abstracts! wave!propagation!in!complex!media!and! applications! · 2012-05-03 · inverse! wave!...

Abstracts

Wave propagation in complex media and applications

Heraklion, Greece

May, 7 – 11, 2012

Abstracts Wave propagation in complex media and applications Heraklion, Greece May, 7 – 11, 2012

Copyrights of abstracts retained by the authors. Printed in Greece. Sponsors

· FP7-‐REGPOT-‐2009-‐1, ACMAC, grant agreement n° 245749 · European Research Council Starting Grant GA 239959 · Marie Curie International Reintegration Grant MIRG-‐CT-‐2007-‐203438

Preface This volume contains the abstracts of the talks and posters presented at the workshop

Wave propagation in complex media and applications held at Heraklion, Greece, on May 7-‐11,2012. The workshop has been organized under the auspices of the Archimedes Center for Modeling, Analysis and Computation (ACMAC), the Department of Applied Mathematics of the University of Crete, and the Institute of Applied and Computational Mathematics (IACM) at the Foundation for Research and Technology Hellas (FORTH). The aim of this workshop is to bring together scientists working on forward and inverse wave propagation problems in different applications ranging from underwater acoustics, geophysics and medical imaging to metamaterials and nanotechnology in order to investigate the role and interconnectivity of the mathematical tools employed for the effective modeling of complex environments. We would like to thank all contributors for submitting their abstracts and presenting their work at the workshop.

Chrysoula Tsogka

Organizing Committee Patrick Joly INRIA – Rocquencourt, France George Papanicolaou Stanford University, USA Chrysoula Tsogka University of Crete & IACM/FORTH

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Table of Contents Monday, May 7, 2012 9:30 – 12:15 Morning Session

Numerical simulation of a grand piano 1 Patrick Joly Novel Boundary Element Methods for Scattering at Composite Objects 1 Ralf Hiptmair

14:15 – 17:00 Afternoon Session

Hybrid numerical-‐asymptotic boundary integral methods for high frequency scattering 2 Simon Chandler-‐Wilde

Retarded potentials and discontinuous Galerkin methods with upwind fluxes for transient wave propagation on unbounded domains 3 Jeronimo Rodriguez

Convergence results of iterative solvers for scattering problems. 4 Nabil Gmati

Tuesday, May 8, 2012

9:30 – 12:15 Morning Session

The imaging of anisotropic media using electromagnetic waves 4 Fioralba Cakoni

A Factorization Method for a Far-‐Field Inverse Scattering Problem in the Time Domain 5 Houssem Haddar

Super–resolution and invisibility in wave imaging 6 Habib Ammari


Can trapped modes occur in open waveguides? 6 Christophe Hazard Riesz bases of Floquet modes in semi-‐infinite periodic waveguides and implications 7 Thorsten Hohage Helmholtz Equation with Artificial Nonlocal Boundary Conditions in a Two-‐Dimensional Waveguide. 7 Dimitris Mitsoudis

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Wednesday, May 9, 2012


Time reversal with partial information for wave refocusing and scatterer identification 8 Dan Givoli Time Reversed Absorbing Condition in the Partial Aperture Case 9 Frederic Nataf Multiple-‐scattering theory and its applications 9 Ying Wu

17:00 -‐ 18:30 Poster session

Numerical modeling of 1D poroelastic waves with dissipative terms involving fractional derivatives 10 Emilie Blanc

Recent advances in numerical study of wave propagation in metamaterials 10 Jichun Li

Propagation of acoustic waves in infinite and fractal trees 11 Adrien Semin Space-‐time focusing on unknown scatterers 11 Cassier Maxence

Thursday, May 10, 2012


On initial-‐boundary-‐value problems for Boussinesq systems 12 Vassilios Dougalis Finite volume schemes for dispersive wave propagation and runup 12 Theodoros Katsaounis

Statistical characterization of underwater acoustic signals with applications in inverse problems of acoustical oceanography. 13 Michalis Taroudakis


The Ultra Weak Variational Formulation of the Time Harmonic Elastic Wave Equation 13 Peter Monk

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Optimal high order elements in H(div) and H(curl) for hexahedra, prisms and pyramids 14 Marc Durufle

Signal to noise ratio estimation in passive correlation based imaging 14 Adrien Semin

