Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications

The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019...

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Vydáno v:	Journal of optimization theory and applications Ročník 181; číslo 1; s. 1 - 22
Hlavní autoři:	Novotny, Antonio André, Sokołowski, Jan, Żochowski, Antoni
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.04.2019 Springer Nature B.V Springer Verlag
Témata:	Algorithms Applications of Mathematics Approximation Asymptotic methods Asymptotic series Calculus of Variations and Optimal Control; Optimization Derivatives Domains Engineering Functionals Inverse reconstruction Invited Paper Iterative methods Mathematics Mathematics and Statistics Newton methods Numerical methods Operations Research/Decision Theory Optimization Robustness (mathematics) Shape optimization Theory of Computation Topology optimization 49J20 Applications in inverse problems 35J15 35Q74 49M15 Second-order method 49N45 Topological derivatives
ISSN:	0022-3239, 1573-2878
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Popis
Shrnutí:	The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019 ), one-term expansions of functionals are required for algorithms of shape-topological optimization. Such an approach corresponds to the simple gradient method in shape optimization. The Newton method of shape optimization can be replaced, for shape-topology optimization, by two-term expansions of shape functionals. Thus, the resulting approximations are more precise and the associated numerical methods are much more complex compared to one-term expansion topological derivative algorithms. In particular, numerical algorithms associated with first-order topological derivatives of shape functionals have been presented in Part II (Novotny et al. 2019 ), together with an account of their applications currently found in the literature, with emphasis on shape and topology optimization. In this last part of the review, second-order topological derivatives are introduced. Second-order algorithms of shape-topological optimization are used for numerical solution of representative examples of inverse reconstruction problems. The main feature of these algorithms is that the method is non-iterative and thus very robust with respect to noisy data as well as independent of initial guesses.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0022-3239 1573-2878
DOI:	10.1007/s10957-018-1420-4