Lothar Banz

Nina Ovcharova, Lothar Banz

In this paper, we couple regularization techniques with the adaptive $hp$-version of the boundary element method ($hp$-BEM) for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions. As a model example we treat the delamination of composite structures with a contaminated interface layer. This problem has a weak formulation in terms of a nonsmooth variational inequality. The resulting hemivariational inequality (HVI) is first regularized and then, discretized by an adaptive $hp$-BEM. We give conditions for the uniqueness of the solution and provide an a-priori error estimate. Furthermore, we derive an a-posteriori error estimate for the nonsmooth variational problem based on a novel regularized mixed formulation, thus enabling $hp$-adaptivity. Various numerical experiments illustrate the behavior, strengths and weaknesses of the proposed high-order approximation scheme.

Lothar Banz, Gregor Milicic, Nina Ovcharova

We improve the residual based stabilization technique for Signorini contact problems with Tresca friction in linear elasticity solved with $hp$-mixed BEM which has been recently analyzed by Banz et al.~in Numer.~Math.~135 (2017) pp.~217--263. The stabilization allows us to circumvent the discrete inf-sup condition and thus the primal and dual sets can be discretized independently. Compared to the above mentioned paper we are able to remove the dependency of the scaling parameter on the unknown Sobolev regularity of the exact solution and can thus also improve the convergence rate in the a priori error estimate. The second improvement is a rigorous a priori and a posteriori error analysis when the boundary integral operators in the stabilization term are approximated. The latter is of fundamental importance to keep the computational time small. We present numerical results in two and three dimensions to underline our theoretical findings, show the superiority of the $hp$-adaptive stabilized mixed scheme and the effect induced by approximating the stabilization term. Moreover, we show the applicability of the proposed method to the Coulomb frictional case for which we extend the a posteriori error analysis

Patrick Bammer, Lothar Banz, Miriam Schönauer et al.

This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational formulation is introduced to resolve the non-differentiablility of the so-called plasticity functional appearing in the weak formulation of the model problem as a variational inequality of the second kind. The discretization of the mixed formulation is then represented as a system of decoupled nonlinear equations which allows the application of an efficient semismooth Newton solver. Finally, an a priori and a posteriori error analysis is given.