Friday, May 11, 2012


Finite element heterogeneous multiscale method for the wave equation 15 Marcus Grote

Absorption of rigid frame porous materials with periodic resonant inclusions and periodic irregularities of the rigid backing 15 Jean Philippe Groby

Reconstruction of 3D images from Boundary Measurements 16 Athanasios Zacharopoulos

http://www.acmac.uoc.gr 1

Numerical simulation of a grand piano

Patrick Joly

INRIA Domaine de Voluceau BP 105 Rocquencourt 78153 Le Chesnay Cedex FRANCE

[email protected]

This presentation will deal with the numerical simulation of a concert piano using physical models. An important part of the talk will be devoted to the derivation of the retained mathematical model as a system of coupled partial differential equations that aims at reaching simultaneously several goals: physical relevance, acoustical pertinence and numerical tractability. This model takes into account all the vibration and propagation phenomena involved in the production of a piano sound, from the excitation of the string by the hammer to the sound radiation in 3D, via the transmission of the string displacement to the soundboard via the bridge. A particular attention will be given to the nonlinear string model that takes into account the stiffness of the string as well as its longitudinal movements, which is necessary to well represent phenomena such as precursors or partial phantoms observed in the analysis of a piano sound. The couplings between the various subsystems (hammer/string, string/bridge/soundboard, soundboard/air) will be described in detail as well as the various dissipation models. The retained numerical method will be presented. It is based on a global variational formulation for space discretization and conservative time stepping. The emphasis will be put on the stability of the method via the conservation (or decay) of a relevant discrete energy. Finally, numerical results will be presented, including comparisons with experiments and synthetic sound examples.

Novel Boundary Element Methods for Scattering at Composite

Objects

Ralf Hiptmair

ETH Zurich Raemistrasse 101, CH-‐8092 Zurich

[email protected]

A task frequently encountered in nano-‐optical simulations is the scattering of time harmonic electromagnetic waves at a penetrable object composed of different linear and homogeneous materials, that is, the material coefficients are supposed to be piecewise constant in sub-‐domains. This setting permits us to use boundary element methods for the approximate computation of both local and far fields. The talk surveys several boundary element methods for this scattering problem, some of them classical, some newly developed with a focus on the conditioning of the linear systems spawned by Ritz-‐Galerkin discretization by means of low-‐order boundary elements (BEM).


• The classical first-‐kind single-‐trace boundary integral formulation also known as PMCHWT formulation in computational electromagnetics. It directly arises from Calderón identities, but gives rise to poorly conditioned linear systems, for which no preconditioner seems to be available so far.

• A new first-‐kind multi-‐trace formulation based on local coupling across sub-‐domain interfaces [5], which is amenable to block-‐diagonal preconditioning.

• Another first-‐kind multi-‐trace formulation that can be obtained by taking a “vanishing gap limit” for the classical single-‐trace equations [2,3]. Calderón preconditioning can be shown to work in this case.

• A second-‐kind single-‐trace approach inspired by applying “sign flipping” of traces in the variational form of the classical single-‐trace formulation. The resulting linear systems will be inherently well conditioned [1, 4].

References [1] X. Claeys, A single trace integral formulation of the second kind for acoustic scattering, Research Report 2011-‐15, SAM, ETH Zürich, Zürich, Switzerland, 2011. [2] X. Claeys and R. Hiptmair, Boundary integral formulation of the first kind for acoustic scattering by composite structures, Report 2011-‐45, SAM, ETH Zürich, Zurich, Switzerland, 2011. Submitted to Comm. Pure Applied Math. [3] ____, Electromagnetic scattering at composite objects: A novel multi-‐trace boundary integral formulation, Report 2011-‐58, SAM, ETH Zürich, Zürich, Switzerland, 2011. Submitted to M2AN. [4] X. Claeys, R. Hiptmair, and E. Spindler, 2nd-‐kind galerkin boundary element method for acoustic scattering at composite objects, report, SAM, ETH Zurich, Switzerland, 2012. In preparation. [5] R. Hiptmair and C. Jerez-‐Hanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems, Adv. Appl. Math., (2011). Published electronically. Hybrid numerical-‐asymptotic boundary integral methods for

