Neurally Integrated Finite Elements for Differentiable Elasticity on Evolving Domains

TL;DR

This paper addresses differentiable elasticity on evolving implicit domains by introducing a neurally integrated finite element framework that combines a neural quadrature rule with a high-order four-field mixed FEM. The neural quadrature learns per-cell quadrature points and weights from corner SDF values, enabling smooth, differentiable integration as geometry evolves. The method extends to a generalized mixed FEM with rotation and symmetric strain, and is integrated into a physics-aware reconstruction pipeline (e.g., FlexiCubes) to jointly optimize geometry and material properties while leveraging adjoint gradients. The resulting approach supports robust forward simulation, interactive editing, and topology/material optimization guided by differentiable rendering, with demonstrated improvements in stability, sub-voxel feature resolution, and physically plausible reconstructions across complex geometries and material contrasts.

Abstract

We present an elastic simulator for domains defined as evolving implicit functions, which is efficient, robust, and differentiable with respect to both shape and material. This simulator is motivated by applications in 3D reconstruction: it is increasingly effective to recover geometry from observed images as implicit functions, but physical applications require accurately simulating and optimizing-for the behavior of such shapes under deformation, which has remained challenging. Our key technical innovation is to train a small neural network to fit quadrature points for robust numerical integration on implicit grid cells. When coupled with a Mixed Finite Element formulation, this yields a smooth, fully differentiable simulation model connecting the evolution of the underlying implicit surface to its elastic response. We demonstrate the efficacy of our approach on forward simulation of implicits, direct simulation of 3D shapes during editing, and novel physics-based shape and topology optimizations in conjunction with differentiable rendering.

Figures (23)

• Figure 1: Illustration in 2D of order-2 quadrature points for a boundary voxel for, from left to right, Full, Clip and Neural quadrature formulae. The neural quadrature points and weights are updated smoothly when the boundary moves, while other formulas experience jumps.
• Figure 2: Learned quadrature points (yellow) for integrating over the part of the unit voxel defined by trilinear interpolation of the corner SDF values (visualized by the green isosurface). Left and middle depict 8-point order-2 quadrature, right is 27 point order-4. Size is proportional to the quadrature point weight.
• Figure 3: Visualization of quadrature points generated for a SDF discretized on a grid at resolution $16^3$, using order-$2$ Full (top left) and Clip (bottom left) quadratures, and our Neural quadrature at order $2$ (top right) and $4$ (bottom right). The non-empty voxels are shown in blue, and the SDF iso-surface (extracted at resolution $64^3$) is shown in purple. For clarity, only one octant is shown.
• Figure 4: Comparison of equilibrium behavior for the dumbbell across several SDF grid resolutions and discretizations. Left, a tetrahedral mesh is first extracted from the grid via the FlexiCubes algorithm, and simulated with linear and quadratic elements. Right, our grid-based simulations are performed with full and clipped quadrature on tri-linear elements, as well as our neural quadrature on both tri-linear and tri-quadratic elements. Color denotes the grid resolution at which the mesh is extracted or the simulation is performed, respectively. The grid simulation is interpolated to a high-resolution surface for visualization.
• Figure 5: At high stiffness ratio, displacement-only FEM suffers from slow convergence, while Mixed FEM does not. Here, the center region of the dumbbell is stiffened by an increasing factor, and in each case the Newton loop is truncated after 250 iterations.