A Posteriori Error Estimates for $hp$-FE Discretizations in Elastoplasticity

TL;DR

This paper develops a residual-based a posteriori error estimator for elastoplasticity with linear kinematic hardening and proves reliability and local efficiency for conforming $hp$ finite element discretizations. It analyzes three closely related discretizations of the model: a variational-inequality discretization and two mixed formulations with different multiplier sets, proving their equivalence via biorthogonal bases. The estimator combines an auxiliary-problem bound with a residual component and includes a minimization-based lower bound over admissible Lagrange multipliers. Numerical experiments in 2D illustrate substantial gains from $h$- and $hp$-adaptivity, especially near singular features like the elastic-plastic boundary, underscoring the practical impact for elastoplastic simulations.

Abstract

In this paper, a reliable a posteriori error estimator for a model problem of elastoplasticity with linear kinematic hardening is derived, which satisfies some (local) efficiency estimates. It is applicable to any discretization that is conforming with respect to the displacement field and the plastic strain. Furthermore, the paper presents $hp$-finite element discretizations relying on a variational inequality as well as on a mixed variational formulation and discusses their equivalence by using biorthogonal basis functions. Numerical experiments demonstrate the applicability of the theoretical findings and underline the potential of $h$- and $hp$-adaptive finite element discretizations for problems of elastoplasticity.

Figures (4)

• Figure 1: Deformation of $\Omega$ magnified by factor 10 for uniform mesh with $h=2^{-7}$ and $p=1$.
• Figure 2: Individual approximation errors vs. degrees of freedom. In the legend $hi$ stands for uniform $h$-mesh refinements with $p=i$, $p$ for uniform $p$-refinements with $h=0.4$, $ai$ for $h$-adaptive refinements with $p=i$ and $hp$ for $hp$-adaptive refinements.
• Figure 3: Individual approximation errors vs. degrees of freedom. In the legend $hi$ stands for uniform $h$-refinements with $p=i$, $p$ for uniform $p$-refinements with $h=0.4$, $ai$ for $h$-adaptive refinements with $p=i$ and $hp$ for $hp$-adaptive refinements.
• Figure 4: Adaptively generated meshes with comparable DOF.

Theorems & Definitions (32)

• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof