Optimality Conditions with Topological Derivatives

  • Antonio André NovotnyEmail author
  • Jan Sokołowski
  • Antoni Żochowski
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 188)


In classical theory of shape optimization the first order necessary optimality conditions account for boundary variations of an optimal domain. On the other hand the relaxed formulation based on homogenization technique is used (Allaire et al, Numer Math 76(1):27–68, 1997, [5], Bendsøe, Optimization of structural topology, shape, and material, 1995, [46], Lewiński and Telega, Plates, laminates and shells, 2000, [173] in the topology optimization of energy functionals, the so called compliance in structural optimization. For such a formulation the coefficients of an elliptic operator are selected in an optimal way and the resulting optimal design takes the form of a composite microstructure rather than any geometrical domain.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Antonio André Novotny
    • 1
    Email author
  • Jan Sokołowski
    • 2
    • 3
    • 4
  • Antoni Żochowski
    • 3
  1. 1.Coordenação de Métodos Matemáticos e ComputacionaisLaboratório Nacional de Computação Científica LNCC/MCTICPetrópolisBrazil
  2. 2.Institut Élie Cartan de Nancy, UMR 7502Université de Lorraine, CNRSVandœuvre-Lès-NancyFrance
  3. 3.Systems Research InstitutePolish Academy of SciencesWarsawPoland
  4. 4.Universidade Federal da Paraiba, Centro de informaticaJoão PessoaBrazil

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