Publications

2017

• A GPU-accelerated nodal discontinuous Galerkin method with high-order absorbing boundary conditions and corner/edge compatibility
  
  • Modave Axel
  • Atle Andreas
  • Chan Jesse
  • Warburton Tim
  International Journal for Numerical Methods in Engineering, Wiley, 2017, 112 (11), pp.1659-1686. Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large-scale wave-propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with non-reflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high-order absorbing boundary conditions (HABCs) for cuboidal computational domains. Compatibility conditions are derived for HABCs intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach. (10.1002/nme.5576)
  DOI : 10.1002/nme.5576
• Acoustic propagation in a vortical homentropic flow
  
  • Mercier Jean-François
  • Mietka Colin
  • Millot Florence
  • Pagneux Vincent
  , 2017. This paper is devoted to the theoretical and the numerical studies of the radiation 4 of an acoustic source in a general homentropic flow. As a linearized model, we consider Goldstein's 5 Equations, which extend the usual potential model to vortical flows. The equivalence between 6 Linearized Euler's Equations with general source terms and Goldstein's Equations is established, 7 and the relations between unknowns, in each model, are analysed. A closed-form relation between 8 the hydrodynamic phenomena and the acoustics is derived. Finally, numerical results are presented 9 and the relevance of using Goldstein's Equations compared to the potential model is illustrated.
• Influence of the neck shape for Helmholtz resonators
  
  • Mercier Jean-François
  • Marigo Jean-Jacques
  • Maurel Agnès
  Journal of the Acoustical Society of America, Acoustical Society of America, 2017, 142 (6), pp.3703 - 3714. The resonance of a Helmholtz resonator is studied with a focus on the influence of the neck shape. This is done using a homogenization approach developed for an array of resonators, and the resonance of an array is discussed when compared to that of a single resonator. The homogenization makes a parameter ℬ appear which determines unambiguously the resonance frequency of any neck. As expected, this parameter depends on the length and on the minimum opening of the neck, and it is shown to depend also on the surface of air inside the neck. Once these three geometrical parameters are known, ℬ has an additional but weak dependence on the neck shape, with explicit bounds. (10.1121/1.5017735)
  DOI : 10.1121/1.5017735
• XLiFE++: a FEM/BEM multipurpose library
  
  • Kielbasiewicz Nicolas
  • Lunéville Éric
  , 2017, pp.1.
• Spectral theory for Maxwell's equations at the interface of a metamaterial. Part I: Generalized Fourier transform
  
  • Cassier Maxence
  • Hazard Christophe
  • Joly Patrick
  Communications in Partial Differential Equations, Taylor & Francis, 2017, 42 (11), pp.1707-1748. We explore the spectral properties of the time-dependent Maxwell's equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. We construct explicitly a generalized Fourier transform which diagonalizes the Hamiltonian that describes the propagation of transverse electric waves. This transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem. It will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles.
• Une méthode hybride couplant la méthode des équations intégrales et la méthode des rayons en vue d'applications au contrôle non destructif ultrasonore.
  
  • Pesudo Laure
  , 2017. Le Contrôle Non Destructif (CND) permet de sonder l’intérieur d’un milieu dans le but desurveiller son intégrité et son vieillissement. Assisté d’outils de simulation il permet de détecter, caractériseret localiser des défauts de structure du milieu inspecté mais sa fiabilité dépend de la précision des méthodesde simulation. Dans le cadre du CND ultrasonore, les méthodes usuelles (numériques et asymptotiques) sontbien souvent inadéquates pour simuler la diffraction par les défauts. On leur préfère des techniques hybrides.On propose dans cette thèse une nouvelle approche hybride pour la simulation numérique de la diffractionhaute fréquence en milieu étendu (configuration critique pour le CND). Combinant la méthode des équationsintégrales et la méthode des rayons, cette approche exploite le caractère multi-échelle du problème hautefréquence en proposant un modèle d’obstacle à deux échelles. Elle permet le calcul précis de la diffraction etla propagation rayon des champs. D’abord mise au point dans le cadre de la diffraction d’ondes acoustiquespar un obstacle de taille inférieure à la longueur d’onde (méthode barycentrique), l’approche est ensuiteétendue à des configurations de diffraction par des obstacles de l’ordre de la longueur d’onde grâce àl’introduction d’un partitionnement de l’unité de sa surface (méthode multi-centres). Pour accélérerl’approche hybride, on propose une procédure de résolution Online-Offline, basée sur un pré-calcul de lamatrice de diffraction associée à un ensemble réduit de directions d’incidence et d’observation et sur uneinterpolation polynomiale de ses vecteurs singuliers pour son évaluation dans des directions quelconquesd’émetteurs et de récepteurs. On étudie ensuite la stratégie dans le cadre de l’acoustique 3D puis on en faitune extension de principe à l’élastodynamique. On donne enfin un ensemble de pistes pour étendre l’approchehybride dans des cas de diffraction par un ou plusieurs obstacles pouvant être proches des bords du milieu.
• High-order absorbing boundary conditions with edge and corner compatibility for the Helmholtz equation
  
