Study On Meshless Radial Basis Reproducing Kernel Particle Method For Elastic Dynamics Problem

Elastic dynamics,as an important branch of solid mechanics,is based on the experimental laws of mechanics and the introduction of mathematical methods to study the dynamic relationship between force and deformation of elastic objects,which has been successfully applied in industries such as railroad engineering,civil engineering,bridges and machinery.The study of elastic dynamics problems can provide theoretical support for structural optimization and has important theoretical significance for practical engineering.Mathematically,the study of elastic dynamics problems can be translated into the problem of solving partial differential equations for the boundary value and initial boundary value.There are mainly analytical and numerical methods to study elastic dynamics problems.The analytical method can only solve elastic dynamics problems with relatively simple geometry,and it is more difficult to solve complex elastic dynamics problems with complex shapes and structures.Therefore,numerical simulation methods are mainly used to study elastic dynamics problems.As the main numerical simulation method,the meshless method establishes the approximation function based on the information of nodes,which effectively avoids the dependence of traditional mesh-based numerical methods on cells and meshes,and shows unique advantages in the study of elastic dynamics problems.At present,the reproducing kernel particle method(RKPM)is a more theoretically complete and more applied lattice-free method.However,the RKPM has a significant drawback that the calculation results are easily influenced by different kernel functions.For this shortcoming of the RKPM,the radial basis reproducing kernel particle method(RRKPM)is constructed by combining the RKPM and the radial basis functions.Moreover,the established RRKPM is applied to analyze the elastic dynamic problems of homogeneous materials and functionally graded materials(FGMs),and an efficient numerical simulation program was prepared in MATLAB.Subsequently,the obtained results are compared to the exact solutions or the reference solutions of the finite element method,the correctness and reliability of the meshless RRKPM for solving elastic dynamic problems of homogeneous materials and FGMs are verified.The effects of kernel function,penalty factor,shaped parameters of radial basis function,control parameters of the radius of influencing domain,loading step size and node distribution on numerical accuracy are discussed in detail with numerical examples,and the values of the optimal parameters for solving elastic dynamic problems are determined.The method developed in this thesis is characterized by easy pre and post processing and low computational errors in the analysis of elastic dynamics problems.Compared with the meshless RKPM,the method established in this thesis reduces the bad impact of different kernel function options on the accuracy of computational results,and exhibits smaller computational errors,faster convergence and higher stability in solving elastic dynamics problems.In addition,the proposed method requires only node information,is not limited by mesh,and does not require mesh reconstruction in the analysis of elastic dynamics problems,providing a new means to carry out research work on elastic dynamics problems.