We present the derivation details necessary for simulation of thin shells with finite strains based on the Kirchhoff-Love assumptions. With an eye towards cloth simulation, we combine this with a nonlinear orthotropic constitutive model framework. We leverage a conforming spatial discretization using Catmull-Clark subdivision surfaces to ensure convergence under refinement, which we confirm by numerical experiments. The dynamics are handled in a fully implicit fashion to allow for large timesteps and solution of quasistatic problems.

Accurate constitutive modeling and parameter estimation for woven fabrics is essential in many fields. To achieve this we first design an experimental protocol for characterizing real fabrics based on commercially available tests. Next, we present a new orthotropic hyperelastic constitutive model for woven fabrics. Finally, we create a method for accurately fitting the material parameters to the experimental data. The last step is accomplished by solving inverse problems using our Catmull-Clark subdivision finite element discretization of the Kirchhoff-Love equations. Through this approach we are able to reproduce the fully nonlinear behavior corresponding to the captured data with a small number of parameters while maintaining all fundamental invariants from continuum mechanics. The resulting constitutive model can be used with any discretization and not just subdivision finite elements, which we demonstrate by providing an alternate implementation based on simple triangle meshes. We illustrate the entire process with results for five types of fabric and compare photo reference of the real fabrics to the simulated equivalents.

This thesis details two numerical methods for the solution of incompressible flow problems using the virtual node framework introduced in (Bedrossian, 2010). The first method is a novel discrete Hodge decomposition for velocity fields defined over irregular domains in two and three dimensions. This new decomposition leads to a sparse, 5-point stencil in 2D (7-point in 3D) at all nodes in the domain, even near the boundary. The corresponding linear system can be factored simply into a weighted product of the standard discrete divergence and gradient operators, is symmetric positive definite, and yields second order accurate pressures and first order velocities in the maximum norm (second order in the 1-norm).

The second method is an extension of the work in (Assenço, 2013), which simulates the Stokes equations in two dimensions, to a method that models the Navier-Stokes equations in two and three spatial dimensions. The extension to three dimensions is partially accomplished by a new approach to discretizing the multiplier term corresponding to the system jump conditions. This method works either on domains with interfacial discontinuities in material quantities such as density and viscosity, or on irregularly shaped domains with Dirichlet, Neumann, or slip boundary conditions. This method leads to a discrete, KKT system solving for velocities and pressures simultaneously, and yields second order accurate velocities in both time and space, and first order pressures.

The Material Point Method (MPM) has shown its high potential for physics-based simulation in the area of computer graphics. In this dissertation, we introduce a couple of improvements to the traditional MPM for different applications and demonstrate the advantages of our methods over the previous methods.

First, we present a generalized transfer scheme for the hybrid Eulerian/Lagrangian method: the Polynomial Particle-In-Cell Method (PolyPIC). PolyPIC improves kinetic energy conservation during transfers, which leads to better vorticity resolution in fluid simulations and less numerical damping in elastoplasticity simulations. Our transfers are designed to select particle-wise polynomial approximations to the grid velocity that are optimal in the local mass-weighted L2 norm. Indeed our notion of transfers reproduces the original Particle-In-Cell Method (PIC) and recent Affine Particle-In-Cell Method (APIC). Furthermore, we derive a polynomial basis that is mass orthogonal to facilitate the rapid solution of the optimality condition. Our method applies to both of the collocated and staggered grid.

As the second contribution, we present a novel method for the simulation of thin shells with frictional contact using a combination of MPM and subdivision finite elements. The shell kinematics are assumed to follow a continuum shell model which is decomposed into a Kirchhoff-Love motion that rotates the mid-surface normals followed by shearing and compression/extension of the material along the mid-surface normal. We use this decomposition to design an elastoplastic constitutive model to resolve frictional contact by decoupling resistance to contact and shearing from the bending resistance components of stress. We show that by resolving frictional contact with a continuum approach, our hybrid Lagrangian/Eulerian approach is capable of simulating challenging shell contact scenarios with hundreds of thousands to millions of degrees of freedom. Without the need for collision detection or resolution, our method runs in a few minutes per frame in these high-resolution examples. Furthermore, we show that our technique naturally couples with other traditional MPM methods for simulating granular and related materials.