  • Modave Axel
  • Mattesi Vanessa
  • Geuzaine Christophe
  , 2017. We deal with the finite element solution of 3D time-harmonic acoustic wave problems defined on unbounded domains, but computed using cuboidal computational domains with artificial boundaries. We combine a standard finite element method for the Helmholtz equation with high-order absorbing boundary conditions (on the faces of the domain) and compatibility relations (on the edges and the corners) that provide an arbitrary high accuracy.
• Asymptotic analysis based modeling of small inhomogeneity perturbation in solids: two computational scenarios
  
  • Marenić Eduard
  • Brancherie D.
  • Bonnet Marc
  , 2017. The presented work is a step towards designing a numerical strategy capable of assessing the nocivity of a small defect in terms of its size and position in the structure with low computational cost, using only a mesh of the defect-free reference structure. We focus here on presenting two computational scenarios allowing to efficiently evaluate flaw criticality. These scenarios are considering either the effect of a fixed flaw for any evalutaion point in solid, or varying flaws on a fixed evaluation point. 1 Motivation, introduction and problem definition The role played by defects in the initiation and development of rupture is crucial and has to be taken into account in order to realistically describe the behavior till complete failure. The difficulties in that context revolve around (i) the fact that the defect length scale is much smaller than the structure length scale, and (ii) the random nature of their position and size. Even in a purely deterministic approach, taking those defects into consideration by standard models imposes to resort to geometrical discretisations at the defect scale, leading to very costly computations and hindering parametric studies in terms of defect location and characteristics. Our current goal is to design an efficient two-scale numerical strategy which can accurately predict the perturbation in terms of stress caused by an inhomogeneity in elastic (back-ground) material. To make it computationally efficient, the analysis uses only a mesh for the defect-free structure, i.e. the mesh size does not depend on the (small) defect scale. We consider a linearly elastic body occupying a smooth bounded domain Ω ⊂ R d (with the spatial dimensionality d = 2 or 3), whose boundary Γ is partitioned as Γ = Γ D ∪Γ N support a prescribed traction ¯ t and a prescribed displacement ¯ u, while a body force density f is applied in Ω. On the basis of this fixed geometrical and loading configuration, we consider two situations, namely (i) a reference solid characterized by a given elasticity tensor C, which defines the background solution u, and (ii) a perturbed solid constituted of the same background material except for a small inhomogeneity whose material is characterized by C , which defines a perturbed solution u a. The aim of this work is to formulate a computational approach allowing to treat case (ii) as a perturbation of the background solution (i), in particular avoiding any meshing at the small inhomogeneity scale. This will be achieved by applying known results on the asymptotic expansion of the displacement
• Theory and implementation of $\mathcal{H}$-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
  
  • Chaillat Stéphanie
  • Desiderio Luca
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2017. In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard $\mathcal{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. $\mathcal{H}^2$-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard $\mathcal{H}$-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the $\mathcal{H}$-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.
• Nonlocal models for interface problems between dielectrics and metamaterials
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  , 2017, pp.3. Consider two materials with permittivities/diffusivities of opposite sign, and separated by an interface with a corner. Then, when solving the classic (local) models derived from electromagnetics theory, strong singularities may appear. For instance the scalar problem may be ill-posed in H1. To address this difficulty, we study here a nonlocal model for scalar problems with sign-changing coefficients. Numerical results indicate that the proposed nonlocal model has some key advantages over the local one.
• Computer-implemented method for reconstructing the topology of a network of cables
  
