A $C^1(\Omega)$ NEM Interpolant for Fourth-Order PDEs

Internal Report

N. Sukumar
Theoretical and Applied Mechanics
Northwestern University
Evanston IL 60208

Date: December 4, 1997

A postscript version of this document.

Abstract:

Farin [1] used the notion of Bézier simplices in Sibson coordinates [2] to propose a $C^1(\Omega)$ interpolant. The $C^1(\Omega)$ interpolant has quadratic precision, and reduces to a cubic polynomial between adjacent nodes on the boundary $\partial \Omega$. In this report, we present the $C^1(\Omega)$ formulation and propose a computational methodology for its implementation in the context of a Galerkin procedure for the solution of partial differential equations (PDEs). The approach involves the transformation of the original Bernstein basis functions $B_{ {\mathbf{i}}}^{3}(\Phi)$ to new shape functions $\Psi$, such that the shape functions $\psi_{3I-2}$, $\psi_{3I-1}$, $\psi_{3I}$ for node $I$ are directly associated with the three nodal degrees of freedom $w_I$, $\theta_{Ix}$, $\theta_{Iy}$, respectively. Such a transformation renders the interpolant amenable to application in a Galerkin scheme for the solution of fourth-order elliptic PDEs. Numerical interpolation results have verified the smoothness and quadratic precision property of the interpolant, and currently its application to the biharmonic equation is in progress.