The Complex Variable Meshless Local Petrov - Galerkin Method Research

The meshless method is a new numerical technique which has beenproposed in recent years. Meshless methods only require a set of scattered nodesto construct the approximate functions. It will not exist the difficulties caused byremeshing or mesh complications. The meshless method has simplerimplementation procedures, and it has good stability and higher computationalprecision. So the meshless method has become a hot research areas and thedevelopment trend of science and engineering problems.The meshless local Petrov-Galerkin method is a new computationaltechnique which is formed by local weak form and the moving least squares. Itis a truly meshless method because the trial function construction and thenumerical integration are not depend on the meshes. However, a disadvantage ofthe conventional least squares method is that the final algebra equations systemis sometimes ill-conditioned, furthermore, the shape function formed with it hasgreat computation cost because of large numbers of nodes selected in thedomain of problem. The advantages of the shape function constructed by thecomplex variable moving least squares is that the trial function of a twodimensional problem is formed with one dimensional basis function. So the trialfunction has fewer undetermined coefficients in the complex variable movingleast squares than in the moving least squares. And then the computationalefficiency of the complex variable moving least square method is higher. So thecomplex variable moving least square method is used to construct the trialfunction and the Heaviside step function is used as the test function.Furthermore, it is applied to solve potential problems, elasticity problems. Themain researches of this thesis are as follows:The shape function, which is constructed by the complex variable movingleast square method, is discussed at first. The complex variable moving leastsquare method is used for the curve fitting and surface fitting. Some selectednumerical examples are presented to illustrate the efficiency of this method. The complex variable meshless local Petrov-Galerkin method is applied totwo-dimensional potential problems, and the complex variable meshless localPetrov-Galerkin method for potential problem is established, and the formulaeof the complex variable meshless local Petrov-Galerkin method are deduced.The complex variable meshless local Petrov-Galerkin method is applied to2D elasticity problem. The discrete equation is deduced by the weak form ofequivalent equation, and then the complex variable meshless localPetrov-Galerkin method for elasticity problem is presented, and the formulae ofthe complex variable meshless local Petrov-Galerkin method are deduced.The corresponding MATLAB programs of the complex variable meshlesslocal Petrov-Galerkin method have been compiled. Several numerical examplesare solved to demonstrate the validity of the present methods.