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http://dx.doi.org/10.4134/JKMS.j190393

A NON-ITERATIVE RECONSTRUCTION METHOD FOR AN INVERSE PROBLEM MODELED BY A STOKES-BRINKMANN EQUATIONS  

Hassine, Maatoug (Department of Mathematics Faculty of Sciences Avenue Monastir University)
Hrizi, Mourad (Department of Mathematics Faculty of Sciences Avenue Monastir University)
Malek, Rakia (Department of Mathematics Faculty of Sciences Avenue Monastir University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.5, 2020 , pp. 1079-1101 More about this Journal
Abstract
This work is concerned with a geometric inverse problem in fluid mechanics. The aim is to reconstruct an unknown obstacle immersed in a Newtonian and incompressible fluid flow from internal data. We assume that the fluid motion is governed by the Stokes-Brinkmann equations in the two dimensional case. We propose a simple and efficient reconstruction method based on the topological sensitivity concept. The geometric inverse problem is reformulated as a topology optimization one minimizing a least-square functional. The existence and stability of the optimization problem solution are discussed. A topological sensitivity analysis is derived with the help of a straightforward approach based on a penalization technique without using the classical truncation method. The theoretical results are exploited for building a non-iterative reconstruction algorithm. The unknown obstacle is reconstructed using a levelset curve of the topological gradient. The accuracy and the robustness of the proposed method are justified by some numerical examples.
Keywords
Inverse problem; Stokes-Brinkmann equations; topological sensitivity analysis;
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1 G. Allaire, Topology optimization with the homogenization and the level-set methods, in Nonlinear homogenization and its applications to composites, polycrystals and smart materials, 1-13, NATO Sci. Ser. II Math. Phys. Chem., 170, Kluwer Acad. Publ., Dordrecht, 2004. https://doi.org/10.1007/1-4020-2623-4_1
2 G. Allaire, F. Jouve, and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys. 194 (2004), no. 1, 363-393. https://doi.org/10.1016/j.jcp.2003.09.032   DOI
3 C. J. S. Alves and A. L. Silvestre, On the determination of point-forces on a Stokes system, Math. Comput. Simulation 66 (2004), no. 4-5, 385-397. https://doi.org/10.1016/j.matcom.2004.02.007   DOI
4 L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
5 S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 11 (2005), no. 3, 401-425. https://doi.org/10.1051/cocv:2005012   DOI
6 S. Amstutz, I. Horchani, and M. Masmoudi, Crack detection by the topological gradient method, Control Cybernet. 34 (2005), no. 1, 81-101.
7 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), no. 5, 2871-2900. https://doi.org/10.1137/070704332   DOI
8 H. Ben Ameur, M. Burger, and B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Inverse Problems 20 (2004), no. 3, 673-696. https://doi.org/10.1088/0266-5611/20/3/003   DOI
9 E. Beretta, C. Cavaterra, J. H. Ortega, and S. Zamorano, Size estimates of an obstacle in a stationary Stokes fluid, Inverse Problems 33 (2017), no. 2, 025008, 29 pp. https://doi.org/10.1088/1361-6420/33/2/025008
10 T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow, Internat. J. Numer. Methods Fluids 41 (2003), no. 1, 77-107. https://doi.org/10.1002/fld.426   DOI
11 V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. https://doi.org/10.1007/978-3-642-61623-5
12 F. Caubet, C. Conca, and M. Godoy, On the detection of several obstacles in 2D Stokes flow: topological sensitivity and combination with shape derivatives, Inverse Probl. Imaging 10 (2016), no. 2, 327-367. https://doi.org/10.3934/ipi.2016003   DOI
