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

Adriano Festa; Simone Göttlich; Marion Pfirsching

doi:10.3934/nhm.2019016

Article Contents

2019, Volume 14, Issue 2: 389-410. Doi: 10.3934/nhm.2019016

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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

^* Corresponding author: Adriano Festa
^* Corresponding author: Adriano Festa

Received: September 30, 2018

Revised: December 31, 2018

Published: April 2019

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).

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Abstract

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.

Keywords:

Production systems,
conservation laws with discontinuous flux,
numerical simulations.

Mathematics Subject Classification: Primary: 90B30; Secondary: 35L65, 65M25.

Citation:

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Figure 1. A conveyor belt in a brewery. Image courtesy of Sidel Blowing & Services SAS

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Figure 2. Characteristics in the non-congested case

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Figure 3. Trajectories in the congested case

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Figure 4. Solution in the congested case: evolution of three shock waves

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Figure 5. Scheme of the two cases considered of one-to-two junction: passive (left) and active (right)

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Figure 6. Choice of the merging parameter $ q $

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Figure 7. Regularized flux function $ f_{ \xi, i} $

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Figure 8. Test 1: non-congested case with $ a_{{1}} = 1 $ and $ a_{{2}} = 2 $

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Figure 9. Test 2: congested case with $ a_{{1}} = 2 $ and $ a_{{2}} = 1 $

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Figure 10. Test 2: space-time diagram for the congested case

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Figure 11. Test 3: "passive" junction with distribution parameter $ \mu = 0.5 $

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Figure 12. Test 4: "active" junction with distribution parameter $ \mu = 0.5 $

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Figure 13. Test 5: merging junction with parameter $ q = 0.3 $

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Table 1. 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

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References

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