high frequency scattering

Simon Chandler-‐Wilde

University of Reading Earley Gate, PO Box 243, Reading RG6 6BB, UK

S.N.Chandler-‐[email protected] In this talk we review recent progress in our research group and internationally in constructing and analysing effective numerical methods for time harmonic wave scattering, based on boundary integral equation formulations, that incorporate information about the phase structure of the solution derived from high frequency asymptotics into the approximation space used. The algorithmic challenges are to construct oscillatory basis functions which capture solution behaviour efficiently and then to efficiently evaluate the oscillatory integrals


which necessarily arise in the Galerkin solution process. The associated theoretical numerical analysis challenges include proving best approximation results with these new approximation spaces, and devising new methods to prove stability or new formulations for which conventional stability analyses can be made to apply. In these analyses a novelty is that we need results and bounds which are explicit in their dependence on the wave number as well as on the dimension of the approximation space. We present classes of scattering problems for which all these aims can be achieved, indeed with a computational cost which depends only logarithmically on the wave number as the wave number increases [1]. References [1] S. N. Chandler-‐Wilde, I. G. Graham, S. Langdon, & E. A. Spence (2012), “Numerical-‐asymptotic boundary integral methods in high-‐frequency acoustic scattering”, Acta Numerica 21, 89-‐305.

Retarded potentials and discontinuous Galerkin methods with upwind fluxes for transient wave propagation on unbounded domains

Jeronimo Rodriguez

Universidade de Santiago de Compostela Campus sur, 15782 Santiago de Compostela, A Coruna, Spain

[email protected] This work deals with the numerical simulation of transient wave propagation on unbounded domains with localized heterogeneities. In this situation we will decompose the computational domain in two non-‐overlapping sub-‐domains; one of them (the interior subdomain) being bounded and containing all the defaults, the other one (the exterior subdomain) assumed to be homogeneous and unbounded. In previous studies [1], in the frame of the scalar wave equation, the authors proposed a hybrid method based on the retarded potential method for the exterior domain and a discontinuous Galerkin (DG) method in space (using centered fluxes) combined with explicit second order finite differences in time (leap frog scheme) in the interior. The coupling was specially built to ensure by construction a discrete energy identity yielding to the stability of the numerical method under the usual CFL condition in the interior domain. Moreover, the coupling technique allowed to use a smaller time step in the interior domain leading to quasi-‐optimal discretization parameters for both methods. Since the DG discretizations based on centered fluxes provide sub-‐optimal rates of convergence, it is desirable to include the possibility of using upwind fluxes. The leap frog scheme used for the previous study being unstable in presence of dissipative terms, the authors propose a time discretization of the interior equations being explicit, conditionally stable and allowing the presence of dissipative terms. Following similar ideas to those presented in [1] we propose a new global discretization being stable by


construction and allowing to couple the novel interior approximation with the retarded potential method. The efficiency of the method will be discussed through some numerical experiments for the scalar wave equation. References [1] T. Abboud et al. Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains. JCP, 230 (2011). Convergence results of iterative solvers for scattering problems

Nabil Gmati

ENIT – LAMSIN B.P. 37, 1002, Tunis Le Belvedere, Tunisia

[email protected]

We are interested with the numerical resolution of a problem of acoustic or electromagnetic scattering in unbounded domains. We use the method of coupling finite elements with integral representation [Jami and Lenoir, 1978] which consists in solving an equivalent problem by imposing on a fictitious border an exact non local condition. We study the convergence of several algorithms allowing the resolution of the obtained linear system. We show the linear convergence of an algorithm of richardson, revisited as Schwarz method with total overlapping [J.Liu and J.M. Jin, 2001], [F.Ben Belgacem, L.Fournié, N.G, F.Jelassi, 2003, 2005]. This first algorithm is also used as preconditioning of a GMRES method [J.Liu and J.M. Jin, 2002]. A detailed analysis of the behavior of this second algorithm proves its superlinear convergence [N.G, B.Philippe, 2008]. For the continuous problem the methodology borrowed from works of [R.Winther, on 1980], based on spectral theory results allows to give the rates of convergence in two and three dimensions [F.Ben Belgacem, N.G, F.Jelassi, 2009, 2010].

