A few remarks on penalty and penalty-duality methods in fluid-structure interactions. (English) Zbl 1466.76019
Summary: In fluid-structure interaction problems, many people use a penalty method for positioning the structure inside the fluid. This is usually performed by considering that the fluid is very stiff or/and very heavy at the place occupied by the structure. These methods are very convenient for the programming point of view but can lead to ill conditioned operators. This is a drawback in the numerical solution methods. In particular the forces applied to the structure by the surrounding flow are not always accurately estimated because the penalty parameter – which is a very large number – appears in their expressions. We suggest in this paper a mathematical analysis of the difficulties encountered and we discuss how the penalty-duality method of D. P. Bertsekas [Math. Program. 9, 87–99 (1975; Zbl 0325.90055)] can be an interesting alternative to overcome them.
MSC:
| 76D55 | Flow control and optimization for incompressible viscous fluids |
| 76M30 | Variational methods applied to problems in fluid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
Keywords:
penalty-duality Bertsekas method; Burgers equation; advection-diffusion equation: fluid-structure interactionCitations:
Zbl 0325.90055References:
| [1] | Angot, P.; Bruneau, C. H.; Fabrie, P., A penalization method to take into account obstacles in incompressible flows, Numer. Math., 81, 4, 497-520 (1999) · Zbl 0921.76168 |
| [2] | Babuska, I.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A. K., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press New York), 5-359 · Zbl 0268.65052 |
| [3] | Belytschko, T.; Lu, Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077 |
| [4] | Benamour, M.; Liberge, E.; Béghein, C., Lattice Boltzmann method for fluid flow around bodies using volume penalization, Int. J. Multiphysics, 9, 3, 299-316 (2015) |
| [5] | Benamour, M.; Liberge, E.; Béghein, C., A volume penalization lattice Boltzmann method for simulating flows in the presence of obstacles, J. Comput. Sci., 39, Article 101050 pp. (2020) |
| [6] | Ben Dhia, H., Problèmes mécaniques multi-échelles: la méthode Arlequin, C. R. Acad. Sci., Sér. IIB, Méc. Phys. Astron., 326, 12, 899-904 (1998) · Zbl 0967.74076 |
| [7] | Ben Dhia, H., Local-global approaches. The Arlequin method, (5ème Colloque National de Calcul des Structures. 5ème Colloque National de Calcul des Structures, Giens 2005 (2005)), (archives ouvertes HAL: 00281019) |
| [8] | Bercovier, M.; Engelman, M., A finite element for the numerical solution of viscous incompressible flows, J. Comput. Phys., 30, 2, 181-201 (1979) · Zbl 0395.76040 |
| [9] | Bertsekas, D., Necessary and sufficient conditions for a penalty method to be exact, Math. Program., 9, 1, 87-99 (1975) · Zbl 0325.90055 |
| [10] | Borker, R.; Huang, D.; Grimberg, S.; Farhat, C.; Avery, P.; Rabinovitch, J., Mesh adaptation framework for embedded boundary methods for computational fluid dynamics and fluid-structure interaction, Int. J. Numer. Methods Fluids, 90, 8, 389-424 (July 2019) |
| [11] | Cea, J., Optimisation: Théorie et algorithmes (1968), Dunod: Dunod Paris · Zbl 0211.17402 |
| [12] | Coquerelle, M.; Cottet, G. H., A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies, J. Comput. Phys., 21, 227, 9121-9137 (2008) · Zbl 1146.76038 |
| [13] | Cottet, G. H.; Maitre, E., A level-set formulation of immersed boundary methods for fluid-structure interaction problems, C. R. Acad. Sci. Paris, 338, 51-56 (2004) |
| [14] | Cottet, G. H.; Gallizio, F.; Magni, A.; Mortazavi, I., A vortex penalization method for flows with moving immersed obstacles, (Proceedings of ASME-JSME-KSME Joint Fluids Engineering Conference. Proceedings of ASME-JSME-KSME Joint Fluids Engineering Conference, July, 2011 (2011), ASME), 3703-3708 |
