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Optimal convex shapes for concave functionals

Dorin Bucur, Ilaria Fragalà, Jimmy Lamboley (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application...

Optimal design in small amplitude homogenization

Grégoire Allaire, Sergio Gutiérrez (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide...

Optimal design of cylindrical shells

Peter Nestler, Werner H. Schmidt (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The present paper studies an optimization problem of dynamically loaded cylindrical tubes. This is a problem of linear elasticity theory. As we search for the optimal thickness of the tube which minimizes the displacement under forces, this is a problem of shape optimization. The mathematical model is given by a differential equation (ODE and PDE, respectively); the mechanical problem is described as an optimal control problem. We consider both the stationary (time independent) and the transient...

Optimal design of laminated plate with obstacle

Ján Lovíšek (1992)

Applications of Mathematics

The aim of the present paper is to study problems of optimal design in mechanics, whose variational form is given by inequalities expressing the principle of virtual power in its inequality form. The elliptic, linear symmetric operators as well as convex sets of possible states depend on the control parameter. The existence theorem for the optimal control is applied to design problems for an elastic laminated plate whose variable thickness appears as a control variable.

Optimal design of the cooling plunger cavity

Petr Salač (2013)

Applications of Mathematics

An axisymmetric system of mould, glass piece, plunger and plunger cavity is considered. The state problem is given as a stationary head conduction process. The system includes the glass piece representing the heat source and is cooled inside the plunger cavity by flowing water and outside by the environment of the mould. The design variable is taken to be the shape of the inner surface of the plunger cavity. The cost functional is the second power of the norm in the weighted space L r 2 of difference...

Optimal design of turbines with an attached mass

Boris P. Belinskiy, C. Maeve McCarthy, Terry J. Walters (2003)

ESAIM: Control, Optimisation and Calculus of Variations

We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.

Optimal design of turbines with an attached mass

Boris P. Belinskiy, C. Maeve McCarthy, Terry J. Walters (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.

Optimal internal dissipation of a damped wave equation using a topological approach

Arnaud Münch (2009)

International Journal of Applied Mathematics and Computer Science

We consider a linear damped wave equation defined on a two-dimensional domain Ω, with a dissipative term localized in a subset ω. We address the shape design problem which consists in optimizing the shape of ω in order to minimize the energy of the system at a given time T . By introducing an adjoint problem, we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation in ω. Expressed as a boundary integral on ∂ω, this derivative is then used as an advection...

Optimal networks for mass transportation problems

Alessio Brancolini, Giuseppe Buttazzo (2005)

ESAIM: Control, Optimisation and Calculus of Variations

In the framework of transport theory, we are interested in the following optimization problem: given the distributions μ + of working people and μ - of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of μ + from μ - with respect to a metric which depends on the transportation network....

Optimal networks for mass transportation problems

Alessio Brancolini, Giuseppe Buttazzo (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In the framework of transport theory, we are interested in the following optimization problem: given the distributions µ+ of working people and µ- of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of µ+ from µ- with respect to a metric which depends on the transportation...

Optimal shape design in a fibre orientation model

Jan Stebel, Raino Mäkinen, Jukka I. Toivanen (2007)

Applications of Mathematics

We study a 2D model of the orientation distribution of fibres in a paper machine headbox. The goal is to control the orientation of fibres at the outlet by shape variations. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Existence of solutions both to the state and the optimization problem is analyzed and sensitivity analysis is performed. Further, discretization is done and a numerical example...

Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem

Giuseppe Buttazzo, Eugene Stepanov (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried...

Optimization of the shape and the location of the actuators in an internal control problem

Antoine Henrot, Hervé Maillot (2001)

Bollettino dell'Unione Matematica Italiana

Consideriamo un corpo Ω sottomesso ad una forza esterna data e del quale vogliamo controllare lo spostamento. Cerchiamo un rinforzo per minimizzare un funzionale che dipende dallo spostamento del corpo. L'insieme delle configurazioni ammissibili è un insieme di funzioni caratteristiche di sottodomini (un rinforzo ammissibile è un sottodominio con una rigidezza uguale ad uno) di volume prescritto. In tal caso, si ha bisogno di una versione rilassata del problema di ottimizzazione e si cerca una densità...

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