Theory Of Hybrid Boundary Node Method And Its Application To Thin Plate Problems

Meshless methods have emerged as a class of effective numerical methods which are capable of avoiding the difficulties encountered in the computational meshed base methods. For example, these include the problems for the complex geometries meshing, mesh distortion due to large deformation and remeshing in the moving boundary problems. At first, the original literature and recent developments of the meshless methods are briefly reviewed, and the application of some numerical methods to plate and shell problems is introduced.The hybrid boundary node method (HBNM) is a boundary type meshless method in the recent years, which requires no cell either for the interpolation of the solution variables or for the numerical integration, and inherits the reduced dimensionality and high accuracy advantages of the BEM. Until now, the governing equation of the problems which have been solved successfully is the second order partial differential equation. For the other physical problems, such as the plate bending, it is need to solve the fourth order partial differential equation. Therefore, the hybrid boundary node method for the fourth order partial differential equation is developed. The present study further improves and develops the theory and application of the HBNM.In the present method, the modified variational principle is employed, and the solution of the original problem is transformed into the local integration equations of the boundary nodes. The key of this method is how to construct the suitable modified functional. For the boundary value problems of the biharmonic equation, the modified functional involves four types of the independent variables, i.e., the potentials inside the domain, the potentials, the normal fluxes and the vorticities on the boundary. For the thin plate bending problems, the modified functional involves five types of the independent variables, i.e., the deflections inside the domain, the deflections and normal slopes on the boundary, the normal bending moments and effective shear forces per unit length on the boundary. The domain variables are interpolated by a linear combination of the fundamental solutions of both the biharmonic equation and Laplace's equation, whereas the boundary variables are approximated using the moving least squares. Based on the above analysis, the formulas of the hybrid boundary node method for the boundary value problems of the fourth order partial differential equation are obtained.For the thin plate bending problems, the dual reciprocity method is introduced to deal with the transverse distributed load and the subgrade reaction. The solution in this method is divided into two parts, i.e., the complementary and particular solutions. The particular one is obtained by the radial basis function interpolation, while the modified boundary conditions are applied in the hybrid boundary node method to solve the complementary one. The thin plates with various shapes and boundary conditions under various transverse loads are analyzed. At the same time, the influences of some computational parameters on the performance of the present method are investigated.Combining the multiple reciprocity method and the hybrid boundary node method, the multiple reciprocity hybrid boundary node method is proposed, and is used to solve the inhomogeneous potential problems. Based on the multiple reciprocity method, the inhomogeneous term of the governing equation is successively operated by differentiation operator. The particular solution is interpolated by the high-order fundamental solutions. No internal points are required in this method. The multiple reciprocity hybrid boundary node method is implemented for solving the elastic torsion problems and heat conductivity problems with heat source.The study shows that not only the boundary value problems of the second order partial differential equation, but also the boundary value problems of the fourth order partial differential equation can be effectively solved by the hybrid boundary node method. This method possesses the high accuracy, the good performance of convergence, and is suitable for analyzing all kinds of plates with arbitrary shapes and complicated boundary conditions and transverse distributed load. Therefore, the hybrid boundary node method is an attractive and prospective method in solving the practical engineering problems.