# American Institute of Mathematical Sciences

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  March 2021 Early access  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 and 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. [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., [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., [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.
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
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 $\Gamma_1$
Density profile $u(s)$ for $s \in [0, \ell_1]$ along the ray $\Gamma_1$ for $c = 1$ and $\alpha = 2/3, 1/3$
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 $x$
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 ${{\bf n}}$.
After a rotation of coordinates, the sunlight comes from the vertical direction. Here the blue lines correspond to the set ${{\mathcal B}}^*$ in (36).
The construction used in the proof of Lemma 3.3.
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.
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.
A more general configuration, considered in Lemma 4.2.
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.
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] Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845 [2] Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations and Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011 [3] Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623 [4] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006 [5] Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich. Isogeometric shape optimization for nonlinear ultrasound focusing. Evolution Equations and Control Theory, 2019, 8 (1) : 163-202. doi: 10.3934/eect.2019010 [6] Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055 [7] Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625 [8] Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825 [9] Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069 [10] Lekbir Afraites, Marc Dambrine, Karsten Eppler, Djalil Kateb. Detecting perfectly insulated obstacles by shape optimization techniques of order two. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 389-416. doi: 10.3934/dcdsb.2007.8.389 [11] Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations and Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331 [12] Afaf Bouharguane, Pascal Azerad, Frédéric Bouchette, Fabien Marche, Bijan Mohammadi. Low complexity shape optimization & a posteriori high fidelity validation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 759-772. doi: 10.3934/dcdsb.2010.13.759 [13] Youness El Yazidi, Abdellatif ELLABIB. A new hybrid method for shape optimization with application to semiconductor equations. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021034 [14] John Sebastian Simon, Hirofumi Notsu. A shape optimization problem constrained with the Stokes equations to address maximization of vortices. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022003 [15] Julius Fergy T. Rabago, Jerico B. Bacani. Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2683-2702. doi: 10.3934/cpaa.2018127 [16] Andrew L. Nevai, Richard R. Vance. The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 101-124. doi: 10.3934/mbe.2008.5.101 [17] Xiaodong Fan, Tian Qin. Stability analysis for generalized semi-infinite optimization problems under functional perturbations. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1221-1233. doi: 10.3934/jimo.2018201 [18] Juan Su, Bing Xu, Lan Zou. Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6783-6815. doi: 10.3934/dcdsb.2019167 [19] Yacine Chitour, Frédéric Grognard, Georges Bastin. Equilibria and stability analysis of a branched metabolic network with feedback inhibition. Networks and Heterogeneous Media, 2006, 1 (1) : 219-239. doi: 10.3934/nhm.2006.1.219 [20] Amin Boumenir. Determining the shape of a solid of revolution. Mathematical Control and Related Fields, 2019, 9 (3) : 509-515. doi: 10.3934/mcrf.2019023

2021 Impact Factor: 1.41

## Tools

Article outline

Figures and Tables