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## Inhaltsverzeichnis

### Dispersive Properties of High Order Finite Elements

We describe recent work on the dispersive properties of finite element methods for wave propagation. In particular, an explicit closed form for the discrete dispersion relation for elements of arbitrary order is given, along with an analysis of the behaviour of the error in the discrete dispersion relation at high wave number as the the order of the method is increased.

Mark Ainsworth

### Computing Solutions for Helmholtz Equation: Domain Versus Boundary Decomposition

When computing solutions of the Helmholtz equation, for a scatterer which boundary shows different geometric features at different locations, needing different kind of discretization or algorithms (integral equations or asymptotic formulas…) one is led to use domain decomposition techniques. This technique can be also useful when the scatterer is too huge, or when distributed computation is the aim. We give here a domain decomposition algorithm to compute acoustics in a non-dissipating bounded cavity. It relies on a Despres type condition on the fictitious interaction boundary between the subdomains. Its convergence does not depend on the (different) type of approximations used for solving the equation in each subdomain.But when computing acoustics outside a bounded scatterer (computing outgoing solutions), the domain decomposition technique leads to infinite fictitious boundaries between subdomains, showing specific features that are not part of the problem to solve, but come in because of the technique used. We introduce a new technique, we call boundary decomposition, and prove its convergence when the scatterer is a disjoint union of subscatterers. It relies on an homological decomposition of the solution, and makes it possible to analyze contribution of each subscatterer. It is specifically suited when the geometric features of the different subscatterers need different approximation algorithms. Tests were made using integral equation algorithm for one subscatterer and Kirchhoff high frequency approximation for another subscatterer: they proved efficency of the method.Proof for convergence of the two algorithms can be found in [2, 3, 4]. Numerical test for both techniques were carried out at EADS — Centre Commun de Recherche — Suresnes.

Mikhaël Balabane

### A Eulerian Geometric Optics Method for High Frequency Electromagnetic Fields Computations in Presence of Fold Caustics

This paper is an abridged version of [1] and [2] which presents: 1) An Eulerian numerical method for the computation of a bi-valued solution of Hamilton-Jacobi type equation in a particular geometric setting. More precisely we consider the fold caustic case. 2) The computation of the amplitudes needed to evaluate the electromagnetic field. 3) A numerical investigation of the asymptotic validity of the obtained “High frequency” GO solution and its energy.We present here the GO Eulerian model and a couple of numerical results and send back the reader to [1, 2] for motivations, a complete reference list and more details, in particular on non trivial issues linked to the numerical resolution of the Eulerian GO model.

Jean-David Benamou, Olivier Lafitte, Rémi Sentis, Ian Solliec

### Exact and High-Order Non-Reflecting Computational Boundaries

Computational methods for the solution of wave problems in unbounded domains have been developed since the 70’s [1]. They have been considered in various fields of application involving wave propagation, such as acoustics, electromagnetics, meteorology and solid geophysics. The main four types of methods that have emerged are: boundary integral methods, infinite element methods, absorbing layer methods and Non-Reflecting (or Absorbing) Boundary Condition (NRBC) methods. Here we concentrate on the latter.

Dan Givoli

### Mathematical Modeling and Numerical Methods for Induction Logging Applications

Magnetic induction tools are used widely in the oil-field service industry to determine the spatial distribution of resistivity in geological formations. Each tool is 2 – 3 m in length and contains transmitting and receiving magnetic dipoles. The tool is inserted into a well and is pulled through it; during this process the tool responses (the magnetic field at receivers) are being logged. Results then are interpreted as follows: a parametric medium model is prescribed, and its parameters are selected so that the modeled tool responses closely parallel real logs. This process demands the numerical simulation of tools in a chosen medium model.

Alexander Bespalov, Yuri A. Kuznetsov, Lev Tabarovsky

### The Inverse Kinematic Problem in Anisotropic Media

We survey recent results on the inverse kinematic problem arising in geophysics. The question is whether one can determine the sound speed (index of refraction) of a medium by measuring the travel times of the corresponding ray paths. We emphasise the anisotropic case.

Gunther Uhlmann

### Aeroacoustics of Moving Compact Bodies Application to the Bullroarer

The bullroarer is an aeromechanical instrument used, for instance, by aboriginal tribesmen in Australia during traditional ceremonies. It sounds like the roaring of a bull, this being the origin of the name, or like a propeller with periodic change in pitch and intensity [2].

Michel Roger, Stéphane Aubert

### Numerical Simulation of a Guitar

The purpose of this study is to present a time-domain numerical modeling of the guitar. The model involves the transverse displacement of the string excited by a force pulse, the flexural motion of the soundboard and the sound radiation in the air. We use a specific spectral method for solving the Kirchhoff-Love’s dynamic plate model for othotropic material, a fictitious domain method for solving the fluid-structure interaction and a conservative scheme for the time discretization.

Eliane Bécache, Antoine Chaigne, Grégoire Derveaux, Patrick Joly

### Challenges and Requirements in the Modelling of Musical Instruments

Modelling musical instruments is a particularly challenging task. For these structures, the complexity of both the continuous model and numerical formulation is mainly due to the high sensitivity of the human ear. An overview of the major difficulties are given, with regard to elastic and acoustic field equations, as well as to initial and boundary conditions. To illustrate these features, some recently obtained results in the modelling of some selected instruments are commented. A number of current advances and research perspectives for the near future are briefly described.

Antoine Chaigne

### A Webster-Lokshin Model for Waves with Viscothermal Losses and Impedance Boundary Conditions: Strong Solutions

Acoustic waves travelling in a duct with viscothermal losses at the wall and radiating conditions at both ends obey a Webster-Lokshin model that involves fractional time-derivatives in the domain and dynamical boundary conditions. This system can be interpreted as the coupling of three subsystems: a wave equation, a diffusive realization of the pseudo-differential time-operator and a dissipative realization of the impedance, thanks to the Kaiman-Yakubovich-Popov lemma.Existence and uniqueness of strong solutions of the system are proved, using the Hille-Yosida theorem.

Houssem Haddar, Thomas Hélie, Denis Matignon

### Numerical Simulation of Acoustic Waveguides for Webster-Lokshin Model Using Diffusive Representations

This paper deals with the numerical simulation of acoustic wave propagation in axisymmetric waveguides with varying cross-section using a Webster-Lokshin model. Splitting the pipe into pieces on which the model coefficients are nearly constant, analytical solutions are derived in the Laplace domain, enabling for the realization of the propagation by concatenating scattering matrices of transfer functions (§2). These functions involve standard differential and delay operators, as well as pseudo-differential operators of diffusive type, induced by both the viscothermal losses and the curvature. These operators are explicitly decomposed thanks to an asymptotic expansion, and the diffusive ones may be defined and classified (§3). Various equivalent diffusive realizations may be proposed, that are deeply linked to choices of cuts in the complex analysis of the transfer functions. Then, finite order approximations are given for their simulation (§4).

Thomas Hélie, Denis Matignon

### Regularization of the Time-Harmonic Galbrun’s Equations

This work concerns the mathematical analysis and the finite element approximation of the time harmonic linearized Galbrun’s equations, which modelize the acoustic propagation in presence of a mean flow [3, 7]. This problem has not been satisfactorily solved until now, particularly in the frequency domain, although it could be a main contribution to the industrial objective of noise reduction, in aeronautics for instance.

Anne-Sophie Bonnet-Ben Dhia, Guillaume Legendre, Eric Lunéville

### Acoustic Transmission Through a Splitter Silencer: A Technique that Avoids “Root Finding”

There are many engineering situations that involve the propagation of sound along a system of ducts of rectangular, circular or elliptical cross-section. The system will usually comprise several sections of duct and a variety of components such as silencers or filters. Mode-matching techniques are an important analytic tool by which this class of problem can be addressed. Although such methods are well established, see for example [1], their impact as an engineering design tool is virtually zero. One problem, that has hitherto posed a significant restraint on their practical use, is the difficulty in determining the roots of the dispersion relation for components with wave-bearing and/or dissipative elements. This article addresses this issue and it is shown that, provided the acoustic field within the component is not required, it is possible to use mode-matching techniques to determine the acoustic transmission through the component without solving the dispersion relation.

Jane B. Lawrie

### Acoustic Waves Tunnelling into Whispering Gallery Waves

Generation and propagation of acoustic interface waves running along curvilinear weakly contrast boundary of two media are stadied by formal highfrequency asymptotic technique. Two types of interface waves are found. Waves of one type resemble creeping waves, on the convex side of the boundary and have ray representation on the concave side. These waves are well excited, but travel with significant attenuation. Waves of the other type have the asymptotics similar to whispering gallery waves on the concave side and are exponentially small on the convex side. Excitation of these waves is small, but they run along the interface almost without attenuation.

Ivan V. Andronov

### High-Order Numerical Simulation of Rocket Launch Noise

The aerodynamic noise propagation from a rocket engine before lift-off has been simulated by solving the axisymmetric Euler equations with a high order difference method. The sound source at the launch table is modelled by a time harmonic velocity perturbation at the dominant frequencies of the Ariane V launch vehicle. Perturbations corresponding to 120 dB sound pressure level lead to a sound pressure level of 112 dB at the nose of the rocket.The numerical method that has been used corresponds to the standard sixth order central difference operator in the interior of the computational domain. Near the boundaries the method is third order accurate and satisfies the Summation by Parts property. This guarantees strict stability for linear problems. The classical explicit fourth order Runge-Kutta method has been used for time integration.

Bernhard Müller, Johan Westerlund

### Lacunae-Based Artificial Boundary Conditions for the Numerical Simulation of Unsteady Waves Governed by Vector Models

Artificial boundary conditions (ABCs) are constructed for the computation of unsteady acoustic and electromagnetic waves. The waves propagate from a source or a scatterer toward infinity, and are simulated numerically on a truncated domain, while the ABCs provide the required closure at the external artificial boundary. They guarantee the complete transparency of this boundary for all the outgoing waves. They are non-local in both space and time but can be implemented efficiently because their temporal non-locality is fixed and limited. The restriction of temporal nonlocality of the proposed ABCs does not come as a result of any model simplification or approximation, but rather as a consequence of a fundamental property of the solutions — the presence of lacunae, or in other words, sharp aft fronts of the waves, in odd-dimension spaces.

Semyon V. Tsynkov

### Active Absorbing Boundary for an Extended Boussinesq-Like Model

A new type of extended Boussinesq model has been developed by decomposing the vertical variations of velocity on a basis of functions. The aim of this paper is firstly to present the theoretical bases of this finite elements model and to study his frequency behaviour. The radiation condition at an open boundary is written for regular waves. For irregular waves, reflection errors coming from an active absorbing boundary are then assessed.

Philippe Sergent, Patrick Gomi, Khouane Meftah

### Stabilized Perfectly Matched Layer for Advective Acoustics

In this paper we present a stabilized Perfectly Matched Layer (PML) for the subsonic aeroacoustics equations. The PML method was introduced by Bérenger [2] to compute the solution of Maxwell’s equations in unbounded domain. This method was adapted by Hu [4] to the aeroacoustics equations. We will first recall his model, considering an horizontal uniform mean flow M. We have to rewrite the acoustics equations whose unknowns are the pressure and the velocity (p, U = (u, v)), (1)$$\left\{ \begin{gathered} \partial _t p + M\partial _x p + \partial _x u + \partial _y v = 0, \hfill \\ \partial _t u + M\partial _x u + \partial _x p = 0, \hfill \\ \partial _t v + M\partial _x v + \partial _y p = 0, \hfill \\ \end{gathered} \right.$$ in a so called “split form” in the unknowns (p x , p y , u,vx,vy), where p x and p y (resp.v x and v y ) are non-physical variables, whose sum gives the pressure (resp. the y-component of the velocity) p = p x + p y (resp.v = v x + v y ). One then obtains the “PML model” by adding a zero-order absorption term proportional to some absorption coefficient σ. Finally, an absorbing layer is obtained by replacing the aeroacoustics equations by the PML model inside a layer of finite width that surrounds the bounded domain of interest for the computations. This absorbing layer has the property to be “perfectly matched”. This means that a wave propagating in the domain of interest does not produce any reflection when it meets the interface with the absorbing layer, whatever its frequency and its angle of incidence are.