In the third part, we present a new hybrid Lagrangian Material Point Method for simulating volumetric objects with frictional contact. The resolution of frictional contact in the thin shell simulation cannot be generalized to the case of volumetric materials directly. Also, even though MPM allows for the natural simulation of hyperelastic materials represented with Lagrangian meshes, it usually coarsens the degrees of freedom of the Lagrangian mesh and can lead to artifacts, e.g., numerical cohesion. We demonstrate that our hybrid method can efficiently resolve these issues. We show the efficacy of our technique with examples that involve elastic soft tissues coupled with kinematic skeletons, extreme deformation, and coupling with various elastoplastic materials. Our approach also naturally allows for two-way rigid body coupling.

We present an extension to the Material Point Method (MPM) for simulating elastic objects with various co-dimensions like hair (1D), thin shells (2D), and volumetric objects (3D). We simulate thin shells with frictional contact using a combination of MPM and subdivision finite elements. The shell kinematics are assumed to follow a continuum shell model which is decomposed into a Kirchhoff-Love motion that rotates the mid-surface normals followed by shearing and compression/extension of the material along the mid-surface normal. We use this decomposition to design an elastoplastic constitutive model to resolve frictional contact by decoupling resistance to contact and shearing from the bending resistance components of stress. We show that by resolving frictional contact with a continuum approach, our hybrid Lagrangian/Eulerian approach is capable of simulating challenging shell contact scenarios with hundreds of thousands to millions of degrees of freedom. Furthermore our technique naturally couples with other traditional MPM methods for simulating granular materials. Without the need for collision detection or resolution, our method runs in a few minutes per frame in these high resolution examples. For the simulation of hair and volumetric elastic objects, we utilize a Lagrangian mesh for internal force computation and an Eulerian mesh for self collision as well as coupling with external materials. While the updated Lagrangian discretization where the Eulerian grid degrees of freedom are used to take variations of the potential energy is effective in simulating thin shells, its frictional contact response strategy does not generalize to volumetric objects. Therefore, we develop a hybrid approach that retains Lagrangian degrees of freedom while still allowing for natural coupling with other materials simulated with traditional MPM. We demonstrate the efficacy of our technique with examples that involve elastic soft tissues coupled with kinematic skeletons, extreme deformation, and coupling with multiple elastoplastic materials. Our approach also naturally allows for two-way rigid body coupling.

We present a novel Material Point Method for simulating complex materials. The method achieves plasticity effects via the temporal evolution of the left elastic Cauchy-Green strain. We recast the upper-convected derivative of the strain in the Oldroyd-B constitutive model as a plastic flow and are able to simulate elastic and viscoelastic effects. Our model provides a volume-preserving rate-based description of plasticity that does not require singular value decompositions. Our semi-implicit discretization allows for high-resolution simulations. We also present novel discretizations of the temporal update of the left elastic Cauchy-Green strain for several constitutive models that preserve symmetry and positive-definiteness of the strain for use in the Material Point Method. A novel modification to a constitutive model is also presented that models material softening under plastic compression.

This dissertation focuses on applications and extensions of the Material Point Method(MPM) in simulating ductile fracture and thermomechanical material behavior for baking and cooking. We conclude the two major contributions as follows:

First, we present novel techniques for simulating and visualizing ductile fracture with MPM. We utilize traditional particle-based MPM [SZS95] as well as the Lagrangian energy formulation of [JSS15] that formulates the deformation gradient and potential energy through a tetrahedron mesh. We model failure and fracture via elasto-plasticity with damage. Material shows elastic behavior until the deformation exceeds a Rankine or von Mises yield criterion, at which point irreversible damage starts to occur. We introduce a softening model that shrinks the yield surface until a damage threshold is reached. Once damaged the material Lam� coefficients are modified to represent failure. We design visualization techniques for rendering the boundary of the material and its intersections with evolving crack surfaces. Our approach uses a simple and efficient element splitting strategy for tetrahedron meshes to represent crack surfaces. For traditional particle-based MPM we use an initial Delaunay tetrahedralization to connect randomly initialized MPM particles and form a reference mesh. Our visualization technique is a postprocess and can be run separately after the MPM simulation for efficiency. We demonstrate the strength of our method with a number of challenging simulations of ductile failure with considerable and persistent self contact.

