The shape derivative of the Gauss curvature

Abstract

We introduce new results about the shape derivatives of scalar-and vector-valued functions. They extend the results from [8] to more general surface energies. In [8] Dogan and Nochetto consider surface energies defined as integrals over surfaces of functions that can depend on the position, the unit normal and the mean curvature of the surface. In this work we present a systematic way to derive formulas for the shape derivative of more general geometric quantities, including the Gauss curvature (a new result not available in the literature) and other geometric invariants (eigenvalues of the second fundamental form). This is done for hyper-surfaces in the Euclidean space of any finite dimension. 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.

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

“…A similar 2 form is given in [43,106], for a smooth function q(H , K ). A generalization to functions that depend on the position and the normal to the surface can be found in [19,26].…”

“…The discrepancy measures how much u deviates from a phase-field with tanh profile as in (19), and is an indicator of good behavior in numerical experiments. In particular, it vanishes for functions with tanh profile.…”

“…In contrast to these models, the polynomial p is here not required to be symmetric in the principal curvatures, which allows the generation of tubules. Other generalizations have been proposed in [43,106] and [19,26].…”

“…which we plot in the curvature diagram. Note that the diffuse curvatures coincide with the curvatures of the levels sets for functions with tanh profile (19), but are only approximations in the general case. Hence, the curvature diagrams shown throughout this article are only approximations of the exact curvature diagrams, most precise for those u having nearly a tanh profile.…”

Cálculo de forma usando formas diferenciais e cálculo exterior

“…uma extensão clássica de alguma função escalar u (campo vetorial v) na superfície Γ (alguma vizinhança de Γ [23]) para todo o espaço R d , por meio da função distância assinada (a função de distância assinada (ou função de distância orientada)…”