  • Beck Geoffrey
  , 2017, pp.https://patents.google.com/patent/US20200363462A1/en. A computer-implemented iterative method for reconstructing the topology of a cable network, includes the steps of: obtaining a measured time reflectogram, obtaining a simulated time reflectogram corresponding to a partial cable network comprising the singular points of the cable network reconstructed in the previous iterations and a matched load at the end of each cable, one endpoint of which is not yet reconstructed, subtracting the simulated time reflectogram from the measured time reflectogram in order to obtain a corrected time reflectogram, reconstructing the topology of the cable network by matching the peaks of the corrected reflectogram with the singular points of the cable network, the matching comprising searching, in the corrected reflectogram, for at least one second peak corresponding to a path of the signal comprising a main reflection off the ambiguous singular point and a secondary reflection off two junctions reconstructed in the previous iterations.
• Equation Level Matching: An Extension of the Method of Matched Asymptotic Expansion for Problems of Wave Propagation
  
  • Faria Luiz
  • Rosales R.
  Studies in Applied Mathematics, Wiley-Blackwell, 2017, 139 (2), pp.265-287. We introduce an alternative to the method of matched asymptotic expansions. In the “traditional” implementation, approximate solutions, valid in different (but overlapping) regions are matched by using “intermediate” variables. Here we propose to match at the level of the equations involved, via a “uniform expansion” whose equations enfold those of the approximations to be matched. This has the advantage that one does not need to explicitly solve the asymptotic equations to do the matching, which can be quite impossible for some problems. In addition, it allows matching to proceed in certain wave situations where the traditional approach fails because the time behaviors differ (e.g., one of the expansions does not include dissipation). On the other hand, this approach does not provide the fairly explicit approximations resulting from standard matching. In fact, this is not even its aim, which to produce the “simplest” setof equations that capture the behavior. (10.1111/sapm.12183)
  DOI : 10.1111/sapm.12183
• The "exterior approach" applied to the inverse obstacle problem for the heat equation
  
  • Bourgeois Laurent
  • Dardé Jérémi
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2017, 55 (4). In this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification. 1. Introduction. This paper deals with the inverse obstacle problem for the heat equation, which can be described as follows. We consider a bounded domain D ⊂ R d , d ≥ 2, which contains an inclusion O. The temperature in the complementary domain Ω = D \ O satisfies the heat equation while the inclusion is characterized by a zero temperature. The inverse problem consists, from the knowledge of the lateral Cauchy data (that is both the temperature and the heat flux) on a subpart of the boundary ∂D during a certain interval of time (0, T) such that the temperature at time t = 0 is 0 in Ω, to identify the inclusion O. Such kind of inverse problem arises in thermal imaging, as briefly described in the introduction of [9]. The first attempts to solve such kind of problem numerically go back to the late 80's, as illustrated by [1], in which a least square method based on a shape derivative technique is used and numerical applications in 2D are presented. A shape derivative technique is also used in [11] in a 2D case as well, but the least square method is replaced by a Newton type method. Lastly, the shape derivative together with the least square method have recently been used in 3D cases [18]. The main feature of all these contributions is that they rely on the computation of forward problems in the domain Ω × (0, T): this computation obliges the authors to know one of the two lateral Cauchy data (either the temperature or the heat flux) on the whole boundary ∂D of D. In [20], the authors introduce the so-called " enclosure method " , which enables them to recover an approximation of the convex hull of the inclusion without computing any forward problem. Note however that the lateral Cauchy data has to be known on the whole boundary ∂D. The present paper concerns the " exterior approach " , which is an alternative method to solve the inverse obstacle problem. Like in [20], it does not need to compute the solution of the forward problem and in addition, it is applicable even if the lateral Cauchy data are known only on a subpart of ∂D, while no data are given on the complementary part. The " exterior approach " consists in defining a sequence of domains that converges in a certain sense to the inclusion we are looking for. More precisely, the nth step consists, 1. for a given inclusion O n , in approximating the temperature in Ω n × (0, T) (Ω n := D \ O n) with the help of a quasi-reversibility method, 2. for a given temperature in Ω n × (0, T), in computing an updated inclusion O n+1 with the help of a level set method. Such " exterior approach " has already been successfully used to solve inverse obstacle problems for the Laplace equation [5, 4, 15] and for the Stokes system [6]. It has also been used for the heat equation in the 1D case [2]: the problem in this simple case might be considered as a toy problem since the inclusion reduces to a point in some bounded interval. The objective of the present paper is to extend the " exterior approach " for the heat equation to any dimension of space, with numerical applications in the 2D case. Let us shed some light on the two steps of (10.1137/16M1093872)
  DOI : 10.1137/16M1093872
• Quantification of the unique continuation property for the heat equation
  
