Boundary Penalty Finite Element Method for Blending Surfaces, II. Superconvergence, Stability and Applications.
by
Zi-Cai Li
National Sun Yat-sen University, Kaohsiung, Taiwan
Coauthors: Chia-Shen Chang

This paper is a continuity of study in [1] by using partial differential equations for blending surfaces. This paper consists of two parts. In Part I, the biharmonic equations are chosen, and the boundary penalty finite method (FEM) using piecewise cubic Hermite elements is employed to seek their approximate solutions, in particular satisfying the derivative and periodical boundary conditions. Theoretical analysis is made to discover that when the penalty power \sigma = 2, 3.5 and 1.5, optimal convergence rate, superconvergence and optimal stability can be achieved, respectively. Moreover, the derivative and periodical conditions of the numerical solutions have, at least, O(h⁴) of convergence rates, where h is the maximal boundary length of quasi-uniform elements. Moreover, a new transformation for the nodal variables used is given to improve numerical stability significantly. To compromise accuracy and stability, \sigma = 2 ~ 3 is suggested. By the techniques proposed in this paper, the finite elements may not be necessarily chosen to be small due to high convergence rates.

In Part II, the blending surfaces in 3 dimensions (3D) are taken into account by parameters, x(r, t), y(r, t) and z(r, t). The boundary penalty techniques are well suited to the complicated tangent (i.e., derivative) in engineering blending. The corresponding theoretical analysis can be obtained from Part I. Moreover, the stiff analysis of 3D blending is given to conduct the linear algebraic equations. Several interesting samples of 3D blending surfaces are provided, to display the effectiveness of the new techniques in this paper, and the advantages: optimal and unique solutions of blending surfaces, ease in handling the complicated boundary constraint conditions, and less CPU time needed.

[1] Z.C. Li, Boundary Penalty Finite Element Methods For Blending Surfaces, I. Basic Theory, accepted by J. Comp. Math.

Date received: February 9, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabp-35.