Throughout many fields of science and engineering, the need for describing waveequations is crucial. Solving the wave equation for high-frequency waves istime-consuming, requires a fine mesh size and memory usage. The main goal wasimplementing and comparing different solution methods for high-frequency waves.Four different methods have been implemented and compared in terms of runtimeand discretization error. From my results, the method which performs the best is thefast sweeping method. For the fast marching method, the time-complexity of thenumerical solver was higher than expected which indicates an error in myimplementation.

<p>This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems.</p><p>Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials.</p><p>First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of <i>escape </i>partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(<i>N</i>³ log <i>N</i>), with<i> N</i>³ being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation.</p><p>Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing.</p><p>We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented.</p>

The present work focuses on the numerical resolution of the acoustic or elastic wave equation in a piece-wise homogeneous medium, splitted by interfaces. We are interested in a high-frequency setting, introduced by strongly oscillating initial conditions, for which one computes the distribution of the energy density by a so-called kinetic approach (based on the use of a Wigner transform). This problem then reduces to a Liouville-type transport equation in a piece-wise homogeneous medium, supplemented by reflection and transmission laws at the interfaces. Several numerical techniques and ranges of application are also reviewed. The transport equation which describes the evolution of the energy density in the phase space positions _ wave vectors is numerically solved by finite differences. This technique raises several difficulties related to the conservation of the total energy in the medium and at the interfaces. They may be alleviated by dedicated numerical schemes allowing to reduce the numerical dissipation by either a global or a local approach. The improvements presented in this thesis concern the interpolation of the energy densities obtained by transmission on the grid of discrete wave vectors, and the correction of the difference of variation scales of the wave celerity on each side of the interfaces. The interest of the foregoing developments is to obtain conservative schemes that also satisfy the usual convergence properties of finite difference methods. The construction of such schemes and their effective implementation constitute the main achievement of the thesis. The relevance of the proposed methods is illustrated by several numerical simulations, that also emphasize their efficiency for rather coarse meshes.

Ce travail de recherche est dédié à la simulation de la réponse transitoire des assemblages de poutres soumis à des chocs. De tels chargements entraînent la propagation d’ondes haute fréquence dans l’ensemble de la structure. L’énergie qu’elles transportent peut être dommageable pour son fonctionnement ou celui des équipements embarqués. Dans des études précédentes, il a été observé sur des structures expérimentales qu’un régime vibratoire diffusif tend à s’installer pour des temps longs. Le but de cette étude est donc de développer un modèle robuste de la réponse transitoire des assemblages de poutres soumis à des chocs permettant de simuler, entre autres, cet état diffusif. Les champs de déplacement étant très oscillants et la densité modale élevée, la simulation numérique de la réponse transitoire à des chocs peut difficilement être menée par une méthode d’éléments finis classique. Une approche utilisant un estimateur de la densité d’énergie de chaque mode de propagation a donc été mise en œuvre. Elle permet d’accéder à des informations locales sur les états vibratoires, et de contourner certaines limitations intrinsèques aux longueurs d’onde courtes. Après avoir comparé plusieurs modèles de réduction cinématique de poutre à un modèle de Lamb de propagation dans un guide d’ondes circulaire, la cinématique de Timoshenko a été retenue afin de modéliser le comportement mécanique haute fréquence des poutres. En utilisant ce modèle dans le cadre de l’approche énergétique évoquée plus haut, deux groupes de modes de propagation de la densité d’énergie vibratoire dans une poutre ont été isolés : des modes longitudinaux regroupant un mode de compression et des modes de flexion, et des modes transversaux regroupant des modes de cisaillement et un mode de torsion. Il peut être également montré que l’´evolution en temps des densités d’énergie associées obéit à des lois de transport. Pour des assemblages de poutres, les phénomènes de réflexion/transmission aux jonctions ont du être pris en compte. Les opérateurs permettant de les décrire en termes de flux d’´energie ont été obtenus grâce aux équations de continuité des déplacements et des efforts aux jonctions. Quelques caractéristiques typiques d’un régime haute fréquence ont été mises en évidence, tel que le découplage entre les modes de rotation et les modes de translation. En revanche, les champs de densité d’énergie sont quant à eux discontinus aux jonctions. Une méthode d’éléments finis discontinus a donc été développée afin de les simuler numériquement comme solutions d’´equations de transport. Si l’on souhaite atteindre le régime diffusif aux temps longs, le schéma numérique doit être peu dissipatif et peu dispersif. La discrétisation spatiale a été faite avec des fonctions d’approximation de type spectrales, et l’intégration temporelle avec des schémas de Runge-Kutta d’ordre élevé du type ”strong stability preserving”. Les simulations numériques ont donné des résultats concluants car elles permettent d’exhiber le régime de diffusion. Il a été remarqué qu’il existait en fait deux limites diffusives différentes : (i) la diffusion spatiale de l’´energie sur l’ensemble de la structure, et (ii) l’équirépartition des densités d’énergie entre les différents modes de propagation. Enfin, une technique de renversement temporel a été développée. Elle pourra être utile dans de futurs travaux sur le contrôle non destructif des assemblages complexes et de grandes tailles. / This research is dedicated to the simulation of the transient response of beam trusses under impulse loads. The latter lead to the propagation of high-frequency waves in such built up structures. In the aerospace industry, that phenomenon may penalize the functioning of the structures or the equipments attached to them on account of the vibrational energy carried by the waves. It is also observed experimentally that high-frequency wave propagation evolves into a diffusive vibrational state at late times. The goal of this study is then to develop a robust model of high-frequency wave propagation within three-dimensional beam trusses in order to be able to recover, for example, this diffusion regime. On account of the small wavelengths and the high modal density, the modelling of high-frequency wave propagation is hardly feasible by classical finite elements or other methods describing the displacement fields directly. Thus, an approach dealing with the evolution of an estimator of the energy density of each propagating mode in a Timoshenko beam has been used. It provides information on the local behavior of the structures while avoiding some limitations related to the small wavelengths of high-frequency waves. After a comparison between some reduced-order beam kinematics and the Lamb model of wave propagation in a circular waveguide, the Timoshenko kinematics has been selected for the mechanical modelling of the beams. It may be shown that the energy densities of the propagating modes in a Timoshenko beam obey transport equations. Two groups of energy modes have been isolated: the longitudinal group that gathers the compressional and the bending energetic modes, and the transverse group that gathers the shear and torsional energetic modes. The reflection/transmission phenomena taking place at the junctions between beams have also been investigated. For this purpose, the power flow reflection/transmission operators have been derived from the continuity of the displacements and efforts at the junctions. Some characteristic features of a high-frequency behavior at beam junctions have been highlighted such as the decoupling between the rotational and translational motions. It is also observed that the energy densities are discontinuous at the junctions on account of the power flow reflection/transmission phenomena. Thus a discontinuous finite element method has been implemented, in order to solve the transport equations they satisfy. The numerical scheme has to be weakly dissipative and dispersive in order to exhibit the aforementioned diffusive regime arising at late times. That is the reason why spectral-like approximation functions for spatial discretization, and strong-stability preserving Runge-Kutta schemes for time integration have been used. Numerical simulations give satisfactory results because they indeed highlight the outbreak of such a diffusion state. The latter is characterized by the following: (i) the spatial spread of the energy over the truss, and (ii) the equipartition of the energy between the different modes. The last part of the thesis has been devoted to the development of a time reversal processing, that could be useful for future works on structural health monitoring of complex, multi-bay trusses.

