Year: 2008
Author: Arnaud Münch
International Journal of Numerical Analysis and Modeling, Vol. 5 (2008), Iss. 2 : pp. 331–351
Abstract
We consider in this paper the homogeneous 2-D wave equation defined on Ω⊂R2. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset ω⊂Ω, the control vω of minimal L2(ω×(0,T))-norm which drives to rest the system at a time T>0 large enough. We address the question of the optimal position of ω which minimize the functional J:ω→||vω||L2(ω×(0,T)). Assuming ω∈C1(Ω), we express the shape derivative of J as a curvilinear integral on ∂ω×(0,T) independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to J. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering its relaxation.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2008-IJNAM-815
International Journal of Numerical Analysis and Modeling, Vol. 5 (2008), Iss. 2 : pp. 331–351
Published online: 2008-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: optimal shape design exact controllability of wave equation level set method numerical schemes relaxation.