Class of Accurate Low Order Finite Elements

Abstract: High accuracy and low computational costs are essential properties for efficient finite element codes. Improvements of the finite element method in order to satisfy these qualities have been the subject for extensive research activities ever since its introduction. In this thesis, a class of improved low-order finite elements is proposed which possesses high accuracy in bending even for coarse meshes. The elements are insensitive to material incompressibility and they are all based upon the Hu-Washizu variational principle. A 4-node plane quadrilateral element as well as a 8-node brick element are proposed and also a 4-node axisymmetric element is presented. The plane element was first formulated by the three-field Hu-Washizu principle, but later it was realized that the same response could be achieved by the assumed strain method. This reformulation leads, among other things, to gained computational efficiency. Consequently, the subsequent axisymmetric element and the brick element were established using the assumed strain method. The assumed strain method can be systematically formulated within the framework of the Hu-Washizu principle which makes the elements variational consistent providing that certain conditions are fulfilled. Due to its simplicity, the assumed strain method is highly rewarding in large-scale analysis and since a strain-driven format is obtained naturally which resembles the standard displacement method, this method is well suited for nonlinear analysis. For simplicity, however, only linear problems are considered here. The approach presented in the appended papers is based upon the fundamental idea in which the element stiffness matrix is divided into two parts. These parts are denoted fundamental stiffness and higher-order stiffness. The fundamental stiffness is the part which guarantees that the element is convergent. This means that the element fulfills the convergence criterion and it can be shown that all elements that are proposed pass the patch test {itshape a priori}. The higher-order stiffness must restore the correct rank to the total element stiffness matrix. In addition, the higher-order stiffness is chosen such that it significantly improves the accuracy. By consistent modifications of the higher-order stiffness, the proposed finite elements provide the exact strain energy for pure bending of regular elements. Sensitivity to mesh distortion is also effectively reduced by use of dimensionless parameters which emerge naturally within this approach which includes use of a coordinate system aligned with the principal axes of inertia.