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Artificial boundaries and formulations for the incompressible Navier-Stokes equations: applications to air and blood flows. (English) Zbl 1314.76022

This article deals with numerical simulations of the air flow in the respiratory system. The physical model consists of the incompressible Navier-Stokes equations in truncated domains with suitable boundary conditions on the artificial boundaries. The author uses the standard formulation for the momentum equation of the Navier-Stokes system based on the basic convective form for the advection term. Mixed Dirichlet-Neumann boundary conditions on each corner of the computational domain lead to numerical difficulties which are investigated throughout this paper.
The paper is organized in 4 sections. After the introduction with physical motivation, the author gives a theoretical overview where the basic notation is introduced and the tool model is described. Furthermore, the mathematical formulation of the tool problem is given with the corresponding weak formulation of the pressure drop problem. Moreover, the author refers to the existence and uniqueness results from the cited literature, which state that there exists a unique local-in-time solution for any data, and a unique global-in-time solution for small data. Using the basic form with natural boundary conditions implies the lack of energy conservation and, as a consequence, one deals with a restriction on the data. It is observed that if one uses the variational form which conserves the energy, existence theorems hold for less restrictive data. On the other side, changing the weak formulation also changes the associated boundary conditions and that can lead to the solutions which are not satisfactory from a physical point of view. Numerical tests are performed using the software package Felisce. When applying too high pressures, the author observes that the GMRES method treating nonlinearities does not converge and even leads to a blow up of the solution. Therefore, different stabilization methods, based on streamline diffusion or on direct handling of kinetic fluxes, are used and reviewed here in detail.

MSC:

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
76Z05 Physiological flows
92C35 Physiological flow

Software:

FELiScE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baffico, L., Grandmont, C., Maury, B.: Multiscale modeling of the respiratory tract. Math. Models Methods Appl. Sci. 20(1), 59–93 (2010) · Zbl 1423.76515
[2] Bardos, C., Bercovier, M., Pironneau, O.: The vortex method with finite elements. Math. Comput. 36(153), 119–136 (1981) · Zbl 0488.65049 · doi:10.1090/S0025-5718-1981-0595046-3
[3] Barth, W.L., Carey, G.F.: On a boundary condition for pressure-driven laminar flow of incompressible fluids. Int. J. Numer. Methods Fluids 54(11), 1313–1325 (2007) · Zbl 1116.76019 · doi:10.1002/fld.1427
[4] Bernard, J.M.: Time-dependent Stokes and Navier–Stokes problems with boundary conditions involving pressure, existence and regularity. Nonlinear Anal. Real World Appl. 4(5), 805–839 (2003) · Zbl 1037.35053 · doi:10.1016/S1468-1218(03)00016-6
[5] Billy, F., Ribba, B., Saut, O., Morre-Trouilhet, H., Colin, T., Bresch, D., Boissel, J.-P., Grenier, E., Flandrois, J.-P.: A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J. Theor. Biol. 260(4), 545–562 (2009) · Zbl 1402.92050 · doi:10.1016/j.jtbi.2009.06.026