The imaging of anisotropic media using electromagnetic waves

Fioralba Cakoni

University of Delaware Ewing 402, Newark, Delaware 19716, USA

[email protected]

We discuss two inverse problems related to anisotropic media for Maxwell's equations. The first one is the inverse scattering problem of determining the anisotropic surface impedance of a bounded obstacle from a knowledge of electromagnetic scattered field due to incident plane waves. Such an anisotropic


boundary condition can arise from surfaces covered with patterns of conducting and insulating patches. We show that the anisotropic impedance is uniquely determined if sufficient data is available, and characterize the non-‐uniqueness present if a single incoming wave is used. We derive an integral equation for the surface impedance in terms of solutions of a certain interior impedance boundary value problem. These solutions can be reconstructed from far field data using the Herglotz theory underlying the Linear Sampling Method. The second problem is to obtain information about matrix index of refraction of an anisotropic media again from a knowledge of electromagnetic scattered field due to incident plane waves. This problem plays a special role in inverse scattering theory due to the fact that the (matrix) index of refraction is not uniquely determined from the scattered fields even if multi-‐frequency data is available. Our imaging tool is a new class of eigenvalues associated with the scattering by inhomogeneous media, known as transmission eigenvalues. In this presentation we describe how transmission eigenvalues can be determined from scattering data and be used to obtain upper and lower bounds on the norm of the index of refraction. Preliminary numerical results will be shown for both problems.

A Factorization Method for a Far-‐Field Inverse Scattering

Problem in the Time Domain

Houssem Haddar

INRIA and Ecole Polytechnique Route de Saclay, 91128 Palaiseau, France

[email protected]

We consider a far-‐field inverse obstacle scattering problem for the scalar wave equation in the time domain. We prove that certain test functions given as far-‐fields of pulse solutions to the wave equation characterize the obstacle by a range criterion: If the source point of the pulse is inside the obstacle, then the test function belongs to the range of the “square root” of the time derivative of the far-‐field operator. If the source point is outside the obstacle, then the test function does not belong to this range. This is hence an explicit characterization of the obstacle by far-‐field measurements of time-‐dependent scattered waves. The proof relies on an operator factorization related to the Factorization method for inverse scattering in the frequency domain, and on the positivity of the time derivative of the inverse of the retarded single-‐layer operator.


Super-‐resolution and invisibility in wave imaging

Habib Ammari

Ecole Normale Superieure DMA, 45 Rue d'Ulm, 75005, Paris, France

[email protected]

The aim of this talk is to show how to achieve on one hand super-‐resolved imaging and on the other hand enhance invisibility based on the new concept of generalized polarization tensors. Fast and efficient procedures for (real-‐time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors will be proposed. Vanishing generalized polarization tensor structures will be designed to achieve near-‐cloaking enhancement using transformation optics.

Can trapped modes occur in open waveguides?

Christophe Hazard

CNRS, ENSTA 32 Boulevard Victor, 75015 Paris, France

[email protected]

Trapped modes in acoustic, elastic or electromagnetic waveguides consist in time-‐harmonic solutions of the propagation equations which have a finite energy localized in a bounded region. For a closed waveguide, that is, a duct which is bounded in the transverse direction by a non-‐penetrable wall, it is now well understood that such modes can occur when considering local perturbations of an infinite cylindrical waveguide (such as a bulge or a defect inside the waveguide). The purpose of this talk is to investigate the case of open waveguides, that is, when the transverse section is unbounded (for instance, optical fibers). We will consider the case of the 3-‐dimensional scalar wave equation (acoustic waveguides) and we will show that trapped modes do not exist in the case of local perturbations of cylindrical waveguides. The case of the junction of two semi-‐infinite cylindrical waveguides will be also mentioned. The basic tool of our approach is the generalized Fourier transform associated with the transverse part of the Helmholtz operator. Its use leads us to represent the acoustic field in a uniform semi-‐infinite part of the waveguide as a superposition of modes which are either propagative or evanescent in the longitudinal direction. When considering trapped modes, such a representation simplifies to a continuous superposition of evanescent modes, since propagative modes do not


allow a confinement of energy. In closed waveguides, the existence of trapped modes is precisely due to the presence of such evanescent components of the field. But in open waveguides, these components must vanish, which follows from an analyticity property with respect to the generalized Fourier variable. This property means that in an open waveguide, propagative and evanescent components of a wave are connected in a subtle but strong way (whereas they are independent in a closed waveguide).