Our second contribution is a Material Point Method for visual simulation of baking breads, cookies, pancakes and similar materials. We develop a novel thermomechanical model using mixture theory to resolve interactions between individual water, gas and solid species. Heat transfer with thermal expansion is used to model thermal variations in material properties. Water based mass transfer is resolved through the porous mixture. Gas represents carbon dioxide produced by leavening agents in the baking process, and the solid component is modeled as a visco-elastoplastic material to represent its varied and complex rheological properties. Water content in the mixture reduces during the baking process according to Fick’s Law which contributes to the drying and cracking of crust at the material boundary. Carbon dioxide gas produced by leavening agents during baking creates internal pressure that causes rising. The visco-elastoplastic model for the dough is temperature dependent and is used to model melting and solidification. We discretize the governing equations using a novel Material Point Method designed to track the solid phase of the mixture.

Physics based simulations often require solving differential equations on domains with irregular geometry. A popular approach to their numerical approximation is the finite element method with unstructured meshes that conform to the domain geometries. However, in many applications such as shape optimization for elastic material and virtual surgery, the simulation domains frequently change shape. Under these cases, continuous remeshing can be a daunting task, and often produces ill-conditioned elements, which will reduce the accuracy and stability of the simulations. Virtual node algorithms(VNA) were invented to overcome this difficulty by embedding the simulation domains on Cartesian grids or structured meshes that do not conform to their geometry. Solutions are then computed on the embedding domains to take advantage of their regularity. The term `virtual nodes" refers to the degrees of freedom that lie outside the actual domains, reflecting the fact that there is no material associated with them. In this dissertation we present two novel virtual node algorithms.

Simulation in computer graphics must accommodate a wide range of material behaviors under large deformation, topological changes, contact or collision. This dissertation focuses on hybrid Lagrangian/Eulerian methods, especially the Material Point Method (MPM) in computer animation, and we list the major contributions here:

First, we simulate sand dynamics using an elastoplastic, continuum assumption. We demonstrate that the Drucker-Prager plastic flow model combined with a Hencky-strain-based hyperelasticity accurately recreates a wide range of visual sand phenomena with moderate computational expense. The Drucker-Prager model naturally represents the frictional relation between shear and normal stresses through a yield stress criterion. We develop a stress projection algorithm used for enforcing this condition with a non-associative flow rule.

We further extend the idea of simulating sand dynamics using an elastoplastic continuum assumption to codimensional objects. Our second contribution is to introduce a novel method for simulation of thin shells with frictional contact using a combination of the MPM and subdivision finite elements. The shell kinematics are assumed to follow a continuum shell model which is decomposed into a Kirchhoff-Love motion that rotates the mid-surface normals followed by shearing and compression/extension of the material along the mid-surface normal. We use this decomposition to decouple resolving contact and shearing from the bending resistance components of stress. Our approach is capable of simulating challenging shell contact scenarios with hundreds of thousands to millions of degrees of freedom with a moderate cost of only a few minutes per frame.

Our third contribution is to introduce a novel transfer scheme for hybrid Lagrangian/Eulerian simulations. Recently the Affine Particle-In-Cell (APIC) Method was introduced to improve the accuracy of the transfers in Particle-In-Cell (PIC) without suffering from the noise present in the historic alternative, Fluid-Implicit-Particle (FLIP). We generalize APIC by augmenting each particle with a more general local function. Our transfers are designed to select particle-wise polynomial approximations to the grid velocity that are optimal in the local mass-weighted $L^2$ norm. With only marginal additional cost, our generalization improves kinetic energy conservation during transfers which leads to better vorticity resolution in fluid simulations and less numerical damping in elastoplasticity simulations.

Simulating physical materials with dynamic movements to photo-realistic resolution has always been one of the most crucial and challenging topics in Computer Graphics. This dissertation considers large-strain elastoplasticity theory applied to the low-to-medium stiffness regime, with topological changes and codimensional objects incorporated. We introduce improvements to the Material Point Method (MPM) for two particular objectives, simulating fracturing ductile materials and incorporation of MPM and Lagrangian Finite Element Method (FEM).

