A Semismooth Newton Solver and its application to an $hp$-FE Discretization in Elastoplasticity

TL;DR

This paper introduces a semismooth Newton solver tailored for nonlinear systems from $hp$-finite element discretizations in elastoplasticity, proving its convergence and demonstrating robustness through numerical tests.

Contribution

The paper develops a specialized semismooth Newton method for $hp$-FEM systems in elastoplasticity, with theoretical convergence analysis and practical numerical validation.

Findings

Proven local convergence of the solver.

Numerical robustness with respect to mesh size and polynomial degree.

Effective application to elastoplasticity problems.

Abstract

In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and prove its well-definedness and local convergence. Thereby, the analysis heavily relies on a special eigenvalue interplay of two matrices involved in the considered nonlinear system. Next, we apply the general results to an $hp$-finite element discretization of a problem in elastoplasticity, which can be formulated as a system of nonlinear equations of the above type by using biorthogonal basis functions. Finally, numerical examples demonstrate the applicability and robustness of the proposed semismooth Newton solver with respect to $h$ and $p$.