Computation of open Willmore-type surfaces

We present a numerical scheme for finding minimizing surfaces of functional s of the form F[S] = integral (s) [alpha + beta (H - H-0)2 - gammaK]d sigma, where H and K are the mean and Gaussian curvatures, respectively, of the su rface S, in the particular case of open surfaces represented as graphs. Suc h surfaces, known as Willmore-type surfaces, arise in various applications, such as biological membranes and shell mechanics. We construct a highly ac curate finite element scheme using the "reduced quintic" finite element, As the problem is nonlinear, we use Newton iterations for subsequently solvin g the resulting nonlinear algebraic system. The initial guess (for starting the iterative process) is provided by solving a Linearized version of the functional F. We illustrate the performance of the scheme by numerical exam ples. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserve d.