Riesz bases of Floquet modes in semi-‐infinite periodic waveguides and implications

Thorsten Hohage

University of Goettingen Lotzestr. 16-‐18, D-‐37083 Goettingen [email protected]‐goettingen.de

Wave propagation in periodic media appears in a number of important applications including photonic crystal structures, metamaterials, and semiconductor nanostructures. The simulation of such devices requires the numerical solution of differential equations in locally perturbed periodic media, which is a challenging task. A basic ingredient for the analytical and numerical study of such problems are time harmonic wave equations in periodic half-‐strips with quasiperiodic boundary conditions. We show that there exists a Riesz basis of the space of solutions to the time-‐harmonic wave equation consisting of Floquet modes. This basis can be chosen such that the translation operator shifting a function by one periodicity length to the left is represented by an infinite Jordan matrix which contains at most a finite number of Jordan blocks of size greater than 1. Moreover, traces of this Riesz basis on the left boundary also form a Riesz basis as long as the corresponding boundary value problems are uniquely solvable. We end by discussing theoretical and algorithmic implications of these results.

Helmholtz Equation with Artificial Nonlocal Boundary

Conditions in a Two-‐Dimensional Waveguide

Dimitris Mitsoudis

ACMAC, University of Crete Voutes campus, 71003, Heraklion, Greece

[email protected] We consider a time-‐harmonic acoustic wave propagation problem in a two-‐ dimensional water waveguide confined between a horizontal surface and a locally varying bottom. We formulate a model based on the Helmholtz equation


coupled with nonlocal Dirichlet-‐to-‐Neumann boundary conditions imposed on two artificial boundaries “near” and “far” from the source. We establish the well-‐posedness of the associated variational problem, under the assumption of a downsloping bottom, by showing stability estimates in appropriate function spaces. Moreover, a priori estimates involving explicit dependence on the frequency and the geometrical parameters of the problem are derived. The analysis is based on utilizing appropriate test functions involving the first order weak derivatives of the solution in the bilinear form and the careful treatment of the nonlocal boundary terms. We present the outcome of several numerical experiments with a code implementing a standard/Galerkin finite element approximation of the variational formulation of the model and in some cases we compare our results with those of standard coupled mode codes. This is joint work with Ch. Makridakis and M. Plexousakis. Time reversal with partial information for wave refocusing and

scatterer identification

Dan Givoli

Technion Dept. of Aerospace Engineering, Haifa 32000, Israel

[email protected]

Time reversal is a well-‐known procedure in application fields involving wave propagation. Among other uses, it can be applied as a computational tool for solving certain inverse problems. The procedure is based on advancing the solution of the relevant wave problem "backward in time". One important use of numerical time-‐reversal is that of refocusing, where a reverse run is performed to recover the location of a source applied at an initial time based on measurements at a later time. Usually, only partial, noisy, information is available, at certain measurement locations, on the field values that serve as data for the reverse run. In this talk, the question concerning the amount and characterization of the available data needed for a successful refocusing is studied for the scalar wave equation. In particular, a simple procedure is proposed which exploits multiple measurement times, and is shown to be very beneficial for refocusing. A tradeoff between availability of spatial and temporal information is discussed. The effect of measurement noise is studied, and the technique is shown to be quite robust, sometimes even in the presence of very high noise levels. The use of the technique as a basis for scatterer identification is also discussed. A numerical study of these effects is presented, employing finite elements in space and a standard explicit marching scheme in time. In contrast to some previous studies, the propagation medium is taken to be homogeneous. This is joint work with Eli Turkel, Tel Aviv University.


Time Reversed Absorbing Condition in the Partial Aperture Case

Frederic Nataf

Laboratoire J.L. Lions, UPMC et CNRS 4 place Jussieu, 75005 Paris

[email protected]

The time-‐reversed absorbing conditions (TRAC) method enables one to “recreate the past” without knowing the source which has emitted the signals that are back-‐propagated. It has been applied to inverse problems for the reduction of the computational domain size and for the determination, from boundary measurements, of the location and volume of an unknown inclusion. The method does not rely on any a priori knowledge of the physical properties of the inclusion. We present the extension of the TRAC method to the partial aperture configuration and to discrete receivers with various spacing. In particular the TRAC method is applied to the differentiation between a single inclusion and a two close inclusion case. Subwavelength resolution can be achieved even with more than 20% noise in the data.

