UCLA

Physics-based simulations are a powerful tool in both computer graphics and engineering applications. Implicit discretization is essential for accurate, stable, and efficient simulations of solids and fluids.

In this thesis, we first present a novel implicit Material Point Method (MPM) discretization of spatially varying surface energies. Our discretization is based on surface energy, enabling implicit time stepping and capturing surface gradients without explicitly resolving them as in traction-condition-based approaches. We include an implicit discretization of thermomechanical material coupling with novel particle-based enforcement of Robin boundary conditions. Lastly, we design a particle resampling approach for perfect conservations of linear and angular momentum with Affine-Particle-In-Cell (APIC) [Jiang et al. 2015].

The second part presents a novel deep-learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. Our method is motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error. We use a deep neural network to accelerate convergence via data-driven improvement of the search direction at each iteration. We demonstrate the efficacy of our approach on discretized Poisson equations with millions of degrees of freedom. Our algorithm can reduce the linear system residual to the target tolerance in a small number of iterations, independent of the problem size, and generalize effectively to various systems beyond those encountered during training.

Finally, we present improvements to Position Based Dynamics (PBD) [Müller et al. 2007] and Extended Position Based Dynamics (XPBD) [Macklin et al. 2016] methods, which are variants of implicit time integrator. PBD/XPBD are powerful methods for the real-time simulation of elastic objects, but they do not always converge. We isolate the root cause in the approximate linearization of the nonlinear backward Euler systems utilized by XPBD. We provide two extensions to XPBD to address the non-convergence and support general hyperelastic models. The following chapter presents a novel position-based nonlinear Gauss-Seidel approach for quasistatic simulations of elastic objects. This approach retains the essential PBD feature of stable behavior with limited computational budgets and allows for convergent behavior when the budgets expand.