Radial Basis Function Meshless Method For Heterogeneous Problems

The problem of inhomogeneous medium is a common practical problem in the seepage field,heat conduction and force analysis.For a long time,the main method is the finite element method for such problems.The finite element method is a domain method.The model of calculation needs to split the entire calculation area.Therefore,for the three-dimensional problem,the time of preprocessing far exceeds the calculation time,especially for some problems with complex computational domains,it may pay a considerable amount.The large computational cost sometimes leads to method troubles.In recent years,the boundary element method has developed rapidly and has been widely applied to various non-homogeneous and non-linear problems such as acoustic propagation,potential flow,elastic static,dynamic and other linear problems as well as functionally graded materials and elastoplastic problems.However,when simulating such problems,the boundary element can only use simple problems,such as the basic solution of the Laplace problem,due to the lack of fundamental solutions,which inevitably leads to the calculation of domain integrals.Because the domain integration term contains unknown quantities,the radial basis function of approximation techniques are needed.Therefore,the choice of radial basis functions(types of basis functions and parameter sizes in the basis functions)plays a crucial role in the accuracy and stability of the method.In addition,for the traditional numerical methods based on Lagrangian interpolation units,such as the finite element method and the boundary element method,there are some inherent defects,such as when the simulation optimization design,metal forming,fluid-structure coupling,etc.involve large deformation or moving boundary problems.The unit may be severely distorted,which not only requires repeated reconstruction of the grid during the calculation process,but also seriously affects the accuracy of the results.The radial basis function technology completely overcomes these shortcomings of the traditional cell-based numerical methods.It has the advantages of simple method,easy implementation,independent of the dimension of the problem and complexity of the problem domain,and becomes more and more in the scientific computing family.The more popular it has become,it is now widely used in various scientific fields such as computer graphics,data processing and economics.However,the effectiveness of radial basis function numerical simulation techniques depends on the type of radial basis function and the size of the shape parameters.In this paper,the radial basis function meshless technique for heterogeneous problems is studied.We have constructed a new radial basis function,which adds a stability term to the original radial basis function,making the radial basis function of shape parameter have a wider choice.The work of this paper is listed as follows:(1)In chapter 3,the problem of two-dimensional anisotropic heat conduction is studied.(2)In Chapter 4,the problem of three-dimensional heterogeneous heat transfer is studied;(3)In chapter 5,the problem of three-dimensional heterogeneity elasticity is studied.