On robustness to a topological perturbation in fluid mechanics

• Autorzy: Abdelbari Merwan , Nachi Khadra , Sokołowski Jan , Novotny Antonio Andre

Identyfikatory

DOI

10.61822/amcs-2025-0006

• Języki publikacji: EN

Abstrakty

EN

The robustness to topological perturbations in geometrical domains filled by a fluid flowing in Stokes-Darcy regime is considered. The cost functional is given by the energy dissipation in the fluid. The topological perturbation is carried out by the nucleation of an infinitesimal circular obstacle, which can be considered as a small measurement device. Our approach is based on the topological derivative method, which has been previously employed in the shape and topology optimization problems. The topological derivative (TD) measures the sensitivity of a given shape functional with respect to topological domain perturbations. The TD is used to determine the location of the small device placement, through a distributed control problem. By taking into account the effect of the disturbance term or uncertain input data in the TD expression, the problem of robustness to topological perturbation for the energy functional can be formulated as a minimax optimization problem with a pointwise observation. Numerical examples illustrate the efficiency of the proposed topological derivative method.

Słowa kluczowe

EN

topological derivative; shape optimization; optimal control; robust control

PL

pochodna topologiczna; optymalizacja kształtu; sterowanie optymalne; sterowanie odporne

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2025
• Tom: Vol. 35, no. 1
• Strony: 69--81
• Opis fizyczny: Bibliogr. 38 poz., rys., tab.

Twórcy

autor

Abdelbari Merwan

• Laboratory of Rheology and Mechanics, Hassiba Ben Bouali University, Hay Salem, 02000 Chlef, Algeria

autor

Nachi Khadra

• Laboratory of Mathematical Analysis and Applications, University of Oran 1 - Ahmed Ben Bella, BP 1524, El M’naouer Oran, Algeria

autor

Sokołowski Jan

• Élie Cartan Institute, University of Lorraine, BP 239, 54506 Vandoeuvre lès Nancy cedex, France
• Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland

autor

Novotny Antonio Andre

• Department of Computational and Applied Mathematics, National Laboratory for Scientific Computing (LNCC/MCT), Av. Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil

