Book review: "Variational Methods in Shape Optimization Problems"
Andrzej Myśliński (2006)
Control and Cybernetics
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Andrzej Myśliński (2006)
Control and Cybernetics
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Timo Tiihonen (1997)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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Raino A.E. Mäkinen, Tuomo Rossi, Jari Toivanen (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers...
Raino A. E. Mäkinen, Tuomo Rossi, Jari Toivanen (2000)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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Bijan Mohammadi (2010)
Control and Cybernetics
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Marc Dambrine, Djalil Kateb (2010)
ESAIM: Control, Optimisation and Calculus of Variations
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The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire , (2004) 363–393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic...
G. Allaire, C. Dapogny, G. Delgado, G. Michailidis (2014)
ESAIM: Control, Optimisation and Calculus of Variations
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We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves...
Antoine Henrot, Grégory Villemin (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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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.