Le présent travail porte sur la résolution numérique de l’équation des ondes acoustiques ou élastiques dans un milieu homogène par morceaux comportant des interfaces. On s’intéresse à un problème haute fréquence, introduit par des conditions initiales fortement oscillantes, pour lequel on détermine la répartition de la densité d’énergie dans le milieu par une approche dite cinétique (fondée sur l’utilisation d’une transformation de Wigner). Le problème considéré est alors réduit à une équation de transport en milieu homogène du type Liouville, complétée par des lois de réflexion et transmission aux interfaces. Différentes méthodes de résolution et d’autres cas d’application sont par ailleurs évoquées. La résolution numérique de l’équation de transport décrivant l’évolution de la densité d’énergie dans l’espace des phases positions vecteurs d’onde est effectuée par différences finies. Cette technique soulève plusieurs difficultés relatives à la conservation de l’énergie totale dans le milieu et aux interfaces. Elles peuvent être corrigées par des schémas numériques adaptés permettant de limiter la dissipation numérique par une approche globale ou locale. Les développements réalisés concernent l’interpolation des densités d’énergie obtenues par transmission sur la grille des vecteurs d’onde discrets, ainsi que la correction de la différence d’échelle de variation de la vitesse des ondes de part et d’autre des interfaces. L’intérêt de ces adaptations est d’obtenir des schémas conservatifs qui satisfont les critères de convergence usuels des méthodes aux différences finies. Leur construction ainsi que leur mise en œuvre effective constituent le principal apport de cette thèse. La pertinence des méthodes utilisées est illustrée par des exemples de simulation, qui montrent également leur efficacité pour des maillages relativement grossiers. / The present work focuses on the numerical resolution of the acoustic or elastic wave equation in a piece-wise homogeneous medium, splitted by interfaces. We are interested in a high-frequency setting, introduced by strongly oscillating initial conditions, for which one computes the distribution of the energy density by a so-called kinetic approach (based on the use of a Wigner transform). This problem then reduces to a Liouville-type transport equation in a piece-wise homogeneous medium, supplemented by reflection and transmission laws at the interfaces. Several numerical techniques and ranges of application are also reviewed. The transport equation which describes the evolution of the energy density in the phase space positions _ wave vectors is numerically solved by finite differences. This technique raises several difficulties related to the conservation of the total energy in the medium and at the interfaces. They may be alleviated by dedicated numerical schemes allowing to reduce the numerical dissipation by either a global or a local approach. The improvements presented in this thesis concern the interpolation of the energy densities obtained by transmission on the grid of discrete wave vectors, and the correction of the difference of variation scales of the wave celerity on each side of the interfaces. The interest of the foregoing developments is to obtain conservative schemes that also satisfy the usual convergence properties of finite difference methods. The construction of such schemes and their effective implementation constitute the main achievement of the thesis. The relevance of the proposed methods is illustrated by several numerical simulations, that also emphasize their efficiency for rather coarse meshes.

This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems. Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials. First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of escape partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(N3 log N), with N3 being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation. Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing. We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented. / QC 20101119

Computer simulation of wave propagation is an active research area as wave phenomena are prevalent in many applications. Examples include wireless communication, radar cross section, underwater acoustics, and seismology. For high frequency waves, this is a challenging multiscale problem, where the small scale is given by the wavelength while the large scale corresponds to the overall size of the computational domain. Research into wave equation modeling can be divided into two regimes: time domain and frequency domain. In each regime, there are two further popular research directions for the numerical simulation of the scattered wave. One relies on direct discretization of the wave equation as a hyperbolic partial differential equation in the full physical domain. The other direction aims at solving an equivalent integral equation on the surface of the scatterer. In this dissertation, we present three new techniques for the frequency domain, boundary integral equations. / text