Julien Diaz, Patrick Joly

### Long-Time Behavior of the Unsplit PML

Recently, Bécache & Joly [2] showed that the split PML has at worst a linearly growing solution in the late-time, i.e., not a genuine instability, and related these equations to those of an unsplit version [4] hereafter referred to as “standard”. At the same time, Abarbanel et. al. [1] have observed and analyzed a late-time linear growth of the field in this standard unsplit two-dimensional PML in rectangular coordinates (these authors also cite their earlier work which discusses this issue for the split PML). Further, [1] offers a remedy which, while removing the observed linear growth as verified with numerical experiments employing the standard PML, nevertheless results in the loss of the perfectly matched property of the air/PML interface.

Eliane Bécache, Peter G. Petropoulos, Stephen D. Gedney

### A New Construction of Perfectly Matched Layers for Hyperbolic Systems with Applications to the Linearized Euler Equations

The radiation of energy to the far field is an important feature of essentially all wave propagation problems. For numerical simulations, this feature necessitates the introduction of an artificial boundary. In recent years, new techniques based on high-order local and nonlocal boundary conditions have been introduced which are both accurate and inexpensive [7]. However, they are also limited in their applicability, requiring homogeneous media (in the far field) and special artificial boundaries.

Thomas Hagstrom

### Discretely Nonreflecting Boundary Conditions for Higher Order Centered Schemes for Wave Equations

Using the framework introduced by Rawley and Colonius [2] we construct a nonreflecting boundary condition for the one-way wave equation spatially discretized with a fourth order centered difference scheme. The boundary condition, which can be extended to arbitrary order accuracy, is shown to be well posed. Numerical simulations have been performed showing promising results.

Daniel Appelö, Gunilla Kreiss

### Diffusive Realization of the Impedance Operator on Circular Boundary for 2D Wave Equation

A new formulation of the perfectly matched operator on circular boundary for 2D wave equation is introduced. It is based on the “diffusive representation”, useful for a wide class of causal operators and which enables exact and easily approximable time-local realisations of dissipative nature.

David Levadoux, Gérard Montseny

### Perfectly Matched Layers for the Convected Helmholtz Equation

Perfectly Matched Layers (PML), introduced by Bérenger [3] in order to design efficient numerical absorbing boundary conditions for Maxwell’s equations in unbounded domains, have been used for the resolution in the time domain of the linearized Euler equations [7, 9, 1] which modelize the acoustic propagation in presence of flow. In that case, it has been observed that perfectly matched layers can lead to instabilities, produced by waves whose phase and group velocities have opposite signs [9] (see [2] for a general analysis of this phenomenon). This has given rise to several models for time domain applications [1, 6].

Eliane Bécache, Anne-Sophie Bonnet-Ben Dhia, Guillaume Legendre

### Laplace Domain Methods for the Construction of Transparent Boundary Conditions for Time-Harmonic Problems

We consider scattering problems governed by the scalar Helmholtz equation (1)$$\Delta u + {k^2}u = 0$$ More general equations involving radially symmetric potentials have been considered in [7, 6, 5]. A proper formulation of such problems on infinite domains must involve a radiation condition at infinity. The standard radiation conditions in ℝd is Sommerfeld’s radiation (2)$${r^{{(d - 1)/2}}}\left( {\frac{{\partial u}}{{\partial r}} - iku} \right) \to 0,\quad r = \left| x \right| \to \infty$$ which holds uniformly for all direction $$\frac{x}{{\left| x \right|}}$$. However, this condition is not valid in general for infinite obstacles and inhomogeneities with infinite support. Special radiation conditions have been devised for particular problems, e.g., for scattering by infinite obstacles, or scattering problems in wave guide structures.

Thorsten Hohage

### Vibrating Systems with Concentrated Masses: On the High Frequencies and the Local Problem

We consider the vibrations of a membrane occupying a domain Ω of ℝ2, that contains many small regions of high density near the boundary, the concentrated masses. Considering rapidly alternating mixed boundary conditions, we study the asymptotic behavior, when ε → 0, of the eigenelements (λε, uε) of the corresponding spectral problem. We show that the computation of correcting terms for certain eigenfunctions uε associated with the high frequencies of (2) is deeply involved with the study of the high frequencies of the microscopic or local problem.

Eugenia Pérez

### Bulk and Surface Waves in Porous Media: Asymptotic Analysis

Surface acoustic waves at a plane interface of an elastic half-space were discovered by Lord Rayleigh [1]. They take different forms and exist in a broad frequency range. Current research extends from seismic waves in the infrasound region (~ 1–100 Hz) to interdigital transducers and laser-generated surface acoustic pulses in the ultrasound region (~ 10–107kHz).

Inna Edelman

### Asymptotical Models for Wave Propagation in Media Including Slots

A number of problems, whether they regard electromagnetics — diffraction through a cracked wall modélisation of a flattened antenna — or acoustics—e.g., car engineering acoustics—, require solving wave propagation problems in domains including narrow slots.

Patrick Joly, Marc Lenoir, Sébastien Tordeux

### Boundary Control of the Maxwell Dynamical System: Lack of Controllability by Topological Reasons

The paper deals with a boundary control problem for the Maxwell dynamical system in a bounbed domain Ω ⊂ R3. Let ΩT ⊂ Ω be a subdomain filled by waves at the moment T, T* the moment at which the waves fill the whole of Ω. The following effect occurs: for small enough T the system is approximately controllable in ΩT whereas for large T < T* a lack of controllability is possible. The subspace of unreachable states is of finite dimension determined by topological characteristics of ΩT.

M. Belishev, A. Glasman

### Global Exact Controllability of Semi-linear Time Reversible Systems in Infinite Dimensional Space

In this paper, we survey the updated available results on global exact controllability problem of some semi-linear time reversible systems in infinite dimensional space. The typical models we will consider are semi-linear wave equations and plate equations, where the nonlinearity is globally Lipschitz continuous, or more generally, satisfies some super-linear growth condition at infinity.

Xu Zhang

### Some Control Problems for the Maxwell Equations Related to Furtivity and Masking Problems in Electromagnetic Obstacle Scattering

Furtivity and masking problems in time dependent electromagnetic obstacle scattering are formulated as control problems for the Maxwell equations. An ingenious way of using the Pontryagin maximum principle makes possible to express the corresponding first order optimality conditions as a system of partial differential equations. This fact offers relevant computational advantages, such that the use of iterative optimization methods in the solution of the optimal control problems can be avoided and highly parallelizable numerical methods can be developed to solve the control problems proposed.

Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli

### The Effect of Group Velocity in the Numerical Analysis of Control Problems for the Wave Equation

In this note we show how the convergence analysis of numerical algorithms for the computation of internal controls for the wave equation depends on the Group Velocity properties of the numerical scheme used to discretize the wave equation. This is done by means of theory of Wigner measures associated to discrete functions developed in [7]. Some results on the convergence of partial controls are also given.

Fabricio Macià

### Simultaneous Interior Controllability of Elastic Strings

We consider the problem of simultaneously controlling two elastic string by means of a control acting on an arbitrarily small region of the strings. We show that, when the densities of the strings are different, the system is exactly controllable in any time larger than the characteristic times of the strings. When the densities coincides, the answer is similar to the case of simultaneous control from one end of the strings: in this case the controllability properties depend on the rational approximation properties of the ratio of the lengths.

René Dáger

### A Study of Numerical Methods for Boundary Control of the Wave Equation in 1D

Consider the linear wave equation in 1D: (1)$$\begin{gathered} u'' - {u_{{xx}}} = 0,\quad (t,x) \in (0,T)x(0,1), \hfill \\ u(t,0) = 0,\quad u(t,1) = v(t),\quad t \in (0,T), \hfill \\ u(0, \cdot ) = {u^0},\quad u'(0, \cdot ) = {u^1}, \hfill \\ \end{gathered}$$ where′ and x denote time and space derivatives respectively, T > 0, v ∈ L2(0, T) and (u0, u1) ∈ H01 (0,1) × L2(0,1). The goal is now to find v(t) such that u(T, ·) = u′(T, ·) = 0. If this is possible for all initial data, the system is called exact controllable. The system (1) is exact controllable if and only if T ≥ T0 = 2.

Jan Marthedal Rasmussen

### A 2-Grid Algorithm for the 1-d Wave Equation

The problem of controlling a semi-discrete 1-d wave equation using a multigrid method is studied. The control function acts on the system through the extreme x = 1 of the space interval (0,1). In this lecture we present a proof of a 2-grid algorithm for the numerical approximation of the control, proposed by R. Glowinski [G].

Mihaela Negreanu, Enrique Zuazua

### Diffraction by a Locally Perturbed Acoustic Grating

Because of the number of fields of application concerned by periodic structures (integrated optics, micro-electronics, coatings,....), several theoretical approaches and numerical methods have been proposed in the last decade to formulate and simulate direct and inverse problems in gratings (see [7] for an integral approach and [1], [3], [5] for variational ones). Nevertheless, the problem that is usually studied in the literature -and which is now well understood- concerns the diffraction of an incident plane wave by a perfectly periodic grating. The diffracted field is then known to be quasi-periodic, and this leads to a natural formulation of the problem in one elementary cell of the grating. Our concern in this work is the following: what is the diffracted field when a non-perfect grating (for instance, a grating perturbed by a bounded obstacle) is illuminated by a plane wave? Answering this question, we will in particular show that the nature of the diffracted field is closely related to the band-gap structure of the spectrum of the grating.

Anne-Sophie Bonnet-Ben Dhia, Karim Ramdani

### A New Class of Integral Equations for Scattering Problems

A new family of integral equations is presented, dedicated to the solution of boundary value problems for the Helmholtz and Maxwell equations in unbounded domains. We show that the factorization of the Calderón projectors by certain approximations of the exterior admittance operator leads to well-posed boundary integral equations, for all frequencies, and which upon discretisation yields well-conditioned systems.

David Levadoux, Bas Michielsen

### Scattering by Inhomogeneities

Acoustic scattering problems are considered when the density and speed of sound are functions of position within a bounded region. An integro-differential equation for the pressure in this region is obtained. Solving this equation is equivalent to solving the scattering problem. Problems of this kind are often solved by regarding the effects of the inhomogeneity as an unknown source term driving a Helmholtz equation, leading to an equation of Lippmann-Schwinger type. It is shown that this approach is erroneous when the density is discontinuous.

P. A. Martin

### Wave Reflection by a Sheet of Sea Ice

The effect of a thin sheet of sea ice, modelled as an elastic plate, on the propagation of surface gravity waves in the ocean has been the subject of extensive study. A classic problem is that of a plane wave obliquely incident from an open ocean of constant finite depth on an ice sheet in the form of a half-plane. This problem was solved using the Wiener-Hopf technique by Evans and Davies [3]. In their report Evans and Davies wrote of part of the solution process “Unfortunately, the determination of the constants... presents enormous computational difficulties...” and ever since their appears to have been a general feeling that the Weiner-Hopf solution to this problem is cumbersome and impractical.

