Publications

2016

• Transient simulation of an electrical rotating machine achieved through model order reduction
  
  • Montier Laurent
  • Henneron Thomas
  • Clenet Stéphane
  • Goursaud Benjamin
  Advanced Modeling and Simulation in Engineering Sciences, Springer, 2016, 3 (1).
• Fiabilité et optimisation des calculs obtenus par des formulations intégrales en propagation d'ondes
  
  • Bakry Marc
  , 2016. Dans cette thèse, on se propose de participer à la popularisation des méthodes de résolution de problèmes de propagation d'onde basées sur des formulations intégrales en fournissant des indicateurs d'erreur a posteriori utilisable dans le cadre d'algorithmes de raffinement autoadaptatif. Le développement de tels indicateurs est complexe du fait de la non-localité des normes associées aux espaces de Sobolev et des opérateurs entrant en jeu. Des indicateurs de la littérature sont étendus au cas de la propagation d'une onde acoustique. On étend les preuves de convergence quasi-optimale (de la littérature) des algorithmes autoadaptatifs associés dans ce cas. On propose alors une nouvelle approche par rapport à la littérature qui consiste à utiliser une technique de localisation des normes, non pas basée sur des inégalités inverses, mais sur l'utilisation d'un opérateur Λ de localisation bien choisi.On peut alors construire des indicateurs d'erreur a posteriori fiables, efficaces, locaux et asymptotiquement exacts par rapport à la norme de Galerkin de l'erreur. On donne ensuite une méthode pour la construction de tels indicateurs. Les applications numériques sur des géométries 2D et 3D confirment l'exactitude asymptotique ainsi que l'optimalité du guidage de l'algorithme autoadaptatif.On étend ensuite ces indicateurs au cas de la propagation d'une onde électromagnétique. Plus précisément, on s'intéresse au cas de l'EFIE. On propose des généralisations des indicateurs de la littérature. On effectue la preuve de convergence quasi-optimale dans le cas d'un indicateur basé sur une localisation de la norme du résidu. On utilise le principe du Λ pour obtenir le premier indicateur d'erreur fiable, efficace et local pour cette équation. On en propose une seconde forme qui est également, théoriquement asymptotiquement exacte.
• On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2016, 322, pp.224-247. We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $\texttt{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{loc}$ if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods. (10.1016/j.jcp.2016.06.037)
  DOI : 10.1016/j.jcp.2016.06.037
• Development and use of higher-order asymptotics to solve inverse scattering problems
  
  • Cornaggia Rémi
  , 2016. The purpose of this work was to develop new methods to address inverse problems in elasticity, taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptotic expansions. The first part is dedicated to the localization and size identification of a buried inhomogeneity B in a 3D elastic domain. In this goal, we focus on the study of functionals J(Ba) quantifying the misfit between B and a trial homogeneity Ba . Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Ba (location, size, mechanical properties ...) to find the best agreement with B. To this end, we produce an expansion of J with respect to the size of Ba , providing a polynomial approximation easier to minimize. This expansion, established up to the sixth order in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space. The main purpose of this second part is to characterize a microstructured two-phases layered 1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigenvalues of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we rely on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method to recover the macroscopic parameters (length and material contrast) of such inclusion. To access the key features of the microstructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.
• On the development and use of higher-order asymptotics for solving inverse scattering problems.
  
  • Cornaggia Rémi
  , 2016. The purpose of this work was to develop new methods to address inverse problems in elasticity,taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptoticexpansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity Bᵗʳᵘᵉ in a 3Delastic domain. In this goal, we focused on the study of functionals J(Bₐ) quantifying the misfit between Bᵗʳᵘᵉ and a trial homogeneity Bₐ. Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Bₐ (location, size, mechanical properties ...) to find the best agreement with Bᵗʳᵘᵉ. To this end, we produced an expansion of J with respect to the size a of Bₐ, providing a polynomial approximation easier to minimize. This expansion, established up to O(a⁶) in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space ℝ³.The main purpose of this second part is to characterize a microstructured two-phases layered1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigen values of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we then relied on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method tore cover the macroscopic parameters (L and material contrast) of such inclusion. To access to the period of themicrostructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.
• Problèmes d'interface en présence de métamatériaux : modélisation, analyse et simulations
  