  • Bourgeois Laurent
  Mathematical Control and Related Fields, AIMS, 2017, 7 (3), pp.347 - 367. In this paper we prove a logarithmic stability estimate in the whole domain for the solution to the heat equation with a source term and lateral Cauchy data. We also prove its optimality up to the exponent of the logarithm and show an application to the identification of the initial condition and to the convergence rate of the quasi-reversibility method. (10.3934/mcrf.2017012)
  DOI : 10.3934/mcrf.2017012
• Stable Perfectly Matched Layers for a Cold Plasma in a Strong Background Magnetic Field
  
  • Bécache Eliane
  • Joly Patrick
  • Kachanovska Maryna
  Journal of Computational Physics, Elsevier, 2017, 341, pp.76-101. This work addresses the question of the construction of stable perfectly matched layers (PML) for a cold plasma in the infinitely large background magnetic field. We demonstrate that the traditional, Bérenger's perfectly matched layers are unstable when applied to this model, due to the presence of the backward propagating waves. To overcome this instability, we use a combination of two techniques presented in the article. First of all, we consider a simplified 2D model, which incorporates some of the difficulties of the 3D case, namely, the presence of the backward propagating waves. Based on the fact that for a fixed frequency either forward or backward propagating waves are present, we stabilize the PML with the help of a frequency-dependent correction. An extra difficulty of the 3D model compared to the 2D case is the presence of both forward and backward waves for a fixed frequency. To overcome this problem we construct a system of equations that consists of two independent systems, which are equivalent to the original model. The first of the systems behaves like the 2D plasma model, and hence the frequency-dependent correction is added to the PML for the stabilization. The second system resembles the Maxwell equations in vacuum, and hence a standard Bérenger's PML is stable for it. The systems are solved inside the perfectly matched layer, and coupled to the original Maxwell equations, which are solved in a physical domain, on a discrete level through an artificial layer. The numerical experiments confirm the stability of the new technique. (10.1016/j.jcp.2017.03.051)
  DOI : 10.1016/j.jcp.2017.03.051
• Trapped modes in thin and infinite ladder like domains. Part 1 : existence results
  
  • Delourme Bérangère
  • Fliss Sonia
  • Joly Patrick
  • Vasilevskaya Elizaveta
  Asymptotic Analysis, IOS Press, 2017, 103(3) (103-134). The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter $\epsilon > 0$) whose limit (as $\epsilon$ tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the $\epsilon$ tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that ε is small enough. Numerical experiments illustrate the theoretical results.
• Eigenvalue problems with sign-changing coefficients
  
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2017, 355 (6), pp.671 - 675. We consider a class of eigenvalue problems involving coefficients changing sign on the domain of interest. We describe the main spectral properties of these problems according to the features of the coefficients. Then, under some assumptions on the mesh, we explain how one can use classical finite element methods to approximate the spectrum as well as the eigenfunctions while avoiding spurious modes. We also prove localisation results of the eigenfunctions for certain sets of coefficients. (10.1016/j.crma.2017.05.002)
  DOI : 10.1016/j.crma.2017.05.002
• Asymptotic analysis for criticality assessment of defects in mechanical structures
  
  • Marenić Eduard
  • Brancherie Delphine
  • Bonnet Marc
  , 2017. The presented work is a step towards designing a numerical strategy capable of assessing the nocivity of a small defect in terms of its size and position in the structure with low computational cost, using only a mesh of the defect-free reference structure. The proposed strategy would allow to assess the criticality of defects by introducing trial micro-defects with varying positions, sizes and me- chanical properties. The main focus of the this work is to present two computational scenarios allowing to efficiently evaluate criticality considering the effect of either a fixed flaw on a region of interest or varying flaws on a fixed evaluation point.
• Calcul des opérateurs d'impédance en Interaction Sol-Structure: méthode éléments de frontière accélérée par méthode multipôle rapide
  
