Particle Grid Hybrid Methods for Multi-Material Dynamics

The Material Point Method (MPM) has shown its capability in simulating multi-physics and multi-material dynamics. In this dissertation, we present an extension to the Material Point Method (MPM) for simulating volumetric collisions of elastic objects, formulate a novel approach for surface tension phenomenon, and a hybrid particle/grid approach for simulating incompressible fluids.First, we present a momentum-conserving hybrid particle/grid iteration method for volumetric elastic contact problems. We use a Lagrangian mesh to discretize the elastic material and an Eulerian grid to provide the collision response at the mesh boundary. The Eulerian grid utilizes Affine-Particle-In-Cell (APIC) transfers which conserve both linear and angular momentum when affine information is provided to the boundary nodes. The transfers are leveraged in terms of performance to capture the impulses needed to prevent a collision. The cohesion that occurs when separating in the APIC transfers is removed through augury iterations. A novel resampling scheme and modifications to the transfers that conserve mass, linear and angular momentum are used to capture the collision on finner Eulerian grids. This iteration can be used to provide faster convergence in impulse-based approaches that are commonly used in graphics. Second, we present an updated Lagrangian discretization of surface tension forces for the simulation of liquids with moderate to extreme surface tension effects. The potential energy associated with surface tension is proportional to the surface area of the liquid. We design discrete forces as gradients of this energy to the motion of the fluid over a time step. We show that this naturally allows for inversion of the Hessian of the potential energy required with the use of Newton's method to solve the systems of nonlinear equations associated with implicit time stepping. We design a novel level-set-based boundary quadrature technique to discretize the surface area calculation in our energy-based formulation. Our approach works most naturally with Particle-In-Cell techniques and we demonstrate our approach with a weakly incompressible model for liquid discretized with the Material Point Method. Lastly, we design a particle resampling approach needed to achieve perfect conservation of linear and angular momentum with APIC. We show that our approach is essential for allowing efficient implicit numerical integration in the limit of materials with variable high surface tension. Last, we present a hybrid particle/grid approach for simulating incompressible fluids on collocated velocity grids. We interchangeably use particle and grid representations of transported quantities to balance efficiency and accuracy. A novel Backward Semi-Lagrangian method is derived to improve the accuracy of grid-based advection. Our approach utilizes the implicit formula associated with solutions of Burgers' equation. We enforce incompressibility over collocated, rather than staggered grids. Our projection technique is variational and designed for B-spline interpolation over regular grids where multi quadratic interpolation is used for velocity and multilinear interpolation for pressure. Despite our use of regular grids, we extend the variational technique to allow for cut-cell definition of irregular flow domains for both Dirichlet and free surface boundary conditions