  • Vinoles Valentin
  , 2016. Nous nous intéressons à des problèmes de transmission entre diélectriques et métamatériaux, milieux présentant des propriétés électromagnétiques inhabituelles comme des caractéristiques effectives négatives à certaines fréquences. Par exemple, ces milieux peuvent être construits comme des assemblages périodiques de microstructures résonantes et dans ce cas la théorie de l'homogénéisation permet de justifier mathématiquement ces propriétés effectives. En régime harmonique et dans des géométries à variables séparables, des calculs analytiques peuvent être menés. Ils révèlent dans des cas dits critiques des difficultés mathématiques: les solutions n'ont pas la régularité standard, voire le problème peut être mal posé.La première partie étudie ces problèmes de transmission en régime temporel pour lequel les métamatériaux sont modélisés par des modèles dispersifs (modèle de Drude ou de Lorentz). Les difficultés résident dans le choix d'un schéma de discrétisation mais surtout dans la construction de conditions absorbantes. La méthode retenue ici est celle des Perfectly Matched Layers (PMLs). Comme les PMLs classiques sont instables pour ces modèles du fait de la présence d'ondes inverses, nous proposons une nouvelle classe de PMLs pour lesquelles nous menons une analyse de stabilité. Cette dernière permet de construire des PMLs stables. Elles sont ensuite utilisées pour simuler le comportement en temps long d'un problème de transmission; nous illustrons alors le fait que le principe d'amplitude limite peut être mis en défaut en raison de résonances d'interface.La deuxième partie vise à pallier ces phénomènes d'interface en régime harmonique en revenant sur le processus d'homogénéisation classique, pour un milieu dissipatif. Pour des problèmes de transmission, il est connu que les modèles issus de cette méthode perdent en précision du fait de la présence de couches limites à l'interface. Nous proposons un modèle enrichi au niveau de l'interface. En combinant la méthode d'homogénéisation double-échelle et celle des développements asymptotiques raccordés, nous construisons des conditions de transmission non standards faisant intervenir des opérateurs différentiels le long de l'interface. Le calcul de ces conditions nécessite la résolution de problèmes de cellule et de problèmes non standards posés dans des bandes périodiques infinies. Une analyse d'erreur confirme l'amélioration de la précision du modèle. Des simulations numériques illustrent l'efficacité de ces nouvelles conditions. Enfin, cette démarche est reproduite formellement dans le cas des matériaux à fort contraste se comportant comme des métamatériaux. Nous montrons alors que ces nouvelles conditions permettent de régulariser le problème de transmission dans les cas critiques.
• New model for datasets citation and extraction reproducibility in VAMDC
  
  • Zwölf Carlo Maria
  • Moreau Nicolas
  • Dubernet Marie-Lise
  Journal of Molecular Spectroscopy, Elsevier, 2016, 327, pp.122-137. (10.1016/j.jms.2016.04.009)
  DOI : 10.1016/j.jms.2016.04.009
• Open periodic waveguides. Theory and computation.
  
  • Vasilevskaya Elizaveta
  , 2016. The present work deals with propagation of acoustic waves in periodic media. These media have particularly interesting properties since the spectrum associated with the underlying wave operator in such media has a band-gap structure: there exist intervals of frequences for which monochromatic waves do not propagate. Moreover, by introducing linear defects in this kind of media, one can create guided modes inside the bands of forbidden frequences. In this work we show that it is possible to create such guided modes in the case of particular periodic media of grid type: more precisely, the periodic domain in question is R2 minus an infinite set of rectangular obstacles periodically spaced in two orthogonal directions (the distance between two neighbour obstacles being ε), which is locally perturbed by diminishing the distance between two columns of obstacles. The results are extended to the 3D case. This work has a theoretical and a numerical aspect. From the theoretical point of view the analysis is based on the fact that, ε being small, the spectrum of the operator associated with our problem is "close" to the spectrum of a problem posed on a graph which is a geometric limit of the domain as ε tends to 0. However, for the limit graph the spectrum can be computed explicitly. Then, we study the spectrum of the non-limit operator using asymptotic analysis. Theoretical results are illustrated by numerical computations obtained with a numerical method developed for study of periodic media: this method is based on the reduction of the initial (linear) eigenvalue problem posed in an unbounded domain to a non-linear problem posed in a bounded domain (using the exact Dirichlet- to-Neumann operator).
• Construction and analysis of an adapted spectral finite element method to convective acoustic equations
  
