Publications

2015

• Tuning the wavelength of spoof plasmons by adjusting the impedance contrast in an array of penetrable inclusions
  
  • Cordero Maria-Luisa
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  • Barra Felipe
  Applied Physics Letters, American Institute of Physics, 2015, 107, pp.084104. While spoof plasmons have been proposed in periodic arrays of sound-hard inclusions, we show that they also exist when inclusions are penetrable. Moreover, we show that their wavelength can be tuned by the impedance mismatch between the inclusion material and the surrounding medium, beyond the usual effect of filling fraction in the array. It is demonstrated that sound-soft materials increase the efficiency in the generation of sub-wavelength plasmons, with much lower wavelengths than sound-hard materials and than a homogeneous slab. An application to the generation of acoustic spoof plasmons by an ultra compact array of air/polydimethylsiloxane inclusions in water is proposed with plasmon wavelength tunable up to deep sub-wavelength scales. (10.1063/1.4929497)
  DOI : 10.1063/1.4929497
• Classical homogenization to analyse the dispersion relations of spoof plasmons with geometrical and compositional effects
  
  • Mercier Jean-François
  • Cordero Maria-Luisa
  • Félix Simon
  • Ourir Abdelwaheb
  • Maurel Agnes
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2015, 471 (2182). We show that the classical homogenization is able to describe the dispersion relation of spoof plasmons in structured thick interfaces with periodic unit cell being at the subwavelength scale. This is because the interface in the real problem is replaced by a slab of an homogeneous birefringent medium, with an effective mass density tensor and an effective bulk modulus. Thus, explicit dispersion relation can be derived, corresponding to guided waves in the homogenized problem. Contrary to previous effective medium theories or retrieval methods, the homogenization gives effective parameters depending only on the properties of the material and on the geometry of the microstructure. Although resonances in the unit cell cannot be accounted for within this low-frequency homogenization, it is able to account for resonances occurring because of the thickness of the interface and thus, to capture the behaviour of the spoof plasmons. Beyond the case of simple grooves in a hard material, we inspect the influence of tilting the grooves and the influence of the material properties. (10.1098/rspa.2015.0472)
  DOI : 10.1098/rspa.2015.0472
• Effective birefringence to analyze sound transmission through a layer with subwavelength slits
  
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  • Ourir Abdelwaheb
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2015, 343 (12). We analyze the transmission of sound through a sound hard film or layer with periodic subwavelength slits. For wavelength comparable to or larger than the slit spacing, the transmission spectra are revisited in terms of the transmission through an equivalent birefringent layer. It is shown that the Fano-type resonances can be understood by means of the dispersion relations of guided waves within the birefringent layer in the homogenized problem, equivalent to spoof plasmons for gratings. This is done by extending the homogenization to the evanescent waves being excited in the near field of the actual perforated layer. (10.1016/j.crme.2015.07.006)
  DOI : 10.1016/j.crme.2015.07.006
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  , 2015. 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.
• Solving the hypersingular boundary integral equation for the Burton and Miller formulation
  
  • Langrenne Christophe
  • Garcia Alexandre
  • Bonnet Marc
  Journal of the Acoustical Society of America, Acoustical Society of America, 2015, 138 (3332-3340). This paper presents an easy numerical implementation of the Burton and Miller (BM) formulation, where the hypersingular Helmholtz integral is regularized by identities from the associated Laplace equation and thus needing only the evaluation of weakly singular integrals. The Helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth. The hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. The method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three-dimensional bodies. Unlike other approaches using direct evaluation of hypersingular integrals, collocation points still coincide with mesh nodes, as is usual when using conforming elements. Using higher-order shape functions (with the boundary element method model size kept fixed) reduces the overall numerical integration effort while increasing the solution accuracy. To reduce the condition number of the resulting BM formulation at low frequencies, a regularized version α = ik/(k2 + λ) of the classical BM coupling factor α = i/k is proposed. Comparisons with the combined Helmholtz integral equation Formulation method of Schenck are made for four example configurations, two of them featuring non-smooth surfaces. (10.1121/1.4935134)
  DOI : 10.1121/1.4935134
• Numerical investigation of acoustic solitons
  
  • Lombard Bruno
  • Mercier Jean-François
  • Richoux Olivier
  Proceedings of the Estonian Academy of Sciences, Estonian Academy Publishers, 2015, 64 (3), pp.304-310. Acoustic solitons can be obtained by considering the propagation of large amplitude sound waves across a set of Helmholtz resonators. The model proposed by Sugimoto and his coauthors has been validated experimentally in previous works. Here we examine some of its theoretical properties: low-frequency regime, balance of energy, stability. We propose also numerical experiments illustrating typical features of solitary waves.
• Uniqueness results for 2D inverse Robin problems with bounded coefficient
  
