求解变分型积分方程的一种新型数值方法——有限变分法  

Finite variation method:a new numerical method for solving variational integral equations

作　　者：卢炎麟[1] 周国斌[1] 贾虹[1] 应富强[1] 傅建钢[1] 

机构地区：[1]浙江工业大学机械制造及自动化教育部重点实验室,杭州310014

出　　处：《计算力学学报》2010年第5期801-808,共8页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金(50675206;10972198)资助项目

摘　　要：将作者提出的多虚拟裂纹扩展法(MVCE法)拓展为求解变分型积分方程问题的一种新型数值方法——有限变分法(FVM)。它的基本思想是,给定有限个(N个)局部变分模式,将所求解的未知量用适当的方法离散化,针对这N个局部变分模式列出N个方程,求解N个未知系数,从而求得未知量。单一未知变量FVM的最终方程组的系数矩阵通常是一个对称的窄带矩阵,对角元是大数,有很好的数值计算性能。用FVM求解了三维I型裂纹前缘的应力强度因子(SIF)分布。利用基于FVM的通用权函数法计算程序,可以高精度和高效率地求解表面力、体积力和温度载荷共同作用情况下三维裂纹前缘SIF的分布及其时间历程。FVM可以被推广到更广泛的领域,是一个求解变分型积分方程问题的普遍适用的新型数值方法。The multiple virtual crack extension（MVCE）method proposed by authors is extended to a new general numerical method-finite variation method（FVM）.Giving finite（N）local variation modes,discretizing the solved variables,writing out the N equations for N local variation modes,the N unknown coefficients in discretization and thus the unknown variables can be solved.The coefficient matrix of the final equations in FVM is usually a symmetrical matrix with small band-width and major diagonals,which has good numerical properties.The distributions of SIFs along 3-D mode I crack fronts are solved by FVM.By means of the programs using the general weight function method based on FVM,the histories of distributions of SIFs along 3-D crack fronts of a body subjected to surface tractions,volume forces and thermal loadings can be determined with high accuracy and efficiency.FVM can be extended to more general areas,which is a widely suitable numerical method for solving the variational integral equations.

关 键 词：有限变分法 变分型积分方程 应力强度因子 三维通用权函数法 多虚拟裂纹扩展法 

分 类 号：O346.1[理学—固体力学]