Multiple-‐scattering theory and its applications

Ying Wu

King Abdullah University of Science and Technology Bldg 1, Room 4104, PO BOX 2855, KAUST, Thuwal, Saudi Arabia

[email protected]

In the past, several methods have been developed to solve wave equations, such as finite-‐difference method, finite-‐element method, etc. In this talk, I will review the multiple-‐scattering method, which is based on the scattering theory and takes full multiple scatterings between any two scatterers into consideration. It solves the wave equation in the frequency domain and is capable of handling systems with large material contrast and can investigate wave scattering and propagating in both periodic and random systems. For a periodic system, it provides a numerical tool to calculate the band structures and also can be utilized to derive the analytic formula of effective medium parameters that can be served as a guide for the design of metamaterials. For a random system, it is capable of calculating the wave field distributions and the wave transport behavior can be obtained. I will show the result of time-‐reversal through a strong scattering media by using the multiple-‐scattering method.


Numerical modeling of 1D poroelastic waves with dissipative terms involving fractional derivatives

Emilie Blanc

LMA 31 chemin Joseph-‐Aiguier, 13402 Marseille cedex 20

[email protected]‐mrs.fr We investigate the propagation of poroelastic waves described by the Biot's model in the time-‐domain. Most of the existing methods have been developed in the low-‐frequency range. The aim of our study is to derive some numerical methods in all the domain of validity of the Biot's model. In the high-‐frequency range, the effects of the viscous boundary layer inside the pores must be taken into account. We use the model of dynamic permeability of Johnson-‐Koplik-‐Dashen (JKD). In this case, some coefficients of the Biot-‐JKD's model are proportional to the square root of the frequency. In the time-‐domain, fractional derivatives are therefore introduced into the evolution partial differential equations. Two strategies exist to calculate these fractional derivatives. The first strategy is to compute the involved convolution integral. However, it requires to store the past of the solution, which is too penalizing in terms of computational memory. The second strategy, which we implement, is based on a diffusive representation of the convolution kernel. The latter is replaced by a finite number of memory variables that satisfy local-‐in-‐time ordinary differential equations. The coefficients of the diffusive representation are determined by optimization on the frequency range of interest. We analyze the properties of the Biot-‐JKD's model with diffusive representation: decay of energy, error of the model. We propose a numerical modeling, based on a splitting strategy: a propagative part is discretized by a fourth-‐order ADER scheme on a Cartesian grid, whereas a diffusive part is solved exactly. We analyze the properties of this algorithm. Numerical solutions are compared to analytical ones, with physical parameters representative of real media.

Recent advances in numerical study of wave propagation in metamaterials

Jichun Li

University of Nevada Las Vegas Las Vegas, Nevada 89154-‐4020, USA

[email protected]

Since the first successful construction of negative index metamaterials in 2000, there is a growing interest in the study of metamaterials due to their potential applications in areas such as design of invisibility cloak and sub-‐wavelength imaging. In this talk, I'll first give a brief introduction to the short history of metamaterials. Then I'll focus on the mathematical modeling of metamaterials,


and discuss some time-‐domain finite element schemes we developed in recent years. Finally, I'll conclude the talk with our cloak simulation and some open issues for further exploration.

Propagation of acoustic waves in infinite and fractal trees

Adrien Semin

University of Crete and IACM FORTH P.O. Box 1385 GR-‐711 10 Heraklion, Crete

[email protected]

We study here a wave propagation problem posed on a network with a great number of branches. After having replaced “a great number of branches” by “an infinity of branches”, we define a functional framework in this kind of geometry and we introduce Helmholtz problem. We study then this problem and we give theoretical results illustrated by numerical results.

