# 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 and 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), 475-499. [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), 209-259. doi: 10.1007/BF00375065. [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), 59-80. [4] F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces, Math. Methods Appl. Sci., 23 (2000), 575-600. doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4. [5] C. Alvarez, C. Conca, L. Friz, O. Kavian and J.-H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003. [6] S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 11 (2005), 401-425. doi: 10.1051/cocv:2005012. [7] S. Amstutz, Topological sensitivity analysis for some nonlinear PDE systems, Journal de mathématiques pures et appliquées, 85 (2006), 540-557. doi: 10.1016/j.matpur.2005.10.008. [8] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., 42 (2003), 1523-1544. doi: 10.1137/S0363012902406801. [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), 2069-2101. doi: 10.1142/S0218202511005660. [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 (2009/10), 2871-2900. doi: 10.1137/070704332. [11] V. Bonnaillie-Noël and M. Dambrine, Interactions between moderately close circular inclusions: The Dirichlet-Laplace equation in the plane, Asymptot. Anal., 84 (2013), 197-227. [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), 1853-1882. doi: 10.1142/S021820250900398X. [13] F. Boyer and P. Fabrie, Éléments d'Analyse pour l'étude de Quelques Modèles d'écoulements de Fluides Visqueux Incompressibles, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-29819-3. [14] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser Boston Inc., Boston, MA, 2005. [15] M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods, J. Comput. Phys., 194 (2004), 344-362. doi: 10.1016/j.jcp.2003.09.033. [16] A. Carpio and M.-L. Rapún, Solving inhomogeneous inverse problems by topological derivative methods, Inverse Problems, 24 (2008), 045014, 32 pp. doi: 10.1088/0266-5611/24/4/045014. [17] A. Carpio and M.-L. Rapún, Topological derivatives for shape reconstruction, in Inverse Problems and Imaging, volume 1943 of Lecture Notes in Math., Springer, (2008), 85-133. doi: 10.1007/978-3-540-78547-7_5. [18] A. Carpio and M.-L. Rapún, Hybrid topological derivative and gradient-based methods for electrical impedance tomography, Inverse Problems, 28 (2012), 095010, 22 pp. doi: 10.1088/0266-5611/28/9/095010. [19] A. Carpio and M.-L. Rapún, Parameter identification in photothermal imaging, J. Math. Imaging Vision, 49 (2014), 273-288. doi: 10.1007/s10851-013-0459-y. [20] F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems, 28 (2012), 105007, 31 pp. doi: 10.1088/0266-5611/28/10/105007. [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), 123-157. doi: 10.3934/ipi.2013.7.123. [22] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection, Comput. Methods Appl. Mech. Engrg., 188 (2000), 713-726. doi: 10.1016/S0045-7825(99)00357-6. [23] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010, 18pp. doi: 10.1088/0266-5611/26/9/095010. [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), 95-121. [25] O. Dorn and D. Lesselier, Level set methods for inverse scattering, Inverse Problems, 22 (2006), R67-R131. doi: 10.1088/0266-5611/22/4/R01. [26] O. Dorn and D. Lesselier, Level set methods for inverse scattering-some recent developments, Inverse Problems, 25 (2009), 125001, 11pp. doi: 10.1088/0266-5611/25/12/125001. [27] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations., Springer-Verlag, New York, 1994. [28] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem, SIAM J. Control Optim., 41 (2002), 1042-1072. doi: 10.1137/S0363012901384193. [29] P. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations, SIAM J. Control Optim., 43 (2004), 1-31. doi: 10.1137/S0363012902411210. [30] M. Hassine, Shape optimization for the Stokes equations using topological sensitivity analysis, ARIMA, 5 (2006), 216-229. [31] M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem, ESAIM Control Optim. Calc. Var., 10 (2004), 478-504. doi: 10.1051/cocv:2004016. [32] L. He, C.