March  2021, 16(1): 1-29. doi: 10.3934/nhm.2020031

A 2-dimensional shape optimization problem for tree branches

1. 

Department of Mathematics, Penn State University, University Park, PA 16802, USA

2. 

Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

* Corresponding author: Alberto Bressan

Received  June 2020 Revised  September 2020 Published  October 2020

The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.

Citation: Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2021, 16 (1) : 1-29. doi: 10.3934/nhm.2020031
References:
[1]

M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Springer Lecture Notes in Mathematics 1955, Berlin, 2009.  Google Scholar

[2]

M. BernotV. Caselles and J.-M. Morel, The structure of branched transportation networks, Calc. Var. Partial Differential Equations, 32 (2008), 279-317.  doi: 10.1007/s00526-007-0139-0.  Google Scholar

[3]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008.  Google Scholar

[4]

A. Brancolini and B. Wirth, Optimal energy scaling for micropatterns in transport networks, SIAM J. Math. Anal., 49 (2017), 311-359.  doi: 10.1137/15M1050227.  Google Scholar

[5]

L. Brasco and F. Santambrogio, An equivalent path functional formulation of branched transportation problems, Discrete Contin. Dyn. Syst., 29 (2011), 845-871.  doi: 10.3934/dcds.2011.29.845.  Google Scholar

[6]

A. BressanS. T. GaltungA. Reigstad and J. Ridder, Competition models for plant stems, J. Differential Equations, 269 (2020), 1571-1611.  doi: 10.1016/j.jde.2020.01.013.  Google Scholar

[7]

A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1.  Google Scholar

[8]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007.  Google Scholar

[9]

A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604.  Google Scholar

[10]

A. Bressan and Q. Sun, Weighted irrigation plans, submitted., Google Scholar

[11]

L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[12]

G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Adv. Nonlinear Stud., 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035.  Google Scholar

[13]

E. N. Gilbert, Minimum cost communication networks., Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x.  Google Scholar

[14]

F. MaddalenaS. Solimini and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85.  Google Scholar

[15]

J.-M. Morel and F. Santambrogio, The regularity of optimal irrigation patterns, Arch. Ration. Mech. Anal., 195 (2010), 499-531.  doi: 10.1007/s00205-008-0210-9.  Google Scholar

[16]

P. PegonF. Santambrogio and Q. Xia, A fractal shape optimization problem in branched transport, J. Math. Pures Appl., 123 (2019), 244-269.  doi: 10.1016/j.matpur.2018.06.007.  Google Scholar

[17]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.  doi: 10.4171/IFB/160.  Google Scholar

[18]

Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.  Google Scholar

[19]

Q. Xia, Interior regularity of optimal transport paths., Calc. Var. Partial Differential Equations, 20 (2004), 283-299.  doi: 10.1007/s00526-003-0237-6.  Google Scholar

[20]

Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.  Google Scholar

show all references

References:
[1]

M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Springer Lecture Notes in Mathematics 1955, Berlin, 2009.  Google Scholar

[2]

M. BernotV. Caselles and J.-M. Morel, The structure of branched transportation networks, Calc. Var. Partial Differential Equations, 32 (2008), 279-317.  doi: 10.1007/s00526-007-0139-0.  Google Scholar

[3]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008.  Google Scholar

[4]

A. Brancolini and B. Wirth, Optimal energy scaling for micropatterns in transport networks, SIAM J. Math. Anal., 49 (2017), 311-359.  doi: 10.1137/15M1050227.  Google Scholar

[5]

L. Brasco and F. Santambrogio, An equivalent path functional formulation of branched transportation problems, Discrete Contin. Dyn. Syst., 29 (2011), 845-871.  doi: 10.3934/dcds.2011.29.845.  Google Scholar

[6]

A. BressanS. T. GaltungA. Reigstad and J. Ridder, Competition models for plant stems, J. Differential Equations, 269 (2020), 1571-1611.  doi: 10.1016/j.jde.2020.01.013.  Google Scholar

[7]

A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1.  Google Scholar

[8]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007.  Google Scholar

[9]

A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604.  Google Scholar

[10]

A. Bressan and Q. Sun, Weighted irrigation plans, submitted., Google Scholar

[11]

L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[12]

G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Adv. Nonlinear Stud., 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035.  Google Scholar

[13]

E. N. Gilbert, Minimum cost communication networks., Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x.  Google Scholar

[14]

F. MaddalenaS. Solimini and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85.  Google Scholar

[15]

J.-M. Morel and F. Santambrogio, The regularity of optimal irrigation patterns, Arch. Ration. Mech. Anal., 195 (2010), 499-531.  doi: 10.1007/s00205-008-0210-9.  Google Scholar

