# American Institute of Mathematical Sciences

February  2013, 7(1): 123-157. doi: 10.3934/ipi.2013.7.123

## A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid

 1 Université de Technologie de Compiègne, Laboratoire de Mathématiques Appliquées, EA 2222, Compiègne, France, France 2 Université de Pau et des pays de l'Adour, Laboratoire de Mathématiques Appliquées, UMR CNRS 5142, Pau, France 3 Université d'Oran, Département de Mathématiques, BP 1524, El-Menaouer, Oran, Algeria

Received  January 2012 Revised  October 2012 Published  February 2013

The aim of our work is to reconstruct an inclusion $\omega$ immersed in a fluid flowing in a larger bounded domain $\Omega$ via a boundary measurement on $\partial\Omega$. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing $\omega$ thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method.
Citation: Fabien Caubet, Marc Dambrine, Djalil Kateb, Chahnaz Zakia Timimoun. A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid. Inverse Problems and Imaging, 2013, 7 (1) : 123-157. doi: 10.3934/ipi.2013.7.123
##### References:
 [1] M. Abdelwahed and M. Hassine, Topological optimization method for a geometric control problem in Stokes flow, Appl. Numer. Math., 59 (2009), 1823-1838. doi: 10.1016/j.apnum.2009.01.008. [2] L. Afraites, M. Dambrine, K. Eppler and D. Kateb, Detecting perfectly insulated obstacles by shape optimization techniques of order two, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 389-416 (electronic). doi: 10.3934/dcdsb.2007.8.389. [3] L. Afraites, M. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography, SIAM J. Control Optim., 47 (2008), 1556-1590. doi: 10.1137/070687438. [4] C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003. [5] C. J. S. Alves, R. Kress and A. L. Silvestre, Integral equations for an inverse boundary value problem for the two-dimensional Stokes equations, J. Inverse Ill-Posed Probl., 15 (2007), 461-481. doi: 10.1515/jiip.2007.026. [6] C. Amrouche and V. Girault, Problèmes généralisés de Stokes, Portugal. Math., 49 (1992), 463-503. [7] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44(119) (1994), 109-140. [8] M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660. [9] A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015, 25 pp. doi: 10.1088/0266-5611/26/12/125015. [10] A. Ballerini, Stable determination of a body immersed in a fluid: The nonlinear stationary case, Appl. Anal., to appear. [11] A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control Optim., 48 (2009/10), 2871-2900. doi: 10.1137/070704332. [12] F. Boyer and P. Fabrie, "Éléments d'Analyse Pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006. [13] D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equations and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005. [14] C. Conca, P. Cumsille, J. Ortega and L. Rosier, Detecting a moving obstacle in an ideal fluid by a boundary measurement, C. R. Math. Acad. Sci. Paris, 346 (2008), 839-844. doi: 10.1016/j.crma.2008.06.007. [15] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010, 18 pp. doi: 10.1088/0266-5611/26/9/095010. [16] M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 96 (2002), 95-121. [17] M. Engliš and J. Peetre, A Green's function for the annulus, Ann. Mat. Pura Appl. (4), 171 (1996), 313-377. doi: 10.1007/BF01759391. [18] K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information, Control Cybernet., 34 (2005), 203-225. [19] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994. [20] S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim., 39 (2001), 1756-1778 (electronic). doi: 10.1137/S0363012900369538. [21] A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques & Applications (Berlin) [Mathematics & Applications], 48, Springer, Berlin, 2005. [22] D. Martin, "Finite Element Library Mélina," Available from: http://anum-maths.univ-rennes1.fr/melina/. [23] V. Maz'ya and T. Shaposhnikova, "Theory of Multipliers in Spaces of Differentiable Functions," Monographs and Studies in Mathematics, 23, Pitman (Advanced Publishing Program), Boston, MA, 1985. [24] F. Murat and J. Simon, "Sur le Contrôle par un Domaine Géométrique," Rapport du L.A. 189, Université de Paris VI, France, 1976. [25] J. Nocedal and S. J. Wright, "Numerical Optimization," Second edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. [26] J. R. Shewchuk, "Mesh Generator Triangle," Available from: http://www.cs.cmu.edu/~quake/triangle.html. [27] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687 (1981). doi: 10.1080/01630563.1980.10120631. [28] J. Simon, Second variations for domain optimization problems, in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1988), Internat. Ser. Numer. Math., 91, Birkhäuser, Basel, (1989), 361-378. [29] J. Simon, Domain variation for drag in Stokes flow, in "Control Theory of Distributed Parameter Systems and Applications" (Shanghai, 1990), Lecture Notes in Control and Inform. Sci., 159, Springer, Berlin, (1991), 28-42. doi: 10.1007/BFb0004434. [30] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272 (electronic). doi: 10.1137/S0363012997323230. [31] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.

