二维变分不等式问题的自适应有限元分析  

SELF-ADAPTIVE FEM ANALYSIS FOR 2D VARIATIONAL INEQUALITY PROBLEMS

作　　者：袁驷[1] 刘泽洲[1] 邢沁妍[1] 

机构地区：[1]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,北京100084

出　　处：《工程力学》2016年第1期11-17,共7页Engineering Mechanics

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)

摘　　要：有限元后处理超收敛计算的EEP(单元能量投影)法以及基于该法的自适应有限元分析已在一维变分不等式问题的求解中取得显著成功。以此为基础,该文对二维变分不等式问题成功地实现了自适应有限元分析。该文提出二维区域二分法和二维C检验技术,有效地提升了松弛迭代的收敛速率,进而应用EEP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。该文给出的典型数值算例表明该文算法高效、稳定、精确,解答可逐点以最大模度量满足精度要求,堪称为数值精确解。The Element Energy Projection（EEP） method for super-convergent calculation in the post-process of FEM and the corresponding EEP-based self-adaptive FEM have achieved remarkable success in 1D variational inequality problems. In this paper, the EEP-based self-adaptive FEM for 2D variational inequality problems has been successfully developed. Two novel techniques, i.e. 2D bisection bounding technique and 2D C-check technique, are proposed, which significantly accelerate the convergence rate of the conventional relaxation iteration in finite element（FE） procedures. Once the converged FE solution is obtained, the super-convergent solution via the EEP method, developed by the first author＇s research team, will be calculated to estimate the FE errors and then to guide mesh refinement. Numerical examples presented show that the proposed algorithm is efficient, reliable, and accurate with the final results being virtually ‘numerically exact＇, i.e. satisfying the error tolerance by the maximum norm.

关 键 词：变分不等式 二维有限元 单元能量投影 超收敛 自适应 

分 类 号：TU311.4[建筑科学—结构工程]