Modified multiplying-factor integration method for solving exponential function dual integrals in crack problems  

改进的求解裂纹问题中指数函数对偶积分的倍数因子积分法

作　　者：Yongjia Song Yannan Lu Hengshan Hu 宋永佳;陆彦楠;胡恒山(Department of Astronautic Science and Mechanics,Harbin Institute of Technology,Harbin 150001,China)

机构地区：[1]Department of Astronautic Science and Mechanics,Harbin Institute of Technology,Harbin 150001,China

出　　处：《Acta Mechanica Sinica》2022年第6期117-127,I0003,共12页力学学报（英文版）

基　　金：This work was supported by the National Natural Science Foundation of China(Grant Nos.11802074,42074057,11972132,and 11734017);China National Postdoctoral Program for Innovative Talents(Grant No.BX201700066);China Postdoctoral Science Foundation(Grant No.2018M630345);the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.2020016)。

摘　　要：Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.裂纹问题通常可归结为对偶积分方程,可通过将位移积分方程展开为类Chebyshev多项式或Jacobi多项式形式的级数来求解.通过构造一个适定的线性代数方程组来确定展开系数的施密特倍数因子积分法是最受欢迎的方法之一.然而,由于线性方程组的矩阵元素是从二重积分计算出来的,施密特方法的数值计算效率较低.本文提出了一种改进方法来构造线性方程组,以便有效地确定展开系数.改进技术是在将某些倍数因子应用于牵引积分方程,然后在“源”区域上积分得到的方程后发展起来的.该操作将矩阵元素简化为单重积分.文章通过数值算例证明了此方法的计算精度高,且效率高.尤其该方法只需要施密特方法计算时间的1/5左右.因此该方法可以用于代替施密特方法并有望在解决裂纹问题中发挥作用.

关 键 词：CRACK Dual integral equations Schmidt’s method Multiplying-factor integration 

分 类 号：O15[理学—数学]