Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincaré type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.

We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace-Beltrami type based metrics are compared with Steklov-Poincaré type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.

Often, the unknown diffusivity in diffusive processes is structured by piecewise constant patches. This paper is devoted to efficient methods for the determination of such structured diffusion parameters by exploiting shape calculus. A novel shape gradient is derived in parabolic processes. Furthermore quasi-Newton techniques are used in order to accelerate shape gradient based iterations in shape space. Numerical investigations support the theoretical results.Key words. Inverse modeling, shape optimization, optimization on shape manifolds. 2.1. Notations and definitions. Let d ∈ N and τ > 0. We will denote by Ω ⊂ R d a bounded domain with Lipschitz boundary Γ := ∂Ω and by J a realvalued functional depending on it. Moreover, let {F t } t∈[0,τ ] be a family of bijective mappings F t : Ω → R d such that F 0 = id. This family transforms the domain Ω into new perturbed domains Ω t := F t (Ω) = {F t (x) : x ∈ Ω} with Ω 0 = Ω and the boundary Γ into new perturbed boundaries Γ t := F t (Γ) = {F t (x) : x ∈ Γ} with

We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. Our aim is a scalable algorithm for application on supercomputers, which can only be achieved by mesh-independent convergence. The impact of discrete approximations of geometrical quantities, like the mean curvature, on a multigrid shape optimization algorithm with quasiNewton updates is investigated. For the purpose of illustration, we consider a complex model for the identification of cellular structures in biology with minimal compliance in terms of elasticity and diffusion equations.

We introduce a novel method for the implementation of shape optimization for non-parameterized shapes in fluid dynamics applications, where we propose to use the shape derivative to determine deformation fields with the help of the $$p-$$ p - Laplacian for $$p > 2$$ p > 2 . This approach is closely related to the computation of steepest descent directions of the shape functional in the $$W^{1,\infty }-$$ W 1 , ∞ - topology and refers to the recent publication Deckelnick et al. (A novel $$W^{1,\infty}$$ W 1 , ∞ approach to shape optimisation with Lipschitz domains, 2021), where this idea is proposed. Our approach is demonstrated for shape optimization related to drag-minimal free floating bodies. The method is validated against existing approaches with respect to convergence of the optimization algorithm, the obtained shape, and regarding the quality of the computational grid after large deformations. Our numerical results strongly indicate that shape optimization related to the $$W^{1,\infty }$$ W 1 , ∞ -topology—though numerically more demanding—seems to be superior over the classical approaches invoking Hilbert space methods, concerning the convergence, the obtained shapes and the mesh quality after large deformations, in particular when the optimal shape features sharp corners.