We introduce a PDE-based node-to-element contact formulation as an alternative to classical, purely geometrical formulations. It is challenging to devise solutions to nonsmooth contact problem with continuous gap using finite element discretizations. We herein achieve this objective by constructing an approximate distance function (ADF) to the boundaries of solid objects, and in doing so, also obtain universal uniqueness of contact detection. Unilateral constraints are implemented using a mixed model combining the screened Poisson equation and a force element, which has the topology of a continuum element containing an additional incident node. An ADF is obtained by solving the screened Poisson equation with constant essential boundary conditions and a variable transformation. The ADF does not explicitly depend on the number of objects and a single solution of the partial differential equation for this field uniquely defines the contact conditions for all incident points in the mesh. Having an ADF field to any obstacle circumvents the multiple target surfaces and avoids the specific data structures present in traditional contact-impact algorithms. We also relax the interpretation of the Lagrange multipliers as contact forces, and the Courant--Beltrami function is used with a mixed formulation producing the required differentiable result. We demonstrate the advantages of the new approach in two- and three-dimensional problems that are solved using Newton iterations. Simultaneous constraints for each incident point are considered.

翻译：我们提出了一种基于偏微分方程的节点到单元接触公式，作为经典纯几何公式的替代方案。使用有限元离散化来设计具有连续间隙的非光滑接触问题的解决方案具有挑战性。本文通过构造固体边界的近似距离函数（ADF）实现了这一目标，并在此过程中获得了接触检测的普适唯一性。单边约束通过结合筛选泊松方程和力单元的混合模型实现，该力单元具有包含额外入射节点的连续体单元拓扑结构。通过求解具有恒定本质边界条件和变量变换的筛选泊松方程获得ADF。ADF不显式依赖于物体数量，且该场的偏微分方程单次解唯一地定义了网格中所有入射点的接触条件。拥有指向任意障碍物的ADF场可规避传统接触-冲击算法中的多个目标曲面及特定数据结构。我们还松动了将拉格朗日乘子解释为接触力的约束，采用Courant-Beltrami函数与混合公式结合，产生所需的可微结果。我们通过牛顿迭代求解的二维和三维问题展示了新方法的优势，并考虑了每个入射点的同时约束。