M2P 2023

Elastodynamic Contact Problems by Energetic Boundary Element Method

  • Aimi, Alessandra (University of Parma)
  • Di Credico, Giulia (University of Parma)
  • Gimperlein, Heiko (University of Innsbruck)

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In this presentation we investigate the numerical solution of unilateral contact problems, which consider the elastodynamic stresses of a body in contact with a rigid obstacle. Such problems arise in numerous mechanical applications, from fracture dynamics and crash tests to rolling car tires. Mathematically, the nonlinear boundary condition corresponding to the contact interaction translates into a variational inequality for the linear elastodynamic equations. While there is a vast computational literature for the time dependent problem, its rigorous theoretical analysis has proven difficult and results about the existence of solutions are only known in simplified situations. Nevertheless, the importance of this study for practical purposes, above all in the time-domain framework, is clear and treatment by Boundary Element Methods represents a natural choice, given that the contact is confined to the boundary while the interior dynamics is linear. For this reason, we propose an Energetic Galerkin Boundary Element Method (E-BEM), that provides an efficient and stable numerical strategy for linear elastodynamics, here adapted for the solution of contact problems. To do that, we start from a weak formulation of the contact problem elaborated as a variational inequality involving the Poincaré-Steklov operator. For its approximation, from the algorithmic point of view, we require the assembly of the E-BEM matrices. This procedure is then followed by an Uzawa method, employed as a solver for the nonlinear problem. Stability, convergence and implementational aspects of the E-BEM are discussed and numerical experiments are presented for a range of 2D geometries, with unilateral contact conditions imposed on part of the boundary (without friction). For example, Figure 1 shows the displacement of a circular elastic body which is hit by a rigid obstacle. The results indicate the stability, convergence and performance of the proposed method for the treatment of elastodynamic contact problems.