A quasi-Newton method in shape optimization for a transmission problem

We consider optimal design problems in stationary diffusion for mixtures of two isotropic phases. The goal is to find an optimal distribution of the phases such that the energy functional is maximized. By following the identity perturbation method, we calculate the first- and second-order shape deri...

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Published inOptimization methods & software Vol. 37; no. 6; pp. 2259 - 2285
Main Authors Kunštek, Petar, Vrdoljak, Marko
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 02.11.2022
Taylor & Francis Ltd
Subjects
Ascent
gradient method
Mathematical analysis
Optimal design
Perturbation methods
Quasi Newton methods
quasi-Newton method
second-order shape derivative
shape derivative
Shape optimization
Online AccessGet full text
ISSN1055-6788
1029-4937
DOI10.1080/10556788.2022.2078823

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Abstract We consider optimal design problems in stationary diffusion for mixtures of two isotropic phases. The goal is to find an optimal distribution of the phases such that the energy functional is maximized. By following the identity perturbation method, we calculate the first- and second-order shape derivatives in the distributional representation under weak regularity assumptions. Ascent methods based on the distributed first- and second-order shape derivatives are implemented and tested in classes of problems for which the classical solutions exist and can be explicitly calculated from the optimality conditions. A proposed quasi-Newton method offers a better ascent vector compared to gradient methods, reaching the optimal design in half as many steps. The method applies well also for multiple state problems.
AbstractList We consider optimal design problems in stationary diffusion for mixtures of two isotropic phases. The goal is to find an optimal distribution of the phases such that the energy functional is maximized. By following the identity perturbation method, we calculate the first- and second-order shape derivatives in the distributional representation under weak regularity assumptions. Ascent methods based on the distributed first- and second-order shape derivatives are implemented and tested in classes of problems for which the classical solutions exist and can be explicitly calculated from the optimality conditions. A proposed quasi-Newton method offers a better ascent vector compared to gradient methods, reaching the optimal design in half as many steps. The method applies well also for multiple state problems.
Author Vrdoljak, Marko
Kunštek, Petar
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  surname: Vrdoljak
  fullname: Vrdoljak, Marko
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  organization: University of Zagreb
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Snippet We consider optimal design problems in stationary diffusion for mixtures of two isotropic phases. The goal is to find an optimal distribution of the phases...
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SubjectTerms Ascent
gradient method
Mathematical analysis
Optimal design
Perturbation methods
Quasi Newton methods
quasi-Newton method
second-order shape derivative
shape derivative
Shape optimization
Title A quasi-Newton method in shape optimization for a transmission problem
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