Bibliografia

• [1] Amstutz, S. (2005). The topological asymptotic for the Navier-Stokes equations, ESAIM: Control, Optimization and Calculus of Variations 11(3): 401-425.
• [2] Amstutz, S. (2006). Topological sensitivity analysis for some nonlinear PDE systems, Journal de Mathématiques Pures et Appliquées 85(4): 540-557.
• [3] Baumann, P. and Sturm, K. (2022). Adjoint-based methods to compute higher-order topological derivatives with an application to elasticity, Engineering Computations 39(1): 60-114.
• [4] Bewley, T.R., Temam, R. and Ziane, M. (2000). A general framework for robust control in fluid mechanics, Physica D: Nonlinear Phenomena 138(3-4): 360-392.
• [5] Bogachev, V.I. and Ruas, M.A.S. (2007). Measure Theory, Vol. 1, Springer, Berlin.
• [6] Boyer, F. and Fabrie, P. (2005). Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, Springer, Berlin.
• [7] Brett, C.E. (2014). Optimal Control and Inverse Problems Involving Point and Line Functionals and Inequality Constraints, PhD thesis, University of Warwick, Coventry.
• [8] Caubet, F. and Dambrine, M. (2012). Localization of small obstacles in Stokes flow, Inverse Problems 28(10): 105007.
• [9] Dáger, R. (2006). Insensitizing controls for the 1-D wave equation, SIAM Journal on Control and Optimization 45(5): 1758-1768.
• [10] Dziri, R., Moubachir, M. and Zolésio, J.-P. (2004). Dynamical shape gradient for the Navier-Stokes system, Comptes Rendus Mathematique 338(2): 183-186.
• [11] Dziri, R. and Zolésio, J.-P. (2011). Drag reduction for non-cylindrical Navier-Stokes flows, Optimization Methods and Software 26(4-5): 575-600.
• [12] Ekeland, I. and Temam, R. (1999). Convex Analysis and Variational Problems, SIAM, Philadelphia.
• [13] Ervedoza, S., Lissy, P. and Privat, Y. (2022). Desensitizing control for the heat equation with respect to domain variations, Journal de l’École Polytechnique Mathématiques 9: 1397-1429.
• [14] Galdi, G. (2011). An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer, New York.
• [15] Garreau, S., Guillaume, P. and Masmoudi, M. (2001). The topological asymptotic for PDE systems: The elasticity case, SIAM Journal on Control and Optimization 39(6): 1756-1778.
• [16] Giusti, S.M., Sokołowski, J. and Stebel, J. (2015). On topological derivatives for contact problems in elasticity, Journal of Optimization Theory and Applications 165: 279-294.
• [17] Guerrero, S. (2007). Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Annales de l’Institut Henri Poincaré C, Analyse non linéaire 24(6): 1029-1054.
• [18] Gueye, M. (2013). Insensitizing controls for the Navier-Stokes equations, Annales de l’Institut Henri Poincaré C, Analyse non linéaire 30(5): 825-844.
• [19] Gugat, M. and Lazar, M. (2023). Optimal control problems without terminal constraints: The turnpike property with interior decay, International Journal of Applied Mathematics and Computer Science 33(3): 429-438, DOI: 10.34768/amcs-2023-0031.
• [20] Gugat, M. and Sokołowski, J. (2023). An aspect of the turnpike property: Long time horizon behavior, Serdica Mathematical Journal 49(1-3): 127-154.
• [21] Guillaume, P. and Hassine, M. (2008). Removing holes in topological shape optimization, ESAIM: Control, Optimisation and Calculus of Variations 14(1): 160-191.
• [22] Hassine, M. and Masmoudi, M. (2004). The topological asymptotic expansion for the quasi-Stokes problem, ESAIM: Control, Optimisation and Calculus of Variations 10(4): 478-504.
• [23] Hlaváček, I., Novotny, A., Sokołowski, J. and Żochowski, A. (2009). On topological derivatives for elastic solids with uncertain input data, Journal of Optimization Theory and Applications 141(3): 569-595.
• [24] Iguernane, M., Nazarov, S.A., Roche, J.-R., Sokolowski, J. and Szulc, K. (2009). Topological derivatives for semilinear elliptic equations, International Journal of Applied Mathematics and Computer Science 19(2): 191-205, DOI: 10.2478/v10006-009-0016-4.
• [25] Kovtunenko, V.A. and Kunisch, K. (2014). High precision identification of an object: Optimality-conditions-based concept of imaging, SIAM Journal on Control and Optimization 52(1): 773-796.
• [26] Krzyżanowski, P., Malikova, S., Mucha, P. B. and Piasecki, T. (2024). Comparative analysis of obstacle approximation strategies for the steady incompressible Navier-Stokes equations, Applied Mathematics & Optimization 89(2): 1-20.
• [27] Leugering, G., Novotny, A.A. and Sokołowski, J. (2022). On the robustness of the topological derivative for Helmholtz problems and applications, Control and Cybernetics 51(2): 227-248.
• [28] Lions, J. (1992). Sentinelles pour les Systmès Distribués à Données Incomplètes, Masson, Paris.
• [29 Logg, A., Mardal, K.-A. and Wells, G. (2012). Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Springer, Berlin.
• [30] Moubachir, M. and Zolesio, J.-P. (2006). Moving Shape Analysis and Control: Applications to Fluid Structure Interactions, Chapman and Hall/CRC, Boca Raton.
• [31] Novotny, A.A. and Sokołowski, J. (2012). Topological Derivatives in Shape Optimization, Springer, Berlin.
• [32] Novotny, A.A., Sokołowski, J. and Żochowski, A. (2019). Applications of the Topological Derivative Method, Springer, Cham.
• [33] Sá, N.L., Amigo, R.R., Novotny, A.A. and Silva, N.E. (2016). Topological derivatives applied to fluid flow channel design optimization problems, Structural and Multidisciplinary Optimization 54: 249-264.
• [34] Sokołowski, J. and Żochowski, A. (1999). On the topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.
• [35] Sokołowski, J. and Zolésio, J.-P. (1992). Introduction to Shape Optimization, Springer, Berlin.
• [36] Sturm, K. (2020). Topological sensitivities via a Lagrangian approach for semilinear problems, Nonlinearity 33(9): 4310.
• [37] Szabó, B. and Babuška, I. (2011). Introduction to Finite Element Analysis: Formulation, Verification and Validation, Wiley, Chichester.
• [38] Tröltzsch, F. (2024). Optimal Control of Partial Differential Equations: Theory, Methods and Applications, American Mathematical Society, Providence.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).