On the far-field operator in elastic obstacle scattering

We investigate the far-field operator for the scattering oftime-harmonic elastic plane waves by either a rigid body, acavity, or an absorbing obstacle. Extending results of Colton& Kress for acoustic obstacle scattering, for the spectrumof the far-field operator we show that there exist an infinitenumber of eigenvalues and determine disks in the complex planewhere these eigenvalues lie. In addition, as counterpart ofan identity in acoustic scattering due to Kress & Päivärinta,we will establish a factorization for the difference of thefar-field operators for two different scatterers. Finally, extendinga sampling method for the approximate solution of the acousticinverse obstacle scattering problem suggested by Kirsch to elasticity,this factorization is used for a characterization of a rigidscatterer in terms of the eigenvalues and eigenelements of thefar-field operator.     

Numerical analysis for the scattering by obstacles in a homogeneous chiral environment

The scattering of time-harmonic electromagnetic waves propagating in a homogeneous chiral environment by obstacles is studied. The problem is simplified to a two-dimensional scattering problem, and the existence and the uniqueness of solutions are discussed by a variational approach. The diffraction problem is solved by a finite element method with perfectly matched absorbing layers. Our computational experiments indicate that the method is efficient.     

Elliptic grid generation methods for scattering from multiple obstacles

A novel finite difference time domain method for acoustic scattering on generalized curvilinear coordinates is briefly described. Scattering over two-dimensional complex regions consisting of multiple scatterers are analyzed. The grid generation algorithm decomposes regions with finite number of holes to contiguous single-hole subregions. Individual grids are obtained for each subregions and they are matched with smoothness across interfaces. The new algorithm is an extension to multiple obstacles of the technique introduced in [V. Villamizar, M. Weber, Boundary-Conforming Cordinates with Grid Line Control for Acoustic Scattering from Complexly Shaped Obstacles, Numer. Meth. Part Differ. Equ. 23 (2007) 1445–1467]. The method is successfully applied to approximate the pressure field resulting from the acoustic scattering of a plane wave from two complexly-shaped obstacles. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low energy π-nucleus elastic scattering analysis

Résumé Un modèle, construit à l'aide de l'équation deKisslinger pour un noyau sphérique de densité nucléaire homogène, est ajusté aux mesures de diffusion élastique ^––C à trois énergies différentes. Les valeurs des potentiels ajustant ces mesures sont indiquées.     

On the far-field operator for electromagnetic scattering in chiral media

We consider time-harmonic electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium where a perfect conductor has been immersed. Assuming that the incident electric field is a superposition of plane incident electric waves, the corresponding scattered field and the far-field pattern are expressed as the superposition of the scattered fields and the far-field patterns respectively. It is also proved that the sets of far-field patterns are complete if and only if there does not exist an eigenfunction to the interior perfect conductor problem that vanishes on the boundary of the scatterer which is an electric Herglotz field. The Left-Circularly Polarized and the Right-Circularly Polarized far-field operators are defined and studied and using them the electric far-field operator is defined too. The properties of the above operators and Herglotz functions are related to the solution of the interior perfect conductor boundary value problem.     

Lens rigidity in scattering by non-trapping obstacles

We prove that if two non-trapping obstacles in \(\mathbb {R}^n\) satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.     

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Seismic data is modeled in the high‐frequency approximation, using the techniques of microlocal analysis. We consider general, anisotropic elastic media. Our methods are designed to allow for the formation of caustics. The data is modeled in two ways. First, we give a microlocal treatment of the Kirchhoff approximation, where the medium is assumed to be piecewise smooth, and reflection and transmission occur at interfaces. Second, we give a refined view on the Born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium. The joint formulation of Born and Kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters. The latter allows the treatment of scattering up to grazing angles. The outcome of the analysis is a characterization of the singular part of seismic data. We obtain a set of pseudodifferential operators that annihilate the data. In the process we construct a Fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function. In our construction this Fourier integral operator becomes invertible. We give the conditions for invertibility for general acquisition geometry. The result is also of interest for inverse scattering in acoustic media. © 2002 John Wiley & Sons, Inc.     

Decoupled field integral equations for electromagnetic scattering from homogeneous penetrable obstacles

We present a new method for the analysis of electromagnetic scattering from homogeneous penetrable bodies. Our approach is based on a reformulation of the governing Maxwell equations in terms of two uncoupled vector Helmholtz systems: one for the electric field and one for the magnetic field. This permits the derivation of resonance-free Fredholm equations of the second kind that are stable at all frequencies, insensitive to the genus of the scatterers, and invertible for all passive materials including those with negative permittivities or permeabilities. We refer to these as decoupled field integral equations.     

The existence of the scattering operator for moving obstacles

For solutions of the wave equation outside a moving obstacle, the scattering operator exists if and only if the local energy decays to zero.     

Domain sensitivity analysis of the acoustic far‐field pattern

We consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd.     

