Study On The Improved Reproducing Kernel Particle Method For Elasticity Problems

In recent years,a class of new numerical method called meshless method has been proposed.Compared with traditional numerical methods,meshless method only needs information at nodes,and doesn't discretize the problem domain into a mesh,which contributes meshless method the merits of simple preprocessing and high accuracy,especially in dealing with elastoplastic,large deformation,fracture and other non-linear problems with unparalleled advantages.This meshless method has become one of the hot-spots and the trend in the research of scientific and engineering computing.Reproducing kernel particle method(RKPM)is one of the most commonly studied and applied meshless methods.The improved reproducing kernel particle method(IRKPM),which is formed by introducing orthogonalization into the RKPM,can effectively solve the disadvantages,such as great computational cost and low computational efficiency.The advantages of the IRKPM is that it can solve inverse matrix conveniently and replace matrix operations with algebraic operations during the process of constructing shape function,which greatly improves the computational efficiency of the shape function and its first-order partial derivative.In this paper,the IRKPM is combined with Galerkin weak form of elasticity problems,then applied to two-dimensional elasticity problems with Lagrange multiplier method applying the essential boundary conditions,and also applied to three-dimensional elasticity problems with penalty method applying the essential boundary conditions.Compared with the RKPM,the advantage of IRKPM is that the computational efficiency of shape function and its first-order partial derivative is greatly improved without reducing the computational accuracy.Element-free Galerkin(EFG)method and improved element-free Galerkin(IEFG)method are also considered when studying the RKPM and IRKPM.Taking the computational accuracy of EFG and IEFG as the reference and the computational efficiency as the contrast.Results show that the accuracy of IRKPM and RKPM is the same,and it is similar to IEFG and EFG.They also show that the efficiency of IRKPM with respect to RKPM is much higher than that of IEFG with respect to EFG.Therefore,the improved reproducing kernel particle method is far more effective.The IRKPM for stress concentration problem is presented in this dissertation.It can flexibly distribute nodes in problem domain due to detachment from elements.It can obtain higher computational accuracy through increasing the number of nodes in the high stress gradient area.Compared with the FEM,not only can the IRKPM obtain higher computational accuracy,but also computational efficiency can be improved greatly.The IRKPM is also used to explore the feasibility and correctness in auxetic honeycomb structure,which lays foundation for studying the physical and mechanical properties of the auxetic honeycomb structure.In order to show the correctness and efficiency of the IRKPM,the MATLAB codes of the IRKPM for two-dimensional elasticity problems and three-dimensional elasticity problems are written.Typical numerical examples are given to demonstrate the correctness and efficiency of the IRKPM for elasticity problems.