A flame propagation model on a network with application to a blocking problem

Fabio Camilli; Elisabetta Carlini; Claudio Marchi

doi:10.3934/dcdss.2018051

Article Contents

2018, Volume 11, Issue 5: 825-843. Doi: 10.3934/dcdss.2018051

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A flame propagation model on a network with application to a blocking problem

1.
Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Italy
2.
Dipartimento di Matematica, "Sapienza" Università di Roma, p.le A. Moro 5, 00185 Roma, Italy
3.
Dip. di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy

* Corresponding author
* Corresponding author

Received: January 31, 2017

Revised: July 31, 2017

Published: June 2018

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Abstract

We consider the Cauchy problem

$\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$

where $\Gamma$ is a network and $H$ is a positive homogeneous Hamiltonian which may change from edge to edge. In the first part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.

Keywords:

Mathematics Subject Classification: Primary: 35D40; Secondary: 35R02, 35F21, 65M06, 49L25.

Citation:

Full Text(HTML)

Figure 1. Test1. Graph structure where $R_0$ is represented by the circle marker and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29), (Top Right) and its 3D view (Bottom).

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Figure 2. Test1. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square marker). 2D view (Left) and 3D view (Right).

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Figure 3. Test1. Optimal blocking strategy $\sigma ^h_{opt}$ (square marker), preserved network region (cross marker) and minimum burnt network region (continuum line) starting from $R_0$ (circle marker).

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Figure 4. Test2. Graph structure where $R_0$ is represented by the circle markers and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29) (Top Right), and its 3D view (Bottom).

Download: Full-size image PowerPoint slide

Figure 5. Test2. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square markers). 2D view(Left) and 3D view (Right).

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Figure 6. Test2. Optimal blocking strategy $\sigma ^h_{opt}$ (square markers), preserved network region (thin line) and minimum burnt network region (thick line) starting from $R_0$ (circle markers).

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Figure 7. Test3. Graph structure where $R_0$ is represented by the circle markers and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29), (Top Right) and its 3D view (Bottom).

Download: Full-size image PowerPoint slide

Figure 8. Test3. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square markers). 2D view(Left) and 3D view (Right).

Download: Full-size image PowerPoint slide

Figure 9. Test3. Optimal blocking strategy $\sigma ^h_{opt}$ (square marker), preserved network region (thin line) and minimum burnt network region (thick line) starting from $R_0$ (circle marker).

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