This paper introduces a general theory for the derivation of the shape functions for the quadrilateral family of finite elements. The first section deals with the Lagrangian shape functions for the cases of uniform and boundary-described elements. Two basic procedures are introduced; the first by linear combinations of sideinterpolations and the second by superposition. The remainder of the paper introduces a theory for the general uniform Hermitian element of any order. Details for quadrilateral elements, with first order derivatives are explained. All of the shape functions presented here were derived in the interval [0,1].
The shape functions, developed by such an engineering approach, have been used successfully in the ABSEA Finite Element System of Cranfield Institute of Technology.
LAGRANGIAN QUADRILATERAL ELEMENTS
The Lagrangian quadrilateral family of finite elements appeared very early in the history of the finite element method. Melosh' derived the 4-node rectangular element. Pian' gave an algorithm for the direct displacement approach with any number of unknown coefficients. The concept of arbitrary-noded elements was described by Irons3 but his work did not include any interpolation details. Argyris4 derived the 8-node parallelogramic element. Ergatoudis' derived the shape function for some Lagrangian and screndipity elements. However, no general theory for serendipity elements was produced. Dunne' showed that two-dimensional shape functions can be complete bivariate polynomials for the mth degree, if the number of element nodes no is given by n, = (rn + l)(m + 2)/2
Uniform Lagrangian element
Assume any quadrilateral element in the x-y Cartesian plane, which is transformed into a square of unit side in the intrinsic 5-q plane. The intrinsic element can be described by thc following set:
Suppose that there exists a field function, in the domain of the element, which satisfies the following interpolatory conditions: p(tr,qs)=pr.s, 1 d r d m , 1 < s < n