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Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications

Author

Listed:
  • Antonio André Novotny

    (Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional)

  • Jan Sokołowski

    (Université de Lorraine
    Systems Research Institute of the Polish Academy of Sciences)

  • Antoni Żochowski

    (Systems Research Institute of the Polish Academy of Sciences)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:180:y:2019:i:3:d:10.1007_s10957-018-1419-x
    DOI: 10.1007/s10957-018-1419-x
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    References listed on IDEAS

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    1. Duan, Xianbao & Li, Feifei, 2015. "Material distribution resembled level set method for optimal shape design of Stokes flow," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 21-30.
    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.
    3. I. Hlaváček & A. A. Novotny & J. Sokołowski & A. Żochowski, 2009. "On Topological Derivatives for Elastic Solids with Uncertain Input Data," Journal of Optimization Theory and Applications, Springer, vol. 141(3), pages 569-595, June.
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    Cited by:

    1. 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.
    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.
    3. 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.
    4. 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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