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##### References:
 [1] M. Abdelwahed and M. Hassine, Topological optimization method for a geometric control problem in Stokes flow, Appl. Numer. Math., 59 (2009), 1823-1838. doi: 10.1016/j.apnum.2009.01.008. [2] L. Afraites, M. Dambrine, K. Eppler and D. Kateb, Detecting perfectly insulated obstacles by shape optimization techniques of order two, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 389-416 (electronic). doi: 10.3934/dcdsb.2007.8.389. [3] L. Afraites, M. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography, SIAM J. Control Optim., 47 (2008), 1556-1590. doi: 10.1137/070687438. [4] C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003. [5] C. J. S. Alves, R. Kress and A. L. Silvestre, Integral equations for an inverse boundary value problem for the two-dimensional Stokes equations, J. Inverse Ill-Posed Probl., 15 (2007), 461-481. doi: 10.1515/jiip.2007.026. [6] C. Amrouche and V. Girault, Problèmes généralisés de Stokes, Portugal. Math., 49 (1992), 463-503. [7] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44(119) (1994), 109-140. [8] M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660. [9] A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015, 25 pp. doi: 10.1088/0266-5611/26/12/125015. [10] A. Ballerini, Stable determination of a body immersed in a fluid: The nonlinear stationary case, Appl. Anal., to appear. [11] A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control Optim., 48 (2009/10), 2871-2900. doi: 10.1137/070704332. [12] F. Boyer and P. Fabrie, "Éléments d'Analyse Pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006. [13] D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equations and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005. [14] C. Conca, P. Cumsille, J. Ortega and L. Rosier, Detecting a moving obstacle in an ideal fluid by a boundary measurement, C. R. Math. Acad. Sci. Paris, 346 (2008), 839-844. doi: 10.1016/j.crma.2008.06.007. [15] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010, 18 pp. doi: 10.1088/0266-5611/26/9/095010. [16] M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 96 (2002), 95-121. [17] M. Engliš and J. Peetre, A Green's function for the annulus, Ann. Mat. Pura Appl. (4), 171 (1996), 313-377. doi: 10.1007/BF01759391. [18] K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information, Control Cybernet., 34 (2005), 203-225. [19] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994. [20] S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim., 39 (2001), 1756-1778 (electronic). doi: 10.1137/S0363012900369538. [21] A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques & Applications (Berlin) [Mathematics & Applications], 48, Springer, Berlin, 2005. [22] D. Martin, "Finite Element Library Mélina," Available from: http://anum-maths.univ-rennes1.fr/melina/. [23] V. Maz'ya and T. Shaposhnikova, "Theory of Multipliers in Spaces of Differentiable Functions," Monographs and Studies in Mathematics, 23, Pitman (Advanced Publishing Program), Boston, MA, 1985. [24] F. Murat and J. Simon, "Sur le Contrôle par un Domaine Géométrique," Rapport du L.A. 189, Université de Paris VI, France, 1976. [25] J. Nocedal and S. J. Wright, "Numerical Optimization," Second edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. [26] J. R. Shewchuk, "Mesh Generator Triangle," Available from: http://www.cs.cmu.edu/~quake/triangle.html. [27] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687 (1981). doi: 10.1080/01630563.1980.10120631. [28] J. Simon, Second variations for domain optimization problems, in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1988), Internat. Ser. Numer. Math., 91, Birkhäuser, Basel, (1989), 361-378. [29] J. Simon, Domain variation for drag in Stokes flow, in "Control Theory of Distributed Parameter Systems and Applications" (Shanghai, 1990), Lecture Notes in Control and Inform. Sci., 159, Springer, Berlin, (1991), 28-42. doi: 10.1007/BFb0004434. [30] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272 (electronic). doi: 10.1137/S0363012997323230. [31] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.
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