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 $\Omega \subset \mathbb{R}^2$. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset $\omega \subset \Omega$, the control $v_{\omega}$ of minimal $L^2 (\omega \times (0, T))$-norm which drives to rest the system at a time $T>0$ large enough. We address the question of the optimal position of $\omega$ which minimize the functional $J : \omega \rightarrow ||v_{\omega}||_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^1(\Omega)$, we express the shape derivative of $J$ as a curvilinear integral on $∂\omega \times (0,T)$ 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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
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.