基于水平集算法的扩展比例边界有限元法研究

扩展比例边界有限元法在裂纹贯穿单元采用Heaviside阶跃函数描述裂纹面两侧的不连续位移，在裂尖则采用半解析的比例边界有限元描述奇异应力场。该方法具有无需预先知道裂尖渐进场的形式，无需采用特殊的数值积分技术直接生成裂尖刚度阵，对多种应力奇异类型可根据定义直接求解广义应力强度因子的特点。该文将扩展比例边界有限元法与水平集方法相结合，进一步发展了扩展比例边界有限元法，并将其应用于解决裂纹扩展的问题。在数值算例中，通过编写完整的MATLAB分析计算程序，求解了单边缺口的三点弯曲梁和四点剪切梁的裂纹扩展问题，计算结果显示扩展比例边界有限元法能有效地预测裂纹轨迹和荷载-位移曲线。通过参数敏感性分析，还可得出该方法具有较低的网格依赖性，且对裂纹扩展步长不敏感。

STUDY ON THE EXTENED SCALED BOUNDARY FINITE ELEMENT METHOD BASED ON LEVEL SET ALGORITHM

In the extended scaled boundary finite element method (X-SBFEM), the Heaviside enrichment is used to represent the jump across the discontinuity surface in split element while the nonsmooth behavior around the crack tip is described by the semi-analytical SBFEM. This method does not require a priori knowledge of the asymptotic fields. It also does not require special numerical integration technique so that the stiffness of the region containing the crack tip is computed directly, while the generalized stress intensity factors of many kinds of singularities could be obtained directly from definition by the consistent formula. In this paper, the X-SBFEM is combined with the level set method so that it can be applied to solve the problem of crack propagation. During its numerical evaluation, the three-point bending specimen and the four-point shear specimen is modeled by a complete MATLAB code. The results show that the X-SBFEM is capable of predicting crack trajectories and load-displacement relations accurately. Based on the sensitivity analysis of parameters, it was found that different mesh densities and crack propagation lengths led to consistent results.