Our first contribution, simulating ductile fracture, utilizes traditional particle-based MPM [SSC13, SCS94] as well as the Lagrangian energy formulation of [JSS15] which uses a tetrahedron mesh, rather than particle-based estimation of the deformation gradient and potential energy. We model failure and fracture via elastoplasticity with damage. The material is elastic until its deformation exceeds a Rankine or von Mises yield condition. At that point, we use a softening model that shrinks the yield surface until it reaches the damage thresh- old. Once damaged, the material Lam ́e coefficients are modified to represent failed material. This approach to simulating ductile fracture with MPM is successful, as MPM naturally captures the topological changes coming from the fracture. However, rendering the crack surfaces can be challenging. We design a novel visualization technique dedicated to rendering the material’s boundary and its intersection with the evolving crack surfaces. Our approach uses a simple and efficient element splitting strategy for tetrahedron meshes to create crack surfaces. It employs an extrapolation technique based on the MPM simulation. For traditional particle-based MPM, we use an initial Delaunay tetrahedralization to connect randomly sampled MPM particles. Our visualization technique is a post-process and can run after the MPM simulation for efficiency. We demonstrate our method with several challenging simulations of ductile failure with considerable and persistent self-contact and applications with thermomechanical models for baking and cooking.

Our second contribution, hybrid MPM–Lagrangian-FEM, aims to simulate elastic objects like hair, rubber, and soft tissues. It utilizes a Lagrangian mesh for internal force computation and a Eulerian grid for self-collision, as well as coupling with external materials. While recent MPM techniques allow for natural simulation of hyperelastic materials represented with Lagrangian meshes, they utilize an updated Lagrangian discretization and use the Eulerian grid degrees of freedom to take variations of the potential energy. It often coarsens the degrees of freedom of the Lagrangian mesh and can lead to artifacts. We develop a hybrid approach that retains Lagrangian degrees of freedom while still allowing for natural coupling with other materials simulated with traditional MPM, e.g., sand, snow, etc. Furthermore, while recent MPM advances allow for resolution of frictional contact with codimensional simulation of hyperelasticity, they do not generalize to the case of volumetric materials. We show that our hybrid approach resolves these issues. We demonstrate the efficacy of our technique with examples that involve elastic soft tissues coupled with kinematic skeletons, extreme deformation, and coupling with various elastoplastic materials. Our approach also naturally allows for two-way rigid body coupling.

We present a discussion on how one can simulate sand as a continuum using elastoplasticity. We showed the efficacy of Drucker-Prager plasticity model and St. Venant Kirchhoff with Hencky strain to model sand. We discretized the continuum equation using Material Point Method (MPM). We also present a multi-species model for the simulation of gravity driven landslides and debris flows with porous sand and water interactions. We use continuum mixture theory to describe individual phases where each species individually obeys conservation of mass and momentum and they are coupled through a momentum exchange term. Water is modeled as a weakly compressible fluid and sand is modeled with an elastoplastic law whose cohesion varies with water saturation. We use Material Point Method to discretize the governing equations. We use two grids, corresponding to water and sand phase. The momentum exchange term in the mixture theory is relatively stiff and we use semi-implicit time stepping to avoid associated small time steps. Our semi-implicit treatment is explicit in plasticity and preserves symmetry of force linearizations. We develop a novel regularization of the elastic part of the sand constitutive model that better mimics plasticity during the implicit solve to prevent numerical cohesion artifacts that would otherwise have occurred. Lastly, we develop an improved return mapping for sand plasticity that prevents volume gain artifacts in the traditional Drucker-Prager model.

Finally, we revisit the problem of redistancing, which is native to the level set paradigms. We used an interesting alternative view that utilizes the Hopf-Lax formulation of the solution to the eikonal equation, as proposed by \cite{lee:2017:revisiting,darbon:2016:algorithms}. In this approach, the signed distance at an arbitrary point is obtained without the need of distance information from neighboring points. We extend the work of Lee et al. \cite{lee:2017:revisiting} to redistance functions defined via interpolation over a regular grid.