  • Hüppe Andreas
  • Cohen Gary
  • Imperiale Sebastien
  • Kaltenbacher Manfred
  Communications in Computational Physics, Global Science Press, 2016. The paper addresses the construction of a non spurious mixed spectral finite element (FE) method to problems in the field of computational aeroacous-tics. Based on a computational scheme for the conservation equations of linear acoustics, the extension towards convected wave propagation is investigated. In aeroacoustic applications, the mean flow effects can have a significant impact on the generated sound field even for smaller Mach numbers. For those convec-tive terms, the initial spectral FE discretization leads to non-physical, spurious solutions. Therefore, a regularization procedure is proposed and qualitatively investigated by means of discrete eigenvalues analysis of the discrete operator in space. A study of convergence and an application of the proposed scheme to simulate the flow induced sound generation in the process of human phonation underlines stability and validity. (10.4208/cicp.250515.161115a)
  DOI : 10.4208/cicp.250515.161115a
• Seismic Wave Amplification in 3D Alluvial Basins: 3D/1D Amplification Ratios from Fast Multipole BEM Simulations
  
  • Meza Fajardo Kristel Carolina
  • Semblat Jean-François
  • Chaillat Stéphanie
  • Lenti Luca
  Bulletin of the Seismological Society of America, Seismological Society of America, 2016, 106 (3), pp.1267-1281. In this work, we study seismic wave amplification in alluvial basins having 3D standard geometries through the Fast Multipole Boundary Element Method in the frequency domain. We investigate how much 3D amplification differs from the 1D (horizontal layering) case. Considering incident fields of plane harmonic waves, we examine the relationships between the amplification level and the most relevant physical parameters of the problem (impedance contrast, 3D aspect ratio, vertical and oblique incidence of plane waves). The FMBEM results show that the most important parameters for wave amplification are the impedance contrast and the so-called equivalent shape ratio. Using these two parameters, we derive simple rules to compute the fundamental frequency for various 3D basin shapes and the corresponding 3D/1D amplification factor for 5% damping. Effects on amplification due to 3D basin asymmetry are also studied and incorporated in the derived rules. (10.1785/0120150159)
  DOI : 10.1785/0120150159
• Couplage BEM-rayon pour le contrôle non destructif par ultrasons
  
  • Pesudo Laure
  • Bonnet Marc
  • Collino Francis
  • Demaldent Edouard
  • Imperiale Alexandre
  , 2016.
• Solitons acoustiques en milieu variable
  
  • Mercier Jean-François
  • Lombard Bruno
  , 2016.
• Modélisation et étude mathématique de réseaux de câbles électriques
  
  • Beck Geoffrey
  , 2016. Cette thèse porte sur la modélisation d'un réseau de câbles coaxiaux et multi-conducteurs. Ce dernier peut être mathématiquement traduit par les équations aux dérivées partielles de Maxwell qui régissent la propagation des ondes électromagnétiques en son sein ou par un modèle type circuit électrique d'inconnues - les potentiels et courants électriques- qui vérifient sur les branches du circuit l'équation des télégraphistes et sur les noeuds les lois de Kirchhoff.Si la première méthode est assez générale pour comprendre toutes sortes de défauts, elle néanmoins trop couteuse pour les applications que nous avons en tête, à savoir le contrôle non destructif. La seconde quant à elle est obtenue par une modélisation non issue de la théorie de Maxwell et est valide que si les câbles sont parfaits (cylindriques, sans pertes...). Nous avons établi diverses modèles 1D venant généraliser l'équation des télégraphistes et les lois de Kirchhoff pour y incorporer diverses défauts (géométrie, pertes, effet de peau, caractéristique des matériaux variables) tant sur les câbles que dans les jonctions. Ceux-ci sont obtenus via des analyses asymptotiques (classiques, multi-échelles, raccordées) des équations 3D de Maxwell en considérant certains paramètres (dimensions transverses des câbles par rapport à leurs longueurs, conductivité du milieu diélectrique par rapport à celle du métal des âmes, petite taille de la zone de jonction par rapport à l'ensemble du réseau) extrêmement petits.Une des difficultés mathématiques tient en ce que les domaines que nous prendrons en compte (sections des câbles, jonctions) ne sont aucunement simplement connexes, nous obligeant ainsi à remanier quelques outils standard tel les décompositions en potentiels.
• Trapped modes in thin and infinite ladder like domains: existence and asymptotic analysis
  
  • Delourme Bérangère
  • Fliss Sonia
  • Joly Patrick
  • Vasilevskaya Elizaveta
  , 2016. We are interested in a 2D propagation medium which is a localized perturbation of a reference homogeneous periodic medium. This reference medium is a "thick graph", namely a thin structure (the thinness being characterized by a small parameter $\varepsilon$ > 0) whose limit (when $\varepsilon$ tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. The question we investigate is whether such a geometrical perturbation is able to produce localized eigenmodes. We have investigated this question when the propagation model is the scalar Helmholtz equation with Neumann boundary conditions . This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We use a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough and construct an asymptotic expansion of the eigenvalues with respect to the small parameter. Following this approach, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung. Using the matched asymptotic expansion method, we obtain an asymptotic expansion at any order of the eigenvalues, which can be used for instance to compute a numerical approximations of these eigenvalues and associated eigenvectors.
• Set-valued solutions for non-ideal detonation
  
