Mixed Elements Of Two And Three Dimensions For Elasticity Problem

In this thesis, we focus on the linear elasticity equation under the Hellinger-Reissner variation formulation. A family of simple and stable rectangular, trian-gular, cubic and tetrahedral finite elements is constructed. The well-posedness, convergence, error estimates of these elements as well as the anisotropic property of rectangular and cubic elements are derived. Meanwhile the numerical tests for the conforming rectangular element and nonconforming triangular element are given. In the third part of this thesis, we construct the simplest anisotropic rectangular element R8 - 2 and cubic element C18 - 3. The R8 - 2 element has eight freedoms for the stress and two freedoms for the displacement. The C18 - 3 element has eighteen freedoms for the stress and three freedoms for the displacement. The BB conditions of the discrete problems are proven and the optimal error estimates are derived on the anisotropic meshes.In the fourth part of this thesis, we further construct a family of higher order rectangular elements. When k≥ 4 the commutativity property of the interpolation and the divergence operators is satisfied, based on this, the well-posedness of the elements, unique solvability of the discrete problem and error estimates are obtained. At the end of this chapter, we construct the corresponding elasticity complexes of these elements.In the fifth part of this thesis, we give a simple method to compute the dimension of the space Mk(K)={τ∈Pk(K; S)|diυτ= 0, τn|(?)k= 0}, which is important on the conforming tetrahedral element construction. Using this method we prove the dimension of the space M3(K) is zero, and give the base functions of the space M4(K).In the last part of this thesis, we present a new family of tetrahedral and a new family of triangular nonconforming elements. It is different from other elements that ours are local, simple, exactly affine equivalent, explicitly expressed and easy to program. They have different shape function spaces with others. We also present two simplified forms for the lowest order elements of two families by using the rigid motion model. The simplified form of triangle element has 12+3 degrees of freedom and the tetrahedral element has 36+6 degrees of freedom. In the same way, ours are local, simple, explicitly expressed and easy to program too.