  • Adnani Zouhair
  • Chaillat Stéphanie
  • Bonnet Marc
  • Nieto Ferro Alex
  • Greffet Nicolas
  , 2017. Les effets de site, qu’ils soient d’origine topographique ou lithologique, influencent la propagation des ondes sismiques et peuvent provoquer une amplification ou atténuation du mouvement sismique, ainsi que la modification de son spectre. Ce travail concerne le développement d’une stratégie de calcul numérique pour la prise en compte des effets de sites dans les calculs d’Interaction Sol-Structure. Il repose sur une nouvelle stratégie de couplage des éléments finis aux éléments de frontière accélérés par la méthode multipôle rapide.
• A nodal discontinuous Galerkin method with high-order absorbing boundary conditions and corner/edge compatibility
  
  • Modave Axel
  • Atle Andreas
  • Chan Jesse
  • Warburton Tim
  , 2017. We present the coupling of a nodal discontinu-ous Galerkin (DG) scheme with high-order absorbing boundary conditions (HABCs) for the simulation of transient wave phenomena. The HABCs are prescribed on the faces of a cuboidal domain in order to simulate infinite space. To preserve accuracy at the corners and the edges of the domain, novel compatibility conditions are derived. The method is validated using 3D computational results.
• Topological derivatives of leading-and second-order homogenized coefficients in bi-periodic media
  
  • Cornaggia Rémi
  • Guzina Bojan B
  • Bonnet Marc
  , 2017. We derive the topological derivatives of the homogenized coefficients associated to a periodic material, with respect of the small size of a penetrable inhomogeneity introduced in the unit cell that defines such material. In the context of an-tiplane elasticity, this work extends existing results to (i) time-harmonic wave equation and (ii) second-order homogenized coefficients, whose contribution reflects the dispersive behavior of the material.
• 3D metric-based anisotropic mesh adaptation for the fast multipole accelerated boundary element method in acoustics
  
  • Amlani Faisal
  • Chaillat Stéphanie
  • Groth Samuel P
  • Loseille Adrien
  , 2017. We introduce a metric-based anisotropic mesh adaptation strategy for the fast multipole accelerated boundary element method (FM-BEM) applied to exterior boundary value problems of the three-dimensional Helmholtz equation. The present methodology is independent of discretiz-ation technique and iteratively constructs meshes refined in size, shape and orientation according to an " optimal " metric reliant on a reconstructed Hessian of the boundary solution. The resulting adaptation is anisotropic in nature and numerical examples demonstrate optimal convergence rates for domains that include geometric singularities such as corners and ridges.
• Magnetization moment recovery using Kelvin transformation and Fourier analysis
  
  • Baratchart Laurent
  • Leblond Juliette
  • Lima Eduardo Andrade
  • Ponomarev Dmitry
  , 2017. In the present work, we consider a magnetization moment recovery problem, that is finding integral of the vector function (over its compact support) whose divergence constitutes a source term in the Poisson equation. We outline derivation of explicit asymptotic formulas for estimation of the net magnetization moment vector of the sample in terms of partial data for the vertical component of the magnetic field measured in the plane above it. For this purpose, two methods have been developed: the first one is based on approximate projections onto spherical harmonics in Kelvin domain while the second stems from analysis in Fourier domain following asymptotic continuation of the data. Recovery results obtained by both methods agree and are illustrated numerically by plotting formulas for net moment components with respect to the size of the measurement area.
• On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations
  
  • Ciarlet Patrick
  • Fliss Sonia
  • Stohrer Christian
  Computers & Mathematics with Applications, Elsevier, 2017, 73 (9), pp.1900-1919. In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell’s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell’s equations in a periodic setting and give numerical experiments in accordance to the stated results. (10.1016/j.camwa.2017.02.043)
  DOI : 10.1016/j.camwa.2017.02.043
• Criticality Computation with Finite Element Method on Non-Conforming Meshes
  
  • Giret L.
  • Ciarlet Patrick
  • Jamelot E.
  , 2017. In this work, we proposed and study a method to use non-conforming meshing for core reactor simulation. This consists in a domain decomposition with Lagrange multipliers of the well known Raviart-Thomas finite element method. Here, we provide an a priori error estimate for criticality computation.