Publications

2013

• Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section
  
  • Imperiale Sébastien
  • Joly Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2013. In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and elec- tric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain "homogenized" telegraphist's model. This model, which includes a non-local in time op- erator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross-section is not homogeneous. (10.1016/j.apnum.2013.03.011)
  DOI : 10.1016/j.apnum.2013.03.011
• Acoustic propagation in non-uniform waveguides: revisiting Webster equation using evanescent boundary modes
  
  • Mercier Jean-François
  • Maurel Agnès
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2013, 469. no abstract (10.1098/rspa.2013.0186)
  DOI : 10.1098/rspa.2013.0186
• On the use of the Linear Sampling Method to identify cracks in elastic waveguides
  
  • Bourgeois Laurent
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2013, 29, pp.025017. We consider the identification of cracks in an elastic 2D or 3D waveguide with the help of a modal version of the linear sampling method. The main objective of our paper is to show that since the usual crack in elasticity is traction free, that is, the boundary condition on the lips of the crack is a priori known to be of Neumann type, we shall adapt the formulation of the sampling method to such a boundary condition in order to improve the efficiency of the method. The need for such adaptation is proved theoretically and illustrated numerically with the help of 2D examples. (10.1088/0266-5611/29/2/025017)
  DOI : 10.1088/0266-5611/29/2/025017
• Strongly oscillating singularities for the interior transmission eigenvalue problem
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  Inverse Problems, IOP Publishing, 2013, 19(10), pp.104004. In this paper, we investigate a two-dimensional interior transmission eigenvalue problem for an inclusion made of a composite material. We consider configurations where the difference between the parameters of the composite material and those of the background changes sign on the boundary of the inclusion. In a first step, under some assumptions on the parameters, we extend the variational approach of the T-coercivity to prove that the transmission eigenvalues form at most a discrete set. In the process, we also provide localization results. Then, we study what happens when these assumptions are not satisfied. The main idea is that, due to very strong singularities that can occur at the boundary, the problem may lose Fredholmness in the natural H1 framework. Using Kondratiev theory, we propose a new functional framework where the Fredholm property is restored. (10.1088/0266-5611/29/10/104004)
  DOI : 10.1088/0266-5611/29/10/104004
• Large Scale Parameter Estimation Problems in Frequency-Domain Elastodynamics Using an Error in Constitutive Equation Functional
  
  • Banerjee Biswanath
  • Walsh Timothy
  • Aquino Wilkins
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2013, 253, pp.60-72. This paper presents the formulation and implementation of an Error in Constitutive Equations (ECE) method suitable for large-scale inverse identification of linear elastic material properties in the context of steady-state elastodynamics. In ECE-based methods, the inverse problem is postulated as an optimization problem in which the cost functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses. Furthermore, in a more recent modality of this methodology introduced by Feissel and Allix (2007), referred to as the Modified ECE (MECE), the measured data is incorporated into the formulation as a quadratic penalty term. We show that a simple and efficient continuation scheme for the penalty term, suggested by the theory of quadratic penalty methods, can significantly accelerate the convergence of the MECE algorithm. Furthermore, a (block) successive over-relaxation (SOR) technique is introduced, enabling the use of existing parallel finite element codes with minimal modification to solve the coupled system of equations that arises from the optimality conditions in MECE methods. Our numerical results demonstrate that the proposed methodology can successfully reconstruct the spatial distribution of elastic material parameters from partial and noisy measurements in as few as ten iterations in a 2D example and fifty in a 3D example. We show (through numerical experiments) that the proposed continuation scheme can improve the rate of convergence of MECE methods by at least an order of magnitude versus the alternative of using a fixed penalty parameter. Furthermore, the proposed block SOR strategy coupled with existing parallel solvers produces a computationally efficient MECE method that can be used for large scale materials identification problems, as demonstrated on a 3D example involving about 400,000 unknown moduli. Finally, our numerical results suggest that the proposed MECE approach can be significantly faster than the conventional approach of L2 minimization using quasi-Newton methods. (10.1016/j.cma.2012.08.023)
  DOI : 10.1016/j.cma.2012.08.023
• An hp-finite element approximation of guided modes in photonic crystal waveguides using transparent boundary conditions
  
  • Klindworth Dirk
  • Schmidt Kersten
  • Fliss Sonia
  Computers & Mathematics with Applications, Elsevier, 2013. no abstract
• The topological derivative in anisotropic elasticity
  