Space-‐time focusing on unknown scatterers

Cassier Maxence

POEMS laboratory 32 Boulevard Victor, 75015 Paris, France maxence.cassier@ensta-‐paristech.fr

We are motivated by the following challenging question: in a propagative medium which contains several unknown scatterers, how can one generate a wave that focuses selectively on one scatterer not only in space, but also in time? In other words, we look for a wave that “hits hard at the right spot”. Such focusing properties have been studied in the frequency domain in the context of the DORT method (“Decomposition of the Time Reversal Operator”). In short, an array of transducers first emits an incident wave which propagates in the medium. This wave interacts with the scatterers and the transducers measure the scattered field. The DORT method consists in doing a Singular Value Decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. It is now well understood that for small and distant enough scatterers, each singular vector associated with a non zero singular value generates a wave which focuses selectively on one scatterer. Can we take advantage of these spatial focusing properties in the frequency domain to find the input signals which generate a time-‐dependent wave which would be also focused in time? Since any frequency superposition of a family of singular vectors associated with a given scatterer leads to a spatial focusing, the main question is to synchronize them by a proper choice of their phases. The method we propose is


based on a particular SVD of the scattering operator related to its symmetry. The signals we obtain do not require the knowledge of the locations of the scatterers. We compare it with some “optimal” signals, which require this knowledge. Our study will be illustrated by a simple two-‐dimensional acoustic model, where both scatterers and transducers are assumed pointlike. Numerical results will be shown. On initial-‐boundary-‐value problems for Boussinesq systems

Vassilios Dougalis

Institute of Applied and Computational Mathematics (IACM), FORTH 100 N. Plastira Ave, 70110 Heraklion, Greece

[email protected]

We review recent theoretical results on the well-‐posedness of initial-‐boundary-‐value problems for various Boussinesq-‐type systems of water wave theory, and solve these systems numerically paying particular attention to the interaction of solitary-‐wave solutions with the boundaries.

Finite volume schemes for dispersive wave propagation and runup

Theodoros Katsaounis

University of Crete Voutes campus, 71003, Heraklion, Greece

[email protected]

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional non-‐linear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-‐breaking long waves.


Statistical characterization of underwater acoustic signals with applications in inverse problems of acoustical oceanography.

Michalis Taroudakis

University of Crete, Department of Mathematics and FORTH, IACM Knossou Ave, 71409 Heraklion, Crete, Greece

[email protected] The presentation concerns a review of the work performed so far for establishing a method for the statistical characterization of an underwater acoustic signal, in relation with specific inverse problems in underwater acoustics. The acoustic signal is characterized using the statistics of the wavelet sub-‐band coefficients, which, as it has been shown, obey a statistical law described by an Alpha-‐Stable distribution. Thus, the signal observables are included in the set of the parameters of the appropriate distributions at the various levels of the signal decomposition. Ocean acoustic tomography and bottom classification have been considered as test cases for the assessment of the method in realistic applications of acoustical oceanography. The inverse problem is formulated as an optimization problem in connection with an appropriate objective function. Various techniques have been applied for directing the inversion process to the most probable solution, including neural networks and genetic algorithms. Recent advances in this study include applications in range dependent but axially symmetric environments, which better represent realistic test sites in the real world. To this end, two types of environments have been considered: Environments with flat bottom with range-‐dependent sound speed profile and environments with irregular bottom but range-‐independent sound speed profile. In the first case, the sound speed structure is to be recovered, while in the second case the geometry of the water-‐bottom interface is the unknown feature of the inverse problem. It is shown that the proposed method is very promising, when some a-‐priori knowledge on the range-‐dependent character of the recoverable parameters is available.

The Ultra Weak Variational Formulation of the Time Harmonic

Elastic Wave Equation

Peter Monk

University of Delaware Department of Mathematical Sciences, Newark DE 19716, USA

[email protected] I shall describe an application of the Ultra Weak Variational Formulation (UWVF) to the time harmonic Navier equation in 3D linear elasticity. In particular applying techniques from the theory of discontinuous Galerkin methods, I shall prove error estimates for the method. Numerical examples will also be presented, as well as comments on the fluid-‐solid interaction problem.


