# Continuous gap contact formulation based on the screened Poisson equation

## TL;DR

This paper presents a novel PDE-based contact formulation using an approximate distance function derived from the screened Poisson equation, enabling continuous gap contact modeling with universal contact detection and improved computational robustness.

## Contribution

It introduces a PDE-based node-to-element contact formulation that uses an approximate distance function, providing a unified, mesh-independent approach to contact detection and enforcement.

## Key findings

- Successfully applied to 2D and 3D problems with Newton iterations.
- Achieves universal contact detection independent of object number.
- Circumvents traditional data structures in contact algorithms.

## Abstract

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.