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On shape optimization with parabolic state equation

Harbrecht, Helmut and Tausch, Johannes. (2014) On shape optimization with parabolic state equation. In: Trends in PDE constrained optimization. Basel, pp. 213-229.

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Abstract

The present paper intends to summarize the main results of [17, 18] on the numerical solution of shape optimization problems for the heat equation. This is carried out by means of a specific problem, namely the reconstruction of a heat source which is located inside the computational domain under consideration from measurements of the heat flux through the boundary. We arrive at a shape optimization prob- lem by tracking the mismatch of the heat flux at the boundary. For this shape functional, the Hadamard representation of the shape gradient is derived by use of the adjoint method. The state and its adjoint equa- tion are expressed as parabolic boundary integral equations and solved using a Nystr ̈om discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. To demonstrate the sim- ilarities to shape optimization problems for elliptic state equations, we consider also the related stationary shape optimization problem which involves the Poisson equation. Numerical results are given to illustrate the theoretical findings.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
UniBasel Contributors:Harbrecht, Helmut
Item Type:Book Section, refereed
Book Section Subtype:Further Contribution in a Book
Bibsysno:Link to catalogue
Publisher:Birkhäuser
ISBN:978-3-319-05082-9
Series Name:International series of numerical mathematics
Issue Number:165
Note:Publication type according to Uni Basel Research Database: Book item
Language:English
Last Modified:31 Dec 2015 10:57
Deposited On:06 Feb 2015 09:59

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