A Conjugate Boundary Element Method For Complex Variable Function Analysis

Complex function analysis plays an important role in anisotropic mechanics,no matter for the isotropic or anisotropic medium,the complex function can usually get a simple mathematical expression as a form solution,but for specific problems,only two kinds of numerical methods,such as finite element method or boundary element method,can be used better,therefore,it is very urgent and necessary to develop the boundary element method based on the complex function.The aim of this paper is to study the complex variable function,but the boundary element theory and method of the classical two-dimensional potential problem are the starting point of our study.In this paper,the interpolation function on the unit group is smoothed by using the unit grouping method in the discrete boundary element space.Then the boundary element method of high precision directional derivative on the boundary is given,on the boundary that is a ?_i-?_i coordinate system can be moved with the node,in this coordinate system the Taylor formula of the two element function is expanded,an inhomogeneous and unequidistant two-dimensional grid,a new difference method for solving the directional derivative of tangent direction is given.Different from the general finite difference method in the formation of the difference scheme,there is no discrete processing of the whole solution domain,but only the discrete nodes are selected on the boundary to carry out the expansion.At last,it can be seen in the example that the calculation accuracy of this method is very high.Then,the paper gives the conjugate boundary element method for the analytic function,an analytic function is equivalent to the two conjugate real harmonic functions within the domain and the two harmonic functions satisfy the Cauchy Riemann condition on the boundary.A set of boundary integral equations for conjugate harmonic functions is obtained by using the theory of weighted residuals,transforming the discrete boundary element into two linear equations.But the Cauchy-Riemann condition on the boundary,then the high precision tangential derivative difference method is applied to a linear equation set,this equation is used as the constraint equation of the first two equations.The approximate solution of the conjugate harmonic function is obtained by solving the large and weak ill conditioned linear equation set up and the Cauchy integral formula is used to solve the problem of the inner point.Finally,the calculation results of the calculation example verify the validity of the method.