  • Bonnet Marc
  • Delgado Gabriel
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2013, 66, pp.557-586. A comprehensive treatment of the topological derivative for anisotropic elasticity is presented, with both the background material and the trial small inhomogeneity assumed to have arbitrary anisotropic elastic properties. A formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are treated; they differ mostly on details in the elastic moment tensor (EMT). Moreover, the main properties of the EMT, a critical component of the topological derivative, are studied for the fully anisotropic case. Then, the topological derivative of strain energy-based quadratic cost functionals is derived, which requires a distinct treatment. Finally, numerical experiments on the numerical evaluation of the EMT and the topological derivative of the compliance cost functional are reported. (10.1093/qjmam/hbt018)
  DOI : 10.1093/qjmam/hbt018
• Two-dimensional Maxwell's equations with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2013. We consider the theoretical study of time harmonic Maxwellʼs equations in presence of sign-changing coefficients, in a two-dimensional configuration. Classically, the problems for both the Transverse Magnetic and the Transverse Electric polarizations reduce to an equivalent scalar Helmholtz type equation. For this scalar equation, we have already studied consequences of the presence of sign-changing coefficients in previous papers, and we summarize here the main results. Then we focus on the alternative approach which relies on the two-dimensional vectorial formulations of the TM or TE problems, and we exhibit some unexpected effects of the sign-change of the coefficients. In the process, we provide new results on the scalar equations. (10.1016/j.apnum.2013.04.006)
  DOI : 10.1016/j.apnum.2013.04.006
• Numerical Microlocal analysis of 2-D noisy harmonic plane and circular waves
  
  • Benamou Jean-David
  • Collino Francis
  • Marmorat Simon
  Asymptotic Analysis, IOS Press, 2013, 83 (1-2), pp.157--187. We present a mathematical and numerical analysis of the stability and accuracy of the NMLA (Numerical MicroLocal Analysis) method [J. Comput. Phys. 199(2) (2004), 717-741] and its discretization.We restrict to homogeneous space and focus on the two simplest cases: (1) Noisy plane wave packets, (2) Noisy point source solutions. A stability result is obtained through the introduction of a new "impedance" observable. The analysis of the point source case leads to a modified second order (curvature dependent) correction of the algorithm. Since NMLA is local, this second order improved version can be applied to general data (heterogeneous media). See [J. Comput. Phys. 231(14) (2012), 4643-4661] for a an application to a source discovery inverse problem. (10.3233/ASY-121157)
  DOI : 10.3233/ASY-121157
• A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides
  
  • Fliss Sonia
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2013, 35 (2), pp.B438 - B461. This work deals with one-dimensional infinite perturbation---namely, line defects---in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. Contrary to existing methods, this one is exact, but there is a price to be paid: the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. © 2013, Society for Industrial and Applied Mathematics (10.1137/12086697X)
  DOI : 10.1137/12086697X
• Domain decomposition for the neutron SPN equations
  
  • Jamelot Erell
  • Ciarlet Patrick
  • Baudron Anne-Marie
  • Lautard Jean-Jacques
  , 2013. Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method applied to the neutron SPN equations, which are an approximation of the transport neutron equation. This method is based on the Schwarz iterative algorithm with optimized Robin interface conditions to handle communications. From a computational point of view, this method is rather easy to implement. We give some numerical results in highly heterogeneous 3D configurations. Computations are carried out with the MINOS solver, which is a multigroup SPN solver of the APOLLO3® neutronics code. Numerical experiments show that the method is robust and efficient, and that our choice of the Robin parameters is satisfactory.no abstract
• Plasmonic cavity modes: black-hole phenomena captured by Perfectly Matched Layers
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  , 2013. no abstract
• Scalar transmission problems between dielectrics and metamaterials: T-coercivity for the Discontinuous Galerkin approach.
  
  • Chung Eric T.
  • Ciarlet Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2013, 239, pp.189--207. no abstract
• Analysis of the factorization method for a general class of boundary conditions
  
  • Chamaillard Mathieu
  • Chaulet Nicolas
  • Haddar Houssem
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013. We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the farfield operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the testing function. (10.1515/jip-2013-0013)
  DOI : 10.1515/jip-2013-0013
• Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: Mathematical justification of the Foldy-Lax model
  
