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.