Trapped-mode and gap-band effects in elastic waveguides with obstacles

Traveling wave propagation in elastic waveguides with obstacles in the form of cracks, voids, inclusions or surface irregularities is considered. The investigation is focused on the trapped–mode phenomena featured by the time–averaged harmonic wave energy localization near the obstacles in the form of energy vortices. The latter results, in particular, in narrow gap bands in the frequency plots of transmission coefficients. The study is carried out using analytically based computer models relying on wave expressions in terms of path Fourier integrals, Green's matrices for the laminate structures, and asymptotics for body and traveling waves derived from those integrals. The connection between the resonance effects and natural frequencies (spectral points of the related boundary value problems) in the complex frequency plane is analyzed as well. Examples of spectral points touching the real axis in the course of varying crack size are presented. The eigenforms associated with such discrete spectral points lying in a continuous spectrum depict strong wave energy localization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonstandard tools of nonsmooth analysis

This is an overview of the basic tools of nonsmooth analysis which are grounded on nonstandard models of set theory. By way of illustration we give a criterion for an infinitesimally optimal path of a general discrete dynamic system.     

On the T-matrix for scattering by small obstacles

Acoustic scattering by bounded obstacles is considered, in both two and three dimensions. Relations between the T-matrix and the far-field pattern are derived, and then used to obtain new approximations for the T-matrix for small obstacles. The problem of scattering by a pair of small sound-soft circular cylinders is also solved, in the Rayleigh approximation, using bipolar coordinates.     

Optimizing human activity patterns using global sensitivity analysis

Implementing realistic activity patterns for a population is crucial for modeling, for example, disease spread, supply and demand, and disaster response. Using the dynamic activity simulation engine, DASim, we generate schedules for a population that capture regular (e.g., working, eating, and sleeping) and irregular activities (e.g., shopping or going to the doctor). We use the sample entropy (SampEn) statistic to quantify a schedule’s regularity for a population. We show how to tune an activity’s regularity by adjusting SampEn, thereby making it possible to realistically design activities when creating a schedule. The tuning process sets up a computationally intractable high-dimensional optimization problem. To reduce the computational demand, we use Bayesian Gaussian process regression to compute global sensitivity indices and identify the parameters that have the greatest effect on the variance of SampEn. We use the harmony search (HS) global optimization algorithm to locate global optima. Our results show that HS combined with global sensitivity analysis can efficiently tune the SampEn statistic with few search iterations. We demonstrate how global sensitivity analysis can guide statistical emulation and global optimization algorithms to efficiently tune activities and generate realistic activity patterns. Though our tuning methods are applied to dynamic activity schedule generation, they are general and represent a significant step in the direction of automated tuning and optimization of high-dimensional computer simulations.     

On the domain derivative for scattering by impenetrable obstacles in chiral media

For scattering of electromagnetic waves in a chiral medium bysome perfectly conducting inclusions, we study the dependenceof the scattered field on the boundary of the inclusions andshow its Fréchet differentiability in appropriate spaces.Further, we derive a characterization of the derivative as asolution to some corresponding chiral boundary value problem.Our proof contains a new approach to rigorously derive thischaracterization.     

Long-wavelength approximation in scattering of elastic waves

We construct the asymptotic form of the solution in the long-wavelength limit for the problem of scattering of plane waves in an elastic medium by a cavity or rigid nonmoving inclusion. The parameters determining the scattered wave at large distances in the first approximation are expressed in terms of the integrated characteristics of the scatterer, such as its volume, the tensor analog of the capacity, the Wiener capacity in the two-dimensional case, and the dipole elastic tensor. Some of these characteristics are new.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 6–19, 1986.     

Tangentially continuous directional derivatives in nonsmooth analysis

In this paper, we introduce a new class of nonsmooth functions in terms of a continuity property of the usual directional derivative. Under this approach, we study the subregular and the semismooth functions. Finally, we give conditions for a marginal function to be subregular and semismooth.This research was supported in part by the Fondo Nacional de Ciencias, Santiago, Chile.The authors thank A. Auslender and L. Thibault for discussions on the subject.     

Proper exhausters and coexhausters in nonsmooth analysis

There exist many tools to analyze nonsmooth functions. For convex and max-type functions, the notion of subdifferential is used, for quasidifferentiable functions – that of quasidifferential. By means of these tools, one is able to solve, e.g. the following problems: to get an approximation of the increment of a functional, to formulate conditions for an extremum, to find steepest descent and ascent directions and to construct numerical methods. For arbitrary directionally differentiable functions, these problems are solved by employing the notions of upper and lower exhausters and coexhausters, which are generalizations of such notions of nonsmooth analysis as sub- and superdifferentials, quasidifferentials and codifferentials. Exhausters allow one to construct homogeneous approximations of the increment of a functional while coexhausters provide nonhomogeneous approximations. It became possible to formulate conditions for an extremum in terms of exhausters and coexhausters. It turns out that conditions for a minimum are expressed by an upper exhauster, and conditions for a maximum are formulated via a lower one. This is why an upper exhauster is called a proper one for the minimization problem (and adjoint for the maximization problem) while a lower exhauster is called a proper one for the maximization problem (and adjoint for the minimization problem). The conditions obtained provide a simple geometric interpretation and allow one to find steepest descent and ascent directions. In this article, optimization problems are treated by means of proper exhausters and coexhausters.