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During this thesis we have investigate two important fields of the anisotropic meshes adaptation problems which are: - Metrics and errors estimators, - the locals modifications of hexahedral and quadrilateral meshes. We propose new algorithms, methods and numerical schemes within this two parts. We add new methods of building errors estimators and metrics in the software Freefem++. We have work about some hessian matrix recovery techniques, as second order error estimator of the Lagrange polynomial interpolation which are: - The last square recovery method, - the method based on the Green formula, - the local approximation of the function by a second degree polynomial. We propose a new recovery technique which uses a local interpolation inside each element of the mesh and a finite difference scheme. we show some properties like con- sistence and convergence of all these methods and numerical results in dimension two of the space. We study the third order derivatives recovery using the least square tech- nique. We also propose a new calculation of the Lagrange interpolation error estimator. This result uses a Taylor development of the third order without any direct calculus of third other derivatives. We also propose an algorithm for building metrics using any given error estimation that can be represented by a close curve. We finally propose a new set of local hex meshes transformations and a study about the existing local quad and hex meshes modifications by showing some new properties., Cette thèse est consacrée aux études théoriques et numériques des problèmes sui- vants qui sont liés à l'adaptation de maillage anisotrope : Les métriques et estimateurs d'erreur, les modifications locales de maillages hexaédriques et quadrilatéraux. Nous procédons à la mise en oeuvre de nouveaux algorithmes, schémas numériques et méthodes dans ces deux parties ; notamment en codant dans le logiciel Freefem++ de nouvelles méthodes de reconstruction d'estimateurs d'erreur et de construction de métriques. Nous étudions trois des méthodes de reconstruction de la matrice hessienne, estimateur d'erreur d'interpolation de Lagrange à l'ordre deux qui sont : La reconstruction de la matrice hessienne par moindres carrés, la méthode basée sur la formule de Green, l'approximation locale de la fonction par un polynôme du second degré. Nous proposons une nouvelle approche basée sur l'interpolation polynomiale locale par maille et un schéma aux différences finies. Nous établissons des propriétés de sta- bilité et de convergence ainsi que des résultats numériques en dimension deux. Nous étudions aussi la reconstruction des dérivées troisièmes par moindres carrés. Nous pro- posons également de nouvelles estimations d'erreur d'interpolation de Lagrange grâce à un développement de Taylor à l'ordre trois sans calcul direct de dérivées troisièmes. Il est aussi proposé un algorithme de construction de métriques à partir d'une estima- tion d'erreur pouvant être représentée localement par une courbe fermée, applicable à l'erreur d'interpolation polynomiale d'ordre supérieur. Enfin, nous proposons de nouvelles façons de raffiner ou dé-raffiner localement les maillages hexaédriques. Nous faisons une étude des techniques existantes en proposant de nouvelles caractérisations des transformations locales de maillages quadrilatéraux et hexaédriques.

cote interne IRCAM: Depalle96a; /; National audience; Many studies have been undertaken on the modeling of physical systems by means of waveguidefilters. These methods consist mainly in simulating the propagation of acoustic waves with digitaldelay lines. These models are constrained to have a spatial step fixed by the sampling rate whichbecomes a serious drawback when a high spatial resolution in the geometry of the model is neededor when the length of the waveguide needs to vary. One can use digital filters for approximatingthe exact fractional delay, but length variations usually induce audible distortions because of localinstabilities or modification of the filter's structure. Lagrange Interpolation theory leads to FIR filters which approximate fractional delays accordingto a maximally flat error criterion. Major drawbacks of current implementations of LagrangeInterpolator Filters (LIF), such as the Farrow structure, are a high computation cost and a lack ofcontrol over the delay which can only vary in a narrow range of values. Furthermore, there is noexplicit method for shrinking or enlarging the fractional delay line. We propose a new implementation for fractional delay lines based on the formal power seriesexpansion of the exact z-transform. We have developed different fast and modular algorithms forfractional delay lines which make them usable for real-time delay-varying applications.Modularity in the structure is a key point here as it enables one to switch between filters ofdifferent order while preserving the continuity of the z-transform. Thus the delay may vary overan unlimited range of values. Furthermore, any arbitrary integer part of the fractional delay can besimulated by a classical delay line so that the actual size of the fractional delay line may bemaintained within reasonable limits. We have written a real-time implementation in a MAX-FTSenvironment. Different examples will illustrate its time-varying properties and its numericalstability.

cote interne IRCAM: Tassart96a; /; National audience; Many studies have been undertaken on the modeling of physical systems by means of waveguidefilters. These methods consist mainly in simulating the propagation of acoustic waves with digitaldelay lines. These models are constrained to have a spatial step determined by the sampling ratewhich is a serious drawback when a high spatial resolution in the geometry of the model is neededor when the length of the waveguide needs to vary. One can use digital filters for approximatingthe exact fractional delay, but length variations usually induce audible distortions because of localinstabilities or modification of the filter's structure. Lagrange Interpolation theory leads to FIR filters which approximate fractional delays accordingto a maximally flat error criterion. Major drawbacks of current implementations of LagrangeInterpolator Filters (LIF) are a high computation cost and a lack of control over the delay whichcan only vary in a narrow range of values. We propose a new implementation of LIF based on a formal power series expansion of the exactz-transform. We have developed different fast and modular algorithms for LIF which make theLIF usable for real-time delay-varying applications. Modularity in the structure is a key pointhere as it enables one to switch between filters of different order while preserving the continuityof the z-transform. Thus the delay may vary over an unlimited range of values. Furthermore, anyarbitrary integer part of the fractional delay can be simulated by a classical delay line so that theactual order of the LIF may be maintained within reasonable limits. This paper will focus on thetime-varying properties of our implementation and its numerical stability over a wide range ofdelays.