[6] Boyer, F., Fabrie, P.: Outflow boundary conditions for the incompressible non-homogeneous Navier–Stokes equations. Discret. Contin. Dyn. Syst. Ser. B 7(2), 219–250 (2007) · Zbl 1388.76046
[7] Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 129–151 (1974)
[8] Bruneau, C.-H.: Boundary conditions on artificial frontiers for incompressible and compressible Navier–Stokes equations. ESAIM Math. Model. Numer. Anal. 34(02), 303–314 (2000) · Zbl 0954.76014 · doi:10.1051/m2an:2000142
[9] Bruneau, C.-H., Fabrie, P.: New efficient boundary conditions for incompressible Navier–Stokes equations : a well-posedness result. ESAIM Math. Model. Numer. Anal. 30(7), 815–840 (1996) · Zbl 0865.76016
[10] Bègue, C., Conca, C., Murat, F., Pironneau, O.: À nouveau sur les équations de Stokes et de Navier–Stokes avec des conditions aux limites sur la pression. Comptes Rendus des Séances de l’Académie des Sciences. Série I. Mathématique 304(1), 23–28 (1987) · Zbl 0613.76029
[11] Bègue, C., Conca, C., Murat, F., Pironneau, O.: Les équations de Stokes et de Navier–Stokes avec des conditions aux limites sur la pression. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. IX (Paris, 1985–1986), vol. 181 of Pitman Res. Notes Math. Ser., pp. 179–264. Longman Sci. Tech., Harlow (1988)
[12] Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22(104), 745–762 (1968) · Zbl 0198.50103 · doi:10.1090/S0025-5718-1968-0242392-2
[13] Clavica, F., Alastruey, J., Sherwin, S. J., Khir, A.W.: One-dimensional modelling of pulse wave propagation in human airway bifurcations in space-time variables. In: Engineering in Medicine and Biology Society, 2009. EMBC 2009. Annual International Conference of the IEEE, pp. 5482–5485 (2009)
[14] Colin, T., Iollo, A., Lombardi, D., Saut, O.: Prediction of the evolution of thyroidal lung nodules using a mathematical model. ERCIM News, Special issue ”Computational Biology” (82) (2010)
[15] Conca, C., Murat, F., Pironneau, O.: The Stokes and Navier–Stokes equations with boundary conditions involving the pressure. Jpn. J. Math. New Ser. 20(2), 279–318 (1994) · Zbl 0826.35093
[16] Conca, C., Parés, C., Pironneau, O., Thiriet, M.: Navier–Stokes equations with imposed pressure and velocity fluxes. Inte. J. Numer. Methods Fluids 20(4), 267–287 (1995) · Zbl 0831.76011 · doi:10.1002/fld.1650200402
[17] Egloffe, A.-C.: Étude de quelques problèmes inverses pour le système de Stokes. Application aux poumons. Ph.D. thesis, Université Pierre et Marie Curie (2012)
[18] Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, NY, USA (2005) · Zbl 1083.76001
[19] Ern, A., Guermond, J.-L.: Theory Pract. Finite Elem. Springer, New York (2004)
[20] Esmaily Moghadam, M., Bazilevs, Y., Hsia, T.-Y., Vignon-Clementel, I.E., Marsden, A.L.: A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Computat. Mech. 48(3), 277–291 (2011) · Zbl 1398.76102 · doi:10.1007/s00466-011-0599-0
[21] Esmaily Moghadam, M., Migliavacca, F., Vignon-Clementel, I.E., Hsia, T.-Y., Marsden, A.: Optimization of shunt placement for the norwood surgery using multi-domain modeling. J. Biomech. Eng. 134(5), 051002 (2012) · doi:10.1115/1.4006814
[22] Esmaily Moghadam, M., Vignon-Clementel, I.E., Figliola, R., Marsden, A.L.: A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations. J. Comput. Phys. 244(0), 63–79 (2012) · Zbl 1377.76041 · doi:10.1016/j.jcp.2012.07.035
[23] FELiScE.: INRIA forge project FELiScE: felisce.gforge.inria.fr (2013)
