Detecting an Immersed Obstacle in Stokes Fluid Flow Using the Coupled Complex Boundary Method
Abstract.
A nonconventional shape optimization approach is introduced to address the identification of an obstacle immersed in a fluid described by the Stokes equation within a larger bounded domain, relying on boundary measurements on the accessible surface. The approach employs tools from shape optimization, utilizing the coupled complex boundary method to transform the overspecified problem into a complex boundary value problem by incorporating a complex Robin boundary condition. This condition is derived by coupling the Dirichlet and Neumann boundary conditions along the accessible boundary. The identification of the obstacle involves optimizing a cost function constructed based on the imaginary part of the solution across the entire domain. The subsequent calculation of the shape gradient of this cost function, rigorously performed via the rearrangement method, enables the iterative solution of the optimization problem using a Sobolev gradient descent algorithm. The feasibility of the method is illustrated through numerical experiments in both two and three spatial dimensions, demonstrating its effectiveness in reconstructing obstacles with pronounced concavities under high-level noise-contaminated data, all without perimeter or volume functional penalization.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments.
JFTR is supported by the JSPS Postdoctoral Fellowships for Research in Japan and by the JSPS Grant-in-Aid for Early-Career Scientists under Japan grant JP23K13012. HN is partially supported by JSPS Grants-in-Aid for Scientific Research under grants JP20KK0058, JP21H04431, and JP20H01823. JFTR and HN are also partially supported by the JST CREST grant JPMJCR2014.
References
1.
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), pp. 2871–2900, https://doi.org/10.1137/070704332.
2.
L. Afraites, A new coupled complex boundary method (CCBM) for an inverse obstacle problem, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 23–40, https://doi.org/10.3934/dcdss.2021069.
3.
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), https://doi.org/10.3934/dcdsb.2007.8.389.
4.
L. Afraites, M. Dambrine, and D. Kateb, On second order shape optimization methods for electrical impedance tomography, SIAM J. Control Optim., 47 (2008), pp. 1556–1590, https://doi.org/10.1137/070687438.
5.
L. Afraites, C. Masnaoui, and M. Nachaoui, Shape optimization method for an inverse geometric source problem and stability at critical shape, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 1–21, https://doi.org/10.3934/dcdss.2021006.
6.
L. Afraites and J. F. T. Rabago, Boundary shape reconstruction with Robin condition: Existence result, stability analysis, and inversion via multiple measurements, Comput. Appl. Math., 43 (2024), https://doi.org/10.1007/s40314-024-02741-3.
7.
L. Afraites and J. F. T. Rabago, Shape optimization methods for detecting an unknown boundary with the Robin condition by a single measurement, Discrete Contin. Dyn. Syst. Ser. S, 18 (2025), pp. 43–76, https://doi.org/10.3934/dcdss.2022196.
8.
C. Alvarez, C. Conca, L. Friz, O. Kavian, and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), pp. 1531–1552, https://doi.org/10.1088/0266-5611/21/5/003.
9.
J. B. Bacani and G. H. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013, 384320, https://doi.org/10.1155/2013/384320.
10.
M. Badra, F. Caubet, and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Methods Appl. Sci., 21 (2011), pp. 2069–2101, https://doi.org/10.1142/S0218202511005660.
11.
F. Boyer and P. Fabrie, Éléments d’Analyse Pour L’étude de Quelques Modèles D’écoulements de Fluides Visqueux Incompressibles, Math. Appl. (Berlin), Springer, Berlin, 2006.
12.
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2012.
13.
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progr. Nonlinear Differential Equations Appl. 65, Birkhàuser-Verlag, Basel, 2005.
14.
F. Caubet, Instability of an inverse problem for the stationary Navier-Stokes equations, SIAM J. Control Optim., 51 (2013), pp. 2949–2975, https://doi.org/10.1137/110836857.
15.
F. Caubet, M. Dambrine, and D. Kateb, Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions, Inverse Problems, 29 (2013), 115011, https://doi.org/10.1088/0266-5611/29/11/115011.
16.
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), pp. 123–157, https://doi.org/10.3934/ipi.2013.7.123.
17.
X. L. Cheng, R. F. Gong, W. Han, and X. Zheng, A novel coupled complex boundary method for solving inverse source problems, Inverse Problems, 30 (2014), 055002, https://doi.org/10.1088/0266-5611/30/5/055002.
18.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th ed., Springer, New York, 2019.
19.
M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Math. RACSAM, 96 (2002), pp. 95–121.
20.
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer, New York, 1998.
21.
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd ed., Adv. Des. Control 22, SIAM, Philadelphia, 2011.
22.
G. Doğan, P. Morin, R. Nochetto, and M. Verani, Discrete gradient flows for shape optimization and applications, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 3898–3914, https://doi.org/10.1016/j.cma.2006.10.046.
23.
K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape hessian information, Control Cybern., 34 (2005), pp. 203–225.
24.
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts Natur. Philos., Springer, New York, 1994.
25.
S. Garreau, P. Guillaume, and M. Masmoudi, The topological asymptotic for pde systems: The elasticity case, SIAM J. Control Optim., 39 (2001), pp. 1756–1778, https://doi.org/10.1137/S0363012900369538.
26.
