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

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Abstract

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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Acknowledgements

This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro). The support is gratefully acknowledged. We also thank Habib Ammari, Alfredo Canelas, Michael Hintermüller, Hyeonbae Kang, Antoine Laurain, Jairo Faria, Ravi Prakash and the former students Lucas Fernandez, Andrey Ferreira, Thiago Machado and Suelen Rocha.

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Authors and Affiliations

  1. Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional, Av. Getúlio Vargas 333, Petrópolis, RJ, 25651-075, Brazil

    Antonio André Novotny

  2. UMR 7502 Laboratoire de Mathématiques, Institut Élie Cartan, Université de Lorraine, Nancy 1, B.P. 239, 54506, Vandoeuvre Lès Nancy Cedex, France

    Jan Sokołowski

  3. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Jan Sokołowski & Antoni Żochowski

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  1. Antonio André Novotny
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Correspondence to Jan Sokołowski.

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Novotny, A.A., Sokołowski, J. & Żochowski, A. Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications. J Optim Theory Appl 181, 1–22 (2019). https://doi.org/10.1007/s10957-018-1420-4

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