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Adaptive finite element method for shape optimization

Pedro Morin, Ricardo H. Nochetto, Miguel S. Pauletti, Marco Verani (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution...

An existence result for an interior electromagnetic casting problem

Mohammed Barkatou, Diaraf Seck, Idrissa Ly (2006)

Open Mathematics

This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C 2. Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem.

An optimal shape design problem for a hyperbolic hemivariational inequality

Leszek Gasiński (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider hemivariational inequalities of hyperbolic type. The existence result for hemivariational inequality is given and the existence theorem for the optimal shape design problem is shown.

An optimum design problem in magnetostatics

Antoine Henrot, Grégory Villemin (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we are interested in finding the optimal shape of a magnet. The criterion to maximize is the jump of the electromagnetic field between two different configurations. We prove existence of an optimal shape into a natural class of domains. We introduce a quasi-Newton type algorithm which moves the boundary. This method is very efficient to improve an initial shape. We give some numerical results.

An Optimum Design Problem in Magnetostatics

Antoine Henrot, Grégory Villemin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we are interested in finding the optimal shape of a magnet. The criterion to maximize is the jump of the electromagnetic field between two different configurations. We prove existence of an optimal shape into a natural class of domains. We introduce a quasi-Newton type algorithm which moves the boundary. This method is very efficient to improve an initial shape. We give some numerical results.

Analysis of a one-dimensional variational model of the equilibrium shapel of a deformable crystal

Eric Bonnetier, Richard S. Falk, Michael A. Grinfeld (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The equilibrium configurations of a one-dimensional variational model that combines terms expressing the bulk energy of a deformable crystal and its surface energy are studied. After elimination of the displacement, the problem reduces to the minimization of a nonconvex and nonlocal functional of a single function, the thickness. Depending on a parameter which strengthens one of the terms comprising the energy at the expense of the other, it is shown that this functional may have a stable absolute...

Approximation of a solidification problem

Rajae Aboulaïch, Ilham Haggouch, Ali Souissi (2001)

International Journal of Applied Mathematics and Computer Science

A two-dimensional Stefan problem is usually introduced as a model of solidification, melting or sublimation phenomena. The two-phase Stefan problem has been studied as a direct problem, where the free boundary separating the two regions is eliminated using a variational inequality (Baiocchi, 1977; Baiocchi et al., 1973; Rodrigues, 1980; Saguez, 1980; Srunk and Friedman, 1994), the enthalpy function (Ciavaldini, 1972; Lions, 1969; Nochetto et al., 1991; Saguez, 1980), or a control problem (El Bagdouri,...

Approximation of maximal Cheeger sets by projection

Guillaume Carlier, Myriam Comte, Gabriel Peyré (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of ℝ d . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.

Approximation of maximal Cheeger sets by projection

Guillaume Carlier, Myriam Comte, Gabriel Peyré (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of ℝ d . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian

Pedro Ricardo Simão Antunes, Pedro Freitas, James Bernard Kennedy (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of minimising the nth-eigenvalue of the Robin Laplacian in RN. Although for n = 1,2 and a positive boundary parameter α it is known that the minimisers do not depend on α, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α. We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with n1/N, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further...

Asymptotics of an optimal compliance-location problem

Giuseppe Buttazzo, Filippo Santambrogio, Nicolas Varchon (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of placing a Dirichlet region made by n small balls of given radius in a given domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.

Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty

Toni Lassila, Andrea Manzoni, Alfio Quarteroni, Gianluigi Rozza (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes...

Boundary integral representations of second derivatives in shape optimization

Karsten Eppler (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral...

Characterization of optimal shapes and masses through Monge-Kantorovich equation

Guy Bouchitté, Giuseppe Buttazzo (2001)

Journal of the European Mathematical Society

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

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