An algorithm for prescribed mean curvature using isogeometric methods

Abstract

We present a Newton type algorithm to find parametric surfaces of prescribed mean curvature with a fixed given boundary. In particular, it applies to the problem of minimal surfaces. The algorithm relies on some global regularity of the spaces where it is posed, which is naturally fitted for discretization with isogeometric type of spaces. We introduce a discretization of the continuous algorithm and present a simple implementation using the recently released isogeometric software library igatools. Finally, we show several numerical experiments which highlight the convergence properties of the scheme.

The shape derivative of the Gauss curvature

“…where b Γ * := b(Ω * ) is the signed distance function corresponding to the domain Ω * whose boundary is Γ * , V = C k c (D) and n * = ∇b Γ * is the normal vector to Γ * . We now present a scheme to approximate the solution of (10.3) using a Newton-type method that generalizes the idea of [5] in at least two ways. First, it uses the language of shape derivatives and secondly, it has the potential to work for a large class of shape functionals, not just the area functional.…”

“…An interesting scheme to approximate the solutions of (10.2) for surfaces of prescribed constant mean curvature was presented in [5]. There, results from numerical experiments document its performance and fast convergence.…”

“…We first observe that, due to the structure Theorem (Theorem 12), Problem (10.2) is equivalent to the following: We now present a scheme to approximate the solution of (10.3) using a Newton-type method that generalizes the idea of [5] in at least two ways. First, it uses the language of shape derivatives and secondly, it has the potential to work for a large class of shape functionals, not just the area functional.…”

“…More precisely, we introduce a new scheme of the Newton-type to find a minimizer of a surface shape functional. It is a mathematically sound generalization of the method used in [5].…”

“…As an application of the results, with relevance for numerical methods in applied problems, we introduce a new scheme of Newton-type to approximate a minimizer of a shape functional. It is a mathematically sound generalization of the method presented in [5]. We finally find the particular formulas for the first and second order shape derivative of the area and the Willmore functional, which are necessary for the Newton-type method mentioned above.…”