[16]

P. PegonF. Santambrogio and Q. Xia, A fractal shape optimization problem in branched transport, J. Math. Pures Appl., 123 (2019), 244-269.  doi: 10.1016/j.matpur.2018.06.007.  Google Scholar

[17]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.  doi: 10.4171/IFB/160.  Google Scholar

[18]

Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.  Google Scholar

[19]

Q. Xia, Interior regularity of optimal transport paths., Calc. Var. Partial Differential Equations, 20 (2004), 283-299.  doi: 10.1007/s00526-003-0237-6.  Google Scholar

[20]

Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.  Google Scholar

Figure 1.  A stem $ \gamma_1 $ perpendicular to the sun rays is optimally shaped to collect the most light. For the stem $ \gamma_2 $ bending toward the light source, the upper leaves put the lower ones in shade
Figure 2.  When the light rays impinge from a fixed direction $ {{\bf n}} $, the optimal distribution of leaves is supported on the two rays $ \Gamma_0 $ and $ \Gamma_1 $
Figure 3.  Density profile $ u(s) $ for $ s \in [0, \ell_1] $ along the ray $ \Gamma_1 $ for $ c = 1 $ and $ \alpha = 2/3, 1/3 $
Figure 4.  According to the definition (31), the set $ \chi^-(x) $ is a curve joining the origin to the point $ x $. The set $ \chi^+(x) $ is a subtree, containing all paths that start from $ x $
Figure 5.  If the set $ \chi^+(x) $ is not contained in the slab $ \Gamma_x $ (the shaded region), by taking the perpendicular projections $ \pi^\sharp $ and $ \pi^\flat $ we obtain another irrigation plan with strictly lower cost, which irrigates a new measure $\tilde{\mu}$ gathering exactly the same amount of sunlight. Notice that here $ P $ is the point in the closed set $ \overline{\chi^+(x)}\cap{{\mathbb R}}{{\bf e}}_1 $ which has the largest inner product with $ {{\bf n}} $.
Figure 6.  After a rotation of coordinates, the sunlight comes from the vertical direction. Here the blue lines correspond to the set $ {{\mathcal B}}^* $ in (36).
Figure 7.  The construction used in the proof of Lemma 3.3.
Figure 8.  The thick portions of the curve $ \gamma $ are the only points where a left bifurcation can occur. If a horizontal branch $ \sigma $ bifurcates from $ C_j $, all the mass on this branch can be shifted downward to another branch $ \sigma^* $ bifurcating from $ C_j^* $. Furthermore, if some portion of the path $ \gamma $ between $ P^* $ and $ Q $ lies above the segment $ \gamma^* $ joining these two points, we can take a projection of $ \gamma $ on $ \gamma^* $. In both cases, the transportation cost is strictly reduced.
Figure 9.  Left: an irrigation plan for a measure $ \mu $ with two masses at $ Q $ and at $ P_1 $. Right: an irrigation plan for a modified measure $ \tilde{\mu} $ with two masses at $ \tilde{Q} $ and at $ P_1 $. The lengths of the segments $ PP^* $ and $ P^* P_1 $ will be denoted by $ \ell_a, \ell_b $, respectively.
Figure 10.  A more general configuration, considered in Lemma 4.2.
Figure 11.  Left: in the shaded region $ \Delta $ above the curve $ \gamma $, the measure $ \mu $ cannot concentrate any mass. Otherwise, by shifting this mass downward until it hits a point on $ \gamma $, we would obtain a second measure $ \tilde{\mu} $ which gathers the same amount of sunlight, but has a lower irrigation cost. As a consequence, by the interior regularity property, the flow out of $ P^* $ is locally supported on a finite number of line segments. Right: the construction used in steps 4–6 of the proof of Theorem 2.5.
Figure 12.  Changing the transport plan, when $ \theta_0>0 $ is very small. If in the irrigation problem the bifurcation angle satisfies $ \theta>\theta_0 +{\pi\over 2} $, then the original configuration, where all the mass is supported along $ \Gamma_0\cup\Gamma_1 $, would not be optimal. The analysis at (102) shows that this situation never happens.
[1]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017

[2]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[3]

Luigi Barletti, Giovanni Nastasi, Claudia Negulescu, Vittorio Romano. Mathematical modelling of charge transport in graphene heterojunctions. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021010

[4]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269

[5]

Jingni Guo, Junxiang Xu, Zhenggang He, Wei Liao. Research on cascading failure modes and attack strategies of multimodal transport network. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2020159

[6]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008

[7]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[8]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[9]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006

[10]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[11]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008

[12]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[13]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

2019 Impact Factor: 1.053

Article outline

Figures and Tables

[Back to Top]