Optimal reinforcing networks for elastic membranes

Giovanni Alberti; Giuseppe Buttazzo; Serena Guarino Lo Bianco; Édouard Oudet; Giovanni Alberti; Giuseppe Buttazzo; Serena Guarino Lo Bianco; Édouard Oudet

doi:10.3934/nhm.2019023

Networks and Heterogeneous Media

2019, Volume 14, Issue 3: 589-615. doi: 10.3934/nhm.2019023

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Optimal reinforcing networks for elastic membranes

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

Received: 01 August 2018 Revised: 01 March 2019
49J45, 49Q10, 35R35, 35J25, 49M05

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.
- Optimal networks,
- elastic membranes,
- reinforcement,
- relaxed solution,
- Golab's semicontinuity theorem
Citation: Giovanni Alberti, Giuseppe Buttazzo, Serena Guarino Lo Bianco, Édouard Oudet. Optimal reinforcing networks for elastic membranes[J]. Networks and Heterogeneous Media, 2019, 14(3): 589-615. doi: 10.3934/nhm.2019023

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Abstract

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.

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