[24] Formaggia, L., Gerbeau, J.-F., Nobile, F., Quarteroni, A.: Numerical treatment of defective boundary conditions for the Navier–Stokes equations. SIAM J. Numer. Anal. 40(1), 376–401 (2002) · Zbl 1020.35070 · doi:10.1137/S003614290038296X
[25] Formaggia, L., Lamponi, D., Quarteroni, A.: One-dimensional models for blood flow in arteries. J. Eng. Math. 47(3–4), 251–276 (2003) · Zbl 1070.76059 · doi:10.1023/B:ENGI.0000007980.01347.29
[26] Franca, L.P., Frey, S.L.: Stabilized finite element methods: II. the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 99(2–3), 209–233 (1992) · Zbl 0765.76048 · doi:10.1016/0045-7825(92)90041-H
[27] Gemci, T., Ponyavin, V., Chen, Y., Chen, H., Collins, R.: Computational model of airflow in upper 17 generations of human respiratory tract. J. Biomech. 41(9), 2047–2054 (2008) · doi:10.1016/j.jbiomech.2007.12.019
[28] Gengenbach, T., Heuveline, V., Krause, M. J.: Numerical simulation of the human lung: a two-scale approach. EMCL Preprint Series, 11 (2011)
[29] Grandmont, C., Maday, Y., Maury, B.: A multiscale/multimodel approach of the respiration tree. In: New trends in continuum mechanics, vol. 3 of Theta Ser. Adv. Math., pp. 147–157. Theta, Bucharest (2005) · Zbl 1187.92030
[30] Grandmont, C., Maury, B., Soualah, A.: Multiscale modelling of the respiratory track: a theoretical framework. ESAIM Proc. 23, 10–29 (2008) · Zbl 1156.92304 · doi:10.1051/proc:082302
[31] Gravemeier, V., Comerford, A., Yoshihara, L., Ismail, M., Wall, W.A.: A novel formulation for Neumann inflow boundary conditions in biomechanics. Int. J. Numer. Methods Biomed. Eng. 28(5), 560–573 (2012) · Zbl 1243.92016 · doi:10.1002/cnm.1490
[32] Gresho, P.M.: Incompressible fluid dynamics: some fundamental formulation issues. Annu. Rev. Fluid Mech. 23(1), 413–453 (1991) · Zbl 0717.76006 · doi:10.1146/annurev.fl.23.010191.002213
[33] Gresho, P.M.: Some current CFD issues relevant to the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 87(2), 201–252 (1991) · Zbl 0760.76018 · doi:10.1016/0045-7825(91)90006-R
[34] Gresho, P.M., Sani, R.L.: On pressure boundary conditions for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 7(10), 1111–1145 (1987) · Zbl 0644.76025 · doi:10.1002/fld.1650071008
[35] Gresho, P. M., Sani, R.L.: Incompressible flow and the finite element method. In: Incompressible Flow and the Finite Element Method-Advection-Diffusion and Isothermal Laminar Flow. John Wiley and Sons (June 1998) · Zbl 0941.76002
[36] Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006) · Zbl 1122.76072 · doi:10.1016/j.cma.2005.10.010
[37] Gunzberger, M.D.: Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press Inc., San Diego (1989)
[38] Halpern, L., Schatzman, M.: Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal. 20(2), 308–353 (1989) · Zbl 0668.76048 · doi:10.1137/0520021
[39] Hannasch, D., Neda, M.: On the accuracy of the viscous form in simulations of incompressible flow problems. Numer. Methods Partial Differ. Equ. 28(2), 523–541 (2012) · Zbl 1410.76175 · doi:10.1002/num.20632
[40] Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 22(5), 325–352 (1996) · Zbl 0863.76016 · doi:10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y
[41] Hughes, T.J.R., Wells, G.N.: Conservation properties for the galerkin and stabilised forms of the advection-diffusion and incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 194(9–11), 1141–1159 (2005) · Zbl 1091.76035 · doi:10.1016/j.cma.2004.06.034
