Space-time boundary elements for frictional contact in elastodynamics

TL;DR

This work develops a space-time boundary element framework for dynamic frictional contact between linear elastic bodies, formulating the problem as a boundary variational inequality via the elastodynamic Poincaré-Steklov operator $\mathcal{S}$ and solving a mixed displacement–traction system with space–time Galerkin discretization. A space–time Uzawa algorithm handles unilateral friction (Tresca and Coulomb) and is extended to two-body contact, with boundary integral representations enabling efficient computations through marching-on-time schemes. Theoretical results include an a priori error estimate for the unilateral Tresca model, while numerical experiments in 2D validate stability, energy conservation, and convergence for realistic materials (concrete/steel) under both unilateral and two-sided friction. The approach demonstrates robust performance for curved and polygonal boundaries and for both symmetric and non-symmetric PS formulations, highlighting its potential for dynamic contact simulations, with future work targeting 3D extensions and space–time adaptivity.

Abstract

This article studies a boundary element method for dynamic frictional contact between linearly elastic bodies. We formulate these problems as a variational inequality on the boundary, involving the elastodynamic Poincaré-Steklov operator. The variational inequality is solved in a mixed formulation using boundary elements in space and time. In the model problem of unilateral Tresca friction contact with a rigid obstacle we obtain an a priori estimate for the resulting Galerkin approximations. Numerical experiments in two space dimensions demonstrate the stability, energy conservation and convergence of the proposed method for contact problems involving concrete and steel in the linearly elastic regime. They address both unilateral and two-sided dynamic contact with Tresca or Coulomb friction.

Figures (14)

• Figure 1: Schematic depiction of unilateral (a) and bilateral (b) contact problems.
• Figure 2: Schematic representation of the Neumann datum.
• Figure 3: Time evolution of $\textbf{u}_{h,\Delta t}$ in the midpoint of each side of $\Gamma$ for Test 1 (a, no friction), Test 2 (b, Tresca friction) and Test 3 (c, Coulomb friction), together with the corresponding vertical Neumann datum.
• Figure 4: Global deformation of the square at several time instants for Test 1 (a, no friction), Test 2 (b, Tresca friction) and Test 3 (c, Coulomb friction)
• Figure 5: Vertical component of $\pmb{\lambda}$ on $(0,T]\times \Gamma_C$, for Test 3 with Coulomb friction (elements from 1 to 20 are in $\Gamma_b$, from 21 to 40 in $\Gamma_l$).

Theorems & Definitions (11)

• Proposition 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• proof : Proof.
• Theorem 5
• proof
• Definition 6