On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

Abstract

In this paper, we couple regularization techniques of nondifferentiable optimization with the h-version of the boundary element method (h-BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality (HVI) with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by a h-BEM. We prove convergence of the h-BEM Galerkin solution of the regularized problem in the energy norm, provide an a-priori error estimate and give a numerical example.

Coupling of finite element and boundary element methods with regularization for a nonlinear interface problem with nonmonotone set-valued transmission conditions

“…Proof. First, similar to the proof of Theorem 5 in [35], in virtue of Lemma 5,(39), further by ( 43) and ( 37), there holds for 0 < h < h 0 modulo a positive constant, independent of h, for all (u…”

Coupling regularization and adaptive hp-BEM for the solution of a delamination problem

“…The hypotheses (H1) and (H2) are due to Glowinski [17] and describe the Mosco convergence [1] of the family K t to K, whereas f t in (H4) is a standard approximation of the linear functional f , for example, by numerical integration. The hypotheses (H5)-(H6) have been verified in [33]. We apply Theorem 4 with t = (ε, h, p) and show convergence for ε → 0 + , and either p → ∞ or h → 0.…”

“…In this section, we discuss the uniqueness of solution of the boundary hemivariational inequality and of the corresponding regularization problem. The main results are presented in Theorem 2 which is based on the abstract uniqueness Theorem 1 from [33] that gives a sufficient condition for uniqueness. Similar uniqueness result but for the regularized problem is derived in Theorem 3.…”

“…We emphasize that in fact the weak convergence of {u ε hp } to u in H 1/2 (Γ ) can be replaced by the strong one. The complete proof of this result for the h-version of BEM can be found in [33].…”

“…In this paper, we extend the approximation scheme from [33] (dealing with the h -version of the BEM) to the hp-version of the BEM for the regularized problem. More precisely, the nonsmooth variational inequality is first regularized and then discretized by hp-adaptive BEM.…”