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.

翻译：本文提出了一种通用的可微分求解器，用于处理含接触与摩擦的时间依赖形变问题。该方法采用有限元离散化结合高阶时间积分器，并耦合近期提出的增量势接触法来处理接触与摩擦力的计算，以求解具有复杂几何场景中受常微分方程及偏微分方程约束的优化问题。本求解器支持静态与动态问题，并能对物理问题描述中涉及的所有物理参数进行微分，这些参数包括形状、材料参数、摩擦参数以及初始条件。我们通过解析推导得到的伴随形式具有高效性，其计算开销相较于正向仿真较小（对于非线性问题通常低于10%），且与正向问题具有诸多相似之处，从而能够复用现有正向仿真器代码的大部分内容。我们在开源PolyFEM库的基础上实现了该方法，并通过仿真结果与物理验证，展示了本求解器在形状设计、初始条件优化以及材料参数估计等任务中的适用性。