Neurally Integrated Finite Elements for Differentiable Elasticity on Evolving Domains

TL;DR

This paper introduces a neural-integrated finite element method for differentiable elasticity simulation on evolving implicit domains, enabling efficient shape optimization and 3D reconstruction.

Contribution

It presents a novel neural quadrature approach combined with finite elements for robust, differentiable elastic simulation on implicit surfaces.

Findings

Effective forward simulation of implicit shapes

Accurate shape editing and deformation

Enabling physics-based shape and topology optimization

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.