Optimal high order elements in H(div) and H(curl) for hexahedra, prisms and pyramids

Marc Durufle

INRIA Bordeaux Sud-‐Ouest 351 cours de la Liberation [email protected]

The discretisation of H(curl) and H(div) spaces is usually performed by using Nédélec and Raviart-‐Thomas elements respectively. However these two finite elements are not optimal for meshes composed of non-‐affine hexahedra and prisms. Indeed, a loss of one order in the convergence of the associated numerical method is observed for H(curl), two orders for H(div). In this talk, it will be explained how to construct optimal finite spaces so that the convergence is eventually in $O(h^r)$ in H(curl) (resp. H(div)) norm. Two conditions of optimality will be explored. The finite element spaces found will be detailed for the hexahedron, the prism and the pyramid. For this last case, we will compare our spaces with other finite element spaces proposed in the literature. Nodal and hierarchical basis functions will be detailed. 3-‐D Numerical experiments will illustrate the good properties of these spaces. Signal to noise ratio estimation in passive correlation based

imaging

Adrien Semin

University of Crete and IACM FORTH P.O. Box 1385 GR-‐711 10 Heraklion, Crete

[email protected] We consider here the problem of imaging using passive incoherent recordings due to ambient noise sources. The first step towards imaging in this configuration is the computation of the cross-‐correlations of the recorded signals. These cross-‐correlations are computed between pairs of sensors (receivers) and they contain very important information about the background medium. Indeed, it was shown both experimentally and theoretically that the Green’s function between two sensors can be retrieved from the cross-‐correlation of passive incoherent recordings at these sensors. Here, we propose to employ these cross-‐correlations for imaging reflectors using a travel time migration method. The signal to noise ratio analysis of the proposed method is carried out.


Finite element heterogeneous multiscale method for the wave equation

Marcus Grote

University of Basel Rheinsprung 21, CH-‐4051 Basel, Switzerland

[email protected] A finite element heterogeneous multiscale method (FE-‐HMM) is proposed for the time dependent wave equation with highly oscillatory, albeit not necessarily periodic, coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective coefficients are computed “on the fly” on sampling domains within each macro finite element at the micro scale ε > 0. Since the sampling domains scale in size with ε, which corresponds to the finest scales in the possibly highly heterogeneous medium, the computational work is independent of ε. We prove optimal error estimates in the energy norm and the L2 norm with respect to the micro and macro scale mesh parameters, h and H, and also convergence to the homogenized solution as ε → 0. Absorption of rigid frame porous materials with periodic resonant

inclusions and periodic irregularities of the rigid backing

Jean Philippe Groby

Laboratoire d'Acoustique de l'Universite du Maine LAUM, UMR6613 CNRS, Av. Olivier Messiaen, F-‐72085 LE MANS Cedex 9, France

Jean-‐Philippe.Groby@univ-‐lemans.fr Air saturated porous materials, which are mainly dedicated to sound absorption, suffer from a lack of efficiency at low frequency, when compared to their absorption properties at higher frequency. The usual way to avoid this problem is by multi-‐layering optimized panels. Other ways consist in taking advantages of additional geometric or material heterogeneities, to combine the high frequency efficiency of porous material with lower frequency phenomena related to these heterogeneities. The basic idea to increase the absorption properties of a complex structure is to excite some mode of this structure. This excitation leads to local field amplification inside the structure and also to an energy entrapment, whose translation in terms of absorption coefficient is a large increase of amplitude. Here, the acoustic properties of a rigidly backed homogeneous rigid frame porous layer in which possibly resonant macroscopic inclusions are periodically embedded is investigated theoretically, numerically, and experimentally. Developments are carried out via either the multipole method together with a mode matching technique or a Finite-‐Element method. The rigid backing could also present some periodic irregularities. The results show that this type of structure exhibits quite large absorption at very low frequency, i.e.,


below the so-‐called quarter-‐wavelength resonance of the initial rigidly backed homogeneous rigid frame porous layer. In particular, the combined excitation of mode of the resonant inclusions together with some peculiar trapped modes enable quasi-‐total absorption of incident wavelength more than 10 times larger than the structure thickness.

Reconstruction of 3D images from Boundary Measurements

Athanasios Zacharopoulos

Foundation for Research and Technology-‐Hellas 100 Nikolaou Plastira str. [email protected]

We will discuss some recent work done on model based tomographic techniques and more specifically on reconstructions for Optical Tomography and Fluorescence Multiwavelength Tomography. Starting with the formation of the forward model for the Diffusion equation using either Boundary Elements or Finite Elements Method and the solution of the Inverse problem on a shape based approach using parametric description for closed surfaces, such as Spherical Harmonics, for Optical Tomography and the Matrix free method for Multiwavelength Fluorescence Tomography.

abstracts! wave!propagation!in!complex!media!and! applications! · 2012-05-03 · inverse! wave!...

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