断裂问题的插值型无单元伽辽金比例边界法与有限元法的耦合研究

Coupled interpolating element-free Galerkin scaled boundary method and finite element method for crack problems

插值型无单元伽辽金比例边界法是一种只需在计算域的边界上采用插值型无单元伽辽金法离散且无需基本解的半解析数值方法,特别适用于求解包含无限域和奇异物理场的问题.本文提出了一种用于断裂分析的插值型无单元伽辽金比例边界法与有限元法的耦合分析方法,更好地发挥插值型无单元伽辽金比例边界法和有限元法各自的优势.裂尖周边一定范围的计算域采用插值型无单元伽辽金比例边界法模拟,而其余区域则采用有限元法模拟.在这两个区域内,分别采用各自相应的位移模式,两者相互独立.利用交界面两侧位移的连续条件,可以方便地建立耦合求解方程,简明有效,易于编程计算.最后给出了两个数值算例验证本文方法的有效性.

The interpolating element-free Galerkin scaled boundary method （IEFG-SBM） is a semi-analytical method which only requires to discretize the boundary by the interpolating element-free Galerkin （EFG） method without fundamental solution. This method is ideally suited to solve problems containing infinite domain and singular physical fields. This study develops a novel method that couples the IEFG-SBM and the finite element method （FEM） for crack analysis in order to take the full advantages of both IEFG-SBM and FEM. The IEFG-SBM is adopted to model the domain close to the crack tip and the FEM is adopted in the rest of the domain. The corresponding displacement interpolation models are employed for each sub-domain respectively. Through continuity conditions on the interface between IEFG-SBM sub-domain and FEM sub-domain, the coupled formula of the proposed method can be easily derived. The proposed method is simple, effective, and easy to be programmed. Finally, two numerical examples are presented to demonstrate the validity of the proposed method.