High performance hybrid mixed elements using orthogonal stress interpolants and scaling of the higher order stiffness

Abstract: During the evolution of the finite element method, formulations based on the principle of minimum potential energy have dominated the scene. These formulations employ an assumed displacement field within each element which must preserve continuity of displacements between adjacent elements. The difficulties in constructing shape functions which satisfy interelement continuity have proved to be a serious obstacle in the search for accurate and robust plate and shell elements. To avoid the problems inherent in the formulations based on the potential energy principle, attention has been drawn to other, alternative ways of developing finite element models. Most prominent among these approaches are the use of mixed and hybrid formulations associated with multi-field variational principles and the so-called free formulation theory which avoids explicit use of energy principles. Regardless of which of these procedures that are adopted, recent finite element research is to a large extent concerned with the development of high performance elements which at the same time are required to be accurate, reliable and simple with relatively few degrees of freedom allowing for easy physical interpretation of the results. The present thesis focuses attention on hybrid mixed formulations pioneered by Pian and co-workers. Both stress and displacement fields are assumed within the element domain. Compatible displacements expressed in terms of nodal values are enriched by a set of incompatible displacement functions. A key ingredient in the formulations is to employ the incompatible displacement field in a procedure for constraining the initially assumed stress field, thereby reducing the number of undetermined stress parameters. These parameters are subsequently eliminated and the resulting formulation is in standard displacement format. The application of hybrid mixed elements to problems of two dimensional elasticity constitutes a framework for the work described herein. The objective is to develop procedures for improving accuracy and efficiency of four-node hybrid mixed finite elements, especially for distorted element geometries. To this end, the use of orthogonal stress interpoIants, analytical integration and scaling of the higher order stiffness emerge as salient features of the present study. A new method for scaling of the higher order stiffness is proposed. The new scaling factor depends on the geometric shape of each finite element in a given mesh. A large number of elements have been programmed and tested. These include quadrilateral elements with translational degrees of freedom only and quadrilaterals having an extra rotational degree of freedom at each corner node. From the extensive numerical testing accomplished in the present study it is concluded that plane. quadrilateral hybrid mixed elements have superior performance over existing elements with identical configurations of nodal degrees of freedom. Furthermore, the use of orthogonal stress interpolants in conjunction with geometrically dependent scaling parameters for the higher order stiffness leads to significant improvements in computational efficiency and reduces distortion sensitivity without affecting the general superior performance of the hybrid mixed elements.