  • Baratchart Laurent
  • Bourgeois Laurent
  • Leblond Juliette
  , 2015. We address in this work the uniqueness issue in the classical Robin inverse problem with the Laplace equation on a Dini-smooth planar domain, with uniformly bounded Robin coefficient and L2 Neumann data. We prove uniqueness of the Robin coefficient on a subpart of the boundary, given Cauchy data on the complementary part.
• Three-dimensional transient elastodynamic inversion using an error in constitutive relation functional
  
  • Bonnet Marc
  • Aquino Wilkins
  Inverse Problems, IOP Publishing, 2015, 31, pp.035010. This work is concerned with large-scale three-dimensional inversion under transient elastodynamic conditions by means of the modified error in constitutive relation (MECR), an energy-based, cost functional. In contrast to quasi-static or frequency-domain contexts, time-domain formulations have so far seen very limited investigation. A peculiarity of time-domain MECR formulations is that each evaluation involves the solution of two elastodynamic problems (one forward, one backward), which moreover are coupled (unlike the case of $L^2$ misfit functionals, where the forward state does not depend on the adjoint state). This coupling creates a major computational bottleneck, making MECR-based inversion difficult for spatially 2D or 3D configurations. To overcome this obstacle, we propose an approach whose main ingredients are (a) setting the entire computational procedure in a consistent time-discrete framework that incorporates the chosen time-stepping algorithm, and (b) using an iterative SOR-like method for the resulting stationarity equations. The resulting MECR-based inversion algorithm is formulated under quite general conditions, allowing for three-dimensional transient elastodynamics, straightforward use of available parallel solvers, a wide array of time-stepping algorithms commonly used for transient structural dynamics, and flexible boundary condition and measurement settings. The proposed MECR algorithm is then demonstrated on computational experiments involving 2D and 3D transient elastodynamics and up to over 500,000 unknown elastic moduli. (10.1088/0266-5611/31/3/035010)
  DOI : 10.1088/0266-5611/31/3/035010
• A method to build non-scattering perturbations of two-dimensional acoustic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Lunéville Éric
  • Mbeutcha Yves
  • Nazarov Sergei
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2015. We are interested in finding deformations of the rigid wall of a two-dimensional acoustic waveguide, which are not detectable in the far field, as they produce neither reflection nor conversion of propagative modes. A proof of existence of such invisible deformations has been presented in a previous paper. It combines elements of the asymptotic analysis for small deformations and a fixed-point argument. In the present paper, we give a systematic presentation of the method, and we prove that it works for all frequencies except a discrete set. A particular attention is devoted to the practical implementation of the method. The main difficulty concerns the building of a dual family to given oscillating functions. Advantages and limits of the method are illustrated by several numerical results. Copyright © 2015 John Wiley & Sons, Ltd. (10.1002/mma.3447)
  DOI : 10.1002/mma.3447
• Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perflectly matched layer methods
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Hazard Christophe
  Journal of Sound and Vibration, Elsevier, 2015, 344, pp.pp.158-178. Among the numerous techniques of non destructive evaluation, elastic guided waves are of particular interest to evaluate defects inside industrial and civil elongated structures owing to their ability to propagate over long distances. However for guiding structures buried in large solid media, waves can be strongly attenuated along the guide axis due to the energy radiation into the surrounding medium, usually considered as unbounded. Hence, searching the less attenuated modes become necessary in order to maximize the inspection distance. In the numerical modeling of embedded waveguides, the main difficulty is to account for the unbounded section. This paper presents a numerical approach combining a semi-analytical finite element method and a perfectly matched layer (PML) technique to compute the so-called trapped and leaky modes in three-dimensional embedded elastic waveguides of arbitrary cross-section. Two kinds of PML, namely the Cartesian PML and the radial PML, are considered. In order to understand the various spectral objects obtained by the method, the PML parameters effects upon the eigenvalue spectrum are highlighted through analytical studies and numerical experiments. Then, dispersion curves are computed for test cases taken from the literature in order to validate the approach. (10.1016/j.jsv.2014.12.032)
  DOI : 10.1016/j.jsv.2014.12.032
• Intrinsic Finite Element Methods for the Computation of Fluxes for Poisson's Equation
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  • Sauter Stefan
  • Simian C
  Numerische Mathematik, Springer Verlag, 2015, pp.30. In this paper we consider an intrinsic approach for the direct computation of the fluxes for problems in potential theory. We develop a general method for the derivation of intrinsic conforming and non-conforming finite element spaces and appropriate lifting operators for the evaluation of the right-hand side from abstract theoretical principles related to the second Strang Lemma. This intrinsic finite element method is analyzed and convergence with optimal order is proved. (10.1007/s00211-015-0730-9)
  DOI : 10.1007/s00211-015-0730-9
• Modal method for 2D wave propagation in heterogeneous anisotropic media
  
