Abstract
Natural neighbor coordinates [20] are optimum weighted-average measures for an irregular arrangement of nodes in R-n. [26] used the notion of Bezier simplices in natural neighbor coordinates Phi to propose a C-1 interpolant. The C-1 interpolant has quadratic precision in Omega subset of R-2, and reduces to a cubic polynomial between adjacent nodes on the boundary partial derivative Omega. We present the C-1 formulation and propose a computational methodology for its numerical implementation (Natural Element Method) for the solution of partial differential equations (PDEs). The approach involves the transformation of the original Bernstein basis functions B-i(3)(Phi) to new shape functions Psi(Phi), such that the shape functions psi(3I-2) (Phi), psi(3I-1)(Phi), and psi(3I)(Phi) for node I are directly associated with the three nodal degrees of freedom w(I), theta(Ix), and theta(Iy), respectively. The C-1 shape functions interpolate to nodal function and nodal gradient values, which renders the interpolant amenable to application in a Galerkin scheme for the solution of fourth-order elliptic PDEs. Results for the biharmonic equation with Dirichlet boundary conditions are presented. The generalized eigenproblem is studied to establish the ellipticity of the discrete biharmonic operator, and consequently the stability of the numerical method. (C) 1999 John Wiley & Sons, Inc.