This Master’s thesis presents a method for the reconstruction of a windshield surface using an image observed through it. This image is distorted because the light rays passing through the windshield undergo two refractions : one on each side of the glass. The distortion depends on the windshield shape and therefore this data is used to solve the problem. The first step is the construction of a vector field in the ambient space, from the deviations of the light rays passing through the windshield. This step relies on the Snell-Descartes refraction law and on simplifying assumptions regarding the curvature and thickness of a windshield. A vector at a point of this field corresponds to a prediction of the surface normal vector at this point, under the hypothesis that this point lies on the surface. The second step is to find a surface that is compatible with the obtained vector field. For this purpose, a minimisation problem is formulated for which the minimized variable is the difference between the surface normal vector and the one deduced from the system’s measurements. This leads to a nonlinear Euler- Lagrange equation for which the Dirichlet boundary conditions are imposed. The graph of the solution is the desired surface. The third step is a fixed-point method to solve the Euler- Lagrange equation. At the center of this method is a sequence of linear Poisson equations, each giving an approximating solution. It is shown that the limit of this sequence of solutions respects the original nonlinear equation. The Banach fixed-point theorem is used to get sufficient existence and uniqueness conditions, that are also sufficient conditions under which the proposed fixed-point method converges.

The core of this thesis consists in the development of a new coupled model for wildfire spread. This model relies on an atmospheric model based on a single constraint for the wind velocity flow. This constraint given by a divergence equation is derived from a low Mach number approximation. The fire model represents the fireline on the topography as an infinitely thin interface which outlines the burned regions. The level set method allows to track the spread of this interface on the topography. The fire and atmosphere models are coupled with a source term in the divergence equation governing the wind velocity field. This source is represented as a singular sink-source pair in order to capture the main features of the atmospheric flow near the fireline. Each singularity is supported on an interface, a codimension-2 manifold. The computation of the amplitude of the source term is achieved with Byram's formula for fire intensity. The derivation and particular characteristics of this coupled model are presented in this work. A regularization technique combined with a rescaling algorithm for a delta function supported on a codimension-2 manifold has been elaborated for this model. A study of the convergence of the solutions of the elliptic problem, associated with the atmospheric model, demonstrates the necessity of this technique to achieve convergence with mesh refinements. This thesis presents the numerical implementation of the coupled model. The simulations conducted with the model are used to study the fire spread regimes with a dimensionless number obtained in a dimensional analysis. The model is compared to the Firetec model with numerical experiments of fire spread over idealized topographies.

This thesis is divided in three chapters. The first explains how to use the level-set method in a rigorous way in the context of forest fire simulation when the physical propagation model for firespread is Richards' ellipse model. The second chapter presents a new semi-implicit scheme with a proof of convergence for the numerical solution of an anisotropic Hamilton-Jacobi partial differential equation. The advantage of this scheme is it allows the use of approximative solutions as initial conditions which reduces the computation time. The third chapter shows how to use the tools introduced in the first two chapters to study the influence of small-scale variations on the wind speed on firespread using the theory of homogenization.

The Projection method and Sasaki's variational technique are two methods allowing one to extract a divergence-free vector field from any vector field. From a high altitude wind speed, a velocity field is generated on a staggered grid over a topography given by an analytical function. The Cartesian grid Embedded Boundary method is used for solving a Poisson equation, obtained from the projection, on an irregular domain with mixed boundary conditions. The solution of this equation gives the correction for the initial velocity field to make it such that it satisfies the conservation of mass and takes into account the effects of the terrain. The incompressible velocity field will be used to spread a wildfire over the topography with the Level Set Method. The algorithm is described for the two and three dimensional cases and convergence tests are done.

We are interested here in the modeling errors of subgrid flamelet models in nonpremixed turbulent combustion. The goal of this thesis is to develop an a posteriori error estimation strategy to determine the best model within a hierarchy, with a numerical cost at most that of using the models in the first place. Firstly, we develop and test a dual-weighted residual estimator strategy on a system of advection-diffusion-reaction equations. Secondly, we test that methodology on another system of equations, where quenching and ignition effects are added. In the absence of advection, a rigorous asymptotic analysis shows the existence of many combustion regimes already observed in numerical simulations. We obtain approximations of the quenching and ignition parameters, alongside the S-shaped curve, a plot of the maximal flame temperature as a function of the Damköhler number, consisting of three branches and two bends. When advection effects are added, we still obtain a S-shaped curve corresponding to the known combustion regimes. We compare the modeling errors of the asymptotic approximations in the two stable regimes and establish new model hierarchies for each combustion regime. These errors are compared with the estimations obtained by using the error estimation strategy. When only one stable combustion regime exists, the error estimator correctly identifies that regime; when two or more regimes are possible, it gives a systematic way of choosing one regime. For regimes where more than one model is appropriate, the error estimator’s predicted hierarchy is correct.

This work presents a forest fire simulation model which uses the Level-Set method. We use a partial differential equation to deform a surface on which our flame front is inscribed. The mathematical foundations of the Level-set method are presented. We then explain a reinitialization method that allows us to treat in a robust way real data and to reduce the calculation time. The effect of the presence of barriers in the fire propagation domain is also studied. Finally, we make an attempt to find the ignition point of a forest fire.

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Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

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Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

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