Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On an optimal AFEM for elastoplasticity

Miriam Schönauer, Andreas Schröder

TL;DR

The paper addresses optimal convergence of adaptive finite element methods for a quasi-static elastoplasticity model with isotropic and linear kinematic hardening, formulated as a variational inequality of the second kind. It uses the axioms of adaptivity framework to verify stability, reduction, quasi-orthogonality, and discrete reliability, and combines these with newest-vertex-bisection refinements to prove convergence and quasi-optimal rates. A key contribution is showing that the AFEM for elastoplasticity satisfies the axioms and achieves optimal convergence without requiring estimator efficiency, contrasting with the ElastoAFEM approach. The work thus provides a robust, generalizable pathway to optimal AFEM performance in nonlinear, variational-inequality settings with practical implications for computational mechanics.

Abstract

In this paper, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [Comput. Math. Appl., 67(6) (2014), 1195-1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [Numer. Math., 132(1) (2016), 131-154], which presents an alternative approach to optimality without explicitly relying on the axioms.

On an optimal AFEM for elastoplasticity

TL;DR

The paper addresses optimal convergence of adaptive finite element methods for a quasi-static elastoplasticity model with isotropic and linear kinematic hardening, formulated as a variational inequality of the second kind. It uses the axioms of adaptivity framework to verify stability, reduction, quasi-orthogonality, and discrete reliability, and combines these with newest-vertex-bisection refinements to prove convergence and quasi-optimal rates. A key contribution is showing that the AFEM for elastoplasticity satisfies the axioms and achieves optimal convergence without requiring estimator efficiency, contrasting with the ElastoAFEM approach. The work thus provides a robust, generalizable pathway to optimal AFEM performance in nonlinear, variational-inequality settings with practical implications for computational mechanics.

Abstract

In this paper, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [Comput. Math. Appl., 67(6) (2014), 1195-1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [Numer. Math., 132(1) (2016), 131-154], which presents an alternative approach to optimality without explicitly relying on the axioms.
Paper Structure (20 sections, 6 theorems, 78 equations)

This paper contains 20 sections, 6 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Model of elastoplasticity
  3. Equilibrium equation
  4. Linear hardening
  5. Weak formulation of variational inequality
  6. Discretization with adaptive finite elements
  7. Discrete solution
  8. Error measure
  9. Refinement indicator and global error estimator
  10. Newest vertex bisection
  11. Adaptive refinements
  12. An optimal adaptive finite element method
  13. Preliminaries
  14. The axioms of adaptivity
  15. Convergence and quasi-optimal convergence
  16. ...and 5 more sections

Key Result

Lemma 1

There exists a constant $C_1>0$ such that for all $\hat{\mathcal{T}}\in\mathbb{T}$, $\mathcal{S}\subseteq \mathcal{T}\cap\hat{\mathcal{T}}$, $z\in X(\mathcal{T})$ and $\hat{z}\in X(\hat{\mathcal{T}})$.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Lemma 1: A1
  • proof
  • Lemma 2: A2
  • proof
  • Lemma 3: A3
  • proof
  • Lemma 4: A4
  • proof
  • ...and 4 more