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[1] T. Arens - The Scattering of Plane Elastic
Waves by a One-dimensional Periodic Surface
Math. Meth. Appl. Sci. 22, 55 - 72 (1999)
The two-dimensional scattering problem for time-harmonic plane waves
in an isotropic elastic medium and an effectively infinite periodic surface
is considered. A radiation condition for quasi-periodic solutions similar
to the condition utilised in the scattering of acoustic waves by one-dimensional
diffraction gratings is proposed. Under this condition, uniqueness of solution
to the first and third boundary value problems is established. We then
proceed by introducing a quasi-periodic free field matrix of fundamental
solutions for the Navier equation. The solution to the first boundary value
problem is sought as a superposition of single- and double-layer potentials
defined utilizing this quasi-periodic matrix. Existence of solution is
established by showing the equivalence of the problem to a uniquely solvable
second kind Fredholm integral equation.
[2] T. Arens - A New Integral Equation
Formulation for the Scattering of Plane Elastic Waves by Diffraction
Gratings
J. Int. Equ. Appl. 11, 275-297 (1999)
The scattering of plane elastic waves by a rigid periodic surface is
considered. The Green's tensor for a half-space with a rigid surface is
introduced and its properties, notably its asymptotic decay rate in horizontal
layers above the plane, are analysed. The Green's tensor is then used to
define single and double layer potentials for a periodic surface, making
use of the generalised stress tensor. Subsequently, a novel integral equation
formulation for the scattering of plane waves by a diffraction grating
is derived using a Brakhage/Werner type ansatz for the solution. Employing
the Fredholm alternative, existence of solution is proved for all angles
of incidence and all wave-numbers.
[3] A. Meier, T. Arens, S.N. Chandler-Wilde and
A. Kirsch - A Nyström Method for a Class of Integral Equations on the Real
Line with Applications to Scattering by Diffraction Gratings and Rough
Surfaces
J. Int. Equ. Appl. 12 , 281-321 (2000)
We propose a Nyström/product integration method for a class of
second kind integral equations on the real line which arise in problems
of two-dimensional scalar and elastic wave scattering by unbounded surfaces.
Stability and convergence of the method is established, with convergence
rates dependent on the smoothness of components of the kernel. The method
is applied to the problem of acoustic scattering by a sound soft one-dimensional
surface which is the graph of a function f, and superalgebraic convergence is
established in the case when f is infinitely smooth. Numerical results are presented
illustrating this behaviour for the case when f is periodic (the diffraction grating case).
The Nyström method for this problem is stable and convergent uniformly
with respect to the period of the grating, in contrast to standard integral equation
methods for diffraction gratings which fail at a countable set of grating periods.
[4] T. Arens - Uniqueness for Elastic Wave Scattering
by Rough Surfaces
SIAM J. Math. Anal. 33, 461--476 (2001).
A zipped postscript version of this paper is available for
download (200 kB).
We consider a 2D elastic wave scattering problem for an unbounded surface
represented as the graph of a C^{1,a} function. The total displacement is
assumed to vanish on the surface. We present a new radiation condition, the
upwards propagating radiation condition, for such problems based on a similar
condition recently introduced for acoustic scattering problems. The relation
between this radiation condition and more commonly used conditions is
discussed. Subsequently we prove uniqueness of solution to the scattering
problem under this radiation condition for a general class of incident fields,
including plane and cylindrical waves.
[5] T. Arens - Linear Sampling Methods for 2D Inverse
Elastic Wave Scattering
Inverse Problems 17, 1445-1464 (2001).
This paper won a 2nd prize at the 2001 Leslie Fox Prize competition.
The problem of determining the shape of a rigid body from the knowledge of the
far field patterns of incident plane compressional and shear waves in two
dimensional elasticity is considered. We discuss the application of the
original linear sampling method and of the related (F^{*}F)^^{1/4}-method
to tackle this problem. It is established that the far field operator is compact
and normal. A theoretical basis for both reconstruction methods is developed
and numerical results are shown, illustrating the excellent quality of
reconstructions attainable.
[6] T. Arens - Existence of Solution in Elastic
Wave Scattering by Unbounded Rough Surfaces
Math. Meth. Appl. Sci. 25, 507-528 (2002).
We consider the two-dimensional problem of the scattering of a time-harmonic
wave, propagating in an homogeneous, isotropic elastic medium, by a rough
surface on which the displacement is assumed to vanish. This surface is
assumed to be given as the graph of a function f Î
C^{1,1}(R). Following up on earlier work establishing uniqueness
of solution to this problem, existence of solution is studied via the boundary integral equation
method. This requires a novel approach to the study of solvability of integral
equations on the real line. The paper establishes the existence of a unique
solution to the boundary integral equation formulation in the space of bounded
and continuous functions as well as in all L^{p} spaces, p
Î [1, ¥] and hence existence
of solution to the elastic wave scattering problem.
