Research On Interpolation Meshless Method Based On Node Integration

Meshless method is currently one of the most popular numerical calculation methods.In the numerical calculation,meshless method only needs the node informations to construct the shape function,which obtains highly valuable and applicable prospect without the constraints of cells and grids.With the merit of Kronecker ? function characteristics,the interpolation meshless method has obvious advantages in dealing with complex boundary problems compared with the majority meshless methods which have difficulty in the implementation of essential boundary condition.Most of the traditional interpolation meshless methods use GAUSS integrals for numerical integration and usually need higher-order GAUSS integrals to obtain high-precision numerical solutions,which is detrimental to the calculation efficiency.Despite some node integration methods have been proposed to improve the calculation efficiency,they are mainly aimed at the research of traditional approximation meshless methods,and the research and application in interpolation meshless methods are relatively limited.Therefore,the interpolation meshless method based on node integrals is studied in this paper.Three kinds of interpolation meshless methods are mainly studied in this thesis,including the radial base point interpolation method,the improved moving least squares interpolation method and the moving least squares interpolation method based on non-singular weight function.Furthermore,the node integrals are introduced on the basis of these methods.This thesis proposed new meshless method inherits the advantages of the interpolation meshless method to facilitate the application of boundary conditions and the nodal integral without the need for a background integral mesh,demonstrating its feasibility,efficiency and correctness in numerical calculations.The basic principles of constructing interpolation shape functions were firstly illustrated in this thesis by radial basis point interpolation method,improved interpolation element-free Galerkin method,and non-singular weight function-based interpolation element-free Galerkin method,respectively.Then,the interpolation meshless method for elastic mechanics problems was constructed for numerical examples to verify the correctness and effectiveness of the three methods.The calculation accuracy was analyzed by the corresponding calculation parameters.Three improved direct node integrations were combined with the three interpolation meshless methods to analyze the numerical examples.The results show that the interpolation meshless methods can effectively improve numerical stability and calculation accuracy,and has higher calculation efficiency than GAUSS integration.The calculation accuracy and efficiency of three interpolation meshless methods based on improved direct node integration were also compared and analyzed the corresponding calculation parameters.Finally,stableilized conforming node integration and three improved stableilized conforming node integration were combined with the three interpolation meshless methods to illustrate that the calculations accuracy of the four stableilized conforming node integration by numerical examples.The results computed by the interpolation meshless methods were equivalent to that of GAUSS integrals,verifying the applicability and effectiveness of the four stableilized conforming node integration in the interpolation meshless method.