Introduction

The ability to develop $C^1(\Omega)$ interpolants over an arbitrary bounded domain $\Omega$ is a much researched and far from trivial task. In the realm of finite elements, one of the first $C^1(\Omega)$ interpolants developed was the Clough-Tocher finite element [3], and in subsequent years many of its variants have emerged [4,5,6,7]. The higher-order smoothness or continuity requirement of interpolants is of interest since such class of trial functions are stipulated in a Galerkin forumulation for the solution of higher-order elliptic partial differential equations-- $C^1(\Omega)$ trial functions for the biharmonic (fourth-order) equation in elasticity, with particular emphasis on thin plate Kirchhoff bending being a notable application and a case in point.

In the interest of brevity, we directly proceed to present Farin's $C^1(\Omega)$ formulation, without touching upon the foundations of Sibson's original natural neighbor interpolant [2]. For some familiarity with the Sibson interpolant and its application to second-order partial differential equations in mechanics, the interested reader can refer to [8] and [9], and the references therein.