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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Popis
Shrnutí: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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1419-x