Related papers: PDE constrained shape optimisation with first-orde…
This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape…
Built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), we present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology together with a suitable…
In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought…
In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape…
Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which where a so-called shape functional is constrained by a partial differential equation (PDE)…
We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of…
This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem.…
The space mapping technique is used to efficiently solve complex optimization problems. It combines the accuracy of fine model simulations with the speed of coarse model optimizations to approximate the solution of the fine model…
We introduce a novel method for the implementation of shape optimziation in fluid dynamics applications, where we propose to use the shape derivative to determine deformation fields with the help of the $p-$ Laplacian for $p > 2$. This…
In this paper we present a shape optimization scheme which utilizes the alternating direction method of multipliers (ADMM) to approximate a direction of steepest descent in $W^{1,\infty}$. The followed strategy is a combination of the…
Second-order methods are of great importance for composite convex optimization problems due to their local super-linear convergence rates (under appropriate assumptions). However, the presence of even a simple nonsmooth function in the…
The novel Riemannian view on shape optimization developed in [Schulz, FoCM, 2014] is extended to a Lagrange-Newton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space…
PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically…
We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a density function. To obtain candidates for local minima, we want to solve the first order optimality system via…
For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the…
In this work, the problem of shape optimization, subject to PDE constraints, is reformulated as an $L^p$ best approximation problem under divergence constraints to the shape tensor introduced in Laurain and Sturm: ESAIM Math. Model. Numer.…
We study optimal design problems involving variational inequalities with unilateral conditions in the domain and pointwise boundary observation. We use regularizing and penalization tehniques in the setting of the Hamiltonian approach to…
We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a…
We present a general numerical approach to shape optimization with state constraints for 2-dimensional geometries, without relaxing the constraints. To do this we reformulate the problem on a fixed reference domain using a conformal…
In this thesis we develop a functional analytic framework for shape optimization with elliptic partial differential equation (PDE) constraints in classical function spaces (H\"older spaces). This approach is motivated by shape optimization…