Researches On The Hybrid Complex Variable Element-Free Galerkin Method

Meshless method is an important numerical method solving science and engineering problems.Compared with traditional numerical methods,such as finite element method and boundary element method,meshless method does not need remeshing technique,then it can obtain the solutions with greater computational precision for some complicated problems,such as large deformation and dynamic crack growth.Now the element-free Galerkin(EFG)method is one of meshless methods studied and applied most widely,and the improved complex variable element-free Galerkin(ICVEFG)method has higher computational efficiency than the EFG method.In this dissertation,combining the dimension splitting method with the ICVEFG method,the hybrid complex variable element-free Galerkin(HCVEFG)method is presented for three-dimensional potential,transient heat conduction,wave equation,advection-diffusion,elasticity and elastoplasticity problems.The HCVEFG method for three-dimensional potential problems is presented.By using the dimension splitting method,a three-dimensional potential problem is transformed into a series of two-dimensional ones which can be solved with the ICVEFG method to obtain the discretized equations.And finite difference method is used in the spliting direction for coupling the two-dimensional discretized equations,then the equation of the HCVEFG method for three-dimensional potential problem can be obtained.The error and the convergence of the solutions of the HCVEFG method for three-dimensional potential problems are analyzed by the numerical examples.Compared with the improved element-free Galerkin(IEFG)method for solving the three-dimensional potential problems,the HCVEFG method presented in this dissertation can improve the computational efficiency greatly.The HCVEFG method for three-dimensional transient heat conduction problems is presented.By using the dimension splitting method,a three-dimensional transient heat conduction problem is transformed into a series of two-dimensional ones which can be solved with the ICVEFG method to obtain the discretized equations.And finite difference method is used in the splitting direction for coupling the two-dimensional discretized equations,and the two-point difference method is selected for the time discretization,then the equation of the HCVEFG method for three-dimensional transient heat conduction problems can be obtained.The error and the convergence of the solutions of the HCVEFG method for three-dimensional transient heat conduction problems are analyzed by the numerical examples.It is shown that the new method has the advantage of improving computational efficiency.The HCVEFG method for three-dimensional wave equations is presented.The dimension splitting method and the ICVEFG method are applied to discretize the three-dimensional space domian,and the central difference method is used for time discretization,then the final discretized equation of the HCVEFG method for three-dimensional wave equations can be derived.The error and the convergence of the solutions of the method are analyzed by the numerical examples.It is shown that the new method can not only has higher computational precision,but also can improve the computational efficiency.The HCVEFG method for three-dimensional advection-diffusion problems is presented.The three-dimensional space domain can be discretized by using dimension splitting method and the ICVEFG method,and two-point difference method is used for the time discretization,then the final discretized equation of the HCVEFG method for three-dimensional advection-diffusion problems can be derived.Numerical examples are presented to discuss the error and the convergence of the solutions of the HCVEFG method for three-dimensional advection-diffusion problems.And it is shown that the HCVEFG method has the advantages of higher computational accuracy and efficiency.The HCVEFG method for three-dimensional elasticity problems is presented.The equilibrium equation of a three-dimensional elasticity problem can be written as three sets of equations,which contains the equilibrium equations of any two directions.By introducing the dimension splitting method,a set of the equilibrium equation is transformed into a series of two-dimensional problems,which can be solved with the ICVEFG method to obtain the discretized system equations.And finite difference method is used in the splitting direction for coupling the two-dimensional discretized equations.Similarly,the discretized equation of another set of the equilibrium equation can be derived.Then by coupling those two discretized system equations,the numerical solutions of the HCVEFG method for three-dimensional elasticity problems can be obtained.The error and the convergence of the solutions of the HCVEFG method are analyzed by the numerical examples,and the advantage of the HCVEFG method has higher computational efficiency is shown.The HCVEFG method for three-dimensional elastoplasticity problems is presented.The equilibrium equation of a three-dimensional elastoplasticity problem can be written as three sets of equations,which contains the equilibrium equations of any two directions.By introducing the dimension splitting method,a set of the equilibrium equation is transformed into a series of two-dimensional problems,which can be solved with the ICVEFG method to obtain the discretized system equations.And finite difference method is used in the splitting direction for coupling the two-dimensional discretized equations.Similarly,the discretized equation of another set of the equilibrium equation can be derived.Then by coupling those two discretized system equations,the numerical solutions of the HCVEFG method for three-dimensional elastoplasticity problems can be obtained.The error and the convergence of the solutions of the HCVEFG method are analyzed by the numerical examples.And the numercial solutions are compared with the ones of ABAQUS and the IEFG method to show that the HCVEFG method has higher computational efficiency.For the HCVEFG methods presented above,the MATLAB codes are written.And some numerical examples are given to show the validity and efficiency of the HCVEFG methods.The HCVEFG methods presented in this dissertation can improve the efficiency of the EFG method for three-dimensional problems,and will promote the applications of mshless methods in engineering.