C. M. Linton, H. Chung

### Modification of Boundary Condition for the Problem of Source Radiation in a Half-Space with a Curvilinear Impedance Surface

In this report we suggest a method allowing one, instead of calculating the field of given source immersed in a domain with a given boundary impedance, to study a simpler auxiliary problem of a calculation of the field in a half-space with the boundary of the same shape but with zero Dirichlet-type boundary conditions. The solution for the vertical electric dipole located in a vacuum half-space bounded by a plane impedance surface was found in a similar way in [1, 2]. Here we suggest an analogous method to find the field excited by the point source located in a half-space with a smoothly distorted impedance boundary. We specify the equation of the boundary shape in the form z = f(x). We assume that inequalities A λ≪ l and a ≪ l, are satisfied, where a and l are, respectively, the vertical and horizontal scales of the boundary irregularities and λ is the wavelength. The surface impedance η0 is assumed constant.

L. P. Kogan, V. V. Tamoikin, T. M. Zaboronkova

### A Domain Imbedding Method with Distributed Lagrange Multipliers for Acoustic Scattering Problems

The numerical computation of acoustic scattering by bounded twodimensional obstacles is considered. A domain imbedding method with Lagrange multipliers is introduced for the solution of the Helmholtz equation with a second-order absorbing boundary condition. Distributed Lagrange multipliers are used to enforce the Dirichlet boundary condition on the scatterer. The saddle-point problem arising from the conforming finite element discretization is iteratively solved by the GMRES method with a block triangular preconditioner. Numerical experiments are performed with a disc and a semi-open cavity as scatterers.

Erkki Heikkola, Tuomo Rossi, Jari Toivanen

### A Galerkin Boundary Element Method for a High Frequency Scattering Problem

In this paper we consider the numerical solution of the Helmholtz equation (1)$$\Delta u + {k^2}u = 0$$ in the upper half-plane U:= {(xi, x2) ∈ R2: x2 > 0}, with impedance boundary condition (2)$$\frac{{\partial u}}{{\partial {x_2}}} + ik\beta u = f$$ on Г:= {(x1,0): x1 ∈ R}, where k > 0 (the wavenumber) is some arbitrary positive constant. This boundary value problem can arise when modelling the acoustic scattering of an incident wave by a planar surface with spatially varying acoustical properties [1]. The total acoustic field ut ∈ C(Ū) ∩C2(U) satisfies (l)–(2) where the wavenumber k = 2πμ/c, with μ being the frequency of the incident wave, c the speed of sound in U, and f ≡ 0. For simplicity of exposition, here we restrict our attention to the case of plane wave incidence, so that the incident field ui is given by (3)$${u^i}(x) = {e^{{ikx \cdot d}}},\quad d({d_1},{d_2}) = (\sin \theta, - \cos \theta )$$ with θ ∈ (—π/2, π/2) being the angle of incidence. The reflected or scattered part of the wave field is u ∈ C(Ū) ∩ C2(U), defined by u = ut - ui. The scattered field then also satisfies (1)–(2) with (4)$$f(s): = ( - \partial {u^i}/\partial {x_2} - ik\beta {u^i})(s,0) = ik{e^{{iks\sin \theta }}}(\cos \theta - \beta (s)),\quad s \in R$$

Simon Chandler-Wilde, Stephen Langdon, Lars Ritter

### Dirichlet-to-Neumann Boundary Condition for Multiple Scattering Problems

A Dirichlet-to-Neumann (DtN) condition is derived for the numerical solution of two-dimensional time-harmonic scattering problems, where the scatterer consists of several obstacles. It is obtained by combining multiple contributions from purely outgoing wave fields. This DtN condition yields an exact artificial boundary condition for the situation, where the computational domain consists of multiple disjoint components. The accuracy of our approach is illustrated by numerical experiments.

Marcus J. Grote, Christoph Kirsch

### Generalized Brakhage-Werner Integral Formulations for the Iterative Solution of Acoustics Scattering Problems

This paper adresses the derivation of well-conditioned generalized Brakhage-Werner integral formulations for the iterative solution of exterior acoustics boundary value problems. These new formulations are suitable to be implemented in a Fast Multipole Method coupled to a Krylov subspace iterative algorithm. Their construction is based on the On-Surface Radiation Condition (OSRC) formalism.

Xavier Antoine, Marion Darbas

### Volume Singular Integral Equations Method for Solving of Diffraction Problem of Electromagnetic Waves in Rectangular Resonator

The method of volume singular integral equations is applied for solving of the problem of electromagnetic diffraction by dielectric body in microwave oven. Using the tensor Green function of parallelepiped new integral equations are presented. Galerkin method is used for numerical solution. The problem of heating in microwave oven is also considered.

Youri Smirnov, Alexei Tsupak

### A Coupling of Spectral and Finite Elements for an Acoustic Scattering Problem

The aim of this paper is to introduce a new fully discrete method for approximating a time-harmonic acoustic wave scattered by a bounded inhomogeneity in the plane. The difficulty related to the unboundness of the domain has been tackled in the literature by different strategies. In particular, the approaches based on finite elements incorporate the far-field effects into the model by means of local (differential) or global absorbing boundary conditions prescribed on an artificial boundary Г enclosing the region of inhomogeneity. Most of the local absorbing boundary conditions are imposed on a circle and they are more exact the larger is the radius; this fact may conduce to large domains or big wave numbers provoking numerical difficulties; cf. [1, 2, 5, 6, 7, 11, 12].

Salim Meddahi, Antonio Márquez, Virginia Selgas

### Transient Scattering from Metallic Enclosures Using 3D Time Domain Methods

In this paper we present efficient models to study the coupling of an electromagnetic wave with electronic equipment enclosures. The 3D Maxwell’s equations are solved using a Finite Element Time Domain method (FETD) and Transmission-Line Modeling Method (TLM). Numerical results are compared with those obtained by measurements or already published results.

J. L. Silveira, S. Benhassine, L. Pichon, A. Raizer

### Plane Wave Basis in Integral Equation for 3D Scattering

The classical boundary element formulation for the Helmholtz equation is rehearsed, and its limitations with respect to the number of variables needed to model a wavelength are explained. A new type of interpolation for the potential is described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions. For a given number of degrees of freedom, the frequency for which accurate results can be obtained, using the new technique, can be up to ten or fifteen times higher than that of the classical method.

Emmanuel Perrey-Debain, Jon Trevelyan, Peter Bettess

### Determination of Green’s Tensor for a Conducting Magneto-Viscoelastic Medium

The interaction of conducting magnetic and elastic or viscoelastic and thermal or thermoelastic fields in an infinite random medium has been under study for some time. Knopoff [1] and Wilson [2] undertook the study of the effect of the presence of magnetic fields in elastic wave propagation. However, the evaluation and application of Green’s functions are essential to the study of wave propagation in interacting magnetic and viscoelastic or elastic fields in random media following J.B. Keller’s perturbation procedure. This is illustrated by the study of wave propagation in random elastic medium by Karal and Keller [3], in random thermoelastic media by Chow [4]. Van Kampen [5] has shwon that the study of the exact solution of wave propagation in a medium with randon refractive index depends on the knowledge of the relevant Green’s function. A knowledge of Green’s function is also essential for the one body scattering problem and the problem of multiple scattering by randomly distributed scatterers (Frisch [6]). In this paper, the components of the Green’s tensor for interacting conducting magnetic and elastic fields in an infinite homogeneous medium is expressed in the form of Fourier integrals by the use of Fourier transforms. It has been possible to evaluate the appropriate integrals approximately for the case of a conducting medium. Two sets of Green’s functions, depending upon high and low frequencies have been presented.

G. Chattopadhyay, R. K. Bhattacharyya

### Numerical Analysis of Transverse Localization of Radiation in Freely Propagating Electromagnetic Bessel Beams

The diffraction broadening of the large-aperture freely propagating Bessel beams is considered. The dependences of integral quantities characterizing the degree of transverse concentration of radiation in such beams on their axial and radial coordinates are obtained. It is shown that the Gaussian beams are better suited for channeled radiation energy transfer in a free space than the Bessel beams.

### Space-Time Mesh Refinement for the Elastodynamic Equations

We are interested in solving the linear elastodynamic equations in a domain with complex geometry using explicit schemes. As it is well known, the ratio $$\frac{{\Delta t}}{{\Delta x}}$$ must be less or equal than a certain constant to obtain the stability of the method. However, it can not be too small to avoid numerical dispersion. For various reasons (singularities of the solution, complex geometry) it could be interesting to do a local space refinement. We propose a technique which uses a local time step in the refined region to have the same ratio between ∆t and ∆x in all the computational domain and that guaranties the stability of the scheme under the usual CFL condition.

Eliane Bécache, Patrick Joly, Jerónimo Rodríguez

### Dealing with Cross-Points in a Non-Overlapping Domain Decomposition Solution of the Helmholtz Equation

When coupling a finite element and a domain decomposition method, a cross-point corresponds to a degree of freedom shared by more than two domains. The problem of dealing with such cross-points is addressed for the case of an usual nodal finite element solution of the Helmholtz equation. An important feature of the approach relies upon its interpretation as an iterative method for solving the discrete problem in the whole domain. The convergence of the iterative procedure is established in the general case and proved to be scalable, that is, to converge at a rate independent of the mesh when the domain decomposition method involves no cross-points.

Y. Boubendir, A. Bendali

### On Handling the Boundary Conditions at the Infinity for Some Exterior Problems by the Alternating Schwarz Method

Partial differential equations modeling physical phenomena occurring in unbounded domains have given rise to lot of work either for their mathematical analysis or for their numerical approximation. To make them accessible to scientific computing it is necessary to write them, in an appropriate way, on truncated domains. Integral equations are affordable tools for the simulation of these problems, although, very often, the associated algebraic system turns out to be heavy to solve. An alternative is to resort to the coupled approach of volumic finite elements/integral representation, which combines the advantages of truncating the domain through the integral formula while avoiding the singularities, and of having a sparse matrix to invert coming from classical finite elements (see [9], [7]). We propose in this note an iterative algorithm to solve the coupled problem in a competitive way. This algorithm may be viewed as an alternating Schwarz method which allows us, for some elliptic exterior problems, to state a geometrical convergence rate.

F. Ben Belgacem, N. Gmati, M. Fournié, F. Jelassi

### Mathematical Modeling of Elastic-Plastic Waves in Granular Media

To describe the process of hetero-modular materials deformation within the framework of rheological approach the classical scheme is completed by a new element — rigid contact. With the help of this element constitutive relationships for granular media possessing elastic, viscous and plastic properties can be presented. In mathematical terms such relationships include sub differentials of convex nondifferentiable potential functions and take a form of variational inequalities having evident procedures of numerical realization. In this paper the process of a shock waves propagation is considered on the basis of the model of elastic-plastic granular medium. Our model uses Mises—Schleiher destruction criterion to describe the granularity of medium and Mises yield condition to determine the plasticity of granules. Under various combinations of phenomenological parameters of the model a percussive adiabats for longitudinal elastic and plastic shock waves of compression and signotones are obtained.

### Wave Excitation, Propagation and Diffraction in Elastic Layered Structures with Obstacles

Mathematical modelling of wave phenomena in layered solids is of increasing interest because it has various applications to seismic and physical acoustics and to non-destructive evaluation. Typically, there are two problems of self-dependent interest: 1) calculation of an incident field u0, excited by a given source, 2) computation of the scattered fields u1, diffracted by some obstacles (cracks, inclusions, rough surface etc.). We deal with low-cost computer implementation of the integral equation method in application to wave phenomena in solids. In this contribution we present some results related to sounding of layered structures and to ultrasonic detection of arbitrarily shaped and oriented cracks.

Evgeny Glushkov, Natalya Glushkova, Alexander Ekhlakov

### On the Oscillation of a Punch Moving on Free Surface of an Elastic Half-Space

The contact problem concerning oscillation of a circular rigid punch moving uniformly at a sub-rayleigh speed along the surface of an elastic half-space is investigated in three-dimensional statement. “Slow” motion of the punch is considered: it is supposed that characteristic time for external loading is much greater than time interval of shear wave propagation along the punch. It is shown that in first approximation the vertical displacement of the punch can be described by equation of dynamics for one degree of freedom system with viscous friction. Coefficients of effective viscosity and stiffness of the elastic half-space are found as functions of speed of the punch and Poisson ratio. The solution for non-stationary problem concerning suddenly applied moving point constant surface load is obtained, which corrects the known result.

