American Institute of Mathematical Sciences

October  2011, 16(3): 767-799. doi: 10.3934/dcdsb.2011.16.767

Shape minimization of the dissipated energy in dyadic trees

 1 Supélec, Plateau de Moulon, 91192 Gif-sur-Yvette, France, Université d’Orléans, Laboratoire MAPMO, CNRS, UMR 6628, Fédération Denis Poisson, FR 2964, Bat. Math., BP 6759, 45067 Orléans cedex 2, France 2 Laboratoire MSC, Université Paris 7, CNRS, 10 rue Alice Domon et Léonie Duguet, F-75205 Paris cedex 13, France 3 ENS Cachan Bretagne, CNRS, Univ. Rennes 1, IRMAR, av. Robert Schuman, F-35170 Bruz, France

Received  July 2010 Revised  September 2010 Published  June 2011

In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks.
Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number $100$) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree.
Citation: Xavier Dubois de La Sablonière, Benjamin Mauroy, Yannick Privat. Shape minimization of the dissipated energy in dyadic trees. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 767-799. doi: 10.3934/dcdsb.2011.16.767
References:
 [1] G. Allaire, "Conception optimale de structures,'' Mathématiques & Applications, 58, Springer-Verlag, Berlin, 2007. [2] A. Bejan, "Shape and Structure, From Engineering to Nature,'' Cambridge University Press, Cambridge, UK, 2000. [3] M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks: Models and Theory,'' Lecture Notes in Mathematics (vol. 1955), Springer, 2008. [4] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces and Free Boundaries, 5 (2003), 301-329. doi: 10.4171/IFB/81. [5] F. Boyer and P. Fabrie, "Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles,'' Mathématiques & Applications, 52, Springer, Berlin, 2006. [6] C.-H. Bruneau and P. Fabrie, Effective downstream boundary conditions for incompressible Navier-Stokes equations, Int. J. for Num. Methods in Fluids, 19 (1994), 693-705. doi: 10.1002/fld.1650190805. [7] C.-H. Bruneau and P. Fabrie, New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result, RAIRO Modél. Math. Anal. Numér., 30 (1996), 815-840. [8] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-289. doi: 10.1016/0022-247X(75)90091-8. [9] M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,'' Advances in Design and Control SIAM, Philadelphia, PA, 2001. [10] G. Dogǧan, P. Morin, R. H. Nochetto and M. Verani, Discrete Gradient Flows for Shape Optimization and Applications, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3898-3914. doi: 10.1016/j.cma.2006.10.046. [11] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'' Volumes 1 and 2, Springer Tracts in Natural Philosophy , Vol. 38, 1998. [12] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45 (2006), 343-367. doi: 10.1137/050624108. [13] A. Henrot and M. Pierre, "Variation et Optimisation de forme,'' Mathématiques et Applications, vol. 48, Springer 2005. [14] A. Henrot and Y. Privat, Une conduite cylindrique n'est pas optimale pour minimiser l'énergie dissipée par un fluide, C. R. Math. Acad. Sci. Paris, 346 (2008), 1057-1061. [15] A. Henrot and Y. Privat, What is the optimal shape of a pipe?, Arch. Ration. Mech. Anal., 196 (2010), 281-302. doi: 10.1007/s00205-009-0243-8. [16] W. R. Hess, Das Prinzip des kleinsten Kraftverbrauchs im Dienste h amodynamischer Forschung, Archiv. Anat. Physiol., (1914). [17] J. