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

TL;DR

This work develops a semismooth Newton solver for a broad class of nonsmooth nonlinear systems arising from $hp$-finite element discretizations in elastoplasticity. By exploiting a structured Clarke subdifferential and the notion of eigencomplementarity among block submatrices, the authors establish well-definedness and local convergence, with a $1+\alpha$-order rate when the nonsmoothities are $\alpha$-order semismooth. They apply the framework to a mixed variational formulation of elastoplasticity with linear kinematic hardening, using biorthogonal bases to decouple constraints and obtain a decoupled nonlinear system that fits the theoretical setting. Numerical experiments on uniform, p-, and hp-adaptive meshes confirm robustness with respect to $h$ and $p$ and corroborate superlinear convergence, with observed rates around $4/3$ in practice.

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 $p$.

Figures (1)

• Figure 1: Iteration number $k$ vs. $q_\text{SSN}(\rho)$.

Theorems & Definitions (19)

• Lemma 1
• proof
• Remark 1
• Definition 1
• Remark 2
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4