S. Gavrilov, G. C. Herman

### On the Antiplane Dynamical Problem of Elasticity Theory in a Domain with Crack

We consider an antiplane dynamical problem in elasticity theory. Let infinite isotropic elastic medium have a cut {(x, y, z): r > 0, α < ∣φ∣ < π, z ∈ R }, where (r, φ) are the polar coordinates on the (x, y) — plane with origin at the corner point. The displacement W satisfies $$\begin{gathered} {\partial^2}W/\partial {t^2} - \Delta W = 0,\quad (x,y) \in K,\quad t > 0, \hfill \\ \partial W/\partial n = 0,\quad (x,y) \in \partial K,\quad t > 0, \hfill \\ W(x,y,0) = f(x,y),\quad {\partial_t}W(x,y,0) = g(x,y) \hfill \\ \end{gathered}$$ where K = {(r, φ): r > 0, 0 < ∣φ∣ < α } is an angle of opening 2α. We consider plane waves travelling from infinity to the corner point (or to the tip of the crack, if α = π). Our main purpose is to study the displacement W after the collision of travelling wave with the tip of the crack. We obtain the asymptotics of W near the corner point. Then we calculate the coefficients of the asymptotics of the displacement W and study their properties after the collision. Besides we find out the range of use of the asymptotics. To this end we compare an explicit solution of some model problem with the asymptotics. Here we essentially use the results of [1], [2], [3]. These articles are devoted to the boundary value problem for the wave equation in domains with edges and conical points under Dirichlet or Neumann boundary condition. The articles contain the theory of solvability of this problem in scales of weighted spaces and asymptotic formulas of the solutions near edges and conical points.

Serguei Matioukevitch

### Diffusive Regime for the High-Frequency Dynamics of Randomly Heterogeneous Plates

We use radiative transfer equations to describe the high-frequency vibrations of randomly heterogeneous plates. The diffusive regime is depicted both numerically and analytically from this model. Numerical simulations are based on the Monte-Carlo method, whereas the analytical model is obtained by an asymptotic analysis. The objectives are to improve the understanding of high-frequency vibrations of complex industrial structures.

Eric Savin

### Mixed Spectral Elements for the Linear Elasticity System in Unbounded Domains

In this paper, we present a mixed formulation of a spectral element approximation of the linear elasticity system. After studying the main features of this approach, we construct PML for modelling unbounded domains. Then, algorithmic issues and numerical results are given.

Gary Cohen, Sandrine Fauqueux

### Element-Based Node Selection Method for Reduction of Eigenvalue Problem

For the condensation of degrees of freedom in large structure problems, various methods have been proposed to approximate the lower eigenvalues that represent the global behavior of the structures. One of the dynamic condensation methods was introduced by Guyan [1]. This method involves elimination of the degrees of freedom which do not give any significant influence on the solution field. O’Callahan has improved the Guyana’s method by considering the first order approximation in the transformation formula of slave degrees of freedom [2]. Zhang has proposed the successive-level approximate reduction method [3]. While these techniques could greatly reduce computational effort by eliminating degrees of freedom, they suffer from nontrivial flaws. Because they ignore several higher order terms, results of these methods are highly dependent on the choice of master degrees of freedom. For this reason, several techniques have been proposed to select the master degrees of freedom effectively. The one of the most effective technique is the sequential elimination of the degrees of freedom of which the ratio of stiffness to mass in the diagonal terms of mass and stiffness matrices is highest. This method has been proposed by Shah and Raymund [4]. Shah method shows the effective selection of master degrees of freedom among others and the prediction of lower mode eigenvalue is very reliable. However, it takes considerable computing time in selecting the master degrees of freedom which is required.

Maenghyo Cho, Hyungi Kim

### A Family of First-Order Conditions for the Long-Time Stability of the Maxwell System

The development of numerical methods for the solution of exterior problems has received special attention in the past. Since the pioneering work of Engquist and Majda [7] in 1977, the method of Artificial Boundary Conditions (ABC) has been widely used for wave problems. It consists in truncating the unbounded propagation domain by introducing an artificial boundary limiting a computational domain. An ABC is then defined from a local operator which links the traces of the wave on the fictitious boundary. From a numerical point of view, ABCs generally result in economical procedures but spurious states can appear in the computational domain. To overcome the pollution of the interior solution, exact non-reflecting boundary conditions have been developed [9]. They are local in the scalar case for spherical boundaries, global in space in the vector waves case. Even if they are easy to implement in a finite element method, they are not well-adapted to inhomogeneous problems and may result in more expensive numerical methods, as compared to the ABC method. Nowadays, the method which seems to be the most attractive consists in using a Perfectly Matched Layer (PML) which was introduced for the first time by Bérenger [5, 6].

Héléne Barucq

### Mass-Lumped Edge Elements for the Lossy Maxwell’s Equations

In this paper, we describe a mass-lumped edge element method for the lossy Maxwell’s equations which enables to save both storage and computing time. Then, we present some comparisons with a Yee scheme which show that this finite element method can do much better than this finite difference method.

Gary Cohen, Xavier Ferrieres, Peter Monk, Sébastien Pernet

### A New Discontinuous Galerkin Method for 3D Maxwell’s Equation on Non-conforming Grids

We propose an explicit scheme based on the Discontinuous Galerkin (DG) formulation introduced by Lesaint and Raviart [3] which is able to deal with structured non-conforming grids. We use a leap-frog time scheme coupled with a centered flux formula depending on a parameter α. Although the refinement rate is high, the dispersion is very small. This method provides the conservation of a discrete energy on non-conforming grids ensuring the stability of the scheme.

Nicolas Canouet, Loula Fézoui, Serge Piperno

### Two Implementations of Nédélec’s Mixed Finite Elements in ℝ3

Here we present two ways to implement Nédélec’s Mixed Finite Elements of tetrahedra in ℝ3 for the space H(curl). We refer the reader to [2] for the proofs.

Kevin Rogovin, Janne Martikainen

### Space-Time Regularity of the Solution to Maxwell’s Equations in Non-Convex Domains

We present various space-time regularity results for Maxwell’s equations. The general results, valid for all Lipschitz domains, are optimal in the absence of singularities. For singular domains that are invariant by translation or rotation, we prove precise results by extending Grisvard’s singularity theory to Maxwell’s equations.

Emmanuelle Garcia, Simon Labrunie

### Maxwell’s Equations in Nonlinear Biperiodic Structures

Significant recent technology advances have been made in nonlinear optics due to rapid developments of laser technology and nonlinear optical materials. Examples of nonlinear optics include laser technology, spectroscopy, optical switching and sensors, parametric amplifiers and oscillators, optical computing, and communications. A remarkable application is to generate powerful coherent radiation at a frequency that is twice that of available lasers, socalled second harmonic generation (SHG). However, nonlinear optical effects are generally very weak without effective enhancement. Recently, a novel idea to enhance the nonlinear interaction between the material and the light by using periodic (or grating) structures has been successfully explored by physicists and engineers. In particular, it has been announced [8], [9] that SHG can be greatly enhanced by using diffraction gratings.

Gang Bao, Aurelia Minut, Zhengfang Zhou

### Homogenization of the Maxwell Equations Using Floquet-Bloch Decomposition

Using Bloch waves to represent the full solution of the Maxwell equations in periodic media, we study the limit process where the material’s period becomes much smaller than the wavelength. It is seen that effective material parameters can be extracted and explicitly represented in terms of the non-vanishing Bloch waves, providing an alternative means of homogenization.

Christian Engström, Gerhard Kristensson, Daniel Sjöberg, David J. L. Wall, Niklas Wellander

### Electromagnetic Wave Propagation Through Small Diameter Tube Bundles

This paper is motivated by the problem of the heating of ceramic filaments by microwaves. For a large bundle of such long tubes one may ask if it is possible for electromagnetic waves to propagate in the axial direction if the tube diameters are small compared to the wavelength of the microwaves? It is the purpose of this paper to present an analytic solution to a specific model problem of this type, and to corroborate the results found recently by Kriegsmann [1] who employed a variational technique. The simple analytical form of the solution presented herein enables the physical structure of the wave field to be revealed explicitly.

I. David Abrahams, Gregory A. Kriegsmann

### Link Between the Parabolic Equation Method and Finding the Gradient Index of Refraction

The use of parabolic equation method from the Helmholtz’s equation has allowed to find a mathematical model including the value of the index of refraction that varies with distance and height throughout its path. Whereever this equation has been applied, results that are close to the experimental results have been achieved, thus surpassing the methods that consider propagation under a standard atmosphere with values suggested by the International Telecommunications Union.

Francisco Varela

### A Fictitious Domain Method for a Unilateral Contact Problem

The aim of this paper is to present a numerical method for solving the diffraction of elastic waves by cracks with unilateral contact (or Signorini’s) boundary conditions. From a mechanical point of view, the crack can partly close or open when such boundary conditions are assumed. Concerning theoretical aspects, there is no existence result to the dynamic unilateral contact problem for purely elastic media, to our knowledge. However, results are known in the scalar case (see [5, 4]) and for viscoelastic media (see [3]).

Eliane Bécache, Patrick Joly, Gilles Scarella

### A Fictitious Domain Method with Operator Splitting for Wave Problems in Mixed Form

We propose a novel operator splitting scheme for time discretization, combined with a new fictitious domain method involving a distributed Lagrange multiplier for the solution of a wave scattering problem. The symmetrized operator splitting scheme decouples the propagation of the wave, and the enforcement of the Dirichlet boundary condition on the obstacle. We employ mixed finite elements for the substeps which propagate the wave. The accuracy of the method is demonstrated via a numerical example.

Vrushali Bokil, Roland Glowinski

### Mathematical Analysis of the Guided Modes of Integrated Optical Guides

The eigenvalue problem for guided modes of integrated optical guides is formulated as a problem for the set of time-harmonic Maxwell’s equations. The original problem is reduced to a strongly-singular domain integral equation, which is often used in practice for computation, and it is proved that the operator of the domain integral equation is a Fredholm operator with zero index. It is also proved that the spectrum of the original problem can only be a set of isolated points.

Evgueni Kartchevski, George Hanson

### Rapidly-Convergent Local-Mode Representations for Wave Propagation and Scattering in Curved-Boundary Waveguides

A new general method for treating wave propagation and scattering problems in partially limited domains with curved boundaries is established, based on an enhanced local-mode series, which includes an additional term accounting for the effects of the curved boundary. The additional mode provides an implicit summation of the slowly convergent part O (n−2) of the local-mode series, rendering the remaining part to converge much faster, like O (n−4), where n is the mode order.

Gerassimos A. Athanassoulis, Konstadinos A. Belibassakis

### Riccati Equation for the Impedance of Waveguides

In this paper, we aim at studying the generalized impedance for the Helmholtz problem in waveguides. A classical method consists in a modal approach (cf. [6]), that is, that the solution is expanded on a basis of eigenfunctions. This usually leads to truncating the analytical field to the lowest-order mode. Whereas the study of the cylindrical guide is accurate, resorting to this method for a guide with a varying section proves more difficult.

Isabelle Champagne, Jacques Henry

### Eigenvalues of the Steklov Problem in an Infinite Cylinder

The Steklov problem is considered in cylindrical domains; the coefficient in the boundary condition has a compact support and is an even function of a coordinate varying along the generators. We study the dependence of eigenvalues on the spacing between two symmetric parts of the coefficient’s support. It is proved that the antisymmetric (symmetric) eigenvalues are monotonically decreasing (increasing) functions of the spacing and formulae for their derivatives are obtained. Application to the sloshing problem in a channel covered by a dock with two equal rectangular gaps is given.

