The material point method (MPM) recently demonstrated its efficacy at simulating many materials and the coupling between them on a massive scale. However, in scenarios containing debris, MPM manifests more dissipation and numerical viscosity than traditional Lagrangian methods. We have two observations from carefully revisiting existing integration methods used in MPM. First, nearby particles would end up with smoothed velocities without recovering momentum for each particle during the particle-grid-particle transfers. Second, most existing integrators assume continuity in the entire domain and advect particles by directly interpolating the positions from deformed nodal positions, which would trap the particles and make them harder to separate. We propose an integration scheme that corrects particle positions at each time step. We demonstrate our method's effectiveness with several large-scale simulations involving brittle materials. Our approach effectively reduces diffusion and unphysical viscosity compared to traditional integrators.

Abstract We present a hybrid particle/grid approach for simulating incompressible fluids on collocated velocity grids. Our approach supports both particle‐based Lagrangian advection in very detailed regions of the flow and efficient Eulerian grid‐based advection in other regions of the flow. A novel Backward Semi‐Lagrangian method is derived to improve accuracy of grid based advection. Our approach utilizes the implicit formula associated with solutions of the inviscid Burgers’ equation. We solve this equation using Newton's method enabled by C1 continuous grid interpolation. We enforce incompressibility over collocated, rather than staggered grids. Our projection technique is variational and designed for B‐spline interpolation over regular grids where multiquadratic 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.

In this paper, we present a mixed explicit and semi-implicit Material Point Method for simulating particle-laden flows. We develop a Multigrid Preconditioned fluid solver for the Locally Averaged Navier Stokes equation. This is discretized purely on a semi-staggered standard MPM grid. Sedimentation is modeled with the Drucker-Prager elastoplasticity flow rule, enhanced by a novel particle density estimation method for converting particles between representations of either continuum or discrete points. Fluid and sediment are two-way coupled through a momentum exchange force that can be easily resolved with two MPM background grids. We present various results to demonstrate the efficacy of our method.

We present a novel method for simulation of thin shells with frictional contact using a combination of the Material Point Method (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.

Recently the Affine Particle-In-Cell (APIC) Method was proposed by Jiang et al.[2015; 2017b] to improve the accuracy of the transfers in Particle-In-Cell (PIC) [Harlow 1964] techniques by augmenting each particle with a locally affine, rather than locally constant description of the velocity. This reduced the dissipation of the original PIC without suffering from the noise present in the historic alternative, Fluid-Implicit-Particle (FLIP) [Brackbill and Ruppel 1986]. We present a generalization of APIC by augmenting each particle with a more general local function. By viewing the grid-to-particle transfer as a linear and angular momentum conserving projection of the particle-wise local grid velocities onto a reduced basis, we greatly improve the energy and vorticity conservation over the original APIC. Furthermore, we show that the cost of the generalized projection is negligible over APIC when using a particular class of local polynomial functions. Lastly, we note that our method retains the filtering property of APIC and PIC and thus has similar robustness to noise.