断裂力学中多维奇异积分方程的高效算法研究

Fracture mechanics is one of significant achievements in solid mechanics since the middle of the twentieth Century. Solving multi-dimensional singular integral equations derived from boundary element methods by Projection Methods?is subjected to the multi-dimensional singularities. This project provides algorithms of high accuracy for these equations, which is so-called mechanical quadrature method (MQM). The MQM is effective in dealing with the multi-dimensional singularities, and decreasing the computational complexities. The main advantages of this new method are as follows: (1) Prior to both the Galerkin and collation methods, it does not require any singular integrations to generate any entry in the discrete matrices. However, the Galerkin method needs an evaluation of a quadruple singular integral and the collation method involves that of a double singular integral; (2) It is numerically stable. The condition number of the linear system is small and only increases linearly with respect to the grid refinement;(3) Numerical solutions are of highly accuracy. Based on the asymptotic error expansions on multi-parameters, efficient extrapolation methods or splitting extrapolation is derived to achieve a higher accuracy; (4) It is self-adaptive and parallel algorithm.

断裂力学是20世纪中叶以来固体力学领域的重大成就之一。用投影法处理断裂力学中由边界元法得到的多维奇异积分方程将受到多维奇异性效应制约。本项目拟研究数值求解多维奇异边界积分方程的高精度算法（也称机械求积法），该算法是克服高维奇异性效应和避免高复杂度的有效方法。主要特点有：（1）离散矩阵的元素生成不需要计算任何奇异积分，计算量少，用配置法每个元素需计算重积分，用有限元Galerkin法每个元素需计算双倍重积分；（2）解方程条件数小，对网格加密时条件数仅线性增长，故算法很稳定；（3）精度高并有多参数误差的渐近展开式，因此使用外推与分裂外推可以得到更高精度；（4）拥有自适应处理功能，是并行算法。

断裂力学是20世纪中叶以来固体力学领域的重大成就之一。 用配置法和有限元Galerkin方法处理断裂力学中由边界元法得到的二维奇异积分方程将受到计算复杂度大和奇异性效应制约。本项目以断裂力学问题为背景，以高精度数值求解多维奇异边界积分方程为目标，采用了理论分析、算法设计与数值实验相结合的方法，旨在进行算法创新，构造高效的求积算法，进一步充实了断裂力学领域数值求解多维积分方程的技巧。本项目按计划顺利进行，主要成果概述如下：（1）建立了两类多维奇异积分的高精度求积公式及得到了误差的Euler-Maclaurin 展开式，有效解决数值解高计算复杂度问题；（2） 建立了解二维奇异积分方程的高精度数值算法；（3） 研究了进一步提高数值解精度和收敛速度以及一些具体物理应用。这些结论和研究方法在一定程度上可以丰富奇异积分方程数值解理论，为我们后面的研究提供了重要的基础。同时在项目执行期间负责人新增培养边界元领域的在读硕士研究生3名。