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 PDE- and ODE-constrained optimization problems on scenes with a complex geometry. It support 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 in physical validations.

翻译：我们提出了一种通用的可微分解法，用于求解含接触与摩擦的时间依赖变形问题。该方法采用有限元离散结合高阶时间积分器，并融合最新提出的增量势接触方法来处理接触与摩擦力，从而解决复杂几何场景中受偏微分方程和常微分方程约束的优化问题。它支持静态与动态问题，并可针对物理问题描述中涉及的所有物理参数（包括形状、材料参数、摩擦参数及初始条件）进行微分。我们基于解析推导的伴随公式具有高效性，其非线性问题的额外开销通常低于正向模拟的10%，且与正向问题高度相似，因此可重用现有正向模拟器的大部分代码。该方法在开源PolyFEM库基础上实现，通过仿真结果与实际物理验证，展示了该求解器在形状设计、初始条件优化及材料参数估计等方面的适用性。