广义有限元方法研究进展

Researching Progress of Generalized Finite Element Method

广义有限元方法是常规有限元方法在思想上的延伸,它基于单位分解方法,通过在结点处引入广义自由度,对结点自由度进行再次插值,从而提高有限元方法的逼近精度,或满足对特定问题的特殊逼近要求。基于广义有限元方法对单元形状函数构造理论的深入研究,具有任意内部特征(空洞、夹杂、裂纹等)及外部特征(凹角、角点、棱边等)的复杂问题,都将在简单、且与区域无关的有限元网格上加以求解。本文主要介绍广义有限元方法的基本思想、主要特征及对重要细节的处理策略,包括线性相关性的处理、局部逼近函数的获取、区域上的数值积分技术以及边界条件的处理。与扩展有限元方法和有限覆盖方法比较,分析它们各自的特点。综述广义有限元方法的研究现状、应用,展望广义有限元方法的未来发展。

Generalized finite element method is the extension of conventional finite element method.Based on the partition of unity method,it improves the approximation accuracy of the finite element method or achieves the special approximation to particular problems by introducing the generalized degrees of freedom and by re-interpolating the nodal degrees of freedom.From the profound study on constructing the shape functions of the generalized finite element method,arbitrarily complex problems with internal features(e.g.void,inclusion and crack)and external features(e.g.re-entrant,corner and edge)can be expected to solve by the simple and domain independent mesh.The essential ideas and corresponding strategies,including treatment of the linear dependency and boundary conditions,capture of the local approximation functions,numerical integration techniques,are introduced in details.The features and connections are analyzed as compared with the extended finite element method and the finite cover method.The progress of the generalized finite element method is reviewed,and then current practical applications are summarized.