Some pre-computed universal numeric arrays for linear convex quadrilateral finite elements

Abstract

In this paper we investigate the finite element analysis for the solution of linear partial differential equations using linear convex quadrilateral elements. We show that a linear polygonal domain can be discretized into a set of special linear convex quadrilateral elements, which generate the same expression for the determinant of the Jacobian matrix under the isoparametric coordinate transformation. Analytical integration of the above element matrices is shown to depend on certain ‘universal pre-computed numeric arrays’, i.e., the arrays which are computed once, stored on a permanent file and then reused in all subsequent applications of the program. We have constructed such arrays for the five commonly used linear quadrilateral elements: Q4, Q8, Q9, Q12 and Q16. One speciality of these pre-computed arrays is that the arrays for lower order elements are already contained in the arrays for higher order elements. The performance of the proposed method is demonstrated by means of a numerical example.