Functionally Graded Materials Meshless Kriging Method

In recent years, a new numerical technique, the meshless method, has been proposed. The meshless method, which was born with the objective of eliminating part of the difficulties associated with reliance on a mesh to construct the approximation, constructs the approximate functions by a set of scattered nodes only. The meshless method overcomes the shortcomings of traditional computational methods, and possesses the advantages of high precision. Another key feature of the present method is that the computational procedures are simple. The meshless method has become one of research topics in computational mechanics.The meshless local Petrov-Galerkin(MLPG) method is a new computational technique which is based on local weak form and the moving least squares approximation (MLS). This method, which does not need mesh, either for purposes of interpolation of the solution variables, or for the numerical integration. However, the MLPG method, which built the shape function based on MLS approximation, have its less efficiency, and that it can form ill-conditioned or singular equations sometimes, and the boundary conditions cannot be enforced accurately. To overcome these shortcomings , combining the local weak form method with the moving Kriging interpolation,the meshless local Kriging method is presented in this paper, in which the Heaviside step function is used as the test function. Furthermore, the present method is proposed for solution of two-dimensional elasticity, elastodynamics and transient heat conduction problems in functionally graded materials, The main researches of this thesis are as follows:The shape function, which constructed based on the moving Kriging interpolation, is discussed at first. The shape function possesses the Kroneckerδproperty and decomposition.Combining the local weak form with the moving Kriging interpolation, and the Heaviside step function is used as the test function,the meshless local Kriging method is developed. The present method has greater computational efficiency than the conventional MLPG method, and the essential boundary conditions can be imposed easily. Furthermore , it is applied to solve two-dimensional elasticity problems in functionally graded materials.The meshless local Kriging method is used for solution of two-dimensional elastodynamics, and the Newmark time integration method is used for time history analyses. The meshless local Kriging method for elastodynamics problems in functionally graded materials is presented.Based on the meshless local Kriging method, the time-domain is discreted by the traditional two-point difference method, the meshless local Kriging method for two-dimensional transient heat conduction problems is presented, and furthermore it is applied to solve two-dimensional transient heat conduction in functionally graded materials.In the dissertation, the corresponding MATLAB codes of the methods above have been written. Some numerical examples are presented to demonstrate the validity of the present methods.