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

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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.

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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 would also like to thank Samuel Amstutz, Sebastián Giusti, Nicolas Van Goethem, Eduardo Neto, Luís Fernando Sá, Emílio Silva, André Torii and the former students Alan Amad, Cinthia Lopes, Vitor Sales, Renatha Santos and Marcel Xavier.

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

    Jan Sokołowski

  3. Systems Research Institute of the 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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Communicated by Marc Bonnet.

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Novotny, A.A., Sokołowski, J. & Żochowski, A. Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications. J Optim Theory Appl 180, 683–710 (2019). https://doi.org/10.1007/s10957-018-1419-x

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