The singular function of a surface and physical optics inverse scattering

It is shown how to recover both the location and the reflection coefficient of a scatterer using only high frequency backscattered data. The result is obtained without use of the far field approximation although a separate identity is derived when this approximation is introduced. This latter result improves upon previously derived physical optics far field inverse scattering identities.     

A comparison and extensions of algorithms for quantitative imaging of laminar damage in plates. I. Point spread functions and near field imaging

Real-time structural health monitoring (SHM) and safety prognostics require quantitative and continuously updated information on damage size and severity. A unified theoretical solution is presented for three distinct approaches that have been used for in situ imaging of structural damage in plate-like structures. These approaches are based on (i) linearised inverse scattering (or generalised diffraction tomography), (ii) beamforming, and (iii) reverse time migration. In all three approaches, the damaged region is regarded as a weak scatterer. Such an approach is appropriate for early damage detection that is of great practical interest. The linearised inverse is based on a rigorous mathematical formulation, whereas beamforming and reverse time migration are based on heuristic arguments, but the latter are more convenient for practical implementation. It is shown that, in the far-field approximation, the three imaging algorithms have a very similar mathematical structure. Analytical expressions are derived for the point spread functions (PSFs), which represent the reconstructed image for a point-like scatterer. Although the analytical expressions for the PSFs are different, the corresponding profiles are virtually identical. Based on these observed mathematical similarities, modified versions of the diffraction tomography and time-reversal algorithms are presented that combine the advantages of the various approaches. These modified algorithms are extensively evaluated using analytical solutions of a circular scatterer. The resulting algorithms are shown to provide accurate estimates for damage size and damage severity over a range of size and severity that is consistent with the weak scatterer approximation.     

Progress on a mathematical inversion technique for non destructive evaluation

In earlier work, computer output was presented for synthetic pulse-echo data which was processed according to a mathematical imaging technique. This technique was based on the physical optics farfield inverse scattering (acronym, POFFIS) formalism for scattering by volume defects. In this paper a number of theoretical advances in the POFFIS formalism are reported, with attendant revisions in the computer algorithm.Firstly, a revised POFFIS formalism was developed in which the surface of the scatterer is directly related to the scatteringi data. In this formalism, aperture limited scattering data yields an image of a corresponding aperture of the scattering surface of the defect. Secondly, this formalism will also yield an image of the scattering surface of a crack. Thirdly, for true amplitude data, the impedance or reflection coefficient may be read directly from the computer output. Related to this last result was the elimination of an “image fading” phenomenon at certain critical angles. Fourthly, the computer algorithm, which was originally designed to process data for a spherically symmetric “trailer hitch”, was modified (and tested) to process data when the range to the center of the coordinate system was different at each observational angle. Fifthly, the algorithm was modified (and tested) to process data when the average propagation speed varied with angle.Implementation on a real data set is discussed.     

High-frequency scattering from a metal-like dielectric lens

The scattering of a plane electromagnetic wave by a dielectric lens which behaves like a metal reflector is considered. At short wavelengths, the leading term of the backscattered field cannot be determined entirely through simple geometrical optics considerations; instead, it is obtained by means of a modified Watson transformation of the exact solution. The difficulties that arise in applying this technique to other lenses are discussed.This research was sponsored by the U.S. Air Force Cambridge Research Laboratories under Contract F 19628-68-C-0071.     

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

We present a novel hybrid numerical–asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.     

Estimates for low-frequency elastic scattering by a rigid body

When a plane elastic wave is scattered by a rigid body the surface integral of the traction, projected along the direction of polarization of the incident wave, provides the leading low-frequency approximation for the scattering amplitudes. Two kinds of lower and upper bounds for the surface traction integral are given. One is based on the geometrical characteristics of the scatterer and is expressed in terms of corresponding values of the best fitting interior and exterior confocal triaxial ellipsoids. The case of best fitting interior and exterior spheres is examined as a special case. These bounds are sharp in the sense that they both become equalities when the scatterer degenerates to an ellipsoid. The other kind of lower and upper bounds involve the capacity of the scatterer. All estimates were obtained by using the generalized Dirichlet and Thomson Principles of Potential Theory in Elastostatics. Furthermore, all constants appearing in the bounds are given in terms of the ratio of the phase velocities for the transverse and the longitudinal wave. An upper bound for scattering by a cube at normal incidence is also included.This work was done while both authors were visiting the Department of Mathematics of The University of Tennessee at Knoxville. The second author wishes to acknowledge partial support from The University of Tennessee Science Alliance.     

