# American Institute of Mathematical Sciences

September  2019, 14(3): 589-615. doi: 10.3934/nhm.2019023

## 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

* Corresponding author

Received  August 2018 Revised  March 2019 Published  May 2019

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.

Citation: Giovanni Alberti, Giuseppe Buttazzo, Serena Guarino Lo Bianco, Édouard Oudet. Optimal reinforcing networks for elastic membranes. Networks & Heterogeneous Media, 2019, 14 (3) : 589-615. doi: 10.3934/nhm.2019023
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##### References:
Approximation of globally optimal reinforcement structures for $m = 0.5$, $L = 1, \, 2$ and $3$. The upper colorbar is related to the weights $\theta$ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
Approximation of globaly optimal reinforcement structures for $m = 0.5$, $L = 4, \, 5$ and $6$. The upper colorbar is related to the weights $\theta$ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
Approximation of globaly optimal reinforcement strucutres for m = 0.5, L = 1.5, 2.5 and 5 for a source consisting of two dirac masses. The upper colorbar is related to the weights θ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
Reinforcement values computed on a fine mesh of $10^6$ elements for classical and computed connected sets for $m = 0.5$
 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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