Subspace Mixed-FEM for Real-Time Heterogeneous Elastodynamics

TL;DR

This paper addresses real-time elastodynamic simulation of heterogeneous materials on large meshes. It introduces a reduced-space MFEM built on a Skinning Eigenmode subspace together with a material-aware cubature scheme, achieving energy-preserving, real-time performance that decouples computation from mesh resolution. The approach delivers robust handling of extreme material and geometric heterogeneities with substantial speedups over full-space MFEM, while analyzing tradeoffs between subspace size and cubature density. The work has practical implications for interactive simulations in biomechanics and engineering, and it highlights directions for integrating global versus local subspaces and extending to contact and inverse design. The combination of rotation-invariant subspaces and efficient cubature forms a versatile toolkit for real-time heterogeneous elastodynamics.

Abstract

We propose a reduced space mixed finite element method (MFEM) built on a Skinning Eigenmode subspace and material-aware cubature scheme. Our solver is well-suited for simulating scenes with large material and geometric heterogeneities in real-time. This mammoth geometry is composed of 98,175 vertices and 531,565 tetrahedral elements and with a heterogenous composition of widely varying materials of muscles ($E= 5\times10^5$ Pa), joints ($E=1\times10^5$ Pa), and bone ($E=1\times10^{10}$ Pa). The resulting simulation runs at 120 frames per second (FPS).

Figures (15)

• Figure 1: A crab with a hard shell (E=1e10 Pa) and soft joints (E=1e6 Pa) is simulated with our subspace MFEM and skinning subspace FEM. With only 2 solver iterations MFEM exhibits correct rotational and elastic behavior, whereas subspace FEM with 4 iterations -- and consequently half the frame rate -- exhibits noticeable damping.
• Figure 2: Material sensitive skinning modes directly lead to richer motion for heterogeneous materials.
• Figure 3: The skinning weights we get from Skinning Eigenmodes are naturally material aware. High frequency modes are concentrated on soft parts of the snail, which are more likely to exhibit rich deformation. In contrast, the stiff shell only has access to a constant skinning weight (shared by all parts of the snail), allowing rigid motion to be producible within our skinning subspace.
• Figure 4: Modal derivatives are not suited for reconstructing rotations on the input shape. Fixing these artifacts typically requires explicitly tracking a rigid frame terzopoulous1988rigidanddeformablecomponents.
• Figure 5: Our cubature points are found as the centroids of each k-means cluster. Note that our centroids are sensitive to the heterogeneity of the Young's modulus. Stiffer regions can have their strain be approximated with fewer cubature points.