Scattering of elastic waves from symmetric inhomogeneities at low frequencies

The problem of the scattering of elastic waves is treated in the low-frequency regime by a systematic expansion in powers of the frequency in the spatial domain of the scatterer. The zeroth and first degree scattering amplitudes vanish if the scatterer is localized in all directions. The second degree scattering amplitude corresponds to the so called Rayleigh regime in which the quasi-static result of Gubernatis et al. is valid. In general, the third and higher degree scattering amplitudes are nonvanishing. However, in the case where the scatterer has inversion symmetry about the origin, it is shown that the third degree scattering amplitude vanishes identically for all incident and scattered directions and for all polarizations. This result implies that the frequency derivative of the phase shift approaches zero at least quadratically as the frequency goes to zero. In other words, at sufficiently low frequencies the effective scattering center of a scatterer with inversion symmetry is its geometrical center. The use of this result in the processing of experimental scattering data is discussed.     

Inverse scattering for the homogeneous dispersive anisotropic slab using transient electromagnetic fields

An inverse scattering problem for a slab containing a homogeneous dispersive anisotropic medium is investigated. The inverse problem is to recover two three-dimensional dyadic susceptibility kernels from knowledge of the scattering kernels. Time domain techniques involving transient electromagnetic plane waves, wave splitting, invariant imbedding and a Green function technique are used. The inverse problem is separated into two parts: The Dynamics Inverse Problem (DIP) and Retrieval of Interior Parameters (RIP). Furthermore, mirror images and the Mirror Image Pair (MIP) are discussed. The DIP is solved numerically by using an inverse algorithm and scattering data from one MIP. The RIP turns out to be well posed (system of Volterra equations of the second kind) and needs in general two MIPs. In the DIP, the equations for initial values using transmission data have in general not a unique solution. Constraints and simplifications for certain classes of media are pointed out. Numerical examples, including noisy data, illustrate the analysis.     

Geometrical optics approximation of light scattering by large air bubbles

For large spherical bubbles in water,geometrical optics approximation is considered a better method for calculating light scattering patterns.In this paper,the basic theory of geometrical optics approximation is clarified.The change of phase for bubbles is calculated when total reflection occurs,which is different from particles with relative refractive indices larger than 1.Verification of the method was achieved by assuming a spherical particle and comparing present results to Mie scattering and Debye calculation.Agreement with the Mie theory was excellent in all directions when the dimensionless size parameter is larger than 50.Limitations of the geometrical optics approximation are also discussed.     

Approximation by Multipoles of the Multiple Acoustic Scattering by Small Obstacles in Three Dimensions and Application to the Foldy Theory of Isotropic Scattering

The asymptotic analysis carried out in this paper for the problem of a multiple scattering in three dimensions of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy’s method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and the taking into account of self-interactions. In this way, it is established that the usual Foldy model may lead to an approximation whose asymptotic behavior is the same than that obtained when the multiple scattering effects are completely neglected.     

One-dimensional direct and inverse scattering in causal space

The conventional exact time-domain formulation of the one-dimensional electromagnetic direct scattering problem is re-derived and transformed into causal space, the axies of which are defined as x ± ct. In this causal space, the direct scattering solution reduces to an exceedingly simple expression. This causal space solution yields the scattered fields incidentally, without computations additional to the basic solution for the current densities. the inverse scattering problem is then solved in this causal space. This solution to the inverse problem also consists of an exceedingly simple expression. Numerico-experimental results for both the direct and inverse causal space solutions are presented, and preliminary results of an error analysis are discussed.     

