# American Institute of Mathematical Sciences

June  2019, 14(2): 389-410. doi: 10.3934/nhm.2019016

## A model for a network of conveyor belts with discontinuous speed and capacity

 1 Institut National des sciences appliquées (INSA) Rouen, Laboratoire de Mathématiques, 685 Avenue de l'Université, 76800 Saint-Étienne-du-Rouvray, France 2 University of Mannheim, Department of Mathematics, A5-6, 68131 Mannheim, Germany

Received  October 2018 Revised  January 2019 Published  April 2019

Fund Project: This work was partially supported by the Haute-Normandie Regional Council via the M2NUM project and the project GO 1920/7-1 by the German Research Foundation (DFG)

We introduce a macroscopic model for a network of conveyor belts with various speeds and capacities. In a different way from traffic flow models, the product densities are forced to move with a constant velocity unless they reach a maximal capacity and start to queue. This kind of dynamics is governed by scalar conservation laws consisting of a discontinuous flux function. We define appropriate coupling conditions to get well-posed solutions at intersections and provide a detailed description of the solution. Some numerical simulations are presented to illustrate and confirm the theoretical results for different network configurations.

Citation: Adriano Festa, Simone Göttlich, Marion Pfirsching. A model for a network of conveyor belts with discontinuous speed and capacity. Networks & Heterogeneous Media, 2019, 14 (2) : 389-410. doi: 10.3934/nhm.2019016
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A conveyor belt in a brewery. Image courtesy of Sidel Blowing & Services SAS
Characteristics in the non-congested case
Trajectories in the congested case
Solution in the congested case: evolution of three shock waves
Scheme of the two cases considered of one-to-two junction: passive (left) and active (right)
Choice of the merging parameter $q$
Regularized flux function $f_{ \xi, i}$
Test 1: non-congested case with $a_{{1}} = 1$ and $a_{{2}} = 2$
Test 2: congested case with $a_{{1}} = 2$ and $a_{{2}} = 1$
Test 2: space-time diagram for the congested case
Test 3: "passive" junction with distribution parameter $\mu = 0.5$
Test 4: "active" junction with distribution parameter $\mu = 0.5$
Test 5: merging junction with parameter $q = 0.3$
Decreasing step sizes (left), decreasing smoothing parameter $\xi$ (right)
 $\Delta x$ $\Delta t$ error $\xi$ error 0.1 $2 \cdot 10^{-4}$ 0.0842 $5 \cdot 10^{-2}$ 0.0051 0.05 $\phantom{2 \cdot }10^{-4}$ 0.0381 $2 \cdot 10^{-2}$ 0.0042 0.01 $5 \cdot 10^{-5}$ 0.0184 $\phantom{1 \cdot} 10^{-2}$ 0.0039 0.01 $2 \cdot 10^{-5}$ 0.0073 $5 \cdot 10^{-3}$ 0.0037 0.005 $\phantom{2 \cdot }10^{-5}$ 0.0057 $2 \cdot 10^{-3}$ 0.0035
 $\Delta x$ $\Delta t$ error $\xi$ error 0.1 $2 \cdot 10^{-4}$ 0.0842 $5 \cdot 10^{-2}$ 0.0051 0.05 $\phantom{2 \cdot }10^{-4}$ 0.0381 $2 \cdot 10^{-2}$ 0.0042 0.01 $5 \cdot 10^{-5}$ 0.0184 $\phantom{1 \cdot} 10^{-2}$ 0.0039 0.01 $2 \cdot 10^{-5}$ 0.0073 $5 \cdot 10^{-3}$ 0.0037 0.005 $\phantom{2 \cdot }10^{-5}$ 0.0057 $2 \cdot 10^{-3}$ 0.0035
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