| 1. | Dipartimento di Matematica, Università di Pisa, l.go B. Pontecorvo 5, 56127 Pisa, Italy |
| 2. | Dipartimento di Matematica e an, 80126 Napoli, Italy |
| 3. | Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 38041 Grenoble, France |
In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal network, and observe that such network carries a multiplicity that in principle can be strictly larger than one. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when the total length becomes large.
| [1] |
G. Alberti and M. Ottolini,
On the structure of continua with finite length and Golab's semicontinuity theorem, Nonlinear Anal., 153 (2017), 35-55.
doi: 10.1016/j.na.2016.10.012. |
| [2] |
E. Acerbi, G. Buttazzo and D. Percivale,
Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math., 386 (1988), 99-115.
doi: 10.1515/crll.1988.386.99. |
| [3] |
M. Beckmann,
A continuous model of transportation, Econometrica, 20 (1952), 643-660.
doi: 10.2307/1907646. |
| [4] |
G. Bouchitté, G. Buttazzo and P. Seppecher,
Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations, 5 (1996), 37-54.
doi: 10.1007/s005260050058. |
| [5] |
L. Brasco, G. Carlier and F. Santambrogio,
Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671.
doi: 10.1016/j.matpur.2010.03.010. |
| [6] |
G. Buttazzo, G. Carlier and S. Guarino Lo Bianco,
Optimal regions for congested transport, ESAIM Math. Model. Numer. Anal., 49 (2015), 1607-1619.
doi: 10.1051/m2an/2015022. |
| [7] |
G. Buttazzo, É. Oudet and B. Velichkov,
A free boundary problem arising in PDE optimization, Calc. Var. Partial Differential Equations, 54 (2015), 3829-3856.
doi: 10.1007/s00526-015-0923-1. |
| [8] |
G. Buttazzo, É. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002, 41–65. |
| [9] |
G. Buttazzo and F. Santambrogio,
Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777.
doi: 10.3934/nhm.2007.2.761. |
| [10] |
G. Buttazzo, F. Santambrogio and N. Varchon,
Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769.
doi: 10.1051/cocv:2006020. |
| [11] |
G. Buttazzo and N. Varchon,
On the optimal reinforcement of an elastic membrane, Riv. Mat. Univ. Parma (Ser. 7), 4 (2005), 115-125.
|
| [12] |
Y.-H. Dai and R. Fletcher,
New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. (Ser. A), 106 (2006), 403-421.
doi: 10.1007/s10107-005-0595-2. |
| [13] |
K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.
|
| [14] |
S. Golab,
Sur quelques points de la théorie de la longueur, Ann. Soc. Polon. Math., 7 (1929), 227-241.
|
| [15] |
S. G. Johnson, The NLopt nonlinear-optimization package., Available from: http://ab-initio.mit.edu/nlopt. |
| [16] |
S. J. N. Mosconi and P. Tilli,
Γ-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.
|
| [17] |
E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980. |
| [18] |
J. G. Wardrop,
Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, 1 (1952), 325-362.
doi: 10.1680/ipeds.1952.11362. |
show all references
| [1] |
G. Alberti and M. Ottolini,
On the structure of continua with finite length and Golab's semicontinuity theorem, Nonlinear Anal., 153 (2017), 35-55.
doi: 10.1016/j.na.2016.10.012. |
| [2] |
E. Acerbi, G. Buttazzo and D. Percivale,
Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math., 386 (1988), 99-115.
doi: 10.1515/crll.1988.386.99. |
| [3] |
M. Beckmann,
A continuous model of transportation, Econometrica, 20 (1952), 643-660.
doi: 10.2307/1907646. |
| [4] |
G. Bouchitté, G. Buttazzo and P. Seppecher,
Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations, 5 (1996), 37-54.
doi: 10.1007/s005260050058. |
| [5] |
L. Brasco, G. Carlier and F. Santambrogio,
Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671.
doi: 10.1016/j.matpur.2010.03.010. |
| [6] |
G. Buttazzo, G. Carlier and S. Guarino Lo Bianco,
Optimal regions for congested transport, ESAIM Math. Model. Numer. Anal., 49 (2015), 1607-1619.
doi: 10.1051/m2an/2015022. |
| [7] |
G. Buttazzo, É. Oudet and B. Velichkov,
A free boundary problem arising in PDE optimization, Calc. Var. Partial Differential Equations, 54 (2015), 3829-3856.
doi: 10.1007/s00526-015-0923-1. |
| [8] |
G. Buttazzo, É. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002, 41–65. |
| [9] |
G. Buttazzo and F. Santambrogio,
Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777.
doi: 10.3934/nhm.2007.2.761. |
| [10] |
G. Buttazzo, F. Santambrogio and N. Varchon,
Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769.
doi: 10.1051/cocv:2006020. |
| [11] |
G. Buttazzo and N. Varchon,
On the optimal reinforcement of an elastic membrane, Riv. Mat. Univ. Parma (Ser. 7), 4 (2005), 115-125.
|
| [12] |
Y.-H. Dai and R. Fletcher,
New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. (Ser. A), 106 (2006), 403-421.
doi: 10.1007/s10107-005-0595-2. |
| [13] |
K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.
|
| [14] |
S. Golab,
Sur quelques points de la théorie de la longueur, Ann. Soc. Polon. Math., 7 (1929), 227-241.
|
| [15] |
S. G. Johnson, The NLopt nonlinear-optimization package., Available from: http://ab-initio.mit.edu/nlopt. |
| [16] |
S. J. N. Mosconi and P. Tilli,
Γ-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.
|
| [17] |
E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980. |
| [18] |
J. G. Wardrop,
Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, 1 (1952), 325-362.
doi: 10.1680/ipeds.1952.11362. |
| Length constraint | Theoretical guesses | Computed optimal networks |
| 1 | -0.179471 (radius) | -0.178873 |
| 2 | -0.165095 (diameter) | -0.161944 |
| 3 | -0.152676 (star) | -0.149601 |
| 4 | -0.141969 (cross) | -0.138076 |
| 5 | - | -0.127661 |
| 6 | - | -0.117140 |
| Length constraint | Theoretical guesses | Computed optimal networks |
| 1 | -0.179471 (radius) | -0.178873 |
| 2 | -0.165095 (diameter) | -0.161944 |
| 3 | -0.152676 (star) | -0.149601 |
| 4 | -0.141969 (cross) | -0.138076 |
| 5 | - | -0.127661 |
| 6 | - | -0.117140 |
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