Abstract
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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Novotny, A.A., Sokołowski, J., Żochowski, A. (2019). Optimality Conditions with Topological Derivatives. In: Applications of the Topological Derivative Method. Studies in Systems, Decision and Control, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-030-05432-8_5
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DOI: https://doi.org/10.1007/978-3-030-05432-8_5
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