  • Semenko R.
  • Faria Luiz
  • Kasimov A.
  • Ermolaev B.
  Shock Waves, Springer Verlag, 2016, 26 (2), pp.141-160. The existence and structure of steady gaseous detonation propagating in a packed bed of solid inert particles are analyzed in the one-dimensional approximation by taking into consideration frictional and heat losses between the gas and the particles. A new formulation of the governing equations is introduced that eliminates the well-known difficulties with numerical integration across the sonic singularity in the reactive Euler equations. The new algorithm allows us to determine that the detonation solutions as the loss factors are varied have a set-valued nature at low detonation velocities when the sonic constraint disappears from the solutions. These set-valued solutions correspond to a continuous spectrum of the eigenvalue problem that determines the velocity of the detonation. (10.1007/s00193-015-0610-3)
  DOI : 10.1007/s00193-015-0610-3
• Dirichlet-to-Neumann operator for diffraction problems in stratified anisotropic acoustic waveguides
  
  • Tonnoir Antoine
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2016. The purpose of this note is to construct a Dirichlet-to-Neumann operator for the diffraction problem in stratified anisotropic acoustic waveguides. The key idea consists in using an adapted change of coordinates that enables to recover the completeness and the orthogonality of the modes on " deformed " cross-sections of the waveguide. To cite this article: (10.1016/j.crma.2015.12.018)
  DOI : 10.1016/j.crma.2015.12.018
• Effective boundary conditions for thin periodic coatings
  
  • Chamaillard Mathieu
  , 2016. We have dealt with the case of the scalar Helmholtz equation. We will try to handle the case of Maxwell's equation. We also will focus on the case of meta-materials. In a first case the permittivity is negative in the thin layer and in the second case is the permeability (1/delta) ^ 2.
• Imaging an acoustic waveguide from surface data in the time domain
  
  • Baronian Vahan
  • Bourgeois Laurent
  • Recoquillay Arnaud
  Wave Motion, Elsevier, 2016, 66, pp.68 - 87. This paper deals with an inverse scattering problem in an acoustic waveguide. The data consist of time domain signals given by sources and receivers located on the boundary of the waveguide. After transforming the data to the frequency domain, the obstacle is then recovered by using a modal formulation of the Linear Sampling Method. The impact of many parameters are analyzed, such as the numbers of sources/receivers and the distance between them, the time shape of the incident wave and the number and the values of the frequencies that are used. Some numerical experiments illustrate such analysis. (10.1016/j.wavemoti.2016.05.006)
  DOI : 10.1016/j.wavemoti.2016.05.006
• Effective transmission conditions for thin-layer transmission problems in elastodynamics. The case of a planar layer model
  
  • Bonnet Marc
  • Burel Aliénor
  • Duruflé Marc
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2016, 50, pp.43-75. This article is concerned with the design, analysis, numerical approximation and implementation of effective transmission conditions (ETCs) for the propagation of elastic waves through a thin planar elastic layer with small uniform thickness η which is embedded in a reference elastic medium, under transient conditions, with both materials assumed to have isotropic properties. A family of ETCs of order k (i.e. whose approximation error is of expected order O(η k+1)) is formulated by deriving and exploiting a formal asymptotic expansion in powers of η of the transmission solution inside the layer. The second-order ETCs are then retained as the main focus for the remainder of the article, and given a full justification in terms of both the stability of the resulting transient elastodynamic problem and the error analysis. The latter is performed by establishing and justifying asymptotic expansions for the solutions of both the exact transmission problem and its approximation based on the second-order ETCs. As a result, the error (in energy norm) between those two solutions is shown to be, as expected, of order O(η 3). Finally, the numerical approximation of the proposed second-order ETC within the framework of spectral element methods is studied, with special attention devoted to the selection of a robust time-stepping scheme that is mostly explicit (and conditionally stable). Among these, a scheme that is implicit only for the interfacial degrees of freedom, termed semi-implicit, is shown to be stable under the same stability condition as for the layer-less configuration. The main theoretical results of this work are illustrated and validated by 2D and 3D numerical experiments under transient elastodynamic conditions. (10.1051/m2an/2015030)
  DOI : 10.1051/m2an/2015030
• Time Domain Simulation of a Piano. Part 2 : Numerical Aspects
  
