The equivalent parallelogram and parallelepiped, and their application to stabilized finite elements in two and three dimensions

Abstract

A new class of quadrilateral and hexahedral elements (four- and eight-noded in two and three dimensions, respectively) is presented. These elements are obtained by combining the concept of the equivalent parallelogram for plane problems, and the equivalent parallelepiped for three-dimensional problems, with the notion of incompatible modes. A key feature of the new elements is that integration of the element stiffness matrices is carried out using one-point integration. The use of affine-equivalent elements (parallelograms and parallelepipeds) permits a closed-form eigenvalue analysis which includes the incompatible modes, and a stabilization procedure based on the eigenvalue analysis ensures the full rank of the stiffness matrix. Numerical results for problems in elasticity and plasticity indicate equivalent or superior performance of the new elements, when compared with various elements based on enhanced strains or incompatible modes.