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 22 (1996), 5. doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. [18] B. Mauroy, 3D Hydronamics in the upper human bronchial tree: interplay between geometry and flow distribution, in "Fractals in Biology and Medicine,'' IV, Birkhauser, (2005). doi: 10.1007/3-7643-7412-8_4. [19] B. Mauroy, M. Filoche, J. S. Andrade and B. Sapoval, Interplay between geometry and flow distribution in an airway tree, Physical Review Letters, 90 (2003), 1-4. doi: 10.1103/PhysRevLett.90.148101. [20] B. Mauroy, M. Filoche, E. R. Weibel and B. Sapoval, An optimal bronchial tree may be dangerous, Nature, 427 (2004), 633-636. doi: 10.1038/nature02287. [21] B. Mauroy and N. Meunier, Optimal Poiseuille flow in a finite elastic dyadic tree, M2AN Math. Model. Numer. Anal., 42 (2008), 507-533. doi: 10.1051/m2an:2008015. [22] B. Maury, N. Meunier, A. Soualah and L. Vial, Outlet dissipative conditions for air flow in the bronchial tree, in "CEMRACS 2004-Mathematics and Applications to Biology and Medicine,'' ESAIM Proc., 14, 201-212, EDP Sci., Les Ulis, (2005). [23] B. Mohammadi and O. Pironneau, "Applied Shape Optimization for Fluids,'' Clarendon Press, Oxford, 2001. [24] F. Murat and J. Simon, Sur le contrôle par un domaine géométrique, Publication du Laboratoire d'Analyse Numérique de l'Université Paris 6, 189 (1976). [25] O. Pironneau, "Optimal Shape Design for Elliptic Systems,'' Springer-Verlag, New York, 1984. [26] B. Protas, T-R Bewley, and G. Hagen, A computational framework for the regularization of adjoint analysis in multiscale PDE systems, J. Comput. Phys., 195 (2004), 49-89. doi: 10.1016/j.jcp.2003.08.031. [27] A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Model. Simul., 1 (2003). [28] M. Raux, M.N. Fiamma, T. Similowski and C. Straus, Contrôle de la ventilation : physiologie et exploration en réanimation, Réanimation, 16 (2007). [29] J. Bello and E. Fernández-Cara, Optimal shape design for Navier-Stokes flow, System modelling and optimization, P. Kall d., Lecture Notes in Control and Inform. Sci., 180 (1992), 481-489. [30] J. Bello and E. Fernández-Cara, The variation of the drag with respect to the domain in Navier-Stokes flow, Optimization, optimal control, partial differential equations, International Series of Numerical Mathematics, 107 (1992), 287-296. [31] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization: Shape Sensitivity Analysis,'' Springer Series in Computational Mathematics, Vol. 16, Springer, Berlin, 1992. [32] R. Temam, "Navier-Stokes Equations,'' North-Holland Pub. Company, 1979. [33] D. Tondeur and L. Luo, Design and scaling laws of ramified fluid distributors by the constructal approach, Chem. Eng. Sci., 59 (2004), 1799-1813. doi: 10.1016/j.ces.2004.01.034. [34] E.R. Weibel, "The Pathway for Oxygen,'' Harvard University Press, Cambridge M A, 1984.

