On the Eikonal Equation in the Pedestrian Flow Problem
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F44555601%3A13440%2F17%3A43893493" target="_blank" >RIV/44555601:13440/17:43893493 - isvavai.cz</a>
Nalezeny alternativní kódy
RIV/00216208:11320/17:10332292 RIV/68407700:21340/17:00318701
Výsledek na webu
<a href="http://dx.doi.org/10.1063/1.4992707" target="_blank" >http://dx.doi.org/10.1063/1.4992707</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1063/1.4992707" target="_blank" >10.1063/1.4992707</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
On the Eikonal Equation in the Pedestrian Flow Problem
Popis výsledku v původním jazyce
We consider the Pedestrian Flow Equations (PFEs) as the coupled system formed by the Eikonal equation and the first order hyperbolic system with the source term. The hyperbolic system consists of the continuity equation and momentum equation of fluid dynamics. Specifying the social and pressure forces in the momentum equation we come to the assumption that each pedestrian is trying to move in a desired direction (e.g. to the exit in the panic situation) with a desired velocity, where his velocity and the direction of movement depend on the density of pedestrians in his neighborhood. In [1] we used the model, where the desired direction of movement is given by the solution of the Eikonal equation (more precisely by the gradient of the solution). Here we avoid the solution of the Eikonal equation, which is the novelty of the paper. Based on the fact that the solution of the Eikonal equation has the meaning of the shortest time to reach the exit, we define explicitly such a function in the framework of the Dijkstra's algorithm for the shortest path in the graph. This is done at the discrete level of the solution. As the graph we use the underlying triangulation, where the norm of each edge is density depending and has the dimension of the time. The numerical examples of the solution of the PFEs with and without the solution of the Eikonal equation are presented.
Název v anglickém jazyce
On the Eikonal Equation in the Pedestrian Flow Problem
Popis výsledku anglicky
We consider the Pedestrian Flow Equations (PFEs) as the coupled system formed by the Eikonal equation and the first order hyperbolic system with the source term. The hyperbolic system consists of the continuity equation and momentum equation of fluid dynamics. Specifying the social and pressure forces in the momentum equation we come to the assumption that each pedestrian is trying to move in a desired direction (e.g. to the exit in the panic situation) with a desired velocity, where his velocity and the direction of movement depend on the density of pedestrians in his neighborhood. In [1] we used the model, where the desired direction of movement is given by the solution of the Eikonal equation (more precisely by the gradient of the solution). Here we avoid the solution of the Eikonal equation, which is the novelty of the paper. Based on the fact that the solution of the Eikonal equation has the meaning of the shortest time to reach the exit, we define explicitly such a function in the framework of the Dijkstra's algorithm for the shortest path in the graph. This is done at the discrete level of the solution. As the graph we use the underlying triangulation, where the norm of each edge is density depending and has the dimension of the time. The numerical examples of the solution of the PFEs with and without the solution of the Eikonal equation are presented.
Klasifikace
Druh
D - Stať ve sborníku
CEP obor
—
OECD FORD obor
10102 - Applied mathematics
Návaznosti výsledku
Projekt
<a href="/cs/project/GA13-00522S" target="_blank" >GA13-00522S: Kvalitativní analýza a numerické řešení problémů proudění v obecně časově závislých oblastech s různými okrajovými podmínkami</a><br>
Návaznosti
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Ostatní
Rok uplatnění
2017
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název statě ve sborníku
PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016)
ISBN
978-0-7354-1538-6
ISSN
0094-243X
e-ISSN
neuvedeno
Počet stran výsledku
4
Strana od-do
"nestrankovano"
Název nakladatele
AMER INST PHYSICS, 2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA
Místo vydání
NY, USA
Místo konání akce
Rhodes, GREECE
Datum konání akce
19. 9. 2016
Typ akce podle státní příslušnosti
EUR - Evropská akce
Kód UT WoS článku
000410159800517