Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

doi:10.1007/s10483-018-2345-7

Applied Mathematics and Mechanics (English Edition) ›› 2018, Vol. 39 ›› Issue (7): 1031-1044.doi: https://doi.org/10.1007/s10483-018-2345-7

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

Si YUAN, Yue WU, Qinyan XING

Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China

收稿日期:2017-12-13 修回日期:2018-02-25 出版日期:2018-07-01 发布日期:2018-07-01
通讯作者: Qinyan XING E-mail:xingqy@tsinghua.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 51378293 and 51078199)

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

Si YUAN, Yue WU, Qinyan XING

Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China

Received:2017-12-13 Revised:2018-02-25 Online:2018-07-01 Published:2018-07-01
Contact: Qinyan XING E-mail:xingqy@tsinghua.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 51378293 and 51078199)

摘要/Abstract

摘要： This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-bydimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson's equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.

关键词: polycrystal, elastic-plastic deformation, constitutive relation, yield, work hardening, dislocation, three-dimensional(3D) problem, generalized one-dimensional(1D) finite element method(FEM), element energy projection(EEP), dimension-by-dimension(D-by-D), super-convergence

Abstract: This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-bydimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson's equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.

Key words: polycrystal, elastic-plastic deformation, constitutive relation, yield, work hardening, dislocation, generalized one-dimensional(1D) finite element method(FEM), super-convergence, element energy projection(EEP), dimension-by-dimension(D-by-D), three-dimensional(3D) problem

中图分类号:

Si YUAN, Yue WU, Qinyan XING. Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(7): 1031-1044.

参考文献

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Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

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