13 F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems 28 (2012), no. 10, 105007, 31 pp. https://doi.org/10.1088/0266-5611/28/10/105007
14 S. Garreau, P. Guillaume, and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim. 39 (2001), no. 6, 1756-1778. https://doi.org/10.1137/S0363012900369538   DOI
15 Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations, SIAM J. Control Optim. 43 (2004), no. 1, 1-31. https://doi.org/10.1137/S0363012902411210   DOI
16 M. Hrizi, M. Hassine, and R. Malek, A new reconstruction method for a parabolic inverse source problem, Appl. Anal. 98 (2019), no. 15, 2723-2750. https://doi.org/10.1080/00036811.2018.1469011   DOI
17 M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem, ESAIM Control Optim. Calc. Var. 10 (2004), no. 4, 478-504. https://doi.org/10.1051/cocv:2004016   DOI
18 H. Heck, G. Uhlmann, and J.-N. Wang, Reconstruction of obstacles immersed in an incompressible fluid, Inverse Probl. Imaging 1 (2007), no. 1, 63-76. https://doi.org/10.3934/ipi.2007.1.63   DOI
19 M. Hrizi and M. Hassine, One-iteration reconstruction algorithm for geometric inverse source problem, J. Elliptic Parabol. Equ. 4 (2018), no. 1, 177-205. https://doi.org/10.1007/s41808-018-0015-4   DOI
20 D. Koster, Numerical simulation of acoustic streaming on surface acoustic wave-driven biochips, SIAM J. Sci. Comput. 29 (2007), no. 6, 2352-2380. https://doi.org/10.1137/060676623   DOI
21 M. Krotkiewski, I. S. Ligaarden, K.-A. Lie, and D. W. Schmid, On the importance of the stokes-brinkman equations for computing effective permeability in karst reservoirs, Commun. Comput. Phys. 10 (2011), 1315-1332.   DOI
22 A. Lechleiter and T. Rienmuller, Factorization method for the inverse Stokes problem, Inverse Probl. Imaging 7 (2013), no. 4, 1271-1293. https://doi.org/10.3934/ipi.2013.7.1271   DOI
23 F. Le Louer and M.-L. Rapun, Detection of multiple impedance obstacles by non-iterative topological gradient based methods, J. Comput. Phys. 388 (2019), 534-560. https://doi.org/10.1016/j.jcp.2019.03.023   DOI
24 N. F. M. Martins, Identification results for inverse source problems in unsteady Stokes flows, Inverse Problems 31 (2015), no. 1, 015004, 17 pp. https://doi.org/10.1088/0266-5611/31/1/015004
25 J. Pommier and B. Samet, The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrarily shaped hole, SIAM J. Control Optim. 43 (2004), no. 3, 899-921. https://doi.org/10.1137/S036301290241616X   DOI
26 M. Masmoudi, The topological asymptotic, computational methods for control applications, ed. h. kawarada and j. periaux, International Series, Gakuto, 2002.
27 M. Masmoudi, J. Pommier, and B. Samet, The topological asymptotic expansion for the Maxwell equations and some applications, Inverse Problems 21 (2005), no. 2, 547-564. https://doi.org/10.1088/0266-5611/21/2/008   DOI
28 A. A. Novotny and J. Soko lowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-35245-4
29 B. Samet, S. Amstutz, and M. Masmoudi, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim. 42 (2003), no. 5, 1523-1544. https://doi.org/10.1137/S0363012902406801   DOI
30 P. Popov, Y. Efendiev, and G. Qin, Multiscale modeling and simulations of flows in naturally fractured Karst reservoirs, Commun. Comput. Phys. 6 (2009), no. 1, 162-184. https://doi.org/10.4208/cicp.2009.v6.p162
31 A. Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien, PhD thesis, Forschungszentrum fur Multidisziplinare Analysen und Angewandte Strukturoptimierung. Institut fur Mechanik und Regelungstechnik, 1996.
32 J. Sokol owski and A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37 (1999), no. 4, 1251-1272. https://doi.org/10.1137/S0363012997323230   DOI
33 W. Yan, M. Liu, and F. Jing, Shape inverse problem for Stokes-Brinkmann equations, Appl. Math. Lett. 88 (2019), 222-229. https://doi.org/10.1016/j.aml.2018.09.003   DOI