Application And Extension Of Finite Element Method Based On Analytic Trial Function

The finite element method(FEM)is an important method in computational mechanics.The research on finite element method has been paid more and more attention.The problem of stress field of singular point is low in calculation accuracy,sensitive to mesh distortion,and the problem of shear locking are common problems in finite element method.Based on analytic function of finite element method(ATF-Q8)is a good way to avoid these problems.Compared with the traditional finite element method,the new finite element method has the advantages of good sensitivity against mesh distortion and high calculation accuracy.On the basis of predecessors' research,this paper studies the finite element method of two-dimensional problem and generalizes it into the calculation of quasicrystal materials.By using the method of generalized complex functions,the exact expression of the n order homogeneous polynomial stress functions of quasicrystal materials is derived,and the independence and completeness of the polynomial of the stress functions obtained are proved.At the same time,the element stiffness matrix of quasicrystal material is obtained based on the minimum complementary energy principle by using the obtained expression and the residual energy functional.Thus,the hybrid stress function element of the analytic trial function is applied to the calculation of the plane problem of quasicrystal materials.The results show that the polynomial stress function of the plane problem of quasicrystal material contains six constants,so that any n-th polynomial stress function contains at most six independent polynomials.The numerical examples in this paper show that the finite element method can solve the traditional finite element problems such as grid distortion sensitive problem and low stress precision,and so on.So that the method can be applied to the plane problem of quasicrystals in multi-field coupling problems.