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References:
 [1] G. Allaire, "Conception optimale de structures,'' Mathématiques & Applications, 58, Springer-Verlag, Berlin, 2007. [2] A. Bejan, "Shape and Structure, From Engineering to Nature,'' Cambridge University Press, Cambridge, UK, 2000. [3] M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks: Models and Theory,'' Lecture Notes in Mathematics (vol. 1955), Springer, 2008. [4] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces and Free Boundaries, 5 (2003), 301-329. doi: 10.4171/IFB/81. [5] F. Boyer and P. Fabrie, "Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles,'' Mathématiques & Applications, 52, Springer, Berlin, 2006. [6] C.-H. Bruneau and P. Fabrie, Effective downstream boundary conditions for incompressible Navier-Stokes equations, Int. J. for Num. Methods in Fluids, 19 (1994), 693-705. doi: 10.1002/fld.1650190805. [7] C.-H. Bruneau and P. Fabrie, New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result, RAIRO Modél. Math. Anal. Numér., 30 (1996), 815-840. [8] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-289. doi: 10.1016/0022-247X(75)90091-8. [9] M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,'' Advances in Design and Control SIAM, Philadelphia, PA, 2001. [10] G. Dogǧan, P. Morin, R. H. Nochetto and M. Verani, Discrete Gradient Flows for Shape Optimization and Applications, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3898-3914. doi: 10.1016/j.cma.2006.10.046. [11] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'' Volumes 1 and 2, Springer Tracts in Natural Philosophy , Vol. 38, 1998. [12] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45 (2006), 343-367. doi: 10.1137/050624108. [13] A. Henrot and M. Pierre, "Variation et Optimisation de forme,'' Mathématiques et Applications, vol. 48, Springer 2005. [14] A. Henrot and Y. Privat, Une conduite cylindrique n'est pas optimale pour minimiser l'énergie dissipée par un fluide, C. R. Math. Acad. Sci. Paris, 346 (2008), 1057-1061. [15] A. Henrot and Y. Privat, What is the optimal shape of a pipe?, Arch. Ration. Mech. Anal., 196 (2010), 281-302. doi: 10.1007/s00205-009-0243-8. [16] W. R. Hess, Das Prinzip des kleinsten Kraftverbrauchs im Dienste h amodynamischer Forschung, Archiv. Anat. Physiol., (1914). [17] J. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 22 (1996), 5. doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. [18] B. Mauroy, 3D Hydronamics in the upper human bronchial tree: interplay between geometry and flow distribution, in "Fractals in Biology and Medicine,'' IV, Birkhauser, (2005). doi: 10.1007/3-7643-7412-8_4. [19] B. Mauroy, M. Filoche, J. S. Andrade and B. Sapoval, Interplay between geometry and flow distribution in an airway tree, Physical Review Letters, 90 (2003), 1-4. doi: 10.1103/PhysRevLett.90.148101. [20] B. Mauroy, M. Filoche, E. R. Weibel and B. Sapoval, An optimal bronchial tree may be dangerous, Nature, 427 (2004), 633-636. doi: 10.1038/nature02287. [21] B. Mauroy and N. Meunier, Optimal Poiseuille flow in a finite elastic dyadic tree, M2AN Math. Model. Numer. Anal., 42 (2008), 507-533. doi: 10.1051/m2an:2008015. [22] B. Maury, N. Meunier, A. Soualah and L. Vial, Outlet dissipative conditions for air flow in the bronchial tree, in "CEMRACS 2004-Mathematics and Applications to Biology and Medicine,'' ESAIM Proc., 14, 201-212, EDP Sci., Les Ulis, (2005). [23] B. Mohammadi and O. Pironneau, "Applied Shape Optimization for Fluids,'' Clarendon Press, Oxford, 2001. [24] F. Murat and J. Simon, Sur le contrôle par un domaine géométrique, Publication du Laboratoire d'Analyse Numérique de l'Université Paris 6, 189 (1976). [25] O. Pironneau, "Optimal Shape Design for Elliptic Systems,'' Springer-Verlag, New York, 1984. [26] B. Protas, T-R Bewley, and G. Hagen, A computational framework for the regularization of adjoint analysis in multiscale PDE systems, J. Comput. Phys., 195 (2004), 49-89. doi: 10.1016/j.jcp.2003.08.031. [27] A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Model. Simul., 1 (2003). [28] M. Raux, M.N. Fiamma, T. Similowski and C. Straus, Contrôle de la ventilation : physiologie et exploration en réanimation, Réanimation, 16 (2007). [29] J. Bello and E. Fernández-Cara, Optimal shape design for Navier-Stokes flow, System modelling and optimization, P. Kall d., Lecture Notes in Control and Inform. Sci., 180 (1992), 481-489. [30] J. Bello and E. Fernández-Cara, The variation of the drag with respect to the domain in Navier-Stokes flow, Optimization, optimal control, partial differential equations, International Series of Numerical Mathematics, 107 (1992), 287-296. [31] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization: Shape Sensitivity Analysis,'' Springer Series in Computational Mathematics, Vol. 16, Springer, Berlin, 1992. [32] R. Temam, "Navier-Stokes Equations,'' North-Holland Pub. Company, 1979. [33] D. Tondeur and L. Luo, Design and scaling laws of ramified fluid distributors by the constructal approach, Chem. Eng. Sci., 59 (2004), 1799-1813. doi: 10.1016/j.ces.2004.01.034. [34] E.R. Weibel, "The Pathway for Oxygen,'' Harvard University Press, Cambridge M A, 1984.
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