  • Cassier Maxence
  • Hazard Christophe
  Wave Motion, Elsevier, 2013, 50 (1), pp.18-28. We are concerned with a two-dimensional problem which models the scattering of a time-harmonic acoustic wave by an arbitrary number of sound-soft circular obstacles. Assuming that their radii are small compared to the wavelength, we propose a mathematical justification of different levels of asymptotic models available in the physical literature, including the so-called Foldy-Lax model. © 2012 Elsevier B.V. (10.1016/j.wavemoti.2012.06.001)
  DOI : 10.1016/j.wavemoti.2012.06.001
• On the Well-Posedness , Stability And Accuracy Of An Asymptotic Model For Thin Periodic Interfaces In Electromagnetic Scattering Problems
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We analyze the well-posedness and stability properties of a parameter dependent problem that models the reflection and transmission of electromagnetic waves at a thin and rapidly oscillating interface. The latter is modeled using approximate interface conditions that can be derived using asymptotic expansion of the exact solution with respect to the small parameter (proportional to the periodicity length of oscillations and the width of the interface). The obtained uniform stability results are then used to analyze the accuracy (with respect to the small parameter) of the proposed model.
• Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Yu Tang Fei
  Journal of Computational Physics, Elsevier, 2013, 235, pp.14--31. In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell's equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee's scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency. (10.1016/j.jcp.2012.10.019)
  DOI : 10.1016/j.jcp.2012.10.019
• On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems
  
  • Blanc F.
  • Touzé Cyril
  • Mercier Jean-François
  • Ege Kerem
  • Bonnet-Ben Dhia Anne-Sophie
  Mechanical Systems and Signal Processing, Elsevier, 2013, 36 (2), pp.520-539. Numerical computation of Nonlinear Normal Modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three di erent mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the center manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve. (10.1016/j.ymssp.2012.10.016)
  DOI : 10.1016/j.ymssp.2012.10.016
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2013, pp.075012. Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods. (10.1088/0266-5611/29/7/075012)
  DOI : 10.1088/0266-5611/29/7/075012
• Obstacles in acoustic waveguides becoming "invisible" at given frequencies
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Nazarov Sergei
  Acoustical Physics / Akusticheskii zhurnal, MAIK Nauka/Interperiodica, 2013, 59(6), pp.633--639. We prove the existence of gently sloping perturbations of walls of an acoustic two-dimensional waveguide, for which several waves at given frequencies pass by the created obstacle without any distortion or with only a phase shift. (10.1134/S1063771013050047)
  DOI : 10.1134/S1063771013050047
• Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces
  
  • Ciarlet Patrick
  Journal of Numerical Mathematics, De Gruyter, 2013, 21 (3), pp.173-180. Since it was originally designed, the Scott-Zhang interpolation operator has been very popular. Indeed, it possesses two keys features: it can be applied to fields without pointwise values and it preserves the boundary condition. However, no approximability properties seem to be available in the literature when the regularity of the field is weak. In this Note, we provide some estimates for such weakly regular fields, measured in Sobolev spaces with fractional order between 0 and 1 (10.1515/jnum-2013-0007)
  DOI : 10.1515/jnum-2013-0007
• Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
  
  • Jamelot Erell
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2013, 241, pp.445--463. no abstract
• Higher-Order Discontinuous Galerkin Method for Pyramidal Elements using Orthogonal Bases
  
  • Bergot Morgane
  • Duruflé Marc
  Numerical Methods for Partial Differential Equations, Wiley, 2013, 29 (1), pp.144-169. We study arbitrarily high-order finite elements defined on pyramids on discontinuous Galerkin methods. We propose a new family of high-order pyramidal finite element using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non-affine elements. Different strategies for the inversion of mass matrix are also considered and discussed. Numerical experiments are conducted for 3-D Maxwell's equations. (10.1002/num.21703)
  DOI : 10.1002/num.21703
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  Applicable Analysis, Taylor & Francis, 2013, pp.1-22. We consider inverse obstacle scattering problems for the wave equation with Robin or Neu- mann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular so- lutions to the wave equation. We analyze this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation. (10.1080/00036811.2013.772583)
  DOI : 10.1080/00036811.2013.772583
• Qualitative identification of cracks using 3D transient elastodynamic topological derivative: formulation and FE implementation
  
  • Bellis Cédric
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2013, 253, pp.89-105. A time-domain topological derivative (TD) approach is developed for transient elastic-wave imaging of buried cracks. The TD, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal trial crack, is expressed in terms of the time convolution of the free field and an adjoint field as a function of that specified location and of the trial crack shape. Following previous studies on cavity identification in similar conditions, the TD field is here considered as a natural and computationally efficient approach for defining a crack location indicator function. This study emphasizes the implementation and exploitation of TD fields using the standard displacement-based FEM, a straightforward exploitation of the relevant sensitivity formulation established here. Results on several numerical experiments on 3D elastodynamic and acoustic configurations are reported and discussed, allowing to assess and highlight many features of the proposed TD-based fast qualitative crack identification, including its ability to identify multiple cracks and its robustness against data noise. (10.1016/j.cma.2012.10.006)
  DOI : 10.1016/j.cma.2012.10.006