扩展有限元方法计算多夹杂问题时圆形夹杂与四边形单元的几何关系

Geometrical relation between circular inclusion and quadrangular element for solving multi-inclusions problem by XFEM

用扩展有限元法XFEM(Extended Finite Element Method)解决夹杂问题时,夹杂与基质的界面把单元分成若干部分。求单元刚度矩阵时,需要分别在这各个部分求积分。找到便于程序编制的描述各积分区域几何形状的方法是亟待解决的问题。本文把各积分区域的形成过程看成是圆对四边形的多次切割。考虑切剩区域与圆的关系时,把不完整的边仍看作完整的边,把切剩区域看成是四边形或是切去一两条边的四边形。采用排列组合的方法,把它们与圆的所有位置关系列了出来。

When solving inclusion problem by the Extended Finite Element Method （XFEM）, an element is split into many regions by the interface between inclusions and matrix. In order to calculate element stiffness matrix, integral in these regions is necessary. The urgent problem to be solved is to find a convenient method to describe integral regions for programming. The process of forming integral regions is taken as circles repeatedly splitting a quadrangle. Geometrical relation between remained region and circles is analyzed. In the process, broken sides are substituted by original sides, and remained region is substituted by quadrangle that discards no or some sides. All possible geometrical relations between circle and remained regions are listed through permutation and combination.