Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixed Finite Elements of Higher-Order in Elastoplasticity

Patrick Bammer, Lothar Banz, Andreas Schröder

Abstract

In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.

Mixed Finite Elements of Higher-Order in Elastoplasticity

Abstract

In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.
Paper Structure (6 sections, 11 theorems, 113 equations, 2 figures, 1 table)

This paper contains 6 sections, 11 theorems, 113 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model Problem and Notation
  3. A Mixed Variational Formulation
  4. An $hp$-Finite Element Discretization of the Mixed Formulation
  5. A Priori Error Analysis
  6. Numerical Results

Key Result

Theorem 1

If $(\mathfrak{u},\boldsymbol{p})\in V\times Q$ solves BBS_eq:variq_second_kind, then $(\mathfrak{u},\boldsymbol{p},\boldsymbol{\lambda})$ with $\boldsymbol{\lambda} := \operatorname{dev}(\boldsymbol{\sigma}(\mathfrak{u},\boldsymbol{p}) - \mathbb{H}\boldsymbol{p})$ is a solution of BBS_eq:mixed_vari

Figures (2)

  • 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. Legend: $hi$ stands for uniform $h$-refinement with $p=i$, $p$ for uniform $p$-refinement with $h=0.4$, $ai$ for $h$-adaptive scheme with $p=i$ and $hp$ for $hp$-adaptive scheme.

Theorems & Definitions (23)

  • Remark
  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • ...and 13 more