Taylor-type Meshless Methods For Higher-order Partial Differential Equation

Generalized finite difference method(GFDM)is a meshless method for solving partial differential equations.It is based on Taylor series expansion and weighted moving least squares approximation,which effectively avoids the complicated work of grid generation and numerical integration.In this thesis,GFDM is used to solve three kinds of higher-order partial differential problems for static models of plate bending with constant thickness,variable thickness and dynamic plate bending model.Firstly,for the bending problem of plates of equal thickness,the overdetermination problem caused by mixed boundary conditions is overcome by introducing "supplementary points" on the boundary,so as to solve this kind of problem efficiently.Secondly,a class of higher order partial differential equations with variable coefficients,namely the bending problem of plates with variable thickness,is solved in this paper.In this paper,the difficulty caused by variable coefficient is solved efficiently and quickly by simple operation,and the bending problem of variable thickness plate with complex geometry is solved.Finally,for the dynamic plate bending problem,this paper combines GFDM with the space-time coupled approach(ST-GFDM),and transforms the two-dimensional transient space-time problem into three-dimensional steady-state space problem by treating time as another dimensional space.This method retains the original advantages of GFDM and reduces the complexity of discrete time and spatial partial derivatives.In this paper,numerical examples with different boundary conditions,different loads,and different shapes such as multi-connected domains are given to fully verify the effectiveness and accuracy of GFDM in solving these three kinds of higher-order partial differential equations.