusingthesameequationastheFEAformulation, butisevaluatedperelement. = DBu B.10 Wherethestresstensorisdenedathesecondorder Cauchy stresscompressedtoasixcomponentvector: =[ x y z x y y z z x ] T .Forvisualization,weconverttheCauchystresstensortothe vonMises stressscalarthroughtheuseofthegeneralmagnitudeequation. = r 1 2 x )]TJ/F21 10.9091 Tf 10.909 0 Td [( y 2 + y )]TJ/F21 10.9091 Tf 10.909 0 Td [( z 2 + z )]TJ/F21 10.9091 Tf 10.909 0 Td [( x 2 +3 2 xy + 2 yz + 2 zx B.11 Thisscalarquantityisthenusedtorepresentthestressthroughthematerialbasedonanassigned color-map.Thisprovidesanintuitivevisualizationoftheinternalbehaviorofthematerialundertheprescribedboundaryconditionsandappliedexternalforces. 127