弱不连续问题扩展有限元法的数值精度研究

STUDY ON NUMERICAL PRECISION OF EXTENDED FINITE ELEMEMT METHODS FOR MODELING WEAK DISCONTINUTIES

主要研究了扩展有限元法(extended finite element method,XFEM)在处理弱不连续问题时不同改进函数形式对XFEM数值求解精度的影响,阐述了各种改进函数影响XFEM求解精度的关键因素,指出校正的扩展有限元法(corrected-XFEM)能够提高数值求解精度的实质在于它拓展了改进结点域,即将常规扩展有限元法(standard-XFEM)的改进结点域增加一层作为corrected-XFEM的改进结点域,文中建议延拓corrected-XFEM的改进结点域,即在corrected-XFEM的改进结点域基础上再增加一层改进结点.利用水平集函数表征材料内部的不连续界面,推导了XFEM求解的支配方程,给出了一种改进单元的数值积分方案以及改进单元处高精度应力的求解方法.含夹杂问题的数值计算结果表明:建议的延拓corrected-XFEM改进结点域的方法能够明显提高XFEM的数值求解精度.

This paper mainly studies the effects of the enrichment functions on the numerical precision of XFEM when the method is used to model weak discontinuities problems. The dominant reasons for the effects of the enrichment functions on the numerical precision of XFEM are discussed in detail. In the corrected-XFEM, the set of enriched nodes equals to the set of enriched nodes in the standard-XFEM plus their neighboring nodes. The fact is thereby indicated that the increase of enriched nodes can improve the numerical precision of the corrected-XFEM. In this paper, the proposed method is to expand the enriched nodes of the corrected-XFEM~ and the expanding enriched nodes is the set of enriched nodes in the corrected-XFEM plus their neighboring nodes. Then, the level set method is used for the description of the inner discontinuous interfaces. The governing equation of XFEM is deduced. A numerical integration scheme is elaborated to integrate the stiffness matrix of enrichment elements. Also a method is introduced for the acquirement of accurate stresses in enrichment elements. The numerical results of two illustrations with single inclusion and multi-inclusions both show that the proposed method of expanding the enriched nodes of the corrected-XFEM can improve evidently the numerical precision of XFEM.