A Study On Average Source Meshless Method For Potential Problems

As is known to all,the FEM is the dominant numerical simulation technique for science and engineering applications.However,the FEM needs to split the whole domain for some complex computing problems,and thus it may to pay a heavy computational cost,sometimes even brings a mathematical trouble.When it solves the infinite domain problems,the FEM needs to make the artificial boundary truncation for limited area,which is artificially limited regional problems instead of the original of infinite region,so that truncations depend entirely on the researchers' experience and all test errors.BEM is an available numerical technique,just make up for the FEM of these shortcomings and reduce the dimension.But the BEM has limits,however,for the three dimensional problem with complex geometry boundary,which element generation is still not an easy thing.In addition,for large-scale computing problem,it consumes too much time.In this paper,we study a new boundary meshless method,the average source method(ASM).The method is based on the "complete" regularized boundary integral equation(CRBIE)and the average source technique(AST).The "complete" means the weak or strong singular integrals are normalized.Through the regularized equation to avoid the singular kernel function,the source and field points are difficult to meet have been eliminated;by using the AST,distributed source can be simplified as a centralized source,the boundary integral is no longer needed.The ASM not only inherits the boundary element method to solve the problem of dimension reduction,but also avoid the boundary collocation,FEM and BEM are in time-consuming mesh,especially for complex geometry simulation.In addition,the meshless method is not related to the concept of element and integral,and has the advantages of simple mathematics,programming,efficiency,precision and so on.Specific work is:In Chapter 1,we mainly introduce the research background and present situation of meshless methods.In Chapter 2,we give the relevant knowledge of the boundary type meshless method.In Chapter 3,we establish a "complete" regularized boundary integral equation for the two dimensional potential problem.On the basis of this,by introducing the concept of average source technology,a new boundary type meshless method,the average source method(ASM)is established.In Chapter 4,we study the method of average source method(ASM)for three-dimensional potential problems.To solve this problem,the "complete" regularized boundary integral equation is derived,and then the average source technique is used to obtain the average source boundary node method(ASBNM).In Chapter 5,we give the summary and prospect of meshless method.