Oleg V. Motygin, Nikolay G. Kuznetsov

### On a Method of Search for Trapped Modes in Domains with Cylindrical Ends

The formally self-adjoint elliptic problems for systems of differential equations of arbitrary order are considered in domains with finitely many cylindrical ends. It is known that one can treat the cylindrical ends as waveguides and introduce families of incoming and outgoing”waves”. The amplitudes of such waves may grow with exponential rate at infinity. Taking into account finitely many waves, one can define unitary scattering matrices. An existence criterion of nontrivial exponentially decaying solutions to the homogeneous problems was obtained in terms of such matrices (an existence criterion of trapped modes in waveguides and surface waves in diffraction gratings) [1, 2]. In the paper, we suggest and justify a method for the computation of the matrices. The class of problems under consideration includes, among others, the systems of equations in elasticity theory, hydrodynamics, and electrodynamics. As an example we give some results on the numerical implementation of the method.The method was first exposed for the Helmholtz equation (to search for surface waves in diffraction gratings) in [3, 4].

Viktor O. Kalvine, Pekka Neittaanmäki, Boris A. Plamenevskii

### The Forced Motion of Structures that Support Trapped or Near-Trapped Water Waves

Trapped water waves are free oscillations of an unbounded fluid with a free surface for which the fluid motion is essentially confined to the vicinity of a fixed structure. Thus the energy of the motion is finite and there is no radiation of energy to infinity. Such modes are non-trivial solutions of the linearized water-wave problem in the frequency domain, they satisfy homogeneous boundary conditions, and contain no waves in the far field (hence there is an absence of any forcing either from incident waves or from an imposed motion of the structure). In recent years trapping structures, that is structures that support trapped modes, have been constructed in both the two- and three-dimensional linear water-wave problems (see [3], and the references therein).

P. McIver, M. McIver

### On the Motion of a Floating Body Supported by an Air Cushion

The question of uniqueness is considered for a couple of new linearized problems in the water-wave theory. The first of these problems describes the radiation and scattering of time-harmonic waves by a fixed body supported in water by an air cushion. In the second problem, the body is assumed to be freely floating. For both problems the method developed by John [2] is applied for proving the uniqueness theorems. The proof is outlined for the first problem and the result is formulated for the more complicated second problem.

Nikolay Kuznetsov

### Long Wave Approximations for Water Waves

We derive a class of symmetric and conservative systems which describe the interaction of long water waves. We prove rigorously that the solutions of the water waves equations can be approximated in terms of the solutions to these symmetric systems. We use this fact to prove rigorously that all the systems of the wide class obtained by J.L. Bona et al. [2] (and in particular the historical Boussinesq system) provide a good approximation to the water waves problem. Our error estimates are better than that obtained in the case of the decoupled KdV-KdV approximation, are valid in the 2-D case for any existing solution to the Euler equations, and remain true in the periodic case.

Jerry Bona, Thierry Colin, David Lannes

### Oblique-Angled Periodicity Lattice in Capillary-Gravity Waves Problems

For the problems of capillary-gravity surface waves the cases with rectangular periodicity lattices were detaily studied in [1, 2, 3, 4], where the most interesting cases of the higher degeneracies were investigated. However the fluid flows with oblique-angled periodicity lattice remain unstudied. Here it is proved that solutions with elementary cells in the form of any parallelograms in oblique lattice are absent. Bifurcation theory methods under group invariance conditions are used [1, 2].

Irina Konopleva, Boris Loginov

### Localized Waves in a Thin Film with Growing Islands

Influence of localized waves to islands growing on a thin film is investigated. The film is modelled as a fluid layer covered by an inertial surface with the variable density of mass and surface tension. The existence of trapped modes for the corresponding frequency-domain problem is established. We show that for the large time wave localization near islands gives some contribution to the growth rate of island.

Dmitry Indeitsev, Yulia Mochalova

### Numerical Schemes for 2D Shallow Water Equations with Variable Depth and Friction Effects

In this work we present a general structure of numerical schemes for 2D Shallow Water Equations with source terms. This family of numerical schemes includes previous extensions to non-homogeneous Shallow Water equations of fluxdifference schemes such as Roe’s and Van Leer’s due to Bermúdez and Vázquez, and other new methods introduced in this work. Also, we introduce the extension of flux-splitting methods such as Vijayasundaram, Steger-Warming, and other new methods of this class. We include in this work a study of linear stability and different numerical tests.

Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto

### Generalized Eigenfunction Expansions; an Application to Linear Water Waves

In this paper, we show a general way of establishing eigenfunction expansions of the transient state of a linear scattering problem, i.e., its decomposition on a continuous family of time-harmonic states which represent the response of the system to a family of time-harmonic plane waves. We illustrate the method in the context of linear water waves.

Christophe Hazard, François Loret

### Shallow Water Wave Modelling Using a High-Order Discontinuous Galerkin Method

We present a high-order discontinuous Galerkin model for solving the two-dimensional shallow water equations on unstructured triangular meshes. The model uses orthogonal Dubiner expansion basis of arbitrary polynomial order in space and a third-order Runge-Kutta scheme in time. Using numerical computations of a standing wave we illustrate that the model exhibits exponential convergence for smooth problems. Finally, an example of computation of harbour resonance is presented.

Claes Eskilsson, Spencer J. Sherwin, Lars Bergdahl

### A Nonlinear Spectral Model for Gravity Waves Generation and Propagation in a Bounded Domain

This paper is devoted to the time accurate simulation of the generation and propagation of gravity waves in a rectangular basin. The flow is described in the framework of nonlinear potential flow theory, with second order Stokes expansion of the boundary conditions at the free surface and at the wavemaker. The resulting initial boundary value problem is solved using a recently developed spectral method. The fully nonlinear potential formulation is first given, and then problems for each order of the perturbation expansion are derived. Details concerning the spectral approach and its numerical implementation follow. Finally, some simulations are briefly described.

Félicien Bonnefoy, David Le Touzé, Pierre Ferrant

### Simulation of Internal Waves in the Strait of Gibraltar Using a Two-layer Shallow-water Model

This paper is concerned with the simulation of the flow of a stratified fluid through a channel with irregular geometry. The channel is supposed to have a straight axis and to be symmetric with regard to a vertical plane passing through its axis. The cross-sections are supposed to be of arbitrary shape. The fluid is assumed to be composed of two shallow layers of immiscible fluids of constant density. Moreover, we assume that the flow is one dimensional, i.e., at every layer the velocities are uniform over the cross-section and the thickness only depend on the coordinate related to the axis and on the time. Therefore, the equations to be solved are a system of two coupled Shallow Water equations with source terms involving depth and breadth functions. Extensions of the Q-schemes of van Leer and Roe are proposed where the coupling and source terms are treated by adapting the techniques developed in [9], [5] and [2]. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar, performing simulations of tidal effects and comparing the results with observed data.

M. J. Castro, J. A. García-Rodríguez, J. M. González-Vida, J. Macías, C. Parés, M. E. Vázquez-Cendón

### Second Order Methods for the Simulation of Pressure Waves and Cavitation of Water

The simulation of pressure waves plays an important role in the development of modern hydraulic tools. Due to its strong influence on the speed of sound cavitation has a great influence on this nonlinear wave propagation. In this presentation we analyze some second order methods used for the simulation of these phenomena. In particular we show that some in the context of gas dynamics well known methods refuse to perform the desired simulations while others give reliable results.

Anselm Berg, Uwe Iben, Andreas Meister

### Raviart-Thomas Approximations for Variational Hydroacoustics Problems

This paper presents numerical solution of hydroacoustic problem in terms of displacements by means of finite-element method. We have proven correctness of the variational problem, constructed Raviart-Thomas approximations on triangular finite elements, performed numerical analysis of the solutions.

Artem Shynkarenko, Vitaly Horlatch

### 1-D Inverse Problem of Thermoelasticity

The paper deals with an inverse dynamic problem for the uncoupled system of thermoelasticity. This problem is reduced to two auxiliary inverse problems: the recovery of the density and Lame functions appearing in the Lame system of elasticity, and then finding the heat extension-function entering in the system of thermoelasticity. Uniqueness and stability estimate theorems are established for each of the auxiliary inverse problem. Existence theorem for the solution of the second auxiliary problem is proved. The numerical methods for the solution of each problem are discussed.

Valery G. Yakhno

### Electromagnetic Propagation in Troposphere: Inverse Problem Using Optimization Approaches

This paper deals with the electromagnetic characterisation of the tropospheric medium (refractive index) above the sea surface. Assuming the emitted signal and the far field are known parameters, our purpose is the determination of the refractive index map.Our methods consist in least square approaches deduced from different optimization algorithms. First, the electromagnetic field at the reciever (far field) is supposed to be recorded without noise. Then, the inverse problem with a strong gaussian noise is considered. To obtain numerical data, a parabolic propagation model simulate the direct problem and synthetic data are computed.

Mohamed Yassine Ayari, Arnaud Coatanhay, Ali Khenchaf

### Inverse Problem for a Nonlinear Helmholtz Equation

This paper is devoted to the uniqueness of the coefficients θ, φ ∈ L∞(ℝ3), and ψ ∈ L∞ (ℝ3, ℝ3) for the nonlinear Helmholtz equations −△v(x) − k2v(x) = θ(x)v(x)F(|v(x) |) and −△v(x) − k2v(x) = (φ(x)v(x) + iψ(x).▽v(x)) ×|▽v(x) |r|v(x) |s. For small values of ⋋, a solution v is uniquely constructed by adding a small outgoing perturbation to the plane wave x → ⋋eikx.d, where |d| = 1 and ⋋ ≥ 0. We write the far-field expansion $$v = v(x,\lambda, d) = \lambda {e^{{ikx.d}}} + u_{\infty }^s(\frac{x}{{\left| x \right|}},d,\lambda )\frac{{{e^{{ik\left| x \right|}}}}}{{\left| x \right|}} + O(\frac{1}{{{{\left| x \right|}^2}}})$$ for large |x|, and we prove, for a fixed k, the uniqueness for the reconstruction of θ, φ and div ψ from the behaviour of $$u_{\infty }^s(\frac{x}{{\left| x \right|}},d,\lambda )$$ when ⋋ → 0.

### On Masking Operations for the Point-Source Method Solving Inverse Boundary Value Problems

We study the point source method for the reconstruction of some field u on parts of a domain Ω from the Cauchy data of the field on the boundary ∂Ω of the domain. Then, the boundary condition for a sound-soft object D in Ω can be used to find the location and shape of some inhomogeneity. The results show that we can detect a sound-soft object from the knowledge of one total field and its normal derivative on ∂Ω by searching the zeros of the total field. In order to find the zeros according to the boundary of the scatterer we are using a masking operation to get a starting guess for the scatterer. Numerical examples are provided.

Klaus Erhard, Roland Potthast

### Some Inverse Problems for the Equation — (pu′)′ = ⋋ru

We will consider the equation —(pu′)′ = ⋋ru on (0, L) where p,r > 0 are piecewise constant real valued functions, ⋋ is a complex number and 0 < ∞

Anton Monsef

### Dynamic Identification of the Deformed Body Parameters

Let us consider a linear elasic body occupying a bounded domain Ω with the boundary ∑. Suppose that the elastic modulus of the material depends on the spatial coordinates x: (1)$${}^4\hat{a} = {}^4\hat{a}(x)$$ where index “4” points out on the 4-th rank of the elastic modulus tensor.

