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Compact gradient tracking in shape optimization

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Abstract

In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions.

We proof that, in contrast to the situation of gradient tracking on the whole domain, the shape Hessian is not strictly H 1/2-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration.

Shape functional and gradient require only knowledge of the Cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed by means of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient algorithm to solve the problem under consideration.

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Authors and Affiliations

  1. Abteilung Mathematik, FB IV—Mathematik, Universität Trier, Gebäude E, 54286, Trier, Germany

    Karsten Eppler

  2. Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Helmut Harbrecht

Authors
  1. Karsten Eppler
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  2. Helmut Harbrecht
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Correspondence to Karsten Eppler.

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Eppler, K., Harbrecht, H. Compact gradient tracking in shape optimization. Comput Optim Appl 39, 297–318 (2008). https://doi.org/10.1007/s10589-007-9061-9

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  • DOI: https://doi.org/10.1007/s10589-007-9061-9

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