[42] Kim, H.J., Figueroa, C., Hughes, T.J.R., Jansen, K.E., Taylor, C.A.: Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundries for three-dimensional finite element simulations of blood flow. Comput. Methods Appl. Mech. Eng. 198(45–46), 3551–3566 (2009) · Zbl 1229.76118 · doi:10.1016/j.cma.2009.02.012
[43] Kuprat, A.P., Kabilan, S., Carson, J.P., Corley, R.A., Einstein, D.R.: A bidirectional coupling procedure applied to multiscale respiratory modeling. J. Comput. Phys. 244, 148–167 (2013) · Zbl 1310.76196 · doi:10.1016/j.jcp.2012.10.021
[44] Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Gordon and Breach (1969) · Zbl 0184.52603
[45] Ladyzhenskaya, O.A.: Mathematical analysis of Navier–Stokes equations for incompressible liquids. Annu. Rev. Fluid Mech. 7(1), 249–272 (1975) · Zbl 0344.76014 · doi:10.1146/annurev.fl.07.010175.001341
[46] Leone, J.M., Gresho, P.M.: Finite element simulations of steady, two-dimensional, viscous incompressible flow over a step. J. Comput. Phys. 41(1), 167–191 (1981) · Zbl 0464.76038 · doi:10.1016/0021-9991(81)90086-3
[47] Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’Hydrodynamique. Journal de Mathématiques Pures et Appliquées 12, 1–82 (1933) · Zbl 0006.16702
[48] Leray, J.: Essai sur les mouvements plans d’un fluide visqueux que limitent des parois. Journal de Mathématiques Pures et Appliquées 13, 331–418 (1934) · JFM 60.0727.01
[49] Ley, S., Mayer, D., Brook, B., van Beek, E., Heussel, C., Rinck, D., Hose, R., Markstaller, K., Kauczor, H.-U.: Radiological imaging as the basis for a simulation software of ventilation in the tracheo-bronchial tree. Eur. Radiol. 12(9), 2218–2228 (2002) · doi:10.1007/s00330-002-1391-5
[50] Limache, A.C., Sánchez, P.J., Dalcín, L.D., Idelsohn, S.R.: Objectivity tests for Navier–Stokes simulations: the revealing of non-physical solutions produced by Laplace formulations. Comput. Methods Appl. Mech. Eng. 197(49), 4180–4192 (2008) · Zbl 1194.76039 · doi:10.1016/j.cma.2008.04.020
[51] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (2002)
[52] Lions, J.-L., Prodi, G.: Un théorème d’existence et d’unicité dans les équations de Navier–Stokes en dimension 2. Comptes Rendus de l’Académie des Sciences de Paris Série I(248), 3519–3521 (1959) · Zbl 0091.42105
[53] Lombardi, D., Colin, T., Iollo, A., Saut, O., Bonichon, F., Palussière, J.: Some models for the prediction of tumor growth: general framework and applications to metastases in the lung. In: Garbey, M. et al. (eds.) Computational Surgery and Dual Training. Springer, New York (2014)
[54] Malvè, M., Chandra, S., López-Villalobos, J.L., Finol, E.A., Ginel, A., Doblaré, M.: CFD analysis of the human airways under impedance-based boundary conditions: application to healthy, diseased and stented trachea. Comput. Methods Biomech. Biomed. Eng. 16(2), 198–216 (2013)
[55] B. Maury. The respiratory system in equations, vol. 7 of MS&amp;A. Modeling, Simulation and Applications. Springer-Verlag Italia, Milan (2013) · Zbl 1312.92005
[56] Maury, B., Meunier, N., Soualah, A., Vial, L.: Outlet dissipative conditions for air flow in the bronchial tree. In: CEMRACS 2004-Mathematics and Applications to Biology and Medicine, vol. 14 of ESAIM Proceedings, pp. 201–212. EDP Sci., Les Ulis (2005) · Zbl 1075.92028
[57] Maz’ya, V., Rossmann, J.: Point estimates for green’s matrix to boundary value problems for second order elliptic systems in a polyhedral cone. J. Appl. Math. Mech. 82(5), 291–316 (2002) · Zbl 1136.35350
[58] Moshkin, N.P., Yambangwai, D.: On numerical solution of the incompressible Navier–Stokes equations with static or total pressure specified on boundaries. Math. Probl. Eng. 2009(1), 1–26 (2009) · Zbl 1207.76051