Alexander Kravchuk, Pekka Neittaanmäki

### Underground Cavity Detection Based on Elastodynamic Boundary Element and Topological Derivative Approaches

Three-dimensional imaging of cavities embedded in a semi-infinite solid using elastic waves is a topic of intrinsic interest in a number of applications ranging from nondestructive material testing to oil prospecting and underground object detection. In situations when detailed mapping of buried objects (defense facilities, buried waste) is required and only a few measurements can be made, the use of surface discretization-based boundary integral equation (BIE) techniques provides the most direct link between the surface measurements and the buried geometrical objects. While such an approach is well established for acoustic problems [2], limited attention has so far been paid to the use of BIE methods in wave-based sensing of elastic solids. This communication reports the development of an analytical and computational framework for the identification of cavities in a semi-infinite solid from surface seismic measurements via an elastodynamic BIE method, as well as preliminary results on the investigation of the usefulness of the topological derivative (e.g. for choosing the initial guess).

Marc Bonnet, Bojan B. Guzina, Sylvain Nintcheu Fata

### Reconstruction of the Original Tsunami Waveform on the Insular Arc Bottom Topography by the Coastal Observations Inversion

The paper is devoted to reconstruction of the original tsunami waveform from the measurement of the arrived wave at the finite set of coastal receivers. The propagation of the wave is described by linearized shallow-water equations when the depth depends on two variables. The direct problem is approximized by explicit-inexplicit finite difference scheme. The ill-posed inverse problem of reconstruction is regularized by means of singular value decomposition, so R-solution is a result of numerical process. Numerical experiments are presented for the model bottom relief having some basic morphological features typical for the island arc regions.

Tatjana Voronina, Veacheslav Gusjakov, Vladimir Tcheverda

### A Numerical Method for Solving a Class of Inverse Eigenvalue Problems. Application to Optical Fibre Design

We propose a solution methodology for designing the refractive index profile of an optical fibre, from the partial knowledge of its corresponding dispersion curves. This methodology is based on the Newton method and uses the exact Fréchét derivatives of the guided modes (propagation constant and its corresponding field) with respect to the refractive index profile. Several numerical results will be presented to illustrate the performances of the proposed computational methodology.

Chokri Bekkey, Rabia Djellouli, Mouna Youssef

### Destabilization of Inertial Waves in a Rotating Cylinder

This study is aimed at performing computations of a new hydrodynamic instability inside a closed rotating gas cylinder, one of whose ends is a piston undergoing small sinusoidal compression. The velocity field was calculated using a new Galerkin spectral approach, allowing a numerical treatment of the corner singularity. The algorithm was tested in several configurations already known in literature. Instability was observed for different values of the relevant dynamic parameters and the overall agreement with theoretical and experimental data is satisfactory.

Y. Duguet, J. F. Scott, L. Le Penven

### Parameter Retrieval in Electron Microscopy by Solving an Inverse Scattering Problem

Instead of using trial-and-error iterative image matching, object data can directly be retrieved from the electron microscope exit wave function via locally linearized analytical inverse solutions of the scattering problem. However, the problem is ill-posed resulting in an ill-conditioned inverse with restricted applicability due to modelling errors. Thus the object retrieval demands suitable generalization and regularization to control the confidence region and the stability of the retrieval procedure.

Kurt Scheerschmidt

### Interior Transmission Problem for Anisotropic Media

In recent years there has been considerable interest in the inverse scattering problem for anisotropic medium, particularly in the case of acoustic waves and electromagnetic waves. Due to the lack of uniqueness in determining the constitutive parameters, traditional methods for solving the inverse scattering problems based on the use of weak scattering approximations or nonlinear optimization techniques are problematic. On the other hand, since the support is uniquely determined [4], the recently developed linear sampling method [1], [2] for determining the support of the scatterer from the knowledge of the far field pattern at a fixed frequency is ideally suited to solving the inverse scattering problem for anisotropic medium. The unique determination of the support and the linear sampling method for the scalar anisotropic scattering problem were first considered in [4] and [1] respectively. The techniques used in the above papers are based on an analysis of a boundary value problem called the interior transmission problem. This analysis is performed only in the case where the norm of the real part of the matrix A that describes the physical properties of the medium is greater than one. The purpose of this paper is to complete the study of the interior transmission problem in the case where the norm of Re(A) is less then one. Our results imply that the uniqueness result of [4] and the validity of the linear sampling method [1] are also valid for the case ∥Re(A)∥ ≤ 1.

Fioralba Cakoni, Houssem Haddar

### A Modified Frozen Newton Method to Identify a Cavity by Means of Boundary Measurements

We propose here a numerical procedure for reconstructing the shape of a cavity in a bounded domain by ultrasounds, from the boundary measurements. The ill posed nonlinear equation for the operator that maps the boundary of the cavity to the trace of the solution in the external surface is solved by a modified frozen Newton method.

Tuong Ha-Duong, Mohamed Jaoua, Faiza Menif

### On the Solution of Inverse Obstacle Acoustic Scattering Problems with a Limited Aperture

We present a computational methodology for retrieving the shape of a rigid obstacle from the knowledge of some acoustic far-field patterns. This methodology is based on the well-known regularized Newton algorithm, but distinguishes itself from similar optimization procedures by using (a) the far field pattern in a limited aperture, (b) a sensitivity-based and frequency-aware multi-stage solution strategy, (c) a computationally efficient usage of the exact sensitivities of the far-field pattern to the specified shape parameters, and (d) a numerically scalable domain decomposition method for the fast solution in a frequency band of direct acoustic scattering problems.

Rabia Djellouli, Radek Tezaur, Charbel Farhat

### Resolution Estimation for Imaging and Time Reversal in Scattering Media

An array of transducers receives the signals emitted by a localized source in a scattering acoustic medium. The recorded signals can be used either in a time-reversal (TR) process in order to refocus acoustic energy back onto the source, or in a matched field imaging (MF) process in order to estimate the source location. In TR the signals are time-reversed and emitted into the physical medium whereas in MF they are time-reversed and back-propagated numerically in a fictitious homogeneous medium. Given the range (distance from the array) of the active source we study here the cross-range resolution in these two processes. As expected, multiple-scattering enhances the refocusing resolution in TR but it degrades resolution in the estimation of the source location with MF. We introduce a robust procedure for estimating the resolution gain (resp. loss) in TR (resp. MF), in weakly scattering random media. Direct numerical simulations in the ultrasound regime show that our estimation method is accurate and effective.

Liliana Borcea, George Papanicolaou, Chrysoula Tsogka

### Inverse Scattering by Rough Surfaces in the Time Domain

In this paper we propose a new method to find the position and shape of buried interfaces using ground penetrating radar(GPR). We discuss the application of the point source method of Potthast [1] for inverse scattering by a bounded obstacle. This is a method for solving the nonlinear inverse problem by directly processing the measurement data to reconstruct the field up to the boundary. The method is carried out in the frequency domain. In this paper we transform the frequency domain reconstructions to obtain the total field in the time domain. We produce a mathematical analysis for a 2D-implementation of the method with a single point source transmitter located above the surface and the field measured on a finite line above the surface. We consider the simplest case where the boundary is perfectly conducting and the field is in TE polarisation.

Claire Lines, Simon Chandler-Wilde

### Global and Selective Focusing Using Acoustic Time Reversal Mirrors in the Frequency Domain

Acoustic time reversal has known in the few last years a significant growth of interest, covering a large number of applications (medical imaging, non destructive testing,…). The main idea of this phenomenon is to take advantage of the reversibility of the wave equation in a non dissipative unknown medium to back-propagate signals to the sources that emitted them. Today, the physical literature (cf. [4] for more details) and mathematical one on this topic are quite rich (see [1] or [2] for time reversal in the time domain, [6] for time reversal in the frequency domain, and [7] for time reversal in random media).

Christophe Hazard, Karim Ramdani

### On the Validation of the Linear Sampling Method in Electromagnetic Inverse Scattering Problems

The Linear Sampling Method (LSM) has gained a growing interest in recent years in inverse scattering problems, due to its effectiveness especially in treating 3-D inverse problems, and also due to its large spectrum of applications. We recall that this algorithm allows the reconstruction of the shape of an obstacle (or a local inhomogeneity) from multistatic data at a fixed frequency. Unlike classical nonlinear methods, it is based on solving independent linear systems and requires no a priori knowledge on the physical properties of the scatterers.

Francis Collino, M’Barek Fares, Houssem Haddar

### Complex Industrial Computations in Electromagnetism Using the Fast Multipole Method

The Fast Multipole Method allows the fast iterative resolution of surfacic integral equations in electromagnetism in the frequency domain. It has been mainly used up to now for computing RCS of perfectly conducting closed objects. The objects had to be p.c., because it was the only kind of material that the FMM could handle, and they had to be closed in order to use a well conditionned equation (such as the CFIE) in the iterative resolution process. We present some improvements both on the FMM itself and on the iterative solver which permit us to solve more complex problems, involving wires, dielectric materials and absorbing media. These développements are fully integrated in AS_ELFIP, EADS integral equation code.

Guillaume Sylvand

### Using the Multipole Techniques for the Far to Near Field Transformation

We are interested in evaluating the coupling between an antenna and a satellite via the only knowledge of the antenna far field pattern. Two approaches, based respectively on a far field approximation or on a multipole expansion of the Green’s kernel, are considered in order to reconstruct the field in the vicinity of the antenna. Some numerical experiments show the improvement of the reconstruction with the multipole techniques.

Nathalie Bartoli, Francis Collino, Fabrice Dodu

### Coupling of a Fast Multipole Method and a Microlocal Discretization for Integral Equations of Electromagnetism

A numerical solution of the integral equations for the 3-D exterior problem of electromagnetism, leads to the solution of dense linear systems. Those systems have generally a bad conditionement and a size that strongly increases with the frequency. We then consider the Després’s Integral Equations to have a well conditioned system ([5], [8]). If Niter is the number of iterations, the classical complexity of the iterative solution is O(Niterκ4), where κ, is the wave number. In order to speed up the solution of the system, we have considered the coupling of two methods, the microlocal discretization method and the fast multipole method (FMM). The microlocal discretization method according to Abboud et al. [1], enables one to consider new systems whose size is O(κ2/3 x κ2/3) instead of O(κ2 × κ2) thanks to a coarse discretization of the unknown for convex geometries. However, due to the geometrical approximation of the surface, the fine mesh of a classical solution is still considered. An other method, the fast multipole method ([6], [7], [4]), is one of the most efficient and robust methods used to speed up iterative solutions. Using a multilevel algorithm, it leads to the cost O(Niterκ2 ln κ2) instead of O(Niterκ4). In this paper, the coupling of both methods, using a multilevel algorithm, enables one to reduce the CPU time efficiently for large wave numbers, with the complexity O(κ8⁄3 ln κ2 + Niterκ4⁄3) (see also [2]).

Alain Bachelot, Eric Darrigrand, Katherine Mer-Nkonga

### Fast Direct Solver for a Time-harmonic Electromagnetic Problem with an Application

A fast direct solution of a periodic problem derived from the time-harmonic Maxwell’s equations is considered. The problem is discretized by low order hexahedral finite elements proposed by Nédélec. The solver is based on the application of FFT, and it has the computational cost O(N log N). An application to scattering of an electromagnetic wave by a periodic structure is presented.

Janne Martikainen, Tuomo Rossi, Kevin Rogovin, Jari Toivanen

### A Finite Element Method for the Solution of a Time-Dependent Nonlinear Wave Propagation Equation

The most important physical effects associated with the transformation of waves in coastal regions can be described by time-dependent nonlinear mild-slope equations. A model of this type was introduced by Nadaoka et al [4] (see also Beji and Nadaoka [1]), and it is a weakly nonlinear and dispersive scalar equation. That equation describes the combined effects of nonlinear refraction and diffraction of surface waves propagating over gently varying depths and can capture backscattered waves as well. Therefore, it seems adequate to describe the wave field outside and inside harbours and sheltered zones.