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  Journal of the Optical Society of America, Optical Society of America, 2015, 32 (5), pp.11. A multimodal method based on a generalization of the admittance matrix is used to analyze wave propagation in heterogeneous two-dimensional anisotropic media. The heterogeneity of the medium can be due to the presence of anisotropic inclusions with arbitrary shapes, to a succession of anisotropic media with complex interfaces between them, or both. Using a modal expansion of the wave field, the problem is reduced to a system of two sets of first-order differential equations for the modal components of the field, similar to the system obtained in the rigorous coupled wave analysis. The system is solved numerically, using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed. The convergence of the method is discussed, considering arrays of anisotropic inclusions with complex shapes, which tend to show that Li’s rules are not concerned within our approach. The method is validated by comparison with a subwavelength layered structure presenting an effective anisotropy at the wave scale. (10.1364/JOSAA.32.000979)
  DOI : 10.1364/JOSAA.32.000979
• Improved multimodal method for the acoustic propagation in waveguides with finite wall impedance
  
  • Félix Simon
  • Maurel Agnes
  • Mercier Jean-François
  Wave Motion, Elsevier, 2015, 54, pp.10. We address the problem of acoustic propagation in waveguides with wall impedance, or Robin, boundary condition. Two improved multimodal methods are developed to remedy the problem of the low convergence of the series in the standard modal approach. In the first improved method, the series is enriched with an additional mode, which is thought to be able to restore the right boundary condition. The second improved method consists in a reformulation of the expansions able to restore the right boundary conditions for any truncation, similar to polynomial subtraction technique. Surprisingly, the first improved method is found to be the most efficient. Notably, the convergence of the scattering properties is increased from N−1 in the standard modal method to N−3 in the reformulation and N−5 in the formulation with a supplementary mode. The improved methods are shown to be of particular interest when surface waves are generated near the impedance wall. (10.1016/j.wavemoti.2014.11.007)
  DOI : 10.1016/j.wavemoti.2014.11.007
• Wave propagation in a waveguide containing restrictions with circular arc shape
  
  • Félix Simon
  • Maurel Agnes
  • Mercier Jean-François
  Journal of the Acoustical Society of America, Acoustical Society of America, 2015, 137 (3), pp.7. A multimodal method is used to analyze the wave propagation in waveguides containing restrictions (or corrugations) with circular arc shapes. This is done using a geometrical transformation which transforms the waveguide with complex geometry in the real space to a straight waveguide in the transformed space, or virtual space. In this virtual space, the Helmholtz equation has a modified structure which encapsulates the complexity of the geometry. It is solved using an improved modal method, which was proposed in a paper by A. Maurel, J.-F. Mercier, and S. Félix [Proc. R. Soc. A 470, 20130743 (2014)], that increases the accuracy and convergence of usual multimodal formulations. Results show the possibility to solve the wave propagation in a waveguide with a high density of circular arc shaped scatterers. (10.1121/1.4913506)
  DOI : 10.1121/1.4913506
• A modified error in constitutive equation approach for frequency-domain viscoelasticity imaging using interior data
  
  • Diaz Manuel I.
  • Aquino Wilkins
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, 296, pp.129-149. This paper presents a methodology for the inverse identification of linearly viscoelastic material parameters in the context of steady-state dynamics using interior data. The inverse problem of viscoelasticity imaging is solved by minimizing a modified error in constitutive equation (MECE) functional, subject to the conservation of linear momentum. The treatment is applicable to configurations where boundary conditions may be partially or completely underspecified. The MECE functional measures the discrepancy in the consti-tutive equations that connect kinematically admissible strains and dynamically admissible stresses, and also incorporates the measurement data in a quadratic penalty term. Regularization of the problem is achieved through a penalty parameter in combination with the discrepancy principle due to Morozov. Numerical results demonstrate the robust performance of the method in situations where the available measurement data is incomplete and corrupted by noise of varying levels. (10.1016/j.cma.2015.07.025)
  DOI : 10.1016/j.cma.2015.07.025