[9] T. Arens - An Approximation Property of Elastic Herglotz Wave Functions and
its Application in the Linear Sampling Method
J. Inverse Ill-Posed Probl. 11, 219-233 (2003)
Herglotz wave functions play an important role in a class of reconstruction
methods for inverse scattering problems known as \textsl{linear sampling
methods}. We here consider these functions in the setting of
linearized elasticity and derive representations in
terms of eigenfunctions to the Navier operator in two spatial dimensions. We
then show the important property that the elastic Herglotz Wave functions are
dense in the space of solutions to the Navier equation with respect to the
[H^{1}(D)]^{2} norm for any bounded Lipschitz domain D.
The proof of this property in three-dimensions, not essentially different from the 2D argument,
is also outlined. The paper is concluded with an application of the approximation property in
the mathematical foundation of the linear sampling method for the
reconstruction of rigid obstacles from the knowledge of the far field operator.
[10] T. Arens and A. Kirsch - The Factorization Method in Inverse
Scattering from Periodic Structures
Inverse Problems 19, 1195 - 1211 (2003)
The application of the Factorization Method, the refined version of
the Linear Sampling Method, to scattering by a periodic surface is
considered. Central to this method is the Near Field Operator
N, mapping incident fields to the corresponding scattered
fields on a horizontal line. A factorization of N forms the
basis for the method. It is shown that the middle operator in this
factorization, the adjoint of the single layer operator, is coercive
if the wave number is small enough. Thus this operator can, for
general wave number, always be written as the sum of a coecive and a
compact operator. We use this property to define an auxiliary positive
operator N_{#} which can be constructed directly from
N and which makes it possible to reconstruct the scattering
surface directly using a simple numerical algorithm.
[11] D. Natroshvili, T. Arens and
S.N. Chandler-Wilde - Uniqueness, Existence and Integral Equation Formulations for Interface Scattering
Problems
Memoirs on Differential Equations and Mathematical Physics 30,
Georgian Academy of Sciences, 105-146 (2003)
We consider a two-dimensional transmission problem in which Hemlotz
equations with different wave numbers hold in adjacent non-locally
perturbed half-planes having a common boundary which is an infinite,
one-dimensional, rough interface line. First a uniqueness theorem for
the interface problem is proved provided that the scatterer is a lossy
obstacle. Afterwards, by potential methods, the non-homogeneous
interface problem is reduced to a system of integral equations and
existence results are established.
[12] T. Arens - Why Linear Sampling Works!
Inverse Problems 20 , 163-173 (2004)
The Linear Sampling Method is a method to reconstruct the shape of
an obstacle in time-harmonic inverse scattering without a priori knowledge
on either the physical properties or the number of disconnected components
of the scatterer. Although it has numerically proven to be a fast and
reliable method in many situations, no mathematical argument has yet been
found to prove why this is so. Using results obtained by Kirsch in deriving
the related Factorization Method, we show in this paper that for a
large class of scattering problems, the Linear Sampling can be interpreted
rigorously as a numerical method to reconstruct the shape of an obstacle,
or in other words that Linear Sampling must work for problems in this
class.
Abstract.We consider the scattering of elastic waves by an unbounded surface on which the displacement vanishes. The wave field is assumed to be time-harmonic and the propagation medium to be homogeneous and isotropic. The scattering surface is assumed to be given as a graph of a bounded function f Î C^{1,a}, but otherwise no assumptions are made.
The problem is formulated as a boundary value problem for the scattered field in the unbounded domain above the scattering surface. This boundary value problem formulation includes a novel radiation condition characterising upward propagating waves. The way in which this radiation condition generalises other radiation conditions commonly employed in elastic wave scattering problems is discussed in detail. It is then shown that the boundary value problem, and thus the scattering problem, admits at most one solution for a general class of incident fields including plane and cylindrical waves.
Existence of solution is established via the boundary integral equation method. The properties of elastic single- and double-layer potentials on rough surfaces are studied with an emphasis on obtaining estimates uniformly for classes of such surfaces. An equivalent formulation of the scattering problem as a boundary integral equation of the second kind is obtained. Since the scatterer is unbounded the integral operator in this equation is not compact, and nor is the equation of a standard singular type previously studied. Thus, a new solvability theory is developed, which establishes solvability of the integral equation in the space of bounded and continuous functions, and also in all L^{p}-spaces, 1 £ p £ ¥.