Time domain boundary elements for dynamic contact problems

TL;DR

This paper develops a boundary element method in the time domain to solve dynamic contact problems for the wave equation, providing a new approach with proven stability and convergence.

Contribution

It introduces a saddle point formulation using boundary elements for the wave equation's boundary variational inequality, including a model for the single layer operator.

Findings

A priori estimates for Galerkin approximations

Numerical experiments show stability and convergence

Method applicable beyond flat contact geometries

Abstract

This article considers a unilateral contact problem for the wave equation. The problem is reduced to a variational inequality for the Dirichlet-to-Neumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using boundary elements in the time domain. As a model problem, also a variational inequality for the single layer operator is considered. A priori estimates are obtained for Galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area, where the existence of solutions to the continuous problem is known. Numerical experiments demonstrate the performance of the proposed mixed method. They indicate the stability and convergence beyond flat geometries.