GORDON’s Transfinite Macroelements

Abstract

This chapter discusses transfinite macroelements, which are based on Gordon interpolation formula. The latter extends Coons interpolation formula (see Chap. 3) considering internal nodes as well. It will be shown that the standard tensor-product elements of Lagrange family constitute a subclass of transfinite elements, while one may generally use more or less internal nodes in several configurations. Moreover, true transfinite elements with different pattern in the arrangement of the internal nodes, as well as degenerated triangular macroelements, are discussed. A class of Cij macroelements is introduced, by influencing the trial functions as well as the blending functions. This class is so wide that can include even an assemblage of conventional bilinear elements in a structured $n_{\xi } \times n_{\eta }$ arrangement. A careful programming of the shape functions and their global partial derivatives resulted in a single subroutine that includes all twelve combinations. The theory is supported by several test cases that refer to potential and elasticity problems in simple domains of primitive shapes where a single macroelement is used. In a couple of cases, somehow more complex domains are successfully treated using two or three Gordon macroelements.