[59] Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numerische Mathematik 38(3), 309–332 (1982) · Zbl 0505.76100 · doi:10.1007/BF01396435
[60] Pironneau, O.: Boundary conditions on the pressure for the Stokes and the Navier–Stokes equations. Comptes Rendus de l’Académie des Sciences Série I Mathématiques 303(9), 403–406 (1986) · Zbl 0613.76028
[61] Pironneau, O.: Finite Element Methods for Fluids. Wiley, London (1990) · Zbl 0748.76003
[62] Porpora, A., Zunino, P., Vergara, C., Piccinelli, M.: Numerical treatment of boundary conditions to replace lateral branches in hemodynamics. Int. J. Numer. Methods Biomed. Eng. 28(12), 1165–1183 (2012) · doi:10.1002/cnm.2488
[63] Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations, 2nd edn. Springer, Berlin (2008) · Zbl 1151.65339
[64] Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems. In: Bank, R., Graham, R.L., Stoer, J., Varga, R., Yserentant, H. (eds.) Springer Series in Computational Mathematics, vol. 24, 2nd edn. Springer-Verlag, Berlin (2008)
[65] Sani, R.L., Gresho, P.M.: Résumé and remarks on the open boundary condition minisymposium. Int. J. Numer. Methods Fluids 18(10), 983–1008 (1994) · Zbl 0806.76072 · doi:10.1002/fld.1650181006
[66] Sapoval, B.: Smaller is better–but not too small: A physical scale for the design of the mammalian pulmonary acinus. Proc. Natl. Acad. Sci. 99(16), 10411–10416 (2002) · doi:10.1073/pnas.122352499
[67] Soualah-Alila, A.: Modélisation mathématique et numérique du poumon humain. Ph.D. thesis, Université Paris Sud (2007)
[68] Temam, R.: Une méthode d’approximation de la solution des équations de Navier–Stokes. Bulletin de la Société Mathématique de France 96, 115–152 (1968) · Zbl 0181.18903
[69] Temam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (II). Arch. Ration. Mech. Anal. 33(5), 377–385 (1969) · Zbl 0207.16904 · doi:10.1007/BF00247696
[70] Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, vol. 2. American Mathematical Society, AMS Chelsea Publishing, Providence, RI (2001) · Zbl 0981.35001
[71] Tezduyar, T.E., Mittal, S., Ray, S.E., Shih, R.: Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Methods Appl. Mech. Eng. 95(2), 221–242 (1992) · Zbl 0756.76048 · doi:10.1016/0045-7825(92)90141-6
[72] Tezduyar, T.E., Osawa, Y.: Finite element stabilization parameters computed from element matrices and vectors. Comput. Methods Appl. Mech. Eng. 190(3–4), 411–430 (2000) · Zbl 0973.76057 · doi:10.1016/S0045-7825(00)00211-5
[73] Tu, J., Inthavong, K., Ahmadi, G.: Computational Fluid and Particle Dynamics in the Human Respiratory System, 1st edn. Springer, Dordrecht (2012)
[74] Vergara, C.: Nitsche’s method for defective boundary value problems in incompressible fluid-dynamics. J. Sci. Comput. 46(1), 100–123 (2011) · Zbl 1237.65116 · doi:10.1007/s10915-010-9389-7
[75] Vignon-Clementel, I.E., Figueroa, C.A., Jansen, K.E., Taylor, C.A.: Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput. Methods Appl. Mech. Eng. 195(29–32), 3776–3796 (2006) · Zbl 1175.76098 · doi:10.1016/j.cma.2005.04.014
[76] Wall, W.A., Wiechert, L., Comerford, A., Rausch, S.: Towards a comprehensive computational model for the respiratory system. Int. J. Numer. Methods Biomed. Eng. 26(7), 807–827 (2010) · Zbl 1193.92068
[77] Weibel, E.R.: Morphometry of the Human Lung. Springer-Verlag, NY, USA (1963)
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