# American Institute of Mathematical Sciences

May  2016, 10(2): 327-367. doi: 10.3934/ipi.2016003

## On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives

 1 Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse Cedex 9, France, France 2 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago

Received  February 2015 Revised  May 2015 Published  May 2016

We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape.
Citation: Fabien Caubet, Carlos Conca, Matías Godoy. On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives. Inverse Problems & Imaging, 2016, 10 (2) : 327-367. doi: 10.3934/ipi.2016003
##### References:
 [1] G. Allaire, Continuity of the Darcy's law in the low-volume fraction limit,, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 18 (1991), 475. Google Scholar [2] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes i. abstract framework, a volume distribution of holes,, Archive for Rational Mechanics and Analysis, 113 (1990), 209. doi: 10.1007/BF00375065. Google Scholar [3] G. Allaire, F. de Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method,, Control Cybernet, 34 (2005), 59. Google Scholar [4] F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces,, Math. Methods Appl. Sci., 23 (2000), 575. doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4. Google Scholar [5] C. Alvarez, C. Conca, L. Friz, O. Kavian and J.-H. Ortega, Identification of immersed obstacles via boundary measurements,, Inverse Problems, 21 (2005), 1531. doi: 10.1088/0266-5611/21/5/003. Google Scholar [6] S. Amstutz, The topological asymptotic for the Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 11 (2005), 401. doi: 10.1051/cocv:2005012. Google Scholar [7] S. Amstutz, Topological sensitivity analysis for some nonlinear PDE systems,, Journal de mathématiques pures et appliquées, 85 (2006), 540. doi: 10.1016/j.matpur.2005.10.008. Google Scholar [8] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmholtz equation,, SIAM J. Control Optim., 42 (2003), 1523. doi: 10.1137/S0363012902406801. Google Scholar [9] 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. doi: 10.1142/S0218202511005660. Google Scholar [10] 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 (): 2871. doi: 10.1137/070704332. Google Scholar [11] V. Bonnaillie-Noël and M. Dambrine, Interactions between moderately close circular inclusions: The Dirichlet-Laplace equation in the plane,, Asymptot. Anal., 84 (2013), 197. Google Scholar [12] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation,, Math. Models Methods Appl. Sci., 19 (2009), 1853. doi: 10.1142/S021820250900398X. Google Scholar [13] F. Boyer and P. Fabrie, Éléments d'Analyse pour l'étude de Quelques Modèles d'écoulements de Fluides Visqueux Incompressibles,, Springer-Verlag, (2006). doi: 10.1007/3-540-29819-3. Google Scholar [14] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems,, Birkhäuser Boston Inc., (2005). Google Scholar [15] M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods,, J. Comput. Phys., 194 (2004), 344. doi: 10.1016/j.jcp.2003.09.033. Google Scholar [16] A. Carpio and M.-L. Rapún, Solving inhomogeneous inverse problems by topological derivative methods,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/4/045014. Google Scholar [17] A. Carpio and M.-L. Rapún, Topological derivatives for shape reconstruction,, in Inverse Problems and Imaging, (2008), 85. doi: 10.1007/978-3-540-78547-7_5. Google Scholar [18] A. Carpio and M.-L. Rapún, Hybrid topological derivative and gradient-based methods for electrical impedance tomography,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095010. Google Scholar [19] A. Carpio and M.-L. Rapún, Parameter identification in photothermal imaging,, J. Math. Imaging Vision, 49 (2014), 273. doi: 10.1007/s10851-013-0459-y. Google Scholar [20] F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/105007. Google Scholar [21] F. Caubet, M. Dambrine, D. