Multi-domain Composite Virtual Boundary Element Method For Solving The Thermo-Elastic Problem And The Elastic Problem Of Orthotropic Bodies

Physics, mechanics, as well as many engineering problems can be regarded as initial value or boundary value in equations of mathematical physics. In which, some simple problems can be solved by analytical method, but for most of complex issues, it is solved by approximation algorithm usually. Boundary element method is a kind of numerical algorithm on boundary; it is based on the boundary integral equation and employs the element discretization technique. Direct boundary element method has relatively broad space of the application, but there are still some shortcomings: calculating some physical quantities near boundary, there is the boundary layer effect due to the quasi-singularity, the accuracy of the solution is reduced; and the coefficient matrix is full and asymmetric which waste too much time in solving process; besides, the adaptability of the nonlinear problem is comparatively bad. In recent years, how to deal with singular integral and avoid the boundary layer effect is a focal problem for researchers. Some scholars do a number of research works on the indirect boundary element method to overcome the shortcomings of direct boundary element method, and some numeric approximation methods were presented accordingly, one of which is virtual boundary element method. Least-squares method and matching collocation method are two frequently used ideas which are applied to virtual boundary element method.Nowadays, the idea which applies multi-domain composite virtual boundary element method (VBEM) to solve potential, fluctuant, electromagnetic and elastic issues etc. is comparatively mature. In order to demonstrate its capacity in engineering application, this paper presents the ideas to solve 2D Thermo-Elastic problem and the elastic problem of orthotropic bodies. For Thermo-Elastic problem, the solving process is divided into two parts, the first part is to solve heat conduction problem with VBEM which is based on assumptive heat source method, and the second part is using VBEM to solve corresponding elastic problem. For the elastic problem of orthotropic bodies, employing the fundamental solution that is suitable not only to orthotropic media but also to isotropic media, a new idea for solving the elastic bodies possessed orthotropic character and composed of different materials is presented. It is worth mentioning that the method has universal significance, it can solve the multi-domain composite problem of orthotropic media, and the issue composed of isotropic and orthotropic media can be solved by the method in this paper, in addition, this method can evolve to solve the problem of isotropic media. It is worth emphasizing that the idea of multi-domain composite virtual boundary element method can evolve to solve the problem of single domain. In this paper, after the theories of solving 2D Thermo-Elastic problem and the elastic problem of orthotropic bodies with virtual boundary element method are given, the program of VBEM is further improved on the basis of the predecessors in my team. And there are some numerical examples to prove the efficiency and calculating accuracy of this method.