Group theoretical formulation of quadrilateral and hexahedral isoparametric finite elements

Abstract

The group supermatrix procedure, developed by Zlokovic, is applied for derivation of stiffness matrices of quadrilateral and hexahedral isoparametric finite elements using the symmetry groups C-2v and D-2h respectively. The group supermatrix procedure introduces monomial shape functions in G-invariant subspaces, as well as nodal coordinates, Jacobians and matrix expressions pertaining to particular subspaces. Decomposition of the spaces of quadrilateral and hexahedral elements into four and eight G-invariant subspaces respectively is accomplished after the isoparametric transformation of the initial elements without symmetry properties into four rectangular and eight rectangular hexahedral elements. The computing of stiffness matrices of these elements by the group supermatrix procedure is programmed in Mathcad and in KOMIPS programs. In comparison with the conventional derivation and computation of stiffness matrices of these elements, the group supermatrix procedure provides substantial reductions in the amount of formulation and calculation, because it deals with monomial instead of polynomial shape functions, shorter expressions and smaller matrices.