| [15] | Destuynder, Ph., Aéroélasticitét Aéroacoustique (2008), Hermès-Lavoisier: Hermès-Lavoisier Paris-Londres · Zbl 1183.74002 |
| [16] | Destuynder, Ph., Aeroelasticity, (An Introduction to Quasi-Static Aeroelasticity. An Introduction to Quasi-Static Aeroelasticity, Lectures Notes (2020), Springer-Verlag: Springer-Verlag Berlin-New-York) |
| [17] | Ph. Destuynder, E. Liberge, Euler-Lagrange technics in fluid-structure modeling for the computation of forces, 2021, research report. |
| [18] | Duvaut, G.; Lions, J. L., Inéquations variationnelles en mécanique et en physique (1972), Dunod: Dunod Paris · Zbl 0298.73001 |
| [19] | Falaise, A.; Liberge, E., A penalty-duality strategy in a fluid structure problem (2021), La Rochelle University, Internal research report LaSIE |
| [20] | Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152, 457-492 (1999) · Zbl 0957.76052 |
| [21] | Formaggia, L.; Quarteroni, A.; Veneziani, A., Cardiovascular Mathematics; Modelling and Simulation of the Circulatory System, vol. 1 (2009), Springer-Verlag: Springer-Verlag Milano · Zbl 1300.92005 |
| [22] | Fortin, M.; Glowinski, R., Résolution numérique de problèmes aux limites par des méthodes de Lagrangien augmenté, Méthodes Mathématiques de l’informatique, vol. 9 (1982), Dunod: Dunod Paris · Zbl 0491.65036 |
| [23] | Fung, Y. C., An Introduction to the Theory of Aeroelasticity, Dover Books on Aeronautical Engineering (1969) · Zbl 1156.74001 |
| [24] | Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics (1986), Springer: Springer Berlin-New-York · Zbl 0585.65077 |
| [25] | Grisvard, P., Singularities in Boundary Value Problems (1992), Masson: Masson Paris · Zbl 0766.35001 |
| [26] | Hovnanian, J., Méthode de frontières immergées pour la mécanique des fluides. Application à la simulation de la nage (2012) |
| [27] | Huang, D. Z.; De Santis, D.; Farhat, C., A family of position- and orientation-independent embedded boundary methods for viscous flow and fluid-structure interaction problems, J. Comput. Phys., 365, 15, 74-104 (July 2018) · Zbl 1395.76044 |
| [28] | James, N.; Maitre, E.; Mortazavi, I., Immersed boundary methods for the numerical simulation of incompressible aerodynamic and fluid-structure interactions, Ann. Math. Blaise Pascal, 20, 1, 139-173 (2013) · Zbl 1426.76543 |
| [29] | Liberge, E.; Béghein, C., Méthode de Lattice Boltzmann couplée avec la pénalisation volumique pour la simulation de problèmes d’interaction fluide structure, (24^ème Congrès Français de Mécanique. 24^ème Congrès Français de Mécanique, Aug 2019, Brest, France (2019)) |
| [30] | Lions, J. L., Contrôlabilité exacte perturbations et stabilisation de systèmes distribués. T.1 (1988), Masson: Masson Paris, RMA n^8 · Zbl 0653.93002 |
| [31] | Lions, J. L.; Magenes, E., Problèmes aux limites non-homogènes et applications, vol. 1 (1968), Dunod: Dunod Paris; Non-Homogeneous Boundary Value Problems and Applications, vol. 1 (1972), Springer: Springer Berlin, and (English version) · Zbl 0165.10801 |
| [32] | Lions, J. L., Perturbations singulièresdans les problèmes aux limites et en contròle optimal, Lectures Notes in Mathematics, vol. 323 (1973), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0268.49001 |
| [33] | Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 1-27 (2005) |
| [34] | Peskin, C., The immersed boundary method, Acta Numer., 11, 1-39 (2002) |
| [35] | dos Santos, Nuno Diniz, Numerical methods for fluid-structure interaction problems with valves (2007), Thesis Université Pierre et Marie Curie - Paris VI, Paris |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
