Coupling of FEM and BEM for Interface Problems in Viscoplasticity and Plasticity with Hardening

Abstract

This paper is concerned with three-dimensional interface (or transmission) problems in solid mechanics which consist of the quasi-static equilibrium condition in a bounded Lipschitz domain $\Omega $, a first order evolution inclusion in $\Omega $, and the homogeneous linear elasticity problem in an unbounded exterior domain $\Omega _2 $. The evolution problem in $\Omega $ models viscoplasticity and Prandtl–Reuß plasticity with hardening. In a former paper [SIAM J. Math. Anal., 25 (1994), pp. 1468–1487] the author stated the interface problem as well as an equivalent formulation and proved existence and uniqueness of solutions. Here, the numerical approximation of the solutions is considered via a symmetric coupling of the FEM (finite element method) and BEM (boundary element method) in space and the generalized midpoint rule in time. Besides existence, uniqueness, and uniform boundedness of the solutions of the discrete problems, error estimates proving strong convergence of the numerical procedure are given. For the implicit Euler method the author also proves strong convergence for the time derivatives. Finally, the Uzawa algorithm is adopted to be used for the computation of the discrete solutions.