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

Antonio André Novotny (), Jan Sokołowski () and Antoni Żochowski ()
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Antonio André Novotny: Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional
Jan Sokołowski: Université de Lorraine
Antoni Żochowski: Systems Research Institute of the Polish Academy of Sciences

Journal of Optimization Theory and Applications, 2019, vol. 180, issue 3, No 1, 683-710

Abstract: 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.

Keywords: Topological derivatives; First-order method; Applications in topology optimization; 35J15; 35Q74; 49J20; 49M05; 49Q10 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10957-018-1419-x

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