A Study On Generalized Average Source Boundary Node Method

The finite element method(FEM)has been widely used in engineering analysis and scientific computing,which is a dominant numerical simulation technology.However,the FEM needs to discretize the interior of the computational domain into a regularized cell.For three-dimensional problems,especially for some problems with complex geometrical calculation domains,the preprocessing of data preparation takes more time than the computation,and sometimes even to bring mathematical problems.When calculating semi-infinite and infinite domain problems such as foundation basis and underground constructions,the FEM needs to artificially give artificial borders without disturbance to truncate the infinite computation domain to a finite computation domain,that is,replace the original semi-infinite or infinite domain problem with artificial finite field problem.It should be pointed out that,the artificial boundary completely relies on the research experience of the researchers,so the calculation accuracy is often unpredictable.In addition,the FEM can lead to misshapen elements when simulating ultra-thin-body and coating structure problems.In recent years,the boundary element method(BEM)has been developed as a very competitive numerical simulation technology,which just makes up for these shortcomings of the FEM and it has the advantage of the dimensionality reduction.It has been widely used in the steady state and transient mechanic problems,unlimited sound field simulations and other linear problems and various heterogeneous materials and nonlinear problems.However,the BEM also has limitations itself.For three-dimensional(3D)problems with complex surface boundaries,it is still not easy to generate boundary elements.In addition,it takes too much time to calculate numerical integration for large-to-medium-scale computational problems.In order to overcome these difficulties,the meshless method has developed rapidly in recent decades.At present,the main boundary-type meshless methods mainly include two categories: one is the meshless method based on the method of fundamental solution,which often leads to ill-conditioned linear systems,and the other type of meshless method is based on moving least square technique and boundary integral equation,the former makes the boundary nodes distribute relatively free,and the latter makes full use of the dimensionality reduction characteristics of the boundary element and the features of the easy-to-handle infinite domain.The disadvantage of the method is that the boundary integral still needs to be calculated.Different from previous research work,the average source boundary node method(ASBNM)is a novel boundary-type meshless method recently proposed.It is based on the coupling of ?completely? regularized boundary integral equations and average source techniques.In the process of solving,this method uses only boundary nodes,without the concept of unit and integration,and has the advantages of simple theory and easy program design.However,this method needs to divide the boundary into many segments,which are different from the elements of the traditional numerical method.Because we do not need to perform interpolation approximation of variables on these subsections,nor do we need to perform numerical integration on the subsections.This paper develops the previous average source technology and regularized boundary integral equations,and proposes the generalized average source boundary node method.This method inherits not only the boundary features of the BEM,but also does not need to calculate any integrals.It is a true meshless method.A large number of numerical experiments show that this method is an efficient,accurate,stable and well-convergent numerical simulation method.