A Scaled Moving Least Squares Approximation And Its Application To The Element-free Galerkin Method

Mesh generation in some situations is arduous,time consuming and fraught with pitfalls.Over the past two decades,meshless(or meshfree)methods,which are approximations based on nodes,can overcome the disadvantage that traditional numerical methods depend on the mesh of the solution domain.Shape functions play an important role in meshless methods.The moving least squares(MLS)approximation is one of the most extensively used methods to construct meshless shape functions.In the MLS,the moment matrix in the MLS may become ill-conditioned.Then,the inversion of the matrix leads to the decrease in computational stability and precision.This drawback also affects the performance of the element-free Galerkin(EFG)method.To overcome the ill-conditioned issue involved in the MLS,a scaled moving least squares(SMLS)approximation is introduced in this paper.Compared with the MLS,the SMLS has higher computational stability and precision.Error estimates of the SMLS are proved theoretically.The theoretical results indicate that the SMLS produces optimal order error estimates for the approximations of the function and its arbitrary order derivatives.In order to improve the stability of the element-free Galerkin method,based on the SMLS,a new EFG method is then developed and analyzed for linear elliptic boundary value problems.Finally,numerical examples are given to demonstrate theoretical error estimates.For all examples,convergent solutions are obtained,and the experimental convergence rates are in excellent agreement with the theoretical rates.This dissertation first reviews the recent developments of some numerical methods,and introduces the research background of the meshless methods and the research progress of the MLS approximation.In the Chapter 2,we describe the MLS approximation and compare it with the least squares method.In the Chapter 3,by using a scaled basis function,the SMLS approximation is described.Error estimates of the SMLS are derived for the approximated function and its arbitrary order derivatives.Finally,some numerical examples are given.Numerical results indicate that the SMLS provides monotonic convergence and high accuracy results with high computational stability.In the Chapter 4,we combine the SMLS with the global Galerkin weak form to develop a new EFG.Error estimates of the new EFG are also established for Robin and Dirichlet boundary value problems.Numerical examples are given to verify the theoretical error results.The last chapter gives some conclusions and prospects.