The Application Of Edge-based Smoothed Finite Element Method In Heat Conduction Problems Of Anisotropic Media

The problem of heat conduction exists in various engineering fields,which has become one of the hottest topics for scholars at home and abroad.Finite element method(FEM)is the most commonly used numerical method to solve such problems.With the deepening of the research,some inherent defects and problems(such as low precision,mesh quality,volume lock,etc.)of FEM are also revealed.The root of these problems in FEM is that all operations of the standard variational principle of the compatible displacement field are limited to the grid.To solve these problems,Liu G.R and his team proposed the smooth finite element method(S-FEM).In S-FEM,the smooth strain replaces the compatible strain in FEM.The smooth strain needs to be completed within the smooth region,and the selection of the smooth region is not necessarily related to the original mesh.As long as they do not overlap with each other and can completely cover the problem domain,they can be smooth areas.To make full use of the finite element mesh and node,successively developed smooth type unit finite element method(CS-FEM)and finite element method of node type smooth smoothed type(NS-FEM),boundary finite element method(ES-FEM)and 3D surface smooth and finite element method(FS-FEM).Compared with the traditional FEM,the S-FEM without introducing any additional degrees of freedom and function derivative calculation form,so there is no need of isoparametric mapping technology,and has better robustness in solving the problem of mesh distortion and the extremely large deformation.In addition,S-FEM has the effect of "weakening",which can weaken the problem of "over-hardness" of traditional FEM stiffness matrix.Therefore,S-FEM has better precision and higher convergence rate than FEM.Some S-FEM,such as NS-FEM,ean solve volume lock problem well.In the case of solid mechanics,ES-FEM always shows the numerical characteristics of super-convergence and high precision.ES-FEM uses easy-generated low-order triangular element to discretize the problem domain.Smooth strain is established by using smooth strain technique and divergence theorem.Because the smooth region of boundary type is beyond the scope of FEM grid,the establishment of interpolation function is not limited to FEM grid.Therefore,S-FEM can be regarded as the outcome of combination of finite element and meshless method.ES-FEM has been applied to many problems,but it is seldom reported in heat conduction problems.In this paper,ES-FEM is used to solve the two-dimensional anisotropic steady state heat conduction problem.Firstly,the current situations of the research of S-FEM at home and abroad are introduced.Then,the theory and formula of ES-FEM are introduced and deduced in detail.The numerical accuracy and super convergence properties of ES-FEM are validated by the problems of cantilever beam and infinite center-hole plate.Finally,ES-FEM is applied to solve the problem of 2D anisotropic steady state heat conduction.The smooth boundary finite element method(ES-FEM)is studied.The numerical results show that ES-FEM has high precision and convergence rate in solving heat conduction problems,and it has wide development space.