R. Gong, X. Cheng, and W. Han, A coupled complex boundary method for an inverse conductivity problem with one measurement, Appl. Anal., 96 (2017), pp. 869–885, https://doi.org/10.1080/00036811.2016.1165215.
27.
J. Haslinger, K. Ito, T. Kozubek, K. Kunisch, and G. H. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 11 (2009), pp. 317–330, https://doi.org/10.4171/ifb/213.
28.
F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), pp. 251–265, https://doi.org/10.1515/jnum-2012-0013.
29.
A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, EMS Tracts Math. 28, European Mathematical Society, Zürich, 2018.
30.
K. Ito, K. Kunisch, and G. H. Peichl, Variational approach to shape derivative for a class of Bernoulli problem, J. Math. Anal. Appl., 314 (2006), pp. 126–149, https://doi.org/10.1016/j.jmaa.2005.03.100.
31.
K. Ito, K. Kunisch, and G. H. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), pp. 517–539, https://doi.org/10.1051/cocv:2008002.
32.
H. Kasumba and K. Kunisch, On shape sensitivity analysis of the cost functional without shape sensitivity of the state variable, Control. Cybern., 40 (2011), pp. 989–1017.
33.
H. Kasumba and K. Kunisch, On computation of the shape Hessian of the cost functional without shape sensitivity of the state variable, J. Optim. Theory Appl., 162 (2014), pp. 779–804, https://doi.org/10.1007/s10957-013-0520-4.
34.
V. A. Kovtunenko and H. Ohtsuka, Inverse problem of shape identification from boundary measurement for Stokes equations: Shape differentiability of Lagrangian, J. Inverse Ill-Posed Probl., 30 (2022), pp. 461–474, https://doi.org/10.1515/jiip-2020-0081.
35.
J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer, Berlin, 1997.
36.
H. Ouaissa, A. Chakib, A. Nachaoui, and M. Nachaoui, On numerical approaches for solving an inverse Cauchy Stokes problem, Appl. Math. Optim., 85 (2022), 37, https://doi.org/10.1007/s00245-022-09833-8.
37.
J. F. T. Rabago, On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems, Math. Control Relat. Fields, 13 (2022), pp. 1362–1398, https://doi.org/10.3934/mcrf.2022041.
38.
J. F. T. Rabago and H. Azegami, Shape optimization approach to defect-shape identification with convective boundary condition via partial boundary measurement, Japan J. Indust. Appl. Math., 31 (2018), pp. 131–176, https://doi.org/10.1007/s13160-018-0337-5.
39.
J. F. T. Rabago and H. Azegami, A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional, Comput. Optim. Appl., 77 (2020), pp. 251–305, https://doi.org/10.1007/s10589-020-00199-7.
40.
J. F. T. Rabago, A. Hadri, L. Afraites, A. S. Hendy, and M. A. Zaky, A robust alternating direction numerical scheme in a shape optimization setting for solving geometric inverse problems, Comput. Math. Appl., 175 (2024), pp. 19–32, https://doi.org/10.1016/j.camwa.2024.08.034.
41.
J. F. T. Rabago and H. Notsu, Numerical solution to a free boundary problem for the Stokes equation using the coupled complex boundary method in shape optimization setting, Appl. Math. Optim., 89 (2024), https://doi.org/10.1007/s00245-023-10065-7.
42.
J. S. H. Simon and H. Notsu, A shape design problem for the Navier-Stokes flow with a convective boundary condition, Comput. Appl. Math., 41 (2022), https://doi.org/10.1007/s40314-022-01876-5.
43.
J. S. H. Simon and H. Notsu, A shape optimization problem constrained with the Stokes equations to address maximization of vortices, Evol. Equ. Control Theory, 11 (2022), pp. 1873–1902, https://doi.org/10.3934/eect.2022003.
44.
J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), pp. 1251–1272, https://doi.org/10.1137/S0363012997323230.
45.
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Ser. Comput. Math., Springer, Berlin, 1992.
46.
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS, Providence, RI, 1984.
47.
X. Zheng, X. Cheng, and R. Gong, A coupled complex boundary method for parameter identification in elliptic problems, Int. J. Comput. Math., 97 (2020), pp. 998–1015, https://doi.org/10.1080/00207160.2019.1601181.
Information & Authors
Information
Published In
SIAM Journal on Control and Optimization
Pages: 822 - 851
DOI: 10.1137/24M1647369
ISSN (online): 1095-7138
Copyright
© 2025 Society for Industrial and Applied Mathematics.
History
Submitted: 18 March 2024
Accepted: 19 December 2024
Published online: 20 March 2025
Keywords
MSC codes
Authors
Funding Information
Japan Society for the Promotion of Science (JSPS): JP23K13012, 24KF0221
Japan Society for the Promotion of Science (JSPS): JP20KK0058, JP21H04431, JP20H01823
Funding: The work of the first author is supported by the JSPS Grant-in-Aid for Early-Career Scientists under Japan grant JP23K13012, JSPS Postdoctoral Fellowships for Research in Japan (under Project Number 24KF0221). The second author is partially supported by JSPS Grants-in-Aid for Scientific Research under grants JP20KK0058, JP21H04431, and JP20H01823. The first and second authors are also partially supported by the JST CREST grant JPMJCR2014.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
There are no citations for this item
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].