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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Published in:Journal of optimization theory and applications Vol. 181; no. 1; pp. 1 - 22
Main Authors: Novotny, Antonio André, Sokołowski, Jan, Żochowski, Antoni
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2019
Springer Nature B.V
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1420-4