On a numerical shape optimization approach for a class of free boundary problems
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On a numerical shape optimization approach for a class of free boundary problems A. Boulkhemair1 · A. Chakib2 · A. Nachaoui1 · A. A. Niftiyev3 · A. Sadik1,2 Received: 18 November 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is devoted to a numerical method for the approximation of a class of free boundary problems of Bernoulli’s type, reformulated as optimal shape design problems with appropriate shape functionals. We show the existence of the shape derivative of the cost functional on a class of admissible domains and compute its shape derivative by using the formula proposed in Boulkhemair (SIAM J Control Optim 55(1):156–171, 2017) and Boulkhemair and Chakib (J Convex Anal 21(1):67–87, 2014), that is, by means of support functions. On the numerical level, this allows us to avoid the tedious computations of the method based on vector fields. A gradient method combined with a boundary element method is performed for the approximation of this problem, in order to overcome the re-meshing task required by the finite element method. Finally, we present some numerical results and simulations concerning practical applications, showing the effectiveness of the proposed approach. Keywords Shape optimization · Free boundary problem · Bernoulli problem · Optimal solution · Shape derivative · Convex domain · Support function · Cost functional
1 Introduction The main target of shape optimization is to provide a common and systematic framework for optimizing structures described by various practical physical or mechanical models; especially, hydrodynamics, elasticity, geophysics and aerodynamics models. Shape optimization problems consist in finding the optimal shape (or domain) In memory of Professor Aghaddin A. Niftiyev. A. A. Niftiyev: Our colleague Aghaddin Aslan Niftiyev from Baku State University passed away after having contributed greatly to this paper. * A. Sadik [email protected]‑nantes.fr Extended author information available on the last page of the article
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which minimizes a certain cost functional under given constraints such as a partial differential equation defined on the variable domain. Since the seventies of the last century, many authors investigated the shape optimization field and remarkable progress has been achieved in shape and topology optimization. In fact, the growing interest in this field reflects a growing sophistication in structural analysis and optimization which allow solving more and more difficult shape optimization problems. However, one may say that no uniform approach to shape optimization problems has yet emerged. The numerical investigation of shape optimization problems is based on the study of the first variation of the cost functional, and in particular on the computation of its gradient. So, as the variation of the domain is characterized by the variation of its boundary, in this process arise both numerical and theoretical difficulties. The method of variation of domains usin
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