C. J. E. M. Fortes, J. L. M. Fernandes, M. A. Vaz dos Santos

### Elimination of First Order Errors in Time Dependent Shock Calculations

First order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions. We use matched asymptotic expansions to analyze the phenomenon for one dimensional time dependent hyperbolic systems and show how to design the artificial viscosity term in order to avoid the first order error. Numerical computations verify that second order accurate solutions are obtained.

Malin Siklosi, Gunilla Kreiss

### On the Numerical Simulation of the Constrained Wave Motion: a Penalty Approach

The main goal of this paper is to address the solution of a constrained wave system which has application in theoretical and applied physics. The numerical methodology relies on: the penalty treatment of the constraints and an energy preserving time discretization leading to a scheme which is essentially unconditionally stable and second order accurate, both being combined with a globally continuous piecewise linear finite element approximation. Numerical experiments confirm the properties of the computational methods discussed here.

Roland Glowinski, Alexander Lapin, Serguei Lapin

### Nonlinear Supratransmission: Energy Transmission by Nonlinear Mode Generation

A nonlinear system possessing a natural forbidden band gap can transmit energy of a signal with a frequency in the gap.This process of nonlinear supratransmission, occuring at a threshold exactly predictable in many cases, is presented and discussed. A simple theory is confronted both to numerical simulations and experiments on a mechanical chain of coupled pendula.

F. Geniet, J. Leon

### A Half Plane Problem Related to the Stability of Viscous Shock Waves

We consider a non linear half plane problemf associated with the stability of viscous shock waves. The half plane problem gives the far field behavior of a perturbation. Estimates of decay in time are given.

Gunilla Kreiss, Heinz-Otto Kreiss

### The Propagation of Nonlinear Water Waves in a Bounded Domain

We review results recently obtained for two-point boundary value problems for integrable evolution PDEs modelling the one-way propagation of small amplitude water waves. The main example is the Korteweg-deVries equation. We give an explicit representation of the solution of the corresponding linearised PDE. This representation differs from the one obtained for the solution of boundary value problems on an infinite interval because of the presence of a discrete sum contribution. The discrete spectrum associated in this way with the linearised equation is due to the boundary conditions. The same discrete structure appears in the representation of the solution of the analogous boundary value problem for the KdV equation; in this case however, the spectral transform has an additional discrete spectrum due to the nonlinearity, and present in the representation of the solution of KdV on any domain. In the case of a finite domain,the nature of the joint discrete spectrum seems to determine the nature of the interaction between the propagating solitons and the boundary, and in particular whether solitons can propagate after the interaction.

Beatrice Pelloni

### Homoclinic Structures for Nonlinear Integrable Wave Equations: New Approach

A method is proposed to generate homoclinic solutions for integrable nonlinear wave equations with periodic boundaries. This approach resembles the dressing method known in the theory of solitons. As an example, we obtain the homoclinic orbit for the modified nonlinear Schrödinger equation solvable by the Wadati-Konno-Ichikawa spectral problem.

E. V. Doktorov, V. M. Rothos

### Nonlinear Coupled Thermomechanical Waves: Modelling Shear Type Phase Transformation in Shape Memory Alloys

Starting from a two-dimensional model approximating the dynamics of cubic-to-tetragonal and tetragonal-to-orthorhombic phase transformations in shape memory materials, it is shown that the Falk model in the one dimensional case is a special case of the formulated model. Computational experiments based on a conservative difference scheme are carried out to analyse thermomechanical wave interactions in a rod with shape memory effect.

Linxiang Wang, Roderick Melnik

### Evans Function for the AKNS Problem with Algebraic Solitons

We study renormalization of the Evans function for the AKNS spectral problem on a real line with algebraically decaying potentials. We show that the Evans function may have pole singularities at embedded eigenvalues of the essential spectrum, which are related to algebraically decaying eigenfunctions. Stability of algebraic solitons in the initial-value problem for the modified KdV equation is studied with the renormalized Evans function.

Dmitry E. Pelinovsky, Vassilis M. Rothos

### Nonlinear Focusing and Rogue Waves in Deep Water

Rogue wave dynamics in deep water are examined using the nonlinear Schrödinger equation, as well as higher order generalizations. We observe that a chaotic regime greatly increases the likelihood or occurence of rogue waves.

C. M. Schober

### Mixed Spectral Elements for the Helmholtz Equation

In this paper, we present a mixed formulation of a spectral element approximation for the Helmholtz equation. This mixed formulation enables us to reduce dramatically the storage and to provide a fast iterative algorithm of resolution.

Gary Cohen, Marc Duruflé

### Three Dimensional Plane Wave Basis Finite Elements for Short Wave Modelling

This work deals with the solution of the Helmholtz equation in three dimensions using finite elements capable of capturing many wavelengths per nodal spacing. This is done by constructing oscillatory shape functions as the product of the usual polynomial shape functions and planar waves. This technique leads to larger elementary matrices but since the mesh contains fewer finite elements, the final system to solve is greatly reduced. The current model is used to solve a problem of a plane wave with a local wavenumber or the so-called evanescent mode problem. The results show the validity of the technique and a significant reduction in the total number of degrees of freedom compared to the classical finite element model.

Omar Laghrouche, Peter Bettess, Jon Trevelyan

### Methods for Computing Synthetic Seismograms and Estimating Their Computational Error

In order to infer the structure of the Earth’s interior and the source parameters of earthquakes, seismologists require the ability to make accurate calculations of the seismic wavefield by numerically solving the elastic equation of motion, which is a linear second order PDE. Although these calculations are straightforward in principle, the computational requirements are large because the the Earth’s interior is highly heterogeneous. Our group has derived a general criterion for optimally accurate numerical operators. This theory can also be used to estimate the accuracy of the numerical solutions before making any calculations, so that the coarsest possible grid that will meet the user’s accuracy requirements can be used.

Robert J. Geller, Nozomu Takeuchi, Hiromitsu Mizutani, Nobuyasu Hirabayashi

### Numerical Simulation of Thermoelastic Wave and Phase-Transition Front Propagation

A thermodynamically consistent form for the finite-volume numerical algorithm for thermoelastic wave and front propagation is proposed in the paper. Such reformulation provides applicability of the Godunov type numerical schemes based on averages of field variables to the description of non-equilibrium situations. The non-equilibrium description uses contact quantities instead of numerical fluxes. These quantities satisfy the thermodynamic consistency conditions which generalize the classical equilibrium conditions.

Arkadi Berezovski, Jüri Engelbrecht, Gerard A. Maugin

### Finite Volume Methods for Wave Phase Conjugation in Active Media

The paper presents a numerical approach of the parametric wave phase conjugation in a magnetostrictive material in the case of a giant amplification in supercritical mode. This transient propagation phenomena can be modelized mathematically by an hyperbolic operator with a source term for the coupling between the elastic and magnetic field. As a consequence the powerfull numerical methods developped in the frame of gas dynamics and shock wave research can be used. As far as qualitative results are sufficient, one dimensional simulations give the most important features of the phenomena and allow physical understanding through the study of the dynamic behaviour of the system. This is particularly important for the identification of the non linear processes between the magnetic and the elastic field. The step between passive and active media is discussed for multidimensional problem.

Alain Merlen, Peter Voinovitch, Vladimir Preobrazhensky, Philippe Pernod

### Convergence of a Collocation Scheme for a Retarded Potential Integral Equation

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations (RPIEs). We outline how Fourier and Laplace transforms can be used to obtain an O(h2) convergence result for a “polar” piecewise—linear collocation approximation of a scalar RPIE on an infinite flat plate.

Dugald B. Duncan, Penny J. Davies

### On the Numerical Stability of Schemes for Time Domain Scattering from a Thin Strip

A numerical algorithm for calculating the current induced on a perfectly conducting thin strip scatterer by a time-dependent electric field is described and analysed. It is known that numerical approximations of the usual models for thin wire scattering typically exhibit (zero frequency) numerical instabilities when the spatial discretisation becomes small relative to the wire radius. Here it is shown that this is not the case for the thin strip algorithm.

Penny J. Davies

### The Use of Nonlinear Solitary Waves for Computing Short Wave Equation Pulses

There are many important problems where thin, concentrated pulses must be numerically convected over long distances. Examples include acoustic and EM pulses scattered or produced by aircraft, rotorcraft and submarines. Often, for these cases, the main interest is in the far field, where the integrated energy at each point along the pulse surface and the motion of the centroid surface are important, rather than the details of the internal pulse structure. Within this scope, there have been many efforts to discretize and solve the time dependent wave equations. The application of these, however, has been limited by the requirement that a sufficient number of grid cells must span the pulse thickness to accurately solve the equations.

John Steinhoff, Meng Fan, Lesong Wang, Min Xiao

### On a Class of Preconditioners for Solving the Discrete Helmholtz Equation

In this paper, we analyse and implement preconditioners for efficiently solving the discrete Helmholtz equation. The preconditioners are based on definite versions of the Helmholtz equation. The computational performance with Bi-CGSTAB is presented for a 2D homogeneous model problem.

Y. A. Erlangga, C. Vuik, C. W. Oosterlee

### Wavelet Analysis in Solving the Cauchy Problem for the Wave Equation in Three-Dimensional Space

We propose a new method for solving the Cauchy problem for the wave equation in three dimensional space. The method is based on continuous waveletanalysis. We show that the exact non-stationary solution of the wave equation with finite energy found in [1] at any fixed moment of time should be regarded as a mother wavelet. This solution was named in [1] as a “Gaussian wave packet”. It is a new three-dimensional axially symmetric wavelet which is given by a simple explicit formula as well as its Fourier transformation. This wavelet has an infinite number of vanishing moments. It is a smooth function, i.e. it has derivatives of any order with respect to spatial coordinates and time. We show that using the wavelet decomposition of the initial data we can find the exact formula for the solution of the Cauchy problem as a linear superposition of “Gaussian packets”.

Maria V. Perel, Mikhail S. Sidorenko

### Backward Error Analysis for a Multi-Symplectic Integrator

In this note we provide a backward error analysis of the numerical solution of a multi-symplectic (MS) Hamiltonian PDE of the form (1)$$Mz_t + Kz_x = \nabla _z S(z),\quad z \in R^n$$ where M, K ∈ Rn×n are skew-symmetric matrices and S(z) is a smooth function of z(x, t) (the vector of state variables). The variational equation associated with (1) is is given by (2)$${\text{Mdz}}_{\text{t}} {\text{ + Kdz}}_{\text{x}} {\text{ = D}}_{{\text{zz}}} {\text{S(z)dz }}$$ Taking the wedge product of (2) with dz, and letting ω = 1/2 dz Λ M dz and κ = 1/2 dz Λ K dz, we obtain the MS conservation law (3)$$\omega _t + \kappa _x = 0$$ The two forms ω and κ define a symplectic structure associated with time and space, respectively. An important consequence of the MS structure is that when the Hamiltonian S(z) is independent of t and x, the PDE has local energy and momentum conservation laws [1, 2] (4)$$E_t + F_x = 0,\quad E = S(z) + z^T Kz_x ,\quad F = - z_t ^TKz$$(5)$$I_t + G_x = 0,\quad G = S(z) + z^T Mz_t ,\quad I = - z_x^T Mz$$ For periodic boundary conditions, the local conservation laws can be integrated in x to obtain global conservation of energy and momentum.

A. L. Islas, C. M. Schober

### Time-Periodic Solutions of Wave Equation via Controllability and Fictitious Domain Methods

We combine the controllability and fictitious domain methods to compute time-periodic solution for a wave equation describing scattering by an obstacle. The fictitious domain method uses distributed Lagrange multipliers to satisfy the Dirichlet boundary condition on the scatterer. The controllability technique leads to an optimization problem which we solve with a preconditioned conjugate gradient method. Numerical experiments are performed with a disc and a semi-open cavity as scatterers.

