Transfinite element methods: Blending-function interpolation over arbitrary curved element domains

Abstract

In order to better conform to curved boundaries and material interfaces, curved finite elements have been widely applied in recent years by practicing engineering analysts. The most well known of such elements are the “isoparametric elements”. As Zienkiewicz points out in [18, p. 132] there has been a certain parallel between the development of “element types” as used in finite element analyses and the independent development of methods for the mathematical description of general free-form surfaces. One of the purposes of this paper is to show that the relationship between these two areas of recent mathematical activity is indeed quite intimate. In order to establish this relationship, we introduce the notion of a “transfinite element” which, in brief, is an invertible mapping$\vec T$ from a square parameter domainJ onto a closed, bounded and simply connected regionℛ in thexy-plane together with a “transfinite” blending-function type interpolant to the dependent variablef defined overℛ. The “subparametric”, “isoparametric” and “superparametric” element types discussed by Zienkiewicz in [18, pp. 137–138] can all be shown to be special cases obtainable by various discretizations of transfinite elements Actual error bounds are derived for a wide class of semi-discretized transfinite elements (with the nature of the mapping$\vec T$:J→ℛ remaining unspecified) as applied to two types of boundary value problems. These bounds for semi-discretized elements are then specialized to obtain bounds for the familiar isoparametric elements. While we consider only two dimensional elements, extensions to higher dimensions is straightforward.