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

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

Adriano Festa 1,, , Simone Göttlich 2, and  Marion Pfirsching 2,

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

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.

Keywords: Production systems, conservation laws with discontinuous flux, numerical simulations.
Mathematics Subject Classification: Primary: 90B30; Secondary: 35L65, 65M25.
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
References:
[1]

D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math., 71 (2011), 1070-1087.  doi: 10.1137/100809374.  Google Scholar

[2]

F. Camilli, A. Festa and S. Tozza, A discrete hughes model for pedestrian flow on graphs, Netw. Heterog. Media, 12 (2017), 93-112.  doi: 10.3934/nhm.2017004.  Google Scholar

[3]

C. d'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, 2010. doi: 10.1137/1.9780898717600.  Google Scholar

[4]

J.-P. Dias and M. Figueira, On the riemann problem for some discontinuous systems of conservation laws describing phase transitions, Commun. Pure Appl. Math., 3 (2004), 53-58.  doi: 10.3934/cpaa.2004.3.53.  Google Scholar

[5]

J.-P. Dias, M. Figueira and J.-F. Rodrigues, Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid Mech., 7 (2005), 153-163.  doi: 10.1007/s00021-004-0113-y.  Google Scholar

[6]

U. S. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50 (2012), 544-573.  doi: 10.1137/110836961.  Google Scholar

[7]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, volume 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016.  Google Scholar

[8]

M. Garavello, R. Natalini, B. Piccoli and A. Terracina, Conservation laws with discontinuous flux, Netw. Heterog. Media, 2 (2007), 159-179.  doi: 10.3934/nhm.2007.2.159.  Google Scholar

[9]

M. Garavello and B. Piccoli, Traffic Flow on Networks, volume 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[10]

S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138.  doi: 10.1137/120882573.  Google Scholar

[11]

M. Herty, C. Joerres and B. Piccoli, Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Anal. Appl., 401 (2013), 510-517.  doi: 10.1016/j.jmaa.2012.12.002.  Google Scholar

[12]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668.  Google Scholar

[13]

J. K. Wiens, J. M. Stockie and J. F. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.  doi: 10.1016/j.jcp.2013.02.024.  Google Scholar

show all references

References:
[1]

D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math., 71 (2011), 1070-1087.  doi: 10.1137/100809374.  Google Scholar

[2]

F. Camilli, A. Festa and S. Tozza, A discrete hughes model for pedestrian flow on graphs, Netw. Heterog. Media, 12 (2017), 93-112.  doi: 10.3934/nhm.2017004.  Google Scholar

[3]

C. d'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, 2010. doi: 10.1137/1.9780898717600.  Google Scholar

[4]

J.-P. Dias and M. Figueira, On the riemann problem for some discontinuous systems of conservation laws describing phase transitions, Commun. Pure Appl. Math., 3 (2004), 53-58.  doi: 10.3934/cpaa.2004.3.53.  Google Scholar

[5]

J.-P. Dias, M. Figueira and J.-F. Rodrigues, Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid Mech., 7 (2005), 153-163.  doi: 10.1007/s00021-004-0113-y.  Google Scholar

[6]

U. S. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50 (2012), 544-573.  doi: 10.1137/110836961.  Google Scholar

[7]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, volume 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016.  Google Scholar

[8]

M. Garavello, R. Natalini, B. Piccoli and A. Terracina, Conservation laws with discontinuous flux, Netw. Heterog. Media, 2 (2007), 159-179.  doi: 10.3934/nhm.2007.2.159.  Google Scholar

[9]

M. Garavello and B. Piccoli, Traffic Flow on Networks, volume 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[10]

S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138.  doi: 10.1137/120882573.  Google Scholar

[11]

M. Herty, C. Joerres and B. Piccoli, Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Anal. Appl., 401 (2013), 510-517.  doi: 10.1016/j.jmaa.2012.12.002.  Google Scholar

[12]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668.  Google Scholar

[13]

J. K. Wiens, J. M. Stockie and J. F. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.  doi: 10.1016/j.jcp.2013.02.024.  Google Scholar

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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$ \mu = 0.5 $">Figure 11.  Test 3: "passive" junction with distribution parameter $ \mu = 0.5 $
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$ \mu = 0.5 $">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}$0.0035
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$ \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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