  • Chabassier Juliette
  • Duruflé Marc
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2016, 50 (1), pp.93-133. This article is the second of a series of two papers devoted to the numerical simulation of the piano. It concerns the numerical aspects of the work, the implementation of a piano code and the presentation of corresponding simulations. The main difficulty is time discretisation and stability is achieved via energy methods. Numerical illustrations are provided for a realistic piano and compared to experimental recordings.
• Improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow
  
  • Mercier Jean-François
  • Maurel Agnes
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2016, 472 (2190). (10.1098/rspa.2016.0094)
  DOI : 10.1098/rspa.2016.0094
• A multi-step solution algorithm for Maxwell boundary integral equations applied to low-frequency electromagnetic testing of conductive objects
  
  • Vigneron Audrey
  • Demaldent Edouard
  • Bonnet Marc
  IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 2016, 52, pp.7005208. We consider the solution, using boundary elements (BE), of the surface integral equation system arising in electromagnetic testing of conducting bodies, with emphasis on situations such that $o(1) \leq \sqrt{\omega\varepsilon_{0}/\sigma} \leq O(1)$, $L \sqrt{\omega\sigma\mu_{0}} =O(1)$ which includes in particular the case of eddy current testing) and assuming $L\omega\sqrt{\varepsilon_0 \mu_{0}}\leq 2\pi$, i.e. low-frequency conditions ($L$: diameter of conducting body). Earlier approaches for dielectric objects at low frequencies are not applicable in the present context. After showing that a simple normalization of the BE system significantly improves its conditioning, we propose a multi-step solution method based on block SOR iterations, which facilitates the use of direct solvers and converges within a few iterations for the considered range of physical parameters. This novel, albeit simple, treatment allows to perform eddy current-type analyses using standard Maxwell SIE formulations, avoiding the adverse consequences of ill-conditioning for low frequencies and high conductivities. Its performance and limitations are studied on three numerical examples involfing low frequencies and high conductivities. (10.1109/TMAG.2016.2584018)
  DOI : 10.1109/TMAG.2016.2584018
• On the approximation of electromagnetic fields by edge finite elements. Part 1: Sharp interpolation results for low-regularity fields
  
  • Ciarlet Patrick
  Computers & Mathematics with Applications, Elsevier, 2016. We propose sharp results on the numerical approximation of low-regularity electromagnetic fields by edge finite elements. We consider general geometrical settings, including topologically non-trivial domains or domains with a non-connected boundary. In the model, the electric permittivity and magnetic per-meability are symmetric, tensor-valued, piecewise smooth coefficients. In all cases, the error can be bounded by h δ times a constant, where h is the mesh-size, for some exponent δ ∈]0, 1] that depends both on the geometry and on the coefficients. It relies either on classical estimates when δ > 1/2, or on a new combined interpolation operator when δ < 1/2. The optimality of the value of δ is discussed with respect to abstract shift theorems. In some simple configurations , typically for scalar-valued permittivity and permeability, the value of δ can be further characterized. This paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements. (10.1016/j.camwa.2015.10.020)
  DOI : 10.1016/j.camwa.2015.10.020
• Mathematical Aspects of variational boundary integral equations for time dependent wave propagation
  
  • Joly Patrick
  • Rodríguez Jerónimo
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2016.
• Study of a Model Equation in Detonation Theory: Multidimensional Effects
  
  • Faria Luiz
  • Kasimov A.
  • Rosales R.
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2016, 76 (3), pp.887-909. We extend the reactive Burgers equation presented in [A. R. Kasimov, L. M. Faria,and R. R. Rosales,Phys. Rev. Lett., 110 (2013), 104104], [L. M. Faria, A. R. Kasimov, and R. R.Rosales,SIAM J. Appl. Math., 74 (2014), pp. 547–570] to include multidimensional effects. Fur-thermore, we explain how the model can be rationally justified following the ideas of the asymptotictheory developed in [L. M. Faria, A. R. Kasimov, and R. R. Rosales,J. Fluid Mech., 784 (2015),pp. 163–198]. The proposed model is a forced version of the unsteady small disturbance transonicflow equations. We show that for physically reasonable choices of forcing functions, traveling wavesolutions akin to detonation waves exist. It is demonstrated that multidimensional effects play animportant role in the stability and dynamics of the traveling waves. Numerical simulations indicatethat solutions of the model tend to form multidimensional patterns analogous to cells in gaseousdetonations. (10.1137/15M1039663)
  DOI : 10.1137/15M1039663