# 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 & 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 (2007). Google Scholar [2] A. Bejan, "Shape and Structure, From Engineering to Nature,'', Cambridge University Press, (2000). Google Scholar [3] M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks: Models and Theory,'', Lecture Notes in Mathematics (vol. 1955), 1955 (2008). Google Scholar [4] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction,, Interfaces and Free Boundaries, 5 (2003), 301. doi: 10.4171/IFB/81. Google Scholar [5] F. Boyer and P. Fabrie, "Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles,'', Mathématiques & Applications, 52 (2006). Google Scholar [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. doi: 10.1002/fld.1650190805. 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Google Scholar [12] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization,, SIAM J. Control Optim., 45 (2006), 343. doi: 10.1137/050624108. Google Scholar [13] A. Henrot and M. Pierre, "Variation et Optimisation de forme,'', Mathématiques et Applications, 48 (2005). Google Scholar [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. Google Scholar [15] A. Henrot and Y. Privat, What is the optimal shape of a pipe?,, Arch. Ration. Mech. Anal., 196 (2010), 281. doi: 10.1007/s00205-009-0243-8. Google Scholar [16] W. R. Hess, Das Prinzip des kleinsten Kraftverbrauchs im Dienste h amodynamischer Forschung,, Archiv. Anat. Physiol., (1914). Google Scholar [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). doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. Google Scholar [18] B. Mauroy, 3D Hydronamics in the upper human bronchial tree: interplay between geometry and flow distribution,, in, (2005). doi: 10.1007/3-7643-7412-8_4. Google Scholar [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. doi: 10.1103/PhysRevLett.90.148101. Google Scholar [20] B. Mauroy, M. Filoche, E. R. Weibel and B. Sapoval, An optimal bronchial tree may be dangerous,, Nature, 427 (2004), 633. doi: 10.1038/nature02287. Google Scholar [21] B. Mauroy and N. Meunier, Optimal Poiseuille flow in a finite elastic dyadic tree,, M2AN Math. Model. Numer. Anal., 42 (2008), 507. doi: 10.1051/m2an:2008015. Google Scholar [22] B. Maury, N. Meunier, A. Soualah and L. Vial, Outlet dissipative conditions for air flow in the bronchial tree,, in, (2005), 201. Google Scholar [23] B. Mohammadi and O. Pironneau, "Applied Shape Optimization for Fluids,'', Clarendon Press, (2001). Google Scholar [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). Google Scholar [25] O. Pironneau, "Optimal Shape Design for Elliptic Systems,'', Springer-Verlag, (1984). Google Scholar [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. doi: 10.1016/j.jcp.2003.08.031. Google Scholar [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). Google Scholar [28] M. Raux, M.N. Fiamma, T. Similowski and C. Straus, Contrôle de la ventilation : physiologie et exploration en réanimation,, Réanimation, 16 (2007). Google Scholar [29] J. Bello and E. Fernández-Cara, Optimal shape design for Navier-Stokes flow,, System modelling and optimization, 180 (1992), 481. Google Scholar [30] J. Bello and E. Fernández-Cara, The variation of the drag with respect to the domain in Navier-Stokes flow,, Optimization, 107 (1992), 287. Google Scholar [31] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization: Shape Sensitivity Analysis,'', Springer Series in Computational Mathematics, 16 (1992). Google Scholar [32] R. Temam, "Navier-Stokes Equations,'', North-Holland Pub. Company, (1979). Google Scholar [33] D. Tondeur and L. Luo, Design and scaling laws of ramified fluid distributors by the constructal approach,, Chem. Eng. Sci., 59 (2004), 1799. doi: 10.1016/j.ces.2004.01.034. Google Scholar [34] E.R. Weibel, "The Pathway for Oxygen,'', Harvard University Press, (1984). Google Scholar

show all references

