内置了最大模误差估计器的自适应有限元法——研究进展与展望  

Adaptive finite element method with built-in error estimator in maximum norm:progress and prospects

作　　者：袁驷[1,2] 袁全 杨帅 王亦平 刘海阳 Yuan Si;Yuan Quan;Yang Shuai;Wang Yiping;Liu Haiyang(School of Civil Engineering,Tsinghua University,Beijing 100084,China;Key Laboratory of Civil Engineering Safety and Durability of the Ministry of Education,Tsinghua University,Beijing 100084,China)

机构地区：[1]清华大学土木水利学院,北京100084 [2]清华大学土木工程安全与耐久教育部重点实验室,北京100084

出　　处：《土木工程学报》2024年第6期43-58,共16页China Civil Engineering Journal

基　　金：国家自然科学基金(51878383,51378293)。

摘　　要：新近提出的降阶单元,将常规单元解答的最高次项视为误差项,当作内置的最大模误差估计器;将降低一次的解作为有限元最终解,从而一举实现了按最大模控制误差的自适应有限元算法。降阶单元最初针对结构时程分析的自适应步长算法提出,同时将其推广到一般的一维初值问题和边值问题。其后,将其推广到二维边值问题,构造了任意四边形和三角形降阶单元,已成功求解弹性薄膜挠度、弹性力学平面问题、平面和反平面裂纹问题、中厚板弯曲问题等各类二维问题。近期,又进一步将其推广到诸如非线性初值问题、薄膜结构找形问题、最小曲面问题、薄膜与地基接触、弹塑性扭转问题等非线性问题。该文对这一系列研究进展做了综述性介绍和展望,并给出各类问题的代表性数值算例以展示本法的有效性、灵活性、通用性和可靠性。In the recently developed reduced element,the highest order term in the conventional finite element(FE)solution is taken as the error term and serves as the built-in error estimator in maximum norm;meanwhile the solution with one order reduced is used as the final FE solution,leading to an adaptive FE algorithm with an error controlled by maximum norm.The reduced element was originally proposed for an adaptive step-size algorithm used for analysis of structural dynamics and was readily extended to general one-dimensional(1D)IVPs and BVPs.Subsequently,it was extended to quadrilateral and triangular reduced elements in 2D BVPs,which have been successfully applied to elastic thin-membrane deflection,elastic plane problems,plane and anti-plane cracks,and moderately thick plate bending problems,etc.Recently,the reduced element has been further extended to those nonlinear problems such as nonlinear IVP,thin-membrane shape finding problem,minimal surface problem,thin-membrane and ground contact problem,and elastic-plastic torsion problem.In this study,a comprehensive survey on the related research progress is given,and representative numerical examples of various problems are provided to demonstrate the validity,flexibility,versatility and reliability of the proposed method.

关 键 词：自适应有限元法 降阶单元 初值问题 边值问题 最大模 

分 类 号：TU311.4[建筑科学—结构工程]