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid,, Inverse Probl. Imaging, 7 (2013), 123. doi: 10.3934/ipi.2013.7.123. Google Scholar [22] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection,, Comput. Methods Appl. Mech. Engrg., 188 (2000), 713. doi: 10.1016/S0045-7825(99)00357-6. Google Scholar [23] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095010. Google Scholar [24] 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. Google Scholar [25] O. Dorn and D. Lesselier, Level set methods for inverse scattering,, Inverse Problems, 22 (2006). doi: 10.1088/0266-5611/22/4/R01. Google Scholar [26] O. Dorn and D. Lesselier, Level set methods for inverse scattering-some recent developments,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/125001. Google Scholar [27] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations.,, Springer-Verlag, (1994). Google Scholar [28] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem,, SIAM J. Control Optim., 41 (2002), 1042. doi: 10.1137/S0363012901384193. Google Scholar [29] P. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations,, SIAM J. Control Optim., 43 (2004), 1. doi: 10.1137/S0363012902411210. Google Scholar [30] M. Hassine, Shape optimization for the Stokes equations using topological sensitivity analysis,, ARIMA, 5 (2006), 216. Google Scholar [31] M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem,, ESAIM Control Optim. Calc. Var., 10 (2004), 478. doi: 10.1051/cocv:2004016. Google Scholar [32] L. He, C.-Y. Kao and S. Osher, Incorporating topological derivatives into shape derivatives based level set methods,, J. Comput. Phys., 225 (2007), 891. doi: 10.1016/j.jcp.2007.01.003. Google Scholar [33] A. Henrot and M. Pierre, Variation et Optimisation de Formes,, Springer, (2005). doi: 10.1007/3-540-37689-5. Google Scholar [34] A. Litman, D. Lesselier and F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,, Inverse Problems, 14 (1998), 685. doi: 10.1088/0266-5611/14/3/018. Google Scholar [35] D. Martin, Finite Element Library Mélina,, , (). Google Scholar [36] V. Maz'ya and A. Movchan, Asymptotic treatment of perforated domains without homogenization,, Math. Nachr., 283 (2010), 104. doi: 10.1002/mana.200910045. Google Scholar [37] V. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I,, Birkhäuser Verlag, (2000). Google Scholar [38] V. Maz'ya and S.V. Poborchi, Differentiable Functions on Bad Domains,, World Scientific Publishing Co. Inc., (1997). Google Scholar [39] F. Murat and J. Simon, Sur le Contrôle par un Domaine Géométrique,, Rapport du L.A. 189, (1976). Google Scholar [40] J. Nocedal and S. J. Wright, Numerical Optimization,, $2^{nd}$ edition, (2006). Google Scholar [41] O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization,, North-Holland Publishing Co., (1992). Google Scholar [42] O. Pantz and K. Trabelsi, Simultaneous shape, topology, and homogenized properties optimization,, Struct. Multidiscip. Optim., 34 (2007), 361. doi: 10.1007/s00158-006-0080-4. Google Scholar [43] A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien,, Ph.D thesis, (1995). Google Scholar [44] K. Sid Idris, Sensibilité Topologique en Optimisation de Forme,, Ph.D thesis, (2001). Google Scholar [45] J. Simon, Differentiation with respect to the domain in boundary value problems,, Numer. Funct. Anal. Optim., 2 (1981), 649. doi: 10.1080/01630563.1980.10120631. Google Scholar [46] J. Simon, Second variations for domain optimization problems,, In Control and estimation of distributed parameter systems (Vorau, 91 (1989), 361. Google Scholar [47] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization,, SIAM J. Control Optim., 37 (1999), 1251. doi: 10.1137/S0363012997323230. Google Scholar [48] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization,, Springer-Verlag, (1992). doi: 10.1007/978-3-642-58106-9. Google Scholar