-Y. Kao and S. Osher, Incorporating topological derivatives into shape derivatives based level set methods, J. Comput. Phys., 225 (2007), 891-909. doi: 10.1016/j.jcp.2007.01.003. [33] A. Henrot and M. Pierre, Variation et Optimisation de Formes, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5. [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), 685-706. doi: 10.1088/0266-5611/14/3/018. [35] D. Martin, Finite Element Library Mélina, https://anum-maths.univ-rennes1.fr/melina/danielmartin/melina/. [36] V. Maz'ya and A. Movchan, Asymptotic treatment of perforated domains without homogenization, Math. Nachr., 283 (2010), 104-125. doi: 10.1002/mana.200910045. [37] V. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I, Birkhäuser Verlag, Basel, 2000. [38] V. Maz'ya and S.V. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co. Inc., River Edge, NJ, 1997. [39] F. Murat and J. Simon, Sur le Contrôle par un Domaine Géométrique, Rapport du L.A. 189, Université de Paris VI, France, 1976. [40] J. Nocedal and S. J. Wright, Numerical Optimization, $2^{nd}$ edition, Springer, New York, 2006. [41] O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland Publishing Co., Amsterdam, 1992. [42] O. Pantz and K. Trabelsi, Simultaneous shape, topology, and homogenized properties optimization, Struct. Multidiscip. Optim., 34 (2007), 361-365. doi: 10.1007/s00158-006-0080-4. [43] A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien, Ph.D thesis, Universität-Gesamthochschule-Siegen, 1995. [44] K. Sid Idris, Sensibilité Topologique en Optimisation de Forme, Ph.D thesis, Institut National des Sciences Apliquées de Toulouse, 2001. [45] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1981), 649-687. doi: 10.1080/01630563.1980.10120631. [46] J. Simon, Second variations for domain optimization problems, In Control and estimation of distributed parameter systems (Vorau, 1988) Birkhäuser, Basel, 91 (1989), 361-378. [47] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272. doi: 10.1137/S0363012997323230. [48] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.

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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), 475-499. [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), 209-259. doi: 10.1007/BF00375065. [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), 59-80. [4] F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces, Math. Methods Appl. Sci., 23 (2000), 575-600. doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4. [5] C. Alvarez, C. Conca, L. Friz, O. Kavian and J.-H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003. [6] S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 11 (2005), 401-425. doi: 10.1051/cocv:2005012. [7] S. Amstutz, Topological sensitivity analysis for some nonlinear PDE systems, Journal de mathématiques pures et appliquées, 85 (2006), 540-557. doi: 10.1016/j.matpur.2005.10.008. [8] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., 42 (2003), 1523-1544. doi: 10.1137/S0363012902406801. [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), 2069-2101. doi: 10.1142/S0218202511005660. [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 (2009/10), 2871-2900. doi: 10.1137/070704332. [11] V. Bonnaillie-Noël and M. Dambrine, Interactions between moderately close circular inclusions: The Dirichlet-Laplace equation in the plane, Asymptot. Anal., 84 (2013), 197-227. [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), 1853-1882. doi: 10.1142/S021820250900398X. [13] F. Boyer and P. Fabrie, Éléments d'Analyse pour l'étude de Quelques Modèles d'écoulements de Fluides Visqueux Incompressibles, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-29819-3. [14] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser Boston Inc., Boston, MA, 2005. [15] M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods, J. Comput. Phys., 194 (2004), 344-362. doi: 10.1016/j.jcp.2003.09.033. [16] A. Carpio and M.-L. Rapún, Solving inhomogeneous inverse problems by topological derivative methods, Inverse Problems, 24 (2008), 045014, 32 pp. doi: 10.1088/0266-5611/24/4/045014. [17] A. Carpio and M.