An Improved Interpolating Boundary Element-Free Method For Anisotropic Potential Problems

Combining the improved interpolating moving least-square method with boundary integral equations,a novel boundary-only meshless method,the improved interpolating boundary element-free method,is developed for solving two-dimensional anisotropic potential problems.This method can reduce the dimension of the required problem and does not require meshing.Firstly,this dissertation introduces the research backgrounds of several traditional numerical methods and the moving least squares approximation,anisotropic potential problems and the improved interpolating boundary element-free method.Then,the fundamental theories of the moving least squares approximation and the improved moving least squares approximation are also introduced.Based on the improved moving least squares approximation,an improved interpolating moving least squares method based on nonsingular weight function is discussed.Because its shape function satisfies interpolating properties,boundary conditions in the associated meshless methods can be applied directly.This dissertation presents an improved interpolating boundary element-free method based on direct boundary integral equation for solving anisotropic potential problems.Firstly,the direct boundary integral equations for anisotropic potential problems is introduced.Then,the interpolation of unknown functions and the discretization of the direct boundary integral equation by the improved interpolating boundary element-free method are proposed.The boundary integral equation for anisotropic potential problems contains weak singular integrals.The calculation method of weak singular integrals is given in detail.Finally,some numerical examples of anisotropic potential problems are given.Numerical results show that the current method is effective and feasible for solving two-dimensional anisotropic potential problems.Compared with the boundary element method,the current method has better accuracy and convergence.In this dissertation,an improved interpolating boundary element-free method based on the single-layer potential is also introduced for solving anisotropic potential problems.Firstly,the two-dimensional anisotropic potential problem is reduced into an indirect boundary integral equation by using the single-layer potential theory.Then,the indirect boundary integral equation is discretized by an improved interpolating boundary element-free method.Finally,numerical examples are given to verify the efficiency of the current method.