# Optimal reinforcing networks for elastic membranes

## TL;DR

This paper investigates the optimal reinforcement of elastic membranes using networks, proving existence of solutions, including overlapping regions, and illustrating the complexity through numerical simulations.

## Contribution

It establishes the existence of optimal reinforcement networks with possible overlaps and demonstrates their complexity via numerical examples.

## Key findings

- Optimal networks may have overlapping regions.
- Existence of solutions is mathematically proven.
- Numerical simulations illustrate complex network structures.

## Abstract

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected onedimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when their total length becomes large.