改进型扩展比例边界有限元法

结合了扩展有限元法(extended finite element methods,XFEM)和比例边界有限元法(scaled boundary finite element methods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improved extended scaled boundary finite element methods,i XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的i XSBFEM的有效性,相比于常规XFEM,i XSBFEM的基于位移范数的相对误差收敛性较好;采用i XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合较好.

IMPROVED EXTENDED SCALED BOUNDARY FINITE ELEMENT METHODS

Combining the main advantages of the extended finite element methods (XFEM) and the scaled boundary finite element methods (SBFEM), improved extended scaled boundary finite element methods ($i$XSBFEM) are proposed. The proposed methods can provide a new way for the simulation of fracture problems. Similar to XFEM, two orthogonal level set functions are used to characterize the internal crack surface in materials, and how the element is partitioned by a crack can be judged by level set functions. These elements partitioned by the crack are treated as a subdomain of SBFEM, and then the element stiffness matrix of these discontinuous elements can be directly solved by SBFEM, thus avoiding the need for further element subdivision to the solution of the discontinuous element stiffness matrix in XFEM. At the same time, with the help of the main idea of XFEM, the real displacement of the intersection point between the crack and the element boundary is considered as the additional degrees of freedom of the element nodes, thereby it gives explicit physical meaning of additional degrees of freedom. For the element containing the crack tip, several layers of elements around the crack tip are selected as a super element, and the super element is used as a subdomain of SBFE to solve the stiffness matrix. The node displacement inside the super element can be obtained by the SBFE displacement approximation. The stress intensity factor can be directly obtained based on the singular displacement or stress at the crack tip, without the need of other numerical methods. Finally, several numerical examples are given to verify the effectiveness of the proposed $i$XSBFEM. Compared with the standard XFEM, the relative error convergence of the $i$XSBFEM based on the displacement norm is better, and the stress intensity factors computed by stress based and displacement based method in $i$XSBFEM both are in good agreement with the analytical solution.