Theoretical Study And Application Of Dual Reciprocity Hybrid Boundary Node Method

Meshless method is a new kind of numerical methods developed in the past decades. They have the advantages that no element is needed totally or partly, and their preprocess is easy. Therefore, meshless method can be widely applied to crack propagation problem, elasto-plastic analysis and large scale three-dimensional problem analysis. The original literature and recent developments of meshless methods are briefly reviewed in this dissertation. Boundary-type numerical method and Hybrid Boundary Node Method(HBNM) are described in detail in this dissertation. HBNM is a kind of boundary-type truly meshless method with many excellent characteristics. But when it is applied to the inhomogeneous problems, such as elasticity problems with body force and dynamic loading and so on, it is evitable to need domain integral. So, Dual Reciprocity Hybrid Boundary Node Method (DRHBNM) is proposed in the study, and it successfully avoids domain integral, expands the application scope of this method to inhomogeneous problems, such as two-dimensional and three-dimensional elasticity with body force, elasto-dynamic problems and nonlinear problems. The dissertation includes the following contents:Firstly, introducing the Dual Reciprocity Method(DRM) into HBNM, and applying radial basis function interpolation, a new boundary-type meshless method—DRHBNM is proposed. Appling DRM, the domain integral of inhomogeneous term of governing equation is transformed into boundary integral. The solution in this method is divided into two parts, i. e., the complementary solution and the particular solution. The complementary solution is solved by HBNM. The particular one is obtained by radial basis function interpolation. At the same time, the modified boundary conditions are applied in hybrid boundary node method. Therefore, this method is a truly meshless method, and it does not require a 'boundary elements mesh', either for the purpose of interpolation of variable, or for the integral of the 'energy'. It only need the data of distributed nodes in the calculation, the preprocess of this method is easy. The nodes in the domain are only for the radial basis function interpolation of DRM, they are no need for the variable interpolation and background integral.Secondly, DRHBNM is implemented successfully for solving problems in two dimensional Poisson's equations, two and three dimensional linear elasticity with body force. The formulation of this method in these problems is developed, and the numerical implementation scheme is obtained. The relative programs are complied. The numerical examples are shown that the present method possesses not only high accuracy, but also good performance of convergence, and it can be widely applied to the practical problems.Thirdly, DRHBNM is applied to elasto-dynamic problems, the formulation of this method in elasto-dynamic problems is developed, and numerical implementation details are given in detail. For the elasto-dynamic problems, only boundary integral equations can not solved the solution of this problem, so the relations between the domain variable and boundary variable are applied to form some new equations. Based on the above analysis, the computer codes are written for elasto-dynamic problem, and the numerical examples are shown that the present method is effective for two dimensional elasto-dynamic problems, and it can achieve high accuracy.Fourthly, DRHBNM is applied to nonlinear potential problems, formulations for these kinds of problems are achieved. As same as the dynamic problems, the relations between the boundary variables and domain variables are applied to obtain the additional equations. Numerical examples are given to show that the present method is effective for the nonlinear potential problems, and it can achieve much high convergence. Applied Generalized Quasilinearization Method(GQM), Quasilinear Hybrid Boundary Node Method(QHBNM) is proposed, and employed to solve nonlinear problems. Theroy and numerical examples are shown that this method is stable and has high accuracy, and the convergence is quadratic.Fifthly, some specialized studies are done for numerical implementation details. A series of integral schemes for weakly singular integral and nearly singular integral are proposed. In order to achieve the universal of the present method, the basic form of particular solution is proposed, so the particular solution can be interpolated by the same form of basic form of particular solution for the same kind of problems. The solving method and the results of the basic form of particular solution are presented in detail in this dissertation.Sixthly, the free parameters of the present method have been studied in this dissertation, and the relations between the influence to the precision and the order of the radial basis function are discussed. Also, the nodes number and the distributed position for the radial basis function interpolation are studied. Finally, some optimization proposals are presented for above problems.DRHBNM is is a new boundary-type truly meshless method with many excellent characteristics. Compared to HBNM, it can be applied to the inhomogeneous problems and without domain integral. It keeps the dimension-reduction advantages of HBNM, and only requires data information of nodes in the calculation. Therefore, the preprocess is easy, and suitable for the large scale complex structure analysis. Besides, DRHBNM is a truly meshless method, it does not require 'elements mesh' not only for variable interpolation, but also for background integral.The study shows that the present method possesses the not only high accuracy, but also excellent performance of convergence. It can be widely applied to the practical problems, such as adaptive problem, crack propagation and contact analysis and so on.