MLS-MPM: Moving Least Squares Material Point Method
This presentation explores the Moving Least Squares Material Point Method, a hybrid computational approach that solves large-deformation solid mechanics by combining particles with background grids. We examine how MLS-MPM uses local moving least squares reconstruction to create smooth, high-order interpolation kernels that eliminate spurious stress oscillations at cell boundaries—a critical advancement for accurate simulation of extreme deformations on unstructured meshes.Script
Standard computational methods for simulating solid mechanics break down when materials undergo extreme deformation—stress calculations spike artificially every time a particle crosses from one grid cell to another. The Moving Least Squares Material Point Method eliminates these cell-crossing errors by constructing interpolation kernels that remain smooth no matter how violently the material deforms.
MLS-MPM is a hybrid Eulerian-Lagrangian scheme. Particles move through space carrying the full material history—stress state, deformation gradient, even plasticity parameters. Meanwhile, a background grid, either structured or fully unstructured simplicial mesh, provides the stage for efficient gradient computation. Every timestep, we rasterize particle data onto grid nodes, update velocities there, and pull the results back to particles.
What makes this transfer smooth is the moving least squares kernel itself.
Here's the core idea. At every particle location, MLS-MPM solves a localized weighted least squares problem to reconstruct the field and its gradient from values at nearby grid vertices. The support region—called the 1-ring—includes not just the containing element's vertices but also their immediate neighbors. A diminishing factor ensures that when a particle crosses an element boundary and the vertex set changes, the weights fade in and fade out smoothly rather than jumping discontinuously.
Compare this to standard Material Point Method. There, piecewise-linear basis functions mean gradients are discontinuous across element faces. Every time a particle hops cells, the gradient field jumps, injecting spurious stress oscillations that degrade accuracy and convergence. MLS-MPM's continuous gradients eliminate these artifacts entirely, restoring second-order convergence even under violent deformation.
Rigorous benchmarks confirm the method works. In a 1D bar vibration test, displacement error stays below 0.55 percent on coarse grids, with clean second-order convergence and energy conservation better than 0.1 percent. Colliding elastic disks in 2D lose less than 1 percent momentum and show no oscillations. And in full 3D—elastoplastic slope collapse, spherical container expansion, fluid-solid coupling—MLS-MPM on unstructured meshes delivers results indistinguishable from gold-standard total-Lagrangian solvers.
MLS-MPM proves that smooth, stable simulation of extreme deformation on arbitrary meshes is not just possible—it's practical. By embedding moving least squares directly into the particle-grid transfer, the method turns a numerical nuisance into a solved problem. To explore MLS-MPM further and create your own videos on computational physics, visit EmergentMind.com.
