Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications

The framework of topological sensitivity analysis in singularly perturbed geometrical domains, presented in the first part of this series of review papers, allows the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturba...

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Vydáno v:	Journal of optimization theory and applications Ročník 180; číslo 3; s. 683 - 710
Hlavní autoři:	Novotny, Antonio André, Sokołowski, Jan, Żochowski, Antoni
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.03.2019 Springer Nature B.V Springer Verlag
Témata:	Algorithms Applications of Mathematics Asymptotic series Calculus of Variations and Optimal Control; Optimization Derivatives Domains Engineering Fracture mechanics Functionals Image processing Inverse problems Invited Paper Mathematical models Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Partial differential equations Sensitivity analysis Theory of Computation Topology optimization 49J20 49M05 35J15 35Q74 Applications in topology optimization 49Q10 First-order method Topological derivatives
ISSN:	0022-3239, 1573-2878
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Abstract	The framework of topological sensitivity analysis in singularly perturbed geometrical domains, presented in the first part of this series of review papers, allows the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, source terms and cracks. This new concept in shape sensitivity analysis generalizes the shape derivatives from the domain boundary to its interior for admissible domains in two and three spatial dimensions. Therefore, the concept of topological derivative is a powerful tool for solving shape–topology optimization problems. There are now applications of topological derivative in many different fields of engineering and physics, such as shape and topology optimization in structural mechanics, inverse problems for partial differential equations, image processing, multiscale material design and mechanical modeling including damage and fracture evolution phenomena. In this second part of the review, a topology optimization algorithm based on first-order topological derivatives is presented. The appropriate level-set domain representation method is employed within the iterations in order to design an optimal shape–topology local solution. The algorithm is successfully used for numerical solution of a wide class of shape–topology optimization problems.
AbstractList	The framework of topological sensitivity analysis in singularly perturbed geometrical domains, presented in the first part of this series of review papers, allows the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, source terms and cracks. This new concept in shape sensitivity analysis generalizes the shape derivatives from the domain boundary to its interior for admissible domains in two and three spatial dimensions. Therefore, the concept of topological derivative is a powerful tool for solving shape–topology optimization problems. There are now applications of topological derivative in many different fields of engineering and physics, such as shape and topology optimization in structural mechanics, inverse problems for partial differential equations, image processing, multiscale material design and mechanical modeling including damage and fracture evolution phenomena. In this second part of the review, a topology optimization algorithm based on first-order topological derivatives is presented. The appropriate level-set domain representation method is employed within the iterations in order to design an optimal shape–topology local solution. The algorithm is successfully used for numerical solution of a wide class of shape–topology optimization problems.
Author	Żochowski, Antoni Novotny, Antonio André Sokołowski, Jan
Author_xml	– sequence: 1 givenname: Antonio André surname: Novotny fullname: Novotny, Antonio André organization: Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional – sequence: 2 givenname: Jan orcidid: 0000-0002-3560-7342 surname: Sokołowski fullname: Sokołowski, Jan email: Jan.Sokolowski@univ-lorraine.fr organization: Institut Élie Cartan, UMR 7502 Laboratoire de Mathématiques, Université de Lorraine, Systems Research Institute of the Polish Academy of Sciences – sequence: 3 givenname: Antoni surname: Żochowski fullname: Żochowski, Antoni organization: Systems Research Institute of the Polish Academy of Sciences
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SubjectTerms	Algorithms Applications of Mathematics Asymptotic series Calculus of Variations and Optimal Control; Optimization Derivatives Domains Engineering Fracture mechanics Functionals Image processing Inverse problems Invited Paper Mathematical models Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Partial differential equations Sensitivity analysis Theory of Computation Topology optimization
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Title	Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications
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