Publications

2017

• Stable perfectly matched layers for a class of anisotropic dispersive models. Part II: Energy estimates
  
  • Kachanovska Maryna
  , 2017. This article continues the stability analysis of the generalized perfectly matched layers for 2D anisotropic dispersive models studied in Part I of the work. We obtain explicit energy estimates for the PML system in the time domain, by making use of the ideas stemming from the analysis of the associated sesquilinear form in the Laplace domain. This analysis is based on the introduction of a particular set of auxiliary unknowns related to the PML, which simplifies the derivation of the energy estimates for the resulting system. For 2D dispersive systems, our analysis allows to demonstrate the stability of the PML system for a constant absorption parameter. For 1D dispersive systems, we show the stability of the PMLs with a non-constant absorption parameter.
• Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations
  
  • Modave Axel
  • Lambrechts Jonathan
  • Geuzaine Christophe
  Computers & Mathematics with Applications, Elsevier, 2017, 73 (4), pp.684-700. This paper deals with the design of perfectly matched layers (PMLs) for transient acoustic wave propagation in generally-shaped convex truncated domains. After reviewing key elements to derive PML equations for such domains, we present two time-dependent formulations for the pressure-velocity system. These formulations are obtained by using a complex coordinate stretching of the time-harmonic version of the equations in a specific curvilinear coordinate system. The final PML equations are written in a general tensor form, which can easily be projected in Cartesian coordinates to facilitate implementation with classical discretization methods. Discontinuous Galerkin finite element schemes are proposed for both formulations. They are tested and compared using a three-dimensional benchmark with an ellipsoidal truncated domain. Our approach can be generalized to domains with corners. (10.1016/j.camwa.2016.12.027)
  DOI : 10.1016/j.camwa.2016.12.027
• H-matrix based Solver for 3D Elastodynamics Boundary Integral Equations
  
  • Desiderio Luca
  , 2017. This thesis focuses on the theoretical and numerical study of fast methods to solve the equations of 3D elastodynamics in frequency-domain. We use the Boundary Element Method (BEM) as discretization technique, in association with the hierarchical matrices (H-matrices) technique for the fast solution of the resulting linear system. The BEM is based on a boundary integral formulation which requires the discretization of the only domain boundaries. Thus, this method is well suited to treat seismic wave propagation problems. A major drawback of classical BEM is that it results in dense matrices, which leads to high memory requirement (O (N 2 ), if N is the number of degrees of freedom) and computational costs.Therefore, the simulation of realistic problems is limited by the number of degrees of freedom. Several fast BEMs have been developed to improve the computational efficiency. We propose a fast H-matrix based direct BEM solver.
• A model for Faraday pilot waves over variable topography
  
  • Faria Luiz
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2017, 811, pp.51-66. (10.1017/jfm.2016.750)
  DOI : 10.1017/jfm.2016.750
• Seismic Wave Amplification in 3D Alluvial Basins: Aggravation factors from Fast Multipole BEM Simulations
  
  • Meza Fajardo Kristel Carolina
  • Semblat Jean-François
  • Chaillat Stéphanie
  • Lenti Luca
  , 2017. In this work, we study seismic wave amplification in alluvial basins having 3D canonical geometries through the Fast Multipole Boundary Element Method in the frequency domain. We investigate how much 3D amplification differs from the 1D (horizontal layering) and the 2D cases. Considering synthetic incident wave-fields, 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 equivalent shape ratio. Using these two parameters, we derive simple rules to compute the fundamental frequency for different 3D basin shapes and the corresponding 3D aggravation factor for 5% damping.Effects on amplification due to 3D basin asymmetry are also studied and incorporated in the derived rules.
• On some extremal problems for analytic functions with constraints on real or imaginary parts
  
  • Leblond Juliette
  • Ponomarev Dmitry
  , 2017, pp.219-236. We study some approximation problems by functions in the Hardy space H 2 of the upper half-plane or by their real or imaginary parts, with constraint on their real or imaginary parts on the boundary. Situations where the criterion acts on subsets of the boundary or of horizontal lines inside the half-plane are considered. Existence and uniqueness results are established, together with novel solution formulas and techniques. As a by-product, we devise a regularized inversion scheme for Poisson and conjugate Poisson integral transforms. (10.1007/978-3-319-62362-7_8)
  DOI : 10.1007/978-3-319-62362-7_8
• Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations
  
  • Cohen Gary
  • Pernet Sebastien
  , 2017. This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem of its spurious-free approximations. Treatment of unbounded domains by Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML) is described and analyzed in a separate chapter. The two last chapters deal with time approximation including local time-stepping and with the study of some complex models, i.e. acoustics in flow, gravity waves and vibrating thin plates. Throughout, emphasis is put on the accuracy and computational efficiency of the methods, with attention brought to their practical aspects.This monograph also covers in details the theoretical foundations and numerical analysis of these methods. As a result, this monograph will be of interest to practitioners, researchers, engineers and graduate students involved in the numerical simulationof waves. (10.1007/978-94-017-7761-2)
  DOI : 10.1007/978-94-017-7761-2
• Manipulating light at subwavelength scale by exploiting defect-guided spoof plasmon modes
  
