Neurally Integrated Finite Elements for
Differentiable Elasticity on Evolving Domains

Our neurally integrated, high-order mixed finite element solver can accurately simulate evolving implicit geometry (including sub-grid features). It is end-to-end differentiable and can be easily combined with other differentiable tools to enable new applications such as image-guided material, 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.

Results

We reconstruct structurally-sound chairs by using our simulator as a physics-prior in FlexiCubes geometry reconstruction pipeline. Right: rest state, Left: simulated – colors show the stress distribution in the simulated geometry. (Note: The jittering seen in the video is due to random perturbations applied at each iteration of the optimization process.)

Interactively simulating chairs reconstructed without our physics-prior (Left) leads to geometry that is not self-supporting and/or deforms excessively under user specified forces. The addition of our physics prior (Right) leads to geometry that is more robust during interaction.

Our solver runs interactively inside a shape sculpting tool. A user can manipulate and sculpt the geometry in real-time, ensuring that it behaves in the desired fashion when simulated.

We reconstruct the lego bulldozer by combining our simulator with nvdiffrec, a photogrammetry pipeline using a differentiable renderer. First we optimize for a self-supporting shape and next we optimize for a self-supporting material distribution. Colors show stiffness of materials.

Citation

@article{10.1145/3727874,
    author = {Daviet, Gilles and Shen, Tianchang and Sharp, Nicholas and Levin, David I.W.},
    title = {Neurally Integrated Finite Elements for Differentiable Elasticity on Evolving Domains},
    year = {2025},
    issue_date = {April 2025},
    publisher = {Association for Computing Machinery},
    address = {New York, NY, USA},
    volume = {44},
    number = {2},
    issn = {0730-0301},
    url = {https://doi.org/10.1145/3727874},
    doi = {10.1145/3727874},
    journal = {ACM Trans. Graph.},
    month = apr,
    articleno = {20},
    numpages = {17},
    keywords = {Differentiable simulation, numerical integration, topology optimization, 
        shape reconstruction}
}