Convergence of dG(1) in elastoplastic evolution
Convergence of dG(1) in elastoplastic evolution
复制标题
弹塑性演化中 dG(1) 的收敛
DOI：
10.1007/s00211-018-0999-6
复制
发表时间：
2019
影响因子：
2.1
通讯作者：
Alberty Jochen
中科院分区：
文献类型：
--
作者：
Carstensen Carsten;Liu Dongjie;Alberty Jochen
不连续伽辽金（dG）方法为演化问题提供了时间离散方案的层次结构，例如具有Prandtl-Reuß流规则的弹塑性问题。Alberty和Carstensen在（CMME 191:4949-4968, 2002）中提出了一种dG时间离散化方法，用于在速率无关的非弹性材料行为背景下的变分不等式，利用凸分析中的对偶性来证明某些跳跃项的合理性。本文首次对具有不连续分段线性多项式的dG(1)格式在时间和最低阶有限元下的空间离散化进行了先验误差分析。与广义中点规则相比，dG(1)公式将物质定律的作用在时间上以变分不等式的形式分布，因而引入了物质定律的误差。这可能会导致dG(1)方案的次优收敛速度，本文表明，norm中的应力误差仅仅是基于一个看似尖锐的误差分析。对具有已知解析解的基准问题的数值研究提供了经验证据，证明dG(1)格式比dG(0)格式具有更高的收敛速度。
The discontinuous Galerkin (dG) methodology provides a hierarchy of time discretization schemes for evolutionary problems such as elastoplasticity with the Prandtl-Reuß flow rule. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in Alberty and Carstensen in (CMME 191:4949–4968, 2002) with the help of duality in convex analysis to justify certain jump terms. This paper establishes the first a priori error analysis for the dG(1) scheme with discontinuous piecewise linear polynomials in the temporal and lowest-order finite elements for the spatial discretization. Compared to a generalized mid-point rule, the dG(1) formulation distributes the action of the material law in the form of the variational inequality in time and so it introduces an error in the material law. This may result in a suboptimal convergence rate for the dG(1) scheme and this paper shows that the stress error in thenorm is merelybased on a seemingly sharp error analysis. The numerical investigation for a benchmark problem with known analytic solution provides empirical evidence of a higher convergence rate of the dG(1) scheme compared to dG(0).