An Algorithm For Smoothed Finite Element Method Based On Bilinear Element

The smoothed finite element method(S-FEM)based on linear element,such as triangular element(T3)and tetrahedral element(T4),has been widely used in solving solid mechanical problems because of these mesh be automatically divided.However there is a problem of low stress accuracy for linear elements.It is well known that the smoothed finite element method based on bilinear elements,such as quadrilateral element(Q4)and eight-node hexahedron element(H8),can overcome the shortcoming of low precision of linear element,but it needs the coordinate mapping which leads to the accuracy of solutions depending on the quality of the mesh greatly and lower efficiency.Therefore,this paper mainly studies the smoothed finite element method based on bilinear element to obtain the exact solution of displacement and strain energy from the aspects of precision and speed for solid mechanics problems.In the 2D problems,a novel method(?SFEM-Q4)is proposed,which can obtain ultra-exact solutions by combining node-based S-FEM(NS-FEM),edge-based S-FEM(ES-FEM)and cell-based S-FEM(CS-FEM).This novel combination makes the best use of the upper bound property of the NS-FEM and the lower bound property of the CS-FEM,and establishes a continuous strain-energy function of a scale factor ?.Our ?SFEM-Q4 also ensures the variational consistence and the compatibility of the displacement field,and hence guarantees to reproduce linear field exactly.The stability,efficiency and ultra-accuracy of the method are verified by theoretical research and various examples of solid mechanics problems.In the 3D problems,the eight-node hexahedral element has the advantage of high precision compared to the tetrahedral element.However,coordinate mapping required in the hexahedron elements of FEM formulation costs huge running time,leading to poorperformance.Besides,the high-quality Jacobian matrices and meshes are required for H8,which affects the accuracy of the strain results greatly.In order to solve these problems,author proposes a novel simplified integration technique based on the smoothed finite element method(S-FEM)for the eight-noded hexahedron elements,where coordinate mapping is not demanded.The proposed new S-FEM-H8 models include simplified NS-FEM-H8(using node-based smoothing domains)and simplified FS-FEM-H8(using face-based smoothing domains).In the work,author divides a quadrilateral surface segment of a smoothing domain into two triangular sub-segments,so that the strain-displacement matrix can be calculated using a Gauss integration technique in each quadrilateral section,thus avoiding complex coordinate mapping.The numerical examples verify the proposed method can obtain the same accuracy and stability as traditional H8 elements.At the same time,the proposed method improves the efficiency of traditional H8 elements greatly..Therefore,by studying the smoothed finite element method based on bilinear element,a novel ?SFEM-Q4 is proposed for two-dimensional problems,and a new simplified smoothed finite element method based on eight-node hexahedron element is proposed for three-dimensional problems.These two methods not only solve the problem of low stress precision based on linear element,but also simplified H8 elements avoid complex coordinate mapping of bilinear element,and reduce the calculation of Jacobian matrix required for bilinear isoparametric elements.