Simulating fluids and solid materials undergoing large deformation remains an important and challenging problem in Computer Graphics. The dynamics of these materials usually involve dramatic topological changes and therefore require sophisticated numerical approaches to achieve sufficient accuracy and visual realism. This dissertation focuses on the Material Point Method (MPM) for simulating solids and fluids for use in computer animation, and it makes four major contributions: First, we introduce new MPM for simulating viscoelastic fluids, foams and sponges. Our second contribution is to introduce a novel technique designed to retain the stability of the original PIC, without suffering from the noise and instability of FLIP. Our third contribution is to introduce a novel material point method for heat transport, melting and solidifying materials. Our fourth contribution is to show that recasting the backward Euler method as a minimization problem allows Newton's method to be stabilized by standard optimization techniques

with some novel improvements of our own.

Accurate and robust simulation of solid and fluid dynamics with their interactions is non-trivial in computer graphics. In this thesis, we present a robust and efficient method for simulating Lagrangian solid-fluid coupling based on a new operator splitting strategy. We use variational formulations to approximate fluid properties and solid-fluid interactions, and introduce a unified two-way coupling formulation for SPH fluids and FEM solids using interior point barrier-based frictional contact. We split the resulting optimization problem into a fluid phase and a solid-coupling phase using a novel time-splitting approach with augmented \emph{contact proxies}, and propose efficient custom linear solvers. Our technique accounts for fluids interaction with nonlinear hyperelastic objects of different geometries and codimensions, while maintaining an algorithmically guaranteed non-penetrating criterion. Comprehensive benchmarks and experiments demonstrate the efficacy of our method.

Physics-based simulation, coupled with the Finite Element Method (FEM), has emerged asa powerful tool in understanding and analyzing the complex behavior of elastic materials. Implicit Discretization is essential for efficiently and accurately simulating elastic solids and cloth. In this thesis, we first explore ways of creating volumetric mesh for embedding surface mesh. The embedded surface mesh has many small self intersections. We devise an efficient and robust way of generating a hexahedron mesh to embed the triangle mesh, so that the self-intersecting regions are correctly duplicated. We then simulate the hexahedron mesh using Finite Element Method and interpolate to the embedded triangle mesh. The second part explores different ways of improving the core simulation solver of Finite Element Method. We improve on the Position Based Dynamics (PBD)/Extended Position Based Dynamics (XPBD) framework. PBD/XPBD are known for their robustness under a very small computational budget. However, they have several limitations. PBD/XPBD does not converge when the computational budget is sufficient. PBD/XPBD also supports limited hyperelastic constitutive models. We devise PXPBD (Primary Extended Position Based Dynamics) to address these issues. The PXPBD methods improves convergence rate of PBD/XPBD and support arbitrary hyperelastic models. PBD/XPBD framework does not support quasistatic simulation either. We note quasistatic simulation is important for generating training data for machine learning based simulation approaches such as QNN (Quasistatic Neural Network). We also notice that the constraint-based Gauss-Seidel approach in PBD/XPBD causes loss of information on the node solve. So we design Position-Based Nonlinear Gauss Seidel (PBNG), where the hyperelastic energy from FEM is optimized per node, instead of per constraint. Doing so not only boosts convergence rate, but also enables high quality quasistatic simulation, which runs much faster than the existing methods such as Newton’s method. The last part of my thesis uses a machine learning-based approach to simulate human musculature effectively. We use a neural network to model the deformation of human soft tissues such as muscle, tendon, fat and skin with high fidelity. By deploying a biomechanicsbased approach, we estimate the activation of single muscle during deformation. The activation parameters are then incorporated into an active neural network (ANN). which adds per-vertex deformation on top of Linear Blend Skinning (LBS). Our neural network achieves more than 1000X speed up as compared to traditional FEM approaches, but it has the same level of visual fidelity.

Machine learning, particularly deep neural networks, has become a powerful tool in scientific research. In this thesis, we demonstrate how neural networks can enhance performance and reduce memory requirements in physics-based simulation applications for computer graphics.In the first part of the thesis we present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search direction at each iteration. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using self-supervised learning with a loss function equal to the $L^2$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations, which arise in computational fluid dynamics applications, with millions of degrees of freedom. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

In the second part we present learning-based implicit shape representations designed for real-time avatar collision queries arising in the simulation of clothing. Signed distance functions (SDFs) have been used for such queries for many years due to their computational efficiency. Recently deep neural networks have been used for implicit shape representations (DeepSDFs) due to their ability to represent multiple shapes with modest memory requirements compared to traditional representations over dense grids. However, the computational expense of DeepSDFs prevents their use in real-time clothing simulation applications. We design a learning-based representation of SDFs for human avatars whose bodies change shape kinematically due to joint-based skinning. Rather than using a single DeepSDF for the entire avatar, we use a collection of extremely computationally efficient (shallow) neural networks that represent localized deformations arising from changes in body shape induced by the variation of a single joint. This requires a stitching process to combine each shallow SDF in the collection together into one SDF representing the signed closest distance to the boundary of the entire body. To achieve this we augment each shallow SDF with an additional output that resolves whether or not the individual shallow SDF value is referring to a closest point on the boundary of the body, or to a point on the interior of the body (but on the boundary of the individual shallow SDF). Our model is extremely fast and accurate and we demonstrate its applicability with real-time simulation of garments driven by animated characters.

We present a comprehensive neural network to model the deformation of human soft tissues including muscle, tendon, fat and skin. Our approach provides kinematic and active correctives to linear blend skinning that enhance the realism of soft tissue deformation at modest computational cost, aiming to revolutionize character animation in the context of metaverse and game development. Our network accounts for deformations induced by changes in the underlying skeletal joint state as well as the active contractile state of relevant muscles. Training is done to approximate quasistatic equilibria produced from physics-based simulation of hyperelastic soft tissues in close contact. We use a layered approach to equilibrium data generation where deformation of muscle is computed first, followed by an inner skin/fascia layer, and lastly a fat layer between the fascia and outer skin. We show that a simple network model which decouples the dependence on skeletal kinematics and muscle activation state can produce compelling behaviors with modest training data burden. Active contraction of muscles is estimated using inverse dynamics where muscle moment arms are accurately predicted using the neural network to model kinematic musculotendon geometry. Results demonstrate the ability to accurately replicate compelling musculoskeletal and skin deformation behaviors over a representative range of motions, including the effects of added weights.Additionally, we present significant advancements in several related areas of the pipeline including a dynamic mode capture paradigm for augmenting secondary effects on top of our network, a robust meshing algorithm for generating volumetric hexahedron meshes from self-intersecting surfaces and a novel position-based nonlinear Gauss-Seidel approach for quasistatic simulation.