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

Author

Listed:
  • Antonio André Novotny

    (Coordenação de Matemática Aplicada e Computacional)

  • Jan Sokołowski

    (Université de Lorraine
    Polish Academy of Sciences)

  • Antoni Żochowski

    (Polish Academy of Sciences)

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.

Suggested Citation

  • Antonio André Novotny & Jan Sokołowski & Antoni Żochowski, 2019. "Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 1-22, April.
  • Handle: RePEc:spr:joptap:v:181:y:2019:i:1:d:10.1007_s10957-018-1420-4
    DOI: 10.1007/s10957-018-1420-4
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    References listed on IDEAS

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    1. Antonio André Novotny & Jan Sokołowski & Antoni Żochowski, 2019. "Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 683-710, March.
    2. Antonio André Novotny & Jan Sokołowski & Antoni Żochowski, 2019. "Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 341-373, February.
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    Cited by:

    1. Michał Nowak & Jan Sokołowski & Antoni Żochowski, 2020. "Biomimetic Approach to Compliance Optimization and Multiple Load Cases," Journal of Optimization Theory and Applications, Springer, vol. 184(1), pages 210-225, January.
    2. Ada Amendola, 2020. "On the Optimal Prediction of the Stress Field Associated with Discrete Element Models," Journal of Optimization Theory and Applications, Springer, vol. 187(3), pages 613-629, December.

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    1. Ada Amendola, 2020. "On the Optimal Prediction of the Stress Field Associated with Discrete Element Models," Journal of Optimization Theory and Applications, Springer, vol. 187(3), pages 613-629, December.
    2. Michał Nowak & Jan Sokołowski & Antoni Żochowski, 2020. "Biomimetic Approach to Compliance Optimization and Multiple Load Cases," Journal of Optimization Theory and Applications, Springer, vol. 184(1), pages 210-225, January.
    3. Antonio André Novotny & Jan Sokołowski & Antoni Żochowski, 2019. "Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 683-710, March.
    4. Antonio André Novotny & Jan Sokołowski & Antoni Żochowski, 2019. "Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 341-373, February.

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