一维变分不等式问题的自适应有限元分析新探

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。

A NEW APPROACH TO SELF-ADAPTIVE FEM FOR ONE-DIMENSIONAL VARIATIONAL INEQUALITY PROBLEMS

Elastic membrane contact problems and elasto-plastic torsion problems are typical variational inequality problems, and the accurate and efficient analyses of these problems, especially in a self-adaptive manner to satisfy the user-preset error tolerance, are still of great challenges. This paper reports a recently developed approach to the self-adaptive finite element (FE) analysis of one-dimensional (1D) variational inequality problems. To obtain the FE solution rapidly based on variational inequalities, the present paper proposed two techniques, i.e. bisection bounding and C-check technique, which greatly accelerate the convergence rate of the conventional relaxation iteration, and in average, it suffices to implement only 4 to 5 steps of linear FE analysis. With the converged FE solution obtained, the super-convergent solution by EEP (Element Energy Projection) method, also developed by the first author’s research team, is calculated to estimate the errors and then to guide a mesh refinement with final results being virtually ‘numerically exact’, i.e. satisfying the error tolerance by the maximum norm. The numerical examples presented show that the proposed approach is efficient, accurate and reliable.