In both continuous and discrete settings, Laplace operators appear quite commonly in the modeling of natural phenomena, in several context: diffusion, heat propagation, porous media, fluid flows through pipes, electricity.... In these contexts, the discrete Laplace operator enjoys all the properties of its continuous counterpart, in particular: self-adjointness, variational formulation, stochastic interpretation, mean value property, maximum principle, ...In a first part, we give a detailed description of the correspondence between these mathematical properties and modeling considerations, in contexts where the continuous and the discrete settings perfectly match. In a second part, we describe a pathological situation, in the context of granular crowd motion models. Accounting for the non-overlapping constraint between hard discs leads to a particular operator acting on a field of Lagrange multipliers, defined on the dual graph of the contact network. This operator is defective in a certain sense: although it is the microscopic counterpart of the macroscopic Laplace operator, this discrete operator indeed lacks some properties, in particular the maximum principle. We investigate here how this very defectivity may explain some paradoxical phenomena that are observed in crowd motions and granular materials, phenomena that are not reproduced by macroscopic models.
Aux niveaux continu et discret, l’opérateur de Laplace intervient de façon très courante dans la modélisation de phénomènes naturels, dans de nombreux contextes : diffusion, propagation de la chaleur, milieux poreux, écoulement de fluides dans des conduits, électricité.... Dans ces contextes, le laplacien discret possède toutes les propriétés de son pendant continu : caractère auto-adjoint, structure variationnelle, interprétation stochastique, propriété de la valeur moyenne, principe du maximum, .... Dans une première partie, nous proposons une description détaillée des liens entre ces propriétés et les aspects de modélisations, dans des contextes où les notion continues et discrètes se correspondent parfaitement. Dans une seconde partie, nous décrivons une situation pathologique, dans le contexte de la modélisation de foules d’un point de vue granulaire. La prise en compte de la contrainte de non recouvrement entre grains rigides conduit à un opérateur particulier qui agit sur les champs de multiplicateurs de Lagrange, définis sur le graphe dual du réseau de contacts. Cet opérateur est déficient dans un certain sens : bien qu’il apparaisse comme le pendant discret du laplacien continu, il ne vérifie pas certaines des propriétés usuelles du laplacien, en particulier le principe du maximum. Nous explorons comment cette déficience permet d’expliquer certains effets paradoxaux observés en mouvements de foules, que les modèles continus ne reproduisent pas.
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Mots-clés : Laplacien discret, principe du maximum, mouvements de foules, effet faster-is-slower
Bertrand Maury 1
CC-BY 4.0
@article{CRMECA_2023__351_S1_615_0,
author = {Bertrand Maury},
title = {Defective {Laplacians} and paradoxical phenomena in crowd motion modeling},
journal = {Comptes Rendus. M\'ecanique},
pages = {615--646},
publisher = {Acad\'emie des sciences, Paris},
volume = {351},
number = {S1},
year = {2023},
doi = {10.5802/crmeca.205},
language = {en},
}
Bertrand Maury. Defective Laplacians and paradoxical phenomena in crowd motion modeling. Comptes Rendus. Mécanique, The scientific legacy of Roland Glowinski, Volume 351 (2023) no. S1, pp. 615-646. doi : 10.5802/crmeca.205. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.205/
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