C-1 natural neighbor interpolant for partial differential equations

Natural neighbor coordinates [20] are optimum weighted-average measures for an irregular arrangement of nodes in R-n. [26] used the notion of Bezier s implices 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 red uces to a cubic polynomial between adjacent nodes on the boundary partial d erivative 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 involve s 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) (P hi), psi(3I-1)(Phi), and psi(3I)(Phi) for node I are directly associated wi th the three nodal degrees of freedom w(I), theta(Ix), and theta(Iy), respe ctively. The C-1 shape functions interpolate to nodal function and nodal gr adient values, which renders the interpolant amenable to application in a G alerkin scheme for the solution of fourth-order elliptic PDEs. Results for the biharmonic equation with Dirichlet boundary conditions are presented. T he generalized eigenproblem is studied to establish the ellipticity of the discrete biharmonic operator, and consequently the stability of the numeric al method. (C) 1999 John Wiley & Sons, Inc.