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Modélisation mathématique et simulation du trafic routier : analyse statistique de modèles d'insertion et simulation probabiliste d'un modèle cinétique / Mathematical modelling and simulation of the road traffic : statistical analysis of merging models and probabilistic simulation of a kinetic modelMint Moustapha, Jyda 13 November 2014 (has links)
La première partie de cette thèse a consisté à proposer des modèles d'insertion de trafic sur une bretelle d'entrée d'autoroute. Deux types de modélisation ont été élaborés. Une approche statistique utilisant les techniques de régression logistique nous a permis de sélectionner les variables jouant un rôle dans le choix par les véhicules provenant de la voie d'accélération du lieu où ils s'insèrent. Dans un second temps, nous effectuons une modélisation comportementale basée sur le principe d'acceptation de créneaux. Les modèles proposés ont été validés à l'aide de données issues d'un site d'observations expérimentales situé près d'Angers, le site SAROT. La seconde partie est consacrée au développement d'une méthode particulaire probabiliste permettant de simuler un modèle mésoscopique de trafic : le modèle cinétique de Paveri-Fontana. La complexité algorithmique de cette méthode proposée comme alternative aux méthodes déterministes couramment utilisées est optimisée. La comparaison des résultats obtenus à ceux d'une méthode déterministe plus standard de différences finies sur des cas-tests bien choisis a permis de valider la méthode particulaire. Ces expériences numériques ont mis en valeur ses qualités notamment sa rapidité (coût numérique) par rapport à la méthode déterministe ainsi que sa capacité à mieux reproduire certains phénomènes observés dans le trafic / The first part of this thesis is dedicated to the optimization of the lengths of acceleration lanes using microscopic data collected from real traffic. The insertions on the highway junctions can indeed be especially dangerous considering the difference between the speeds on the on ramp merge lane and those on the highway lanes. We develop and analyse some microscopic merging models. We first propose a statistical model based on the logistic regression techniques. Statistical hypothesis tests allow to select the most significant descriptive variables in the merging decision process. A behavioural modelling taking those variables into account is next proposed to better capture the interactions by including some thresholds on the gaps between the merging vehicles and freeway vehicles. The models are validated using real traffic data collected at the SAROT site near Angers. Secondly, traffic simulation at the mesoscopic scale is mostly based on deterministic numerical schemes. However, these methods have a high computational cost. The objective of the second part of this thesis is to present a new method to simulate the Paveri-Fontana kinetic model through a probabilistic approach. We interpret the evolution equation in this model as a Fokker-Planck equation and deduce an approximation based on a system of interacting particles. The algorithmic complexity of this method is optimized. We have performed a numerical comparison between the probabilistic method and a deterministic method on some cases study. The qualitative analysis highlights the benefits of the particle method such as its computation cost and its ability to reproduce some typical traffic effects
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[en] POISSON EQUATION AND THE HELMHOLTZ-HODGE DECOMPOSITION WITH SPH OPERATORS / [pt] A EQUAÇÃO DE POISSON E A DECOMPOSIÇÃO DE HELMHOLTZ-HODGE COM OPERADORES SPHFABIANO PETRONETTO DO CARMO 29 August 2008 (has links)
[pt] A equação diferencial parcial de Poisson é de fundamental
importância em várias áreas de pesquisa, dentre elas:
matemática, física e engenharia. Para resolvê-la
numericamente utilizam-se vários métodos, tais como os
já tradicionais métodos das diferenças finitas e dos
elementos finitos. Este trabalho propõe um método para
resolver a equação de Poisson, utilizando
uma abordagem de sistema de partículas conhecido como SPH,
do inglês Smoothed Particles Hydrodynamics. O método
proposto para a solução da equação de Poisson e os
operadores diferenciais discretos definidos no
método SPH, chamados de operadores SPH, são utilizados neste
trabalho em duas aplicações: na decomposição de campos
vetoriais; e na simulação numérica de escoamentos de fluidos
monofásicos e bifásicos utilizando a equação de Navier-Stokes. / [en] Poisson`s equation is of fundamental importance in many
research areas in engineering and the mathematical and
physical sciences. Its numerical solution uses several
approaches among them finite differences and finite
elements. In this work we propose a method to solve
Poisson`s equation using the particle method known as SPH
(Smoothed Particle Hydrodynamics). The proposed method
together with an accurate analysis of the discrete
differential operators defined by SPH are applied in two
related situations: the Hodge-Helmholtz vector field
decomposition and the numerical simulation of the
Navier-Stokes equations.
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