Numerische Mathematik

, Volume 140, Issue 1, pp 35–94 | Cite as

A consistent relaxation of optimal design problems for coupling shape and topological derivatives

  • Samuel Amstutz
  • Charles Dapogny
  • Àlex Ferrer


In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation scheme, which endows the intermediate density regions with fictitious material properties. Under mild hypotheses on the smoothing operator and on the interpolation scheme, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. In particular, the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective. These results shed new light on the connections between these two different notions of sensitivities for functions of the domain, and they give rise to different numerical algorithms which are illustrated by several experiments.

Mathematics Subject Classification

74P05 35Q93 49Q10 65N85 49M20 



C.D. was partially supported by the ANR OptiForm. A.F. has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 320815 (ERC Advanced Grant Project “Advanced tools for computational design of engineering materials” COMP-DES-MAT).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LMAUniversité d’AvignonAvignon Cedex 9France
  2. 2.Laboratoire Jean Kuntzmann, CNRSUniversité Grenoble-AlpesGrenoble Cedex 9France
  3. 3.CIMNE Centre Internacional de Metodes Numerics en EnginyeriaCampus Nord UPCBarcelonaSpain

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