Differentiable solver for time-dependent deformation problems with contact

TL;DR

We address PDE-constrained optimization for time-dependent elastodynamics with contact and friction by introducing a general differentiable solver that combines high-order spatial and temporal discretization with the Incremental Potential Contact formulation. The core advancement is an analytically derived adjoint framework that yields gradients with respect to shape, material, friction, and initial conditions at low overhead (often under $10\%$ extra cost) and reuses forward-solver components for efficiency. The method supports static and dynamic problems, remeshing, and differentiability across a broad set of objectives, including shape optimization, material estimation, and initial-condition synthesis, demonstrated across numerous simulated and physically validated scenarios. This differentiable solver, implemented on PolyFEM, enables robust, accurate PDE-constrained optimization for complex geometries and paves the way for practical design and identification tasks in engineering and robotics.

Abstract

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations.

Figures (26)

• Figure 1: Notation for domains and maps we use, see Table \ref{['tab:notation']}.
• Figure 2: Domain perturbation $\theta$, see Table \ref{['tab:notation']}.
• Figure 3: An example of remeshing in the shape optimization. The quality is shown for each triangle. Triangles with bad quality have higher values.
• Figure 4: Static: Bridge With Fabricated Solution. The result of the shape optimization (blue surface) matches the target shape (wire-frame).
• Figure 5: Static: Bridge. Result of shape optimization to minimize the average stress.