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##### References:
 [1] G. Allaire, Continuity of the Darcy's law in the low-volume fraction limit,, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 18 (1991), 475. Google Scholar [2] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes i. abstract framework, a volume distribution of holes,, Archive for Rational Mechanics and Analysis, 113 (1990), 209. doi: 10.1007/BF00375065. Google Scholar [3] G. Allaire, F. de Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method,, Control Cybernet, 34 (2005), 59. Google Scholar [4] F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces,, Math. Methods Appl. Sci., 23 (2000), 575. doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4. Google Scholar [5] C. Alvarez, C. Conca, L. Friz, O. Kavian and J.-H. Ortega, Identification of immersed obstacles via boundary measurements,, Inverse Problems, 21 (2005), 1531. doi: 10.1088/0266-5611/21/5/003. Google Scholar [6] S. Amstutz, The topological asymptotic for the Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 11 (2005), 401. doi: 10.1051/cocv:2005012. Google Scholar [7] S. Amstutz, Topological sensitivity analysis for some nonlinear PDE systems,, Journal de mathématiques pures et appliquées, 85 (2006), 540. doi: 10.1016/j.matpur.2005.10.008. Google Scholar [8] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmholtz equation,, SIAM J. Control Optim., 42 (2003), 1523. doi: 10.1137/S0363012902406801. Google Scholar [9] 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. doi: 10.1142/S0218202511005660. Google Scholar [10] 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 (): 2871. doi: 10.1137/070704332. Google Scholar [11] V. Bonnaillie-Noël and M. Dambrine, Interactions between moderately close circular inclusions: The Dirichlet-Laplace equation in the plane,, Asymptot. Anal., 84 (2013), 197. Google Scholar [12] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation,, Math. Models Methods Appl. Sci., 19 (2009), 1853. doi: 10.1142/S021820250900398X. Google Scholar [13] F. Boyer and P. Fabrie, Éléments d'Analyse pour l'étude de Quelques Modèles d'écoulements de Fluides Visqueux Incompressibles,, Springer-Verlag, (2006). doi: 10.1007/3-540-29819-3. Google Scholar [14] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems,, Birkhäuser Boston Inc., (2005). Google Scholar [15] M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods,, J. Comput. Phys., 194 (2004), 344. doi: 10.1016/j.jcp.2003.09.033. Google Scholar [16] A. Carpio and M.-L. Rapún, Solving inhomogeneous inverse problems by topological derivative methods,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/4/045014. Google Scholar [17] A. Carpio and M.-L. Rapún, Topological derivatives for shape reconstruction,, in Inverse Problems and Imaging, (2008), 85. doi: 10.1007/978-3-540-78547-7_5. Google Scholar [18] A. Carpio and M.-L. Rapún, Hybrid topological derivative and gradient-based methods for electrical impedance tomography,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095010. Google Scholar [19] A. Carpio and M.-L. Rapún, Parameter identification in photothermal imaging,, J. Math. Imaging Vision, 49 (2014), 273. doi: 10.1007/s10851-013-0459-y. Google Scholar [20] F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/105007. Google Scholar [21] F. Caubet, M. Dambrine, D. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid,, Inverse Probl. Imaging, 7 (2013), 123. doi: 10.3934/ipi.2013.7.123. Google Scholar [22] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection,, Comput. Methods Appl. Mech. Engrg., 188 (2000), 713. doi: 10.1016/S0045-7825(99)00357-6. Google Scholar [23] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095010. Google Scholar [24] 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. Google Scholar [25] O. Dorn and D. Lesselier, Level set methods for inverse scattering,, Inverse Problems, 22 (2006). doi: 10.1088/0266-5611/22/4/R01. Google Scholar [26] O. Dorn and D. Lesselier, Level set methods for inverse scattering-some recent developments,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/125001. Google Scholar [27] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations.,, Springer-Verlag, (1994). Google Scholar [28] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem,, SIAM J. Control Optim., 41 (2002), 1042. doi: 10.1137/S0363012901384193. Google Scholar [29] P. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations,, SIAM J. Control Optim., 43 (2004), 1. doi: 10.1137/S0363012902411210. Google Scholar [30] M. Hassine, Shape optimization for the Stokes equations using topological sensitivity analysis,, ARIMA, 5 (2006), 216. Google Scholar [31] M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem,, ESAIM Control Optim. Calc. Var., 10 (2004), 478. doi: 10.1051/cocv:2004016. Google Scholar [32] L. He, C.-Y. Kao and S. Osher, Incorporating topological derivatives into shape derivatives based level set methods,, J. Comput. Phys., 225 (2007), 891. doi: 10.1016/j.jcp.2007.01.003. Google Scholar [33] A. Henrot and M. Pierre, Variation et Optimisation de Formes,, Springer, (2005). doi: 10.1007/3-540-37689-5. Google Scholar [34] A. Litman, D. Lesselier and F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,, Inverse Problems, 14 (1998), 685. doi: 10.1088/0266-5611/14/3/018. Google Scholar [35] D. Martin, Finite Element Library Mélina,, , (). Google Scholar [36] V. Maz'ya and A. Movchan, Asymptotic treatment of perforated domains without homogenization,, Math. Nachr., 283 (2010), 104. doi: 10.1002/mana.200910045. Google Scholar [37] V. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I,, Birkhäuser Verlag, (2000). Google Scholar [38] V. Maz'ya and S.V. Poborchi, Differentiable Functions on Bad Domains,, World Scientific Publishing Co. Inc., (1997). Google Scholar [39] F. Murat and J. Simon, Sur le Contrôle par un Domaine Géométrique,, Rapport du L.A. 189, (1976). Google Scholar [40] J. Nocedal and S. J. Wright, Numerical Optimization,, $2^{nd}$ edition, (2006). Google Scholar [41] O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization,, North-Holland Publishing Co., (1992). Google Scholar [42] O. Pantz and K. Trabelsi, Simultaneous shape, topology, and homogenized properties optimization,, Struct. Multidiscip. Optim., 34 (2007), 361. doi: 10.1007/s00158-006-0080-4. Google Scholar [43] A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien,, Ph.D thesis, (1995). Google Scholar [44] K. Sid Idris, Sensibilité Topologique en Optimisation de Forme,, Ph.D thesis, (2001). Google Scholar [45] J. Simon, Differentiation with respect to the domain in boundary value problems,, Numer. Funct. Anal. Optim., 2 (1981), 649. doi: 10.1080/01630563.1980.10120631. Google Scholar [46] J. Simon, Second variations for domain optimization problems,, In Control and estimation of distributed parameter systems (Vorau, 91 (1989), 361. Google Scholar [47] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization,, SIAM J. Control Optim., 37 (1999), 1251. doi: 10.1137/S0363012997323230. Google Scholar [48] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization,, Springer-Verlag, (1992). doi: 10.1007/978-3-642-58106-9. Google Scholar
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