改进型XFEM进展

扩展有限元法(XFEM)在诞生后的十几年时间里,引起学术界和工业界的广泛关注,并已经成为目前裂纹分析的主流数值方法。然而,在实际应用中该法一直受到两方面的困扰,(1)总体方程高度病态;刚度阵条件数随网格尺寸呈h-6变化(普通有限元为h-2)。(2)裂尖强化插值由于能量一致性问题无法直接推广应用于动力学计算。前者表现在XFEM稳态问题的迭代求解收敛慢或难以收敛,后者长期以来导致XFEM裂纹扩展动力学计算实施困难。本文认为XFEM目前遇到的种种困难,均与单位分解引入的额外自由度相关。为此,提出了无额外自由度的单位分解插值格式,基于此格式,进一步构造出改进型扩展有限元方法。改进型XFEM具有如下特点,(1)可以消除原有XFEM的线性依赖性和总体方程病态的问题。(2)避免动力学问题中额外自由度引起的质量集中、零临界时间步长问题以及裂纹扩展过程中的能量一致性问题。本文结合静动力学测试问题综述上述改进。

Recent progresses on improved XFEM

As an unprecedentedly successful tool for crack analysis XFEM gains broad attention in both academy and industries.However,there are still two daunting issues hindering the practical applications of the method.One is the linear dependence issue reflected by the ill-conditioning of global matrices in steady problems.The other is the issue of generalized mass terms associated with the extra dofs in time-dependent problems.The paper presents an improved XFEM,which features:(1) crack tip enrichment without introducing extra dofs; (2) being free of linear dependence and avoiding the singularity of (global) stiffness matrices; (3) well-conditioned.The condition number of the stiffness matrix increases as h^-2,which is the same as the standard FEM,whereas the standard XFEM increases as h^-6 in our tests; (4) overcoming the mass lumping issue of generalized mass and energy inconsistency issue asso-ciated with extra dofs in dynamic analyses.Recent progresses of the new method are summarized in this short paper.The method is tested using selected numerical examples in both static and dynamic problems.It is shown that the new method is the same accuracy as the existing XFEM,but it offers additionally excellent conditioning stability,good computationally efficiency,and implemtnational straifht-forwardness in dynamic problems.