Researches On The Dimension Splitting Element-free Galerkin Method For Three-dimensional Problems

Meshless method is an effective numerical method developed after the finite element method.Compared with finite element method and boundary element method which are based on the mesh,meshless method only relies on the information of nodes in the problem domain when the approximation function or interpolation function is established,and does not need to reconstruct tye mesh,which results in the method has obvious advantages when dealing with crack propagation and large deformation problems and so on.It has become an indispensable research method of scientific and engineering computing.At present,the element-free Galerkin(EFG)method based on the moving leastsquares approximation is one of the most popular meshless methods.And the computational accuracy is high,but the amount of calculation is large.Especially when solving three-dimensional problems,the computational efficiency of the shape function is low which results in long CPU time is spent.In this paper,in order to improve the computational efficiency of the EFG method,by introducing the dimension splitting method,the dimension splitting element-free Galerkin(DSEFG)method for three-dimensional problems is proposed.Using the dimension splitting method,the three-dimensional potential problem is transformed into a series of the corresponding two-dimensional potential problems in the splitting direction.And the two-dimensional problems are solved with the improved element-free Galerkin(IEFG)method.Then the DSEFG method for three-dimensional potential problem is presented.Numerical examples illustrate that the DSEFG method has higher computational accuracy and faster computational speed compared with the IEFG method solving the three-dimensional problem directly.For the three-dimensional transient heat conduction problem,it can be transformed into a series of the corresponding two-dimensional heat conduction problems in the splitting direction by using the dimension splitting method.Finally,the DSEFG method for three-dimensional transient heat conduction problem is obtained.Numerical examples are given to illustrate the effectiveness and superiority of the DSEFG method.For the three-dimensional wave equation,it can be transformed into a series of the corresponding two-dimensional wave equations in the splitting direction by using the dimension splitting method.Finally,the DSEFG method for three-dimensional wave equation is obtained.Numerical examples are given to illustrate the effectiveness and superiority of the DSEFG method.For the three-dimensional advection-diffusion problem,it can be transformed into a series of the corresponding two-dimensional advection-diffusion problems in the splitting direction by using the dimension splitting method.Finally,the DSEFG method for three-dimensional advection-diffusion problem is obtained.Numerical examples are given to illustrate the effectiveness and superiority of the DSEFG method.For the three-dimensional elasticity problem,the governing equation is rewritten into three systems of equivalent equations.Numerical solutions can be obtained by solving any two systems of equations.By using the dimension splitting method,any system of equations can be transformed into a series of corresponding two-dimensional elasticity problems in the splitting direction.Then the final equations derived from two systems of equations are coupled.Finally,the DSEFG method for three-dimensional elasticity problem is obtained.Numerical examples are given to illustrate the effectiveness and superiority of the DSEFG method.For the DSEFG method presented above for each problem,the corresponding algorithm implementation process is given,and the MATLAB codes is developed.The DSEFG method presented in this thesis is one of fast and effective meshless methods to solve three-dimensioanl problems.