Roland Glowinski, Jacques Periaux, Jari Toivanen

### Asymptotics of Bound States, Bands and Resonances for Waveguides and Layers Coupled Through Small Windows

Asymptotics of resonances (quasi-bound states), bound states and bands close to the threshold are obtained. The technique is matching of asymptotic expansions of the solutions.

Igor Yu. Popov

### Spectral Theory of Resonances in Electromagnetic Gratings

Anomalies in the diffracted efficiencies of periodic gratings often occur near resonances. Using the min-max principle for the eigenvalues of the singular, non-compact, integral operator of the electromagnetic scattering problem, the problem of the existence of resonances is studied and conical incident angles for which resonances exist are constructed.

H. Paul Urbach

### Absence of Positive Eigenvalues for the Linearized Elasticity System. The Half-space Case

In this paper, we deal with the linearized and isotropic elasticity system defined on a domain Ω given by a local perturbation of the half-space. We prove that it has no positive eigenvalues. We give a method based on a “pseudo-decomposition” using the operations div and curl in the horizontal direction and the partial Fourier transform. The reduced problems depend strongly on the dual Fourier variable which do not unable us to use known techniques as the limiting absorption technique. To study these reduced problems, we use the analytic theory of linear operators.

### Optimal Design of Waveguide-Grating Resonances

The anomalies of optical diffraction gratings have been of interest since they were originally discovered by Wood in 1902. They manifest themselves as rapid variation in the intensity of the various diffracted spectral orders within certain narrow frequency bands. There are two principal types of anomalous effects: the Rayleigh type which is the classical Wood’s anomaly, and the less known resonant type [8]. The Rayleigh type is due to one of the spectral orders appearing (or disappearing) at the grazing angle (propagating along the surface); while the resonant type anomaly is due to possible guided modes supportable by the waveguide grating. In this paper, we focus on the design of the waveguide-grating resonances, also known as guided mode grating resonance filters (GMGRF).

Gang Bao, Kai Huang

### Resonances of an Elastic Plate in a Confined Flow

We present a theoretical study of the resonances of an elastic plate placed in a duct in presence of a flow. We show that finding resonances amounts to solving a family of 2-D eigenvalue problems. It is well known that resonances exist for a rigid plate and without flow, corresponding to fluid vibrations trapped around the plate. Using the min-max principle for non-compact self-adjoint operators, we extend the study to the cases of an elastic plate and/or a uniform flow, and we prove the existence of resonances versus the characteristics of the plate and of the fluid.

A. S. Bonnet-Ben Dhia, J.-F. Mercier

### The Scattering Amplitude for the Schrödinger Operator in a Layer

The scattering amplitude is a wellknown way to solve some inverse scattering problem. In the litterature we actually find the definition of the scattering amplitude in the whole homogeneous perturbed space [6], [4] or half space [2]. For a stratified space, this definition is more complicated [3]. The situation in the layer, not yet studied, introduces also difficulties. In this paper we are looking for a definition of the scattering amplitude in the layer. Such wave guides can modelize problems of wave propagation in geophysic, underwater acoustic and so on.

Michel Cristofol, Patricia Gaitan

### Adaptive Absorbing Boundary Conditions for Schrödinger-type Equations

Let us consider the Schrödinger-type equation given by (1)$$\partial _t u(x,t) = \frac{{ - i}} {c}(\partial _{xx} u(x,t) + Vu(x,t)),\quad x \in R,\quad t \in 0$$ with c > 0. In order to obtain a numerical solution of the initial value problem for this equation, it is essential to consider a finite spatial subdomain [x l , x r ] and to use artificial boundary conditions. One option is to develop local absorbing boundary conditions (ABC) that allow only small reflections of the solution and are constructed by approximating the transparent or reflection-free boundary conditions (TBC) which are nonlocal.

I. Alonso-Mallo, N. Reguera

### Shallow Potential Wells for the Discrete Schrödinger Equation

It is well known that the Schrödinger equation (1)$$( - \Delta + U)\Psi = E\Psi$$ in the case when U describes a shallow potential well (i.e., U = εV(x), V(x) ∈ C<Stack><Subscript>0</Subscript><Superscript>∞</Superscript></Stack> (Rn), ε → 0) has exactly one eigenvalue E0 = − β2, β ∈ R, below the essential spectrum [0, ∞) in the case when ∫ R nV(x)dx ≤ 0 and the dimension n of the configuration space is 1 or 2. This was established for n = 1 and in the radially symmetric case for n = 2 already in the famous textbook of Landau & Lifshitz [4] and later was demonstrated in the general case in dimension 2 by Simon [6]. Close results on the limiting behavior of the resolvent can be found in [1], [3]. In [7] a different method was used in order to obtain the asymptotics of eigenfunctions (which do not appear at all in Simon’s approach). It is based on an explicit construction of approximate eigenfunctions. It turned out that this construction is completely elementary when one passes to the momentum representation. Moreover, this method is equally efficient for the Schrödinger equation and the problem of water waves trapped by a submarine ridge, which, as it is known in the folklore, is analogous to the Schrödinger equation with a potential well. Our goal here is to apply this method to the discrete Schrödinger equation.

Joel A. Rodriguez-Ceballos, Peter N. Zhevandrov

### Paraxial Approximation in a Tilted Frame for Laser Wave Propagation

We study the Schrödinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. After a mathematical analysis, a numerical method for this problem is sketched. Numerical results are presented.

Remi Sentis, Marie Doumic, François Golse

### Wavepacket Dynamics in Superfluid Helium

Bulk superfluid 4He at 0 K constitutes a high dimensional quantum system that follows Bose statistics. The large zero-point amplitude of the atoms overcomes the weak He-He interaction and therefore classical theory cannot describe the nuclear motion properly. Empirical density functional theory (DFT) has been developed in order to reduce the problem dimension. This is achieved by writing the total action functional in three spatial and one time variable. Minimization of the action with respect to the effective one particle wave function leads to the corresponding Euler-Lagrange equation. Original formulation of the energy functional was given in the fluid dynamical variable (density ρ and velocity υ) representation whereas we propose to apply the complex wave function form [1]. Structure of the resulting equation corresponds to a time dependent non-linear Schrödinger equation with no known analytic solutions. An example of such approach in highly symmetric geometries is described in Refs. [2] and [3].

Lauri Lehtovaara, Jussi Eloranta

### Mathematical Analysis of Diffusion Models in Poro-Elastic Media

We consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity as well as a coupled quasi-static problem in thermoelasticity. The pioneering work of M. A. Biot [1, 2, 3, 4] has given rise to numerous extensions. Among them, R. E. Showalter [7] developed an existence, uniqueness and regularity theory for the degenerate quasi-static system using the theory of linear degenerate evolution equations in Hilbert space while C.M. Dafermos studied the classical coupled thermo-elasticity system which describes the flow of heat through an elastic structure [5]. Herein, we choose to consider the full dynamic coupled system which is a more general model including the previous ones and which describes a more complete phenomenon of consolidation. As far as applications are concerned, Biot’s theory is generally involved for the study of a soil under load but it also provides a good theory for the ultrasonic propagation in fluid-saturated porous media like cancellous bone. Our objective here is to develop the existence-uniqueness theory for the one dimensional systems both in the linear and nonlinear cases using classical functional arguments in the Sobolev background. In the linear case, we present a fixed point method based on J. L. Lions’ results [6] to establish the existence of a solution to the problem while in the nonlinear one, our approach involves Galerkin approximations coupled with a monotonicity method.

Hélène Barucq, Monique Madaune-Tort, Patrick Saint-Macary

### Seismic Wave Modeling with the Generalized Screen Propagator

To test the seismic imaging algorithms, it is helpfull to solve the direct problem. From a known velocity model we produce some synthetic data which can be the input of the seismic imaging algorithms. Also the knowledge of the propagation wave phenomena is increased, which provides better interpretation of the collected data and therefore a better determination of the subsurface velocity model.

Frank Prat, Helene Barucq, Bertrand Duquet

### An Upgrade in Linearized Modeling of Wave Propagation

Quantitative migration or linearized inversion techniques rely on a linearization in the wave equation: the accuracy of the result depends on the accuracy of the linearization. The kinematics of events modeled using such a linearization depends on the reference velocity model. In this paper we show that, for acoustic wave propagation, the amplitudes of the reflected events modeled by such a linearization technique are drastically influenced by the choice of the parameters that describe the model perturbations. We propose a choice for these parameters that makes the linearization error small. This is demonstrated by means of analytic calculations for plane waves propagating in a two-layer model as well as by comparison with a finite difference modeling.

Florence Delprat-Jannaud, Laure Pellé, Patrick Lailly

### Spectral Laguerre Method for Viscoelastic Seismic Modeling

The paper presents an efficient algorithm, based on the spectral Laguerre transform to the problem of seismic wave propagation in the heterogeneous viscoelastic medium.

Boris Mikhailenko, Alexander Mikhailov, Galina Reshetova

### Numerical Modeling of P-wave AVOA in Media Containing Vertical Fractures

This paper investigates the use of seismic anisotropy and amplitude variation with offset and azimuth (AVOA) for fracture characterization. The paper looks at P to P-wave reflectivity at the interface between an isotropic half-space overlying a transversely isotropic half-space with a horizontal symmetry axis (HTI medium). The reflecting HTI medium is chosen to be long wave equivalent a medium with aligned vertical cracks (a single crack orientation) in an isotropic host rock. Azimuthal variation of AVO gradient is investigated for the three models: with fluid-filled cracks, with dry cracks and for the case of saturated cracks in an porous host rock — in different situations of changes in the elastic parameters in the upper and lower media. It is shown that P-P AVOA holds significant information about fracturing but potential ambiguity in the interpretation of these data is observed that could lead to incorrect determination of fracture infill (gas o water) and fracture orientation.

Tatiana Chichinina, Vladimir Sabinin, Gerardo Ronquillo Jarillo

### Uniqueness of the Solution to the Reflection-Transmission Problem in a Viscoelastic Layer

Uniqueness is investigated for the solution to the reflection-transmission problem in a viscoelastic layer sandwiched between elastic half spaces. The layer and the half-spaces are isotropic but are allowed to be uniaxially inhomogeneous.

Angelo Morro

### A Time Domain Method for Modeling Wave Propagation Phenomena in Viscoacoustic Media

In many applications, realistic propagation media disperse and attenuate the acoustic waves. This behavior can be taken into account by a viscoacoustic model. Since in general, the viscoacoustic modulus is a function of frequency, incorporating this into time domain computations is practically non-feasible with classical discretization methods. However, one can use an approximation of the viscoacoustic modulus by a low-order rational function of frequency We use here, such an approximation and show how it can be incorporated in the velocity-pressure formulation for viscoacoustic wave propagation. For the discretization in space we use a finite-element method and for the time discretization a second order centered finite difference scheme.

Jean-Philippe Groby, Chrysoula Tsogka

### Mathematical and Numerical Modeling of Wave Popagation in Linear Viscoelastic Media

In this paper, we are interested in the mathematical and numerical modeling of propagation of high frequency waves in underground and undersea. These media dissipate energy when they are subjected to deformations [6]. For simulating the physical experiments, it is important to take into account the waves absorption. This phenomenon of absorption is due to a viscoelastic property of the propagation medium.

Eliane Bécache, Abdelaâziz Ezziani, Patrick Joly

### Numerical Model of Seismic Wave Propagation in Viscoelastic Media

Oil and gas reservoirs have a high seismic attenuation that is caused by their viscoelastic properties. Mathematical models for viscoelastic media differ by physical assumptions and solution methods [1, 2, 3]. The approach [1] seems attractive for its mathematical simplicity. Below is further development of it.

Vladimir Sabinin, Tatiana Chichinina, Gerardo Ronquillo Jarillo

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