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Shape sensitivity of optimal control for the Stokes problem

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Języki publikacji
EN
Abstrakty
EN
In this article, we study the shape sensitivity of optimal control for the steady Stokes problem. The main goal is to obtain a robust representation for the derivatives of optimal solution with respect to smooth deformation of the flow domain. We introduce in this paper a rigorous proof of existence of the material derivative in the sense of Piola, as well as the shape derivative for the solution of the optimality system. We apply these results to derive the formulae for the shape gradient of the cost functional; under some regularity conditions the shape gradient is given according to the structure theorem by a function supported on the moving boundary, then the numerical methods for shape optimization can be applied in order to solve the associated optimization problems.
Rocznik
Strony
11--40
Opis fizyczny
Bibliogr. 34 poz., rys.
Twórcy
  • Université Oran1, Ahmed Ben Bella, Laboratoire de Mathématique et ses applications BP 1524, El M’naouer, Oran, Algérie
  • Université Hassiba Ben Bouali, Département de Mathématiques, Chlef, Algérie
autor
  • Université Oran1, Ahmed Ben Bella, Laboratoire de Mathématique et ses applications BP 1524, El M’naouer, Oran, Algérie
  • Université de Lorraine-Nancy, Institut Elie Cartan, Laboratoire de Mathématiques, B.P. 239, 54506, Vandoeuvre lès Nancy cedex, France
  • Systems Research Institute of the Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland
  • Departamento de Computa,cão Científica, Centro de Informática, Universidade Federal da Paraíba, Rua dos Escoteiros s/n, Mangabeira, João Pessoa, PB 58058-600, Brasil
  • Systems Research Institute of the Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland
Bibliografia
  • Abdelwahab, M. and Hassine, M. (2009) Topological optimization method for a geometric control problem in Stokes flow. Applied Numerical Mathematics 59. Elsevier, 1823-1838.
  • Allaire, G. (2007) Conception optimale de structures. Mathématiques & Applications (Berlin) [Mathematics & Applications], 58, Springer-Verlag, Berlin.
  • Ammari, H. (2002) An inverse initial boundary value problem for the wave equation in the presence of imperfections of small volume. SIAM J. Control Optim. 41, 1194-1211.
  • Amstutz, S. (2005) The topological asymptotic for the Navier-Stokes equations. ESAIM: COCV, 11, 401-425, .
  • Berggren, M. (2010) A unified discrete-continuous sensitivity analysis method for shape optimization. In: W. Fitzgibbon et al. (eds.), Applied and Numerical Partial Differential Equations. Computational Methods in Applied Sciences, 15, Springer, 2539.
  • Boisgérault, S. (2000) Shape Optimisation: Nonlinear Systems and Fluid Mechanics. PhD Thesis, Ecole des Mines de Paris https://eul.ink/shapeoptimization/
  • Boyer, F. and Fabrie, P. (2006) Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin.
  • Caubet, F. and Dambrine, M. (2012) Localization of small obstacles in Stokes flow. Inverse Problems, 28(10), 105007, 31.
  • Consiglieri, L., Nečasovà, Š. and Sokolowski, J. (2010) New approach to the incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity. J. Differential Equations 249, 3052-3080.
  • Delfour, M. C. and Zolesio, J. P. (2012) Shapes and Geometries, second ed., Advances in Design and Control, 22. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, Metrics, Analysis, Differential Calculus, and Optimization. MR 2731611.
  • Fremiot, G. (2000) Structure de la semi-dérivée eulérienne dans le cas de domaines fissurés et quelques applications. PhD Thesis of University Henri Poincaré-Nancy 1.
  • Fursikov, A. V. (1999) Optimal Control of Distributed Systems. Theory and application. AMS, Providence.
  • Galdi, G.P. (1994) An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I and II, Springer Tracts in Natural Philosophy, 38 and 39. Springer-Verlag, New York.
  • Guillaume, Ph. and Hassine, M. (2008) Removing holes in topological shape optimization. ESAIM: COCV, 14, 1, 160-191.
  • Gurtin, M. E. (1981) An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, 158. Academic Press, New York.
  • Hadamard, J. (1907) Mémoire sur le problème dánalyse relatif à l’équilibre des plaques éastiques encastrées. Bull. Soc. Math. France.
  • Henrot, A. and Pierre, M. (2005) Variation et optimisation de formes. Mathématiques et Applications, 48, Springer-Verlag, Berlin.
  • Lasiecka, I. (2002) Mathematical Control Theory of Coupled PDEs. SIAM, Philadelphia.
  • Lasiecka, I., Szulc, K. and Zochowski, A. (2018) Boundary control of small solutions to fluid structure interactions arising in coupling of elasticity with Navier stokes equation under mixed boundary conditions. Nonlinear Analysis: Real World Applications, 44, 54-85.
  • Laurain, A. (2006) Domaines singulièrement perturbés en optimisation de formes. PhD Thesis, Nancy; https://tel.archives-ouvertes.fr/tel-01748151v2/document
  • Leugering, G., Novotny, A. A., Perla Menzala, G. and Sokołowski, J. (2011) On shape optimization for an evolution coupled system. Appl. Math. Optim. 64:441-466.
  • Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin.
  • Mohammadi B. and Pironneau, O. (2010) Applied Shape Optimization for Fluids, second ed. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford. MR 2567067.
  • Moubachir, M. and Zolesio, J.P. (2006) Moving Shape Analysis and Control Applications to Fluid Structure Interactions. Chapman & Hall/CR.
  • Murat, F. and Simon, J. (1976) Sur le contrôle par un domaine géométrique. Publication du Laboratoire d’Analyse Numérique de l’Université Paris, 6 189.
  • Novotny, A. A. and Sokołowski, J. (2013) Topological Derivatives in the Shape Optimization. Series: Interaction of Mechanics and Mathematics. Springer-Verlag.
  • Pironneau, O. (1984) Optimal Shape Design for Elliptic Systems. Springer, Berlin.
  • Plotnikov, P. and Sokołowski, J. (2012) Compressible Navier Stokes Equations. Theory and shape optimization. Birkhäuser.
  • Plotnikov, P. and Sokołowski, J. (2010) Shape derivative of drag functional. SIAM J. Control Optim. 48, 4680-4706.
  • Rösch, A. and Vexler, B. (2006) Optimal control of the Stokes equations, a priori error analysis for finite element. SIAM J. Numer. Anal. 44, 1903-1920.
  • Sokołowski, J. and Zolesio, J.P. (1992) Introduction to Shape Optimization, Shape Sensitivity Analysis. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, New York.
  • Sokołowski, J. and Zochowski, A. (2005) Topological derivatives for contact problems. Numer. Math. 102, 1, 145-179.
  • Tröltzsch, F. (2000) Optimal Control of Partial Differential Equations. Theory, Methods and Applications. AMS.
  • Zhu, S. and Gao, Z. (2019) Convergence analysis of mixed finite element approximations to shape gradients in Stokes equation. Comput. Methods Appl. Mech. Engrg., 343 (127-150).
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-de384035-22f3-4928-83a5-b53649d856ce
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