一维变分不等式问题的自适应有限元分析新探

发布时间：2018-07-08 16:52

本文选题：变分不等式 + 有限元　；参考：《工程力学》2015年07期

【摘要】：结构工程中的弹性薄膜接触和杆件弹塑性扭转等问题是典型的变分不等式问题,对其高效精确求解,特别是满足给定精度要求下的自适应求解,是挑战性课题。该文作者新近成功实现了一维变分不等式问题的自适应有限元分析,该文对此进展作一报道。对于变分不等式的有限元求解,该文提出区域二分法和C检验技术,极大提升了松弛迭代的收敛速度,一般4次~5次线性解即可得到收敛的有限元解答,进而采用作者提出的EEP(单元能量投影)超收敛公式计算超收敛解答,用其检验误差并指导网格细分,逐步得到堪称为数值精确解的解答,亦即得到按照最大模度量逐点满足精度要求的解答。该文给出的数值算例表明所提出的算法具有高效、可靠、精确的优良特性。
[Abstract]:The problems of elastic film contact and elastic-plastic torsion of members in structural engineering are typical variational inequality problems. It is a challenging task to solve them efficiently and accurately, especially to solve them adaptively under given precision requirements. In this paper, the authors have recently successfully implemented adaptive finite element analysis for one-dimensional variational inequality problems. For the finite element solution of variational inequalities, a domain dichotomy method and a C-test technique are proposed in this paper. The convergence rate of relaxation iteration is greatly improved, and the convergent finite element solution can be obtained by a linear solution of 4 times or 5 times. Then the superconvergence solution is calculated by using the EEP (element energy projection) superconvergence formula proposed by the author, and the error is checked and the mesh subdivision is guided, and the solution which is called numerical exact solution is obtained step by step. In other words, the solution is obtained to satisfy the precision requirement point by point according to the maximum norm metric. The numerical examples given in this paper show that the proposed algorithm has the advantages of high efficiency, reliability and accuracy.
【作者单位】：清华大学土木工程系土木工程安全与耐久教育部重点实验室;
【基金】：国家自然科学基金项目(51378293,51078199,50678093,50278046) 长江学者和创新团队发展计划项目(IRT00736)
【分类号】：TU311.4

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