Boundary Element-free Method For The 2-D Helmholtz Equation

For the propagation of electromagnetic waves,water waves,elastic waves,acoustic waves,etc.,and the study of vibration,radiation problems will often be attributed to solving the Helmholtz equation.Therefore,study of its numerical solution not only has important theoretical significance but also has broader practical significance.In the meshless method,there have been many research results before.However,considering that such problems can often be defined in infinite space,at this time,it is more effective to solve the boundary integral equation than to use the regional method.In this paper,the boundary element-free method is used,and the improved moving least square method is combined with the boundary integral.The real solution of the nodal variables is used as the basic unknown in the formed boundary integral equations.It is easier to introduce the boundary conditions,and it has high accuracy.This paper mainly studies the processing of strong singular integrals in kernel functions.On the one hand,we establish its regularization form in different boundary integral equations,which effectively avoids strong singular integrals.On the other hand,the power series expansion method(PSEM)in the boundary element method is used to deal with strong singular integrals in the meshless method,and its idea is also used for rounding error processing.The specific content is as follows: In the first chapter,the traditional numerical calculation method is introduced.The development of the boundary element-free method in the meshless method is reviewed,and the Helmhlotz equation is introduced in detail.In chapter 2,the boundary element-free method for solving 2-D Helmhlotz equations based on direct boundary integral equations is given.The linear integral grid method(LIGM),regularization method(RM)and power series expansion method(PSEM)are used to solve the strong singularity integral in boundary integral equations.In Chapter 3,the regularized form of the indirect boundary integral equation is given,and the limit form of the spatial derivative of the distance and the normal derivative are deduced in detail.The fourth chapter gives a linear combination of monolayer potentials and double-layer potentials.The boundary element-free method for the Dirichlet exterior boundary value problem of the 2-D Helmhlotz equation with arbitrary wave number.The regularized form of the corresponding boundary integral equation is deduced,and the power series expansion method in Chapter 2 is used for the strong singularity integral.The last chapter gives a summary and outlook.