-L. Rapún, Topological derivatives for shape reconstruction, in Inverse Problems and Imaging, volume 1943 of Lecture Notes in Math., Springer, (2008), 85-133. doi: 10.1007/978-3-540-78547-7_5. [18] A. Carpio and M.-L. Rapún, Hybrid topological derivative and gradient-based methods for electrical impedance tomography, Inverse Problems, 28 (2012), 095010, 22 pp. doi: 10.1088/0266-5611/28/9/095010. [19] A. Carpio and M.-L. Rapún, Parameter identification in photothermal imaging, J. Math. Imaging Vision, 49 (2014), 273-288. doi: 10.1007/s10851-013-0459-y. [20] F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems, 28 (2012), 105007, 31 pp. doi: 10.1088/0266-5611/28/10/105007. [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), 123-157. doi: 10.3934/ipi.2013.7.123. [22] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection, Comput. Methods Appl. Mech. Engrg., 188 (2000), 713-726. doi: 10.1016/S0045-7825(99)00357-6. [23] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010, 18pp. doi: 10.1088/0266-5611/26/9/095010. [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), 95-121. [25] O. Dorn and D. Lesselier, Level set methods for inverse scattering, Inverse Problems, 22 (2006), R67-R131. doi: 10.1088/0266-5611/22/4/R01. [26] O. Dorn and D. Lesselier, Level set methods for inverse scattering-some recent developments, Inverse Problems, 25 (2009), 125001, 11pp. doi: 10.1088/0266-5611/25/12/125001. [27] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations., Springer-Verlag, New York, 1994. [28] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem, SIAM J. Control Optim., 41 (2002), 1042-1072. doi: 10.1137/S0363012901384193. [29] P. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations, SIAM J. Control Optim., 43 (2004), 1-31. doi: 10.1137/S0363012902411210. [30] M. Hassine, Shape optimization for the Stokes equations using topological sensitivity analysis, ARIMA, 5 (2006), 216-229. [31] M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem, ESAIM Control Optim. Calc. Var., 10 (2004), 478-504. doi: 10.1051/cocv:2004016. [32] L. He, C.-Y. Kao and S. Osher, Incorporating topological derivatives into shape derivatives based level set methods, J. Comput. Phys., 225 (2007), 891-909. doi: 10.1016/j.jcp.2007.01.003. [33] A. Henrot and M. Pierre, Variation et Optimisation de Formes, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5. [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), 685-706. doi: 10.1088/0266-5611/14/3/018. [35] D. Martin, Finite Element Library Mélina, https://anum-maths.univ-rennes1.fr/melina/danielmartin/melina/. [36] V. Maz'ya and A. Movchan, Asymptotic treatment of perforated domains without homogenization, Math. Nachr., 283 (2010), 104-125. doi: 10.1002/mana.200910045. [37] V. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I, Birkhäuser Verlag, Basel, 2000. [38] V. Maz'ya and S.V. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co. Inc., River Edge, NJ, 1997. [39] F. Murat and J. Simon, Sur le Contrôle par un Domaine Géométrique, Rapport du L.A. 189, Université de Paris VI, France, 1976. [40] J. Nocedal and S. J. Wright, Numerical Optimization, $2^{nd}$ edition, Springer, New York, 2006. [41] O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland Publishing Co., Amsterdam, 1992. [42] O. Pantz and K. Trabelsi, Simultaneous shape, topology, and homogenized properties optimization, Struct. Multidiscip. Optim., 34 (2007), 361-365. doi: 10.1007/s00158-006-0080-4. [43] A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien, Ph.D thesis, Universität-Gesamthochschule-Siegen, 1995. [44] K. Sid Idris, Sensibilité Topologique en Optimisation de Forme, Ph.D thesis, Institut National des Sciences Apliquées de Toulouse, 2001. [45] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1981), 649-687. doi: 10.1080/01630563.1980.10120631. [46] J. Simon, Second variations for domain optimization problems, In Control and estimation of distributed parameter systems (Vorau, 1988) Birkhäuser, Basel, 91 (1989), 361-378. [47] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272. doi: 10.1137/S0363012997323230. [48] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.
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