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MPM Lite: Linear Kernels and Integration without Particles

Xiang Feng, Yunuo Chen, Chang Yu, Hao Su, Demetri Terzopoulos, Yin Yang, Joe Masterjohn, Alejandro Castro, Chenfanfu Jiang

TL;DR

MPM Lite introduces a grid-centric, hybrid Lagrangian/Eulerian discretization that eliminates particle-based quadrature at solve time by resampling particle state onto fixed cell-centered quadrature points. It replaces particle-driven force assembly with an Incremental Potential formulation solved on a voxel FEM grid using a rotation-free stretch reference, enabling standard nonlinear solvers and boundary handling while preserving particle history. The approach achieves substantial speedups in implicit settings (up to ~15.9× at high PPC) and improved explicit performance, while maintaining robustness across elastoplastic, plastic, and visco-plastic materials, including multi-material coupling with efficient memory use. The method provides strong momentum conservation properties, hourglass stability immunity, and a versatile framework compatible with existing solvers and preconditioners, with explicit avenues for future work on anisotropy and adaptive meshes.

Abstract

In this paper, we introduce MPM Lite, a new hybrid Lagrangian/Eulerian method that eliminates the need for particle-based quadrature at solve time. Standard MPM practices suffer from a performance bottleneck where expensive implicit solves are proportional to particle-per-cell (PPC) counts due to the the choices of particle-based quadrature and wide-stencil kernels. In contrast, MPM Lite treats particles primarily as carriers of kinematic state and material history. By conceptualizing the background Cartesian grid as a voxel hexahedral mesh, we resample particle states onto fixed-location quadrature points using efficient, compact linear kernels. This architectural shift allows force assembly and the entire time-integration process to proceed without accessing particles, making the solver complexity no longer relate to particles. At the core of our method is a novel stress transfer and stretch reconstruction strategy. To avoid non-physical averaging of deformation gradients, we resample the extensive Kirchhoff stress and derive a rotation-free deformation reference solution, which naturally supports an optimization-based incremental potential formulation. Consequently, MPM Lite can be implemented as modular resampling units coupled with an FEM-style integration module, enabling the direct use of off-the-shelf nonlinear solvers, preconditioners, and unambiguous boundary conditions. We demonstrate through extensive experiments that MPM Lite preserves the robustness and versatility of traditional MPM across diverse materials while delivering significant speedups in implicit settings and improving explicit settings at the same time. Check our project page at https://mpmlite.github.io.

MPM Lite: Linear Kernels and Integration without Particles

TL;DR

MPM Lite introduces a grid-centric, hybrid Lagrangian/Eulerian discretization that eliminates particle-based quadrature at solve time by resampling particle state onto fixed cell-centered quadrature points. It replaces particle-driven force assembly with an Incremental Potential formulation solved on a voxel FEM grid using a rotation-free stretch reference, enabling standard nonlinear solvers and boundary handling while preserving particle history. The approach achieves substantial speedups in implicit settings (up to ~15.9× at high PPC) and improved explicit performance, while maintaining robustness across elastoplastic, plastic, and visco-plastic materials, including multi-material coupling with efficient memory use. The method provides strong momentum conservation properties, hourglass stability immunity, and a versatile framework compatible with existing solvers and preconditioners, with explicit avenues for future work on anisotropy and adaptive meshes.

