一类求解含静态裂缝线弹性问题的预条件扩展有限元方法

A Class of Preconditioners for Static Elastic Crack Problems Modeled by Extended Finite Element Method

基于几何型扩展有限元离散方法,研究含静态裂缝线弹性问题的高效区域分解预条件算法。为了构造Schwarz型预条件算法,采用一种特殊的裂尖型区域分解策略,将计算区域分解为包含所有分支增强自由度的裂尖子区域和仅包含标准有限元自由度与Heaviside增强自由度的常规子区域。基于该区域分解策略,推导一类高效的乘性和限制型乘性Schwarz区域分解预条件子,对裂尖子问题进行精确求解,而对常规子问题则非精确求解。数值实验验证了算法的有效性。

This paper mainly discusses some effective domain decomposition preconditioners for static elastic crack problems modeled by geometrical extended finite element method.To construct the Schwarz type preconditioners,we adopt a special crack-tip domain decomposition strategy.The finite element mesh is decomposed into“crack tip”subdomains,which contain all the degrees of freedom(DOFs)of the branch enrichment functions,and“regular”subdomains,which contain the standard DOFs and the DOFs of the Heaviside enrichment functions.Based on the crack-tip domain decomposition strategy,an effective class of multiplicative and restrict multiplicative Schwarz preconditioners are derived.In the preconditioners,the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers.Numerical experiments demonstrate the effectiveness of the preconditioners.