  • Ourir Abdelwaheb
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  • Fink Mathias
  Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2017, 96 (12). We study the defect-guided modes supported by a set of metallic rods structured at the subwavelength scale. Following the idea of photonic crystal waveguide, we show that spoof plasmon surface waves can be manipulated at subwavelength scale. We demonstrate that these waves can propagate without leakage along a row of rods having a different length than the surrounding medium and we provide the corresponding dispersion relation. The principle of this subwavelength colored guide is validated experimentally. This allows us to propose the design of a wavelength demultiplexer whose efficiency is illustrated in the microwave regime. (10.1103/PhysRevB.96.125133)
  DOI : 10.1103/PhysRevB.96.125133
• Mathematical models for dispersive electromagnetic waves: An overview
  
  • Cassier Maxence
  • Joly Patrick
  • Kachanovska Maryna
  Computers & Mathematics with Applications, Elsevier, 2017, 74 (11), pp.2792-2830. In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background. (10.1016/j.camwa.2017.07.025)
  DOI : 10.1016/j.camwa.2017.07.025
• Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics
  
  • Chaillat Stéphanie
  • Darbas Marion
  • Le Louër Frédérique
  Journal of Computational Physics, Elsevier, 2017, 341, pp.429 - 446. The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles. (10.1016/j.jcp.2017.04.020)
  DOI : 10.1016/j.jcp.2017.04.020
• A modified volume integral equation for anisotropic elastic or conducting inhomogeneities. Unconditional solvability by Neumann series
  
  • Bonnet Marc
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2017, 29, pp.271-295. This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e. solvable by Neumann series, implying the well-posedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast (1999) on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background, and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i) unconditionally for static problems and (ii) on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.
• Application of asymptotic analysis to the two-scale modeling of small defects in mechanical structures
  
  • Marenić Eduard
  • Brancherie Delphine
  • Bonnet Marc
  International Journal of Solids and Structures, Elsevier, 2017, 128, pp.199-209. This work aims at designing a numerical strategy towards assessing the nocivity of a small defect in terms of its size and position in a structure, at low computational cost, using only a mesh of the defect-free reference structure. The modification of the fields induced by the presence of a small defect is taken into account by using asymptotic corrections of displacements or stresses. This approach helps determining the potential criticality of defects by considering trial micro-defects with varying positions, sizes and mechanical properties, taking advantage of the fact that parametric studies on defect characteristics become feasible at virtually no extra computational cost. The proposed treatment is validated and demonstrated on two numerical examples involving 2D elastic configurations. (10.1016/j.ijsolstr.2017.08.029)
  DOI : 10.1016/j.ijsolstr.2017.08.029
• Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability
  
  • Bécache Eliane
  • Kachanovska Maryna
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2017, 51 (6), pp.2399-2434. In this work we consider a problem of modelling of 2D anisotropic dispersive wave propagation in unbounded domains with the help of perfectly matched layers (PML). We study the Maxwell equations in passive media with a frequency-dependent diagonal tensor of dielectric permittivity and magnetic permeability. An application of the traditional PMLs to this kind of problems often results in instabilities. We provide a recipe for the construction of new, stable PMLs. For a particular case of non-dissipative materials, we show that a known necessary stability condition of the perfectly matched layers is also sufficient. We illustrate our statements with theoretical and numerical arguments. (10.1051/m2an/2017019)
  DOI : 10.1051/m2an/2017019
• Fonctions d'une variable complexe (avec des applications et de nombreux dessins)
  
  • Lenoir Marc
  , 2017.
• A two-way model for nonlinear acoustic waves in a non-uniform lattice of Helmholtz resonators
  
  • Mercier Jean-François
  • Lombard Bruno
  Wave Motion, Elsevier, 2017, 72, pp.260-275. Propagation of high amplitude acoustic pulses is studied in a 1D waveguide connected to a lattice of Helmholtz resonators. An homogenized model has been proposed by Sugimoto (J. Fluid. Mech., \textbf{244} (1992)), taking into account both the nonlinear wave propagation and various mechanisms of dissipation. This model is extended here to take into account two important features: resonators of different strengths and back-scattering effects. An energy balance is obtained, and a numerical method is developed. A closer agreement is reached between numerical and experimental results. Numerical experiments are also proposed to highlight the effect of defects and of disorder. (10.1016/j.wavemoti.2017.04.004)
  DOI : 10.1016/j.wavemoti.2017.04.004
• Higher order topological derivatives for three-dimensional anisotropic elasticity
  
  • Bonnet Marc
  • Cornaggia Rémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2017 (51), pp.2069-2092. This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function J is expanded in powers of the characteristic size a of a single small inhomogeneity. The O(a 6) approximation of J is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed. (10.1051/m2an/2017015)
  DOI : 10.1051/m2an/2017015
• Domain decomposition methods for the diffusion equation with low-regularity solution
  
  • Ciarlet Patrick
  • Jamelot Erell
  • Kpadonou Félix D.
  Computers & Mathematics with Applications, Elsevier, 2017. We analyze matching and non-matching domain decomposition methods for the numerical approximation of the mixed diffusion equations. Special attention is paid to the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. The domain decomposition method can be non-matching in the sense that the traces of the finite elements spaces may not fit at the interface between subdomains. We prove well-posedness of the discrete problem, that is solvability of the corresponding linear system, provided two algebraic conditions are fulfilled. If moreover the conditions hold independently of the discretization, convergence is ensured. (10.1016/j.camwa.2017.07.017)
  DOI : 10.1016/j.camwa.2017.07.017