Abstract

In this paper, we introduce MPM Lite, a new hybrid Lagrangian/Eulerian method that eliminates the need for particle-based quadrature at solve time. Standard MPM practices suffer from a performance bottleneck where expensive implicit solves are proportional to particle-per-cell (PPC) counts due to the the choices of particle-based quadrature and wide-stencil kernels. In contrast, MPM Lite treats particles primarily as carriers of kinematic state and material history. By conceptualizing the background Cartesian grid as a voxel hexahedral mesh, we resample particle states onto fixed-location quadrature points using efficient, compact linear kernels. This architectural shift allows force assembly and the entire time-integration process to proceed without accessing particles, making the solver complexity no longer relate to particles. At the core of our method is a novel stress transfer and stretch reconstruction strategy. To avoid non-physical averaging of deformation gradients, we resample the extensive Kirchhoff stress and derive a rotation-free deformation reference solution, which naturally supports an optimization-based incremental potential formulation. Consequently, MPM Lite can be implemented as modular resampling units coupled with an FEM-style integration module, enabling the direct use of off-the-shelf nonlinear solvers, preconditioners, and unambiguous boundary conditions. We demonstrate through extensive experiments that MPM Lite preserves the robustness and versatility of traditional MPM across diverse materials while delivering significant speedups in implicit settings and improving explicit settings at the same time. Check our project page at https://mpmlite.github.io.
Paper Structure (42 sections, 3 theorems, 118 equations, 16 figures, 2 tables, 4 algorithms)

This paper contains 42 sections, 3 theorems, 118 equations, 16 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Unloading/Loading Particle Information
  4. Kinematic Transfers
  5. Stress Transfers
  6. Becoming Finite Elements
  7. Error Analysis
  8. Incremental Potential Formulation
  9. Problem with Implicit Stress
  10. A Rotation-Free Stretch Reference Solution
  11. Stretch Reconstruction
  12. StVK with Hencky
  13. Split Neo–Hookean (deviatoric–volumetric form)
  14. Material Mixture
  15. Water
  16. ...and 27 more sections

Key Result

theorem 1

Let $i$ be a grid node of spacing $\Delta x$ and let particles $p$ carry $(m_p,v_p,G_p,x_p)$. Define the two weight families with normalizations $m_i^{(2\mathrm{hop})}:=\sum_p \alpha_p$ and $m_i^{(\beta)}:=\sum_p \beta_p$. Set Assume the continuum velocity $v$ is $C^{2,1}$ on the compact stencil that contributes to node $i$ (i.e. the componentwise Hessian is bounded and Lipschitz there) and that

Figures (16)

  • Figure 1: Particle-to-center transfer and center-to-grid transfer. In the particle-to-center transfer stage, particles unload their mass and momentum to the cell centers. Each cell center additionally accumulates volume and Kirchhoff stress. In the subsequent center-to-grid transfer, grid nodes gather mass, velocities, and forces from neighboring cell centers through a multilinear kernel. Owing to the constant kernel weights and purely gather-based formulation, this transfer can be parallelized without race conditions.
  • Figure 2: Jelly Falling. We simulate two jelly-like objects falling onto a third soft, elastic jelly using CK-MPM and our proposed MPM Lite, respectively.
  • Figure 3: Cantilever Beams. We present both quantitative and visual comparisons between MPM Lite and traditional implicit MPM. (Left) The elastic response produced by MPM Lite closely matches the theoretical predictions reported in romero2021physical. (Right) MPM Lite and traditional implicit MPM yield visually consistent deformation results for cantilever beams with varying stiffness.
  • Figure 4: Speedup curve with respect to PPC. A Faceless object is twisted using MPM Lite and traditional implicit MPM under varying particles-per-cell (PPC) settings. The total runtime of each simulation is reported in the figure. MPM Lite achieves up to a $15.9\times$ speedup at 24 PPC. Notably, traditional MPM often requires relatively large PPC ($\ge 20$) to prevent numerical fracture.
  • Figure 5: Stuffed Toys. A total of 18 stuffed toys are dropped into a glass container. All toys share the same hyperelastic material model. The scene contains 5.22M particles in total, and MPM Lite coupled with VBD simulates the system at $0.22$s per time step.
  • ...and 11 more figures

Theorems & Definitions (6)

  • theorem 1: P2G $v$ is second-order accurate
  • proof
  • theorem 2: G2P $v$ is second order accurate
  • proof
  • theorem 3: G2P $G$ is second order accurate
  • proof