比例边界有限元二阶灵敏度设计及断裂力学分析

比例边界有限元是一种只需在边界上划分网格且无需基本解的半解析方法,能有效处理应力奇异性和无边界问题.论文提出了一种比例边界有限元的二阶灵敏度分析方法,可以准确而高效地求解响应关于参数的二阶梯度.首先通过建立仅需右特征向量的哈密顿矩阵特征灵敏度分析方程,发展了一种改进的比例边界有限元一阶灵敏度分析方法;其次,进一步通过构建二阶哈密顿矩阵特征灵敏度分析方程,并对比例边界有限元系统方程进行一系列二次直接微分,提出了一种半解析形式的比例边界有限元二阶灵敏度分析方法.该方法被应用于线弹性裂纹结构的形状灵敏度分析和不确定性传播分析.最后,给出了两个数值算例验证论文方法的有效性.

THE SCALED BOUNDARY FINITE ELEMENT SECOND ORDER SENSITIVITY DESING AND FRACTURE MECHANICS ANALYSIS

The scaled boundary finite element method is a semi-analytical method that only needs to mesh on the boundary without fundamental solution, which makes it powerful to deal with singular and unbounded problem. This paper aims to propose a second order sensitivity analysis method for the scaled boundary finite element method, which can calculate the second order gradients of the responses with respect to the parameters accurately and efficiently. An improved first order sensitivity analysis method is presented through establishing a new Hamilton eigenproblem equation with only right eigenvectors. The second order Hamilton eigenproblem equation is constructed and the semi-analytical sensitivities of displacements and stresses are further obtained by a series of differential equation. The proposed method is then applied to the shape sensitivity analysis of linear cracked structures and corresponding uncertainty propagation analysis. Finally, two numerical examples are investigated to demonstrate the validity of the proposed method.