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
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 |
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.
Mathematics Subject Classification:
49Q10, 49Q20.
Citation:
Alberto Bressan, Sondre Tesdal Galtung.
A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, doi: 10.3934/nhm.2020031
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References:
| [1] |
M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Springer Lecture Notes in Mathematics 1955, Berlin, 2009. |
| [2] |
M. Bernot, V. 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. |
| [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. |
| [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. |
| [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. |
| [6] |
A. Bressan, S. T. Galtung, A. Reigstad and J. Ridder,
Competition models for plant stems, J. Differential Equations, 269 (2020), 1571-1611.
doi: 10.1016/j.jde.2020.01.013. |
| [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. |
| [8] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
F. Maddalena, S. Solimini and J.-M. Morel,
A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.
doi: 10.4171/IFB/85. |
| [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. |
| [16] |
P. Pegon, F. 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. |
| [17] |
F. Santambrogio,
Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.
doi: 10.4171/IFB/160. |
| [18] |
Q. Xia,
Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.
doi: 10.1142/S021919970300094X. |
| [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. |
| [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. |
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. |
| [2] |
M. Bernot, V. 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. |
| [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. |
| [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. |
| [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. |
| [6] |
A. Bressan, S. T. Galtung, A. Reigstad and J. Ridder,
Competition models for plant stems, J. Differential Equations, 269 (2020), 1571-1611.
doi: 10.1016/j.jde.2020.01.013. |
| [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. |
| [8] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
F. Maddalena, S. Solimini and J.-M. Morel,
A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.
doi: 10.4171/IFB/85. |
| [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. |
| [16] |
P. Pegon, F. 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. |
| [17] |
F. Santambrogio,
Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.
doi: 10.4171/IFB/160. |
| [18] |
Q. Xia,
Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.
doi: 10.1142/S021919970300094X. |
| [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. |
| [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. |
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.
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