Related papers: Min max method, shape, topological derivatives, av…
We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to…
This paper is concerned with a numerical simulation of shape optimization in a two-dimensional viscous incompressible flow governed by Navier--Stokes equations with mixed boundary conditions containing the pressure. The minimization problem…
We introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for…
In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of…
In this paper we prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either E f ($\Omega$),…
This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate…
A class of convex constrained minimization problems over polyhedral cones for geometry-dependent quadratic objective functions is considered in a functional analysis framework. Shape differentiability of the primal minimization problem…
Deformations of the computational mesh arising from optimization routines usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize classical shape…
Although stress-constrained topology optimization has been extensively studied in structural design, the development of optimization frameworks to enable the creation of metamaterials with optimal mechanical performance is still an open…
We propose to differentiate a general curvature functional with two different approaches. In the first one we compute the derivative with the tools of shape optimization and in the second one we compute the derivative of a volumic…
In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worst-case sense, i.e. the worst possible material distribution over a given uncertainty set is taken…
This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints…
We use a gap function in order to compare the torsional performances of different reinforced plates under the action of external forces. Then, we address a shape optimization problem, whose target is to minimize the torsional displacements…
We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves $\gamma$ in a differentiable manifold $M$ that are everywhere tangent to a smooth distribution $\mathcal…
The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical…
The minimum and maximum cuts of an undirected edge-weighted graph are classic problems in graph theory. While the Min-Cut Problem can be solved in P, the Max-Cut Problem is NP-Complete. Exact and heuristic methods have been developed for…
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin…
This work develops an algorithm for PDE-constrained shape optimization based on Lipschitz transformations. Building on previous work in this field, the $p$-Laplace operator is utilized to approximate a descent method for Lipschitz shapes.…
We consider a constrained optimization problem arising from the study of the Helmholtz equation in unbounded domains. The optimization problem provides an approximation of the solution in a bounded computational domain. In this paper we…
In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both…