##### References:
 [1] G. Allaire, "Conception optimale de structures,'', Mathématiques & Applications, 58 (2007). Google Scholar [2] A. Bejan, "Shape and Structure, From Engineering to Nature,'', Cambridge University Press, (2000). Google Scholar [3] M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks: Models and Theory,'', Lecture Notes in Mathematics (vol. 1955), 1955 (2008). Google Scholar [4] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction,, Interfaces and Free Boundaries, 5 (2003), 301. doi: 10.4171/IFB/81. Google Scholar [5] F. Boyer and P. Fabrie, "Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles,'', Mathématiques & Applications, 52 (2006). Google Scholar [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. doi: 10.1002/fld.1650190805. Google Scholar [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. Google Scholar [8] D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. doi: 10.1016/0022-247X(75)90091-8. Google Scholar [9] M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,'', Advances in Design and Control SIAM, (2001). Google Scholar [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. doi: 10.1016/j.cma.2006.10.046. Google Scholar [11] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'', Volumes 1 and 2, 38 (1998). Google Scholar [12] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization,, SIAM J. Control Optim., 45 (2006), 343. doi: 10.1137/050624108. Google Scholar [13] A. Henrot and M. Pierre, "Variation et Optimisation de forme,'', Mathématiques et Applications, 48 (2005). Google Scholar [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. Google Scholar [15] A. Henrot and Y. Privat, What is the optimal shape of a pipe?,, Arch. Ration. Mech. Anal., 196 (2010), 281. doi: 10.1007/s00205-009-0243-8. Google Scholar [16] W. R. Hess, Das Prinzip des kleinsten Kraftverbrauchs im Dienste h amodynamischer Forschung,, Archiv. Anat. Physiol., (1914). Google Scholar [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). doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. Google Scholar [18] B. Mauroy, 3D Hydronamics in the upper human bronchial tree: interplay between geometry and flow distribution,, in, (2005). doi: 10.1007/3-7643-7412-8_4. Google Scholar [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. doi: 10.1103/PhysRevLett.90.148101. Google Scholar [20] B. Mauroy, M. Filoche, E. R. Weibel and B. Sapoval, An optimal bronchial tree may be dangerous,, Nature, 427 (2004), 633. doi: 10.1038/nature02287. Google Scholar [21] B. Mauroy and N. Meunier, Optimal Poiseuille flow in a finite elastic dyadic tree,, M2AN Math. Model. Numer. Anal., 42 (2008), 507. doi: 10.1051/m2an:2008015. Google Scholar [22] B. Maury, N. Meunier, A. Soualah and L. Vial, Outlet dissipative conditions for air flow in the bronchial tree,, in, (2005), 201. Google Scholar [23] B. Mohammadi and O. Pironneau, "Applied Shape Optimization for Fluids,'', Clarendon Press, (2001). Google Scholar [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). Google Scholar [25] O. Pironneau, "Optimal Shape Design for Elliptic Systems,'', Springer-Verlag, (1984). Google Scholar [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. doi: 10.1016/j.jcp.2003.08.031. Google Scholar [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). Google Scholar [28] M. Raux, M.N. Fiamma, T. Similowski and C. Straus, Contrôle de la ventilation : physiologie et exploration en réanimation,, Réanimation, 16 (2007). Google Scholar [29] J. Bello and E. Fernández-Cara, Optimal shape design for Navier-Stokes flow,, System modelling and optimization, 180 (1992), 481. Google Scholar [30] J. Bello and E. Fernández-Cara, The variation of the drag with respect to the domain in Navier-Stokes flow,, Optimization, 107 (1992), 287. Google Scholar [31] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization: Shape Sensitivity Analysis,'', Springer Series in Computational Mathematics, 16 (1992). Google Scholar [32] R. Temam, "Navier-Stokes Equations,'', North-Holland Pub. Company, (1979). Google Scholar [33] D. Tondeur and L. Luo, Design and scaling laws of ramified fluid distributors by the constructal approach,, Chem. Eng. Sci., 59 (2004), 1799. doi: 10.1016/j.ces.2004.01.034. Google Scholar [34] E.R. Weibel, "The Pathway for Oxygen,'', Harvard University Press, (1984). Google Scholar
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