Scattering by penetrable acoustic targets

An acoustic target of constant density ?_t and variable index of refraction is imbedded in a surrounding acoustic fluid of constant density ?_a. A time harmonic wave propagating in the surrounding fluid is incident on the target. We consider two limiting cases of the target where the parameter ε ≡ ?_a/?_t → 0 (the nearly rigid target) or ε → ∞ (the nearly soft target). Wh en the frequency of the incident wave is bounded away from the ‘in-vacuo’ resonant frequencies of the target, the resulting scattered field is essentially the field scattered by the rigid target for ε = 0 or the soft target if ε → ∞. However, when the frequency of the incident wave is near a resonant frequency,the target oscillates and its interaction with the surrounding fluid produces peaks in the scattered field amplitude. In this paper we obtain asymptotic expansions of the solutions of the scattering problems for the nearly rigid and the nearly soft targets as ε → 0 or ε → ∞, respectively, that are uniformly valid in the incident frequency. The method of matched asymptotic expansions is used in the analysis. The outer and inner expansions correspond to the incident frequencies being far or near to the resonant frequencies, respectively. We have applied the method only to simple resonant frequencies, but it can be extended to multiple resonant frequencies. The method is applied to the incidence of a plane wave on a nearly rigid sphere of constant index of refraction. The far field expressions for the scattered fields, including the total scattering cross-sections, that are obtained from the asymptotic method and from the partial wave expansion of the solution are in close agreement for sufficiently small values of ε.     

Diffraction by a metallic wedge covered with a dielectric material

In the literature, the usual formulation which assumes a constant impedance boundary condition does not permit one to express both the value of the geometrical optics field and the actual excited surface wave, since the impedance associated with the material for the reflection and for the surface wave are generally different. To obtain these elements in the field expression, we replace the notion of constant impedance by a differential operator. The general solution presented here preserves enough degrees of freedom to satisfy the continuity of fields at the internal junction of the two materials which cover each face of the wedge. The numerical calculations are based upon a function, related to the Maliuzhinets function, in a form which is easily computable.     

The penetrable coated sphere embedded in a point source excitation field

A low frequency acoustic wave field emanates from a given point and fills up the whole space. A penetrable lossy sphere with a coeccentric spherical core, which is also penetrable and lossy but characterized by different physical parameters, disturbs the given point source field. We obtain zeroth- and first-order low frequency solutions of this scattering problem in the interior of the spherical core, within the spherical shell, and in the exterior medium of propagation. We also derive the leading nonvanishing terms of the normalized scattering amplitude, the scattering cross-section as well as the absorption cross-section. The special case of a penetrable sphere is recovered either by equating the physical parameters that characterize the media in the shell and in the exterior, or by reducing the radius of the core sphere to zero. By letting the compressional viscosity of the medium in the interior sphere, or in the shell, go to zero, we obtain corresponding results for the lossless case. The incident point source field is so modified as to be able to obtain the corresponding results for plane wave incidence in the limit as the source point approaches infinity. It is observed that a small scatterer interacts stronger with a point source generated field than with a plane wave. A detailed analysis of the influence that the geometrical and the physical parameters of the problem have on the scattering process is also included. An interesting conclusion is that if the point source is located at a distance more than five radii of the scatterer away from it, then no significant changes with the plane excitation case are observed.     

Interface effects on the electromagnetic radiation emanating from an embedded piezoelectric nano-fiber incident upon by SH-waves

The multiphysics process of the electromagnetic (EM) radiation induced from an embedded nanofiber scatterer subjected to SH-waves is of interest. By discarding the commonly employed electroquasistatic approximation, the fully-coupled elastodynamics and Maxwell’s equations have been solved simultaneously within the mathematical framework of piezoelectric surface elasticity theory. Certain subtleties regarding the introduced interface characteristic lengths that make the examination of the size effect on the EM radiated power, scattering cross section, fundamental resonance frequency, and distribution of the magnetic field possible will be discussed. The obtained results can be helpful for designing electrically small antennas, sensors, and resonators using piezoelectric nanofibers.     

An explicit asymptotic model for the Bleustein–Gulyaev waveUn modèle explicite pour l'onde de Bleustein–Gulyaev

The antiplane motion of a transversely isotropic piezoelectric half-space is considered. An explicit asymptotic model is derived for the far field of the surface wave. It involves, in particular, a 1D hyperbolic equation for surface shear deformation propagating with the finite wave speed predicted for the first time by J.L. Bleustein and Yu.V. Gulyaev. Neumann and Dirichlet problems are formulated to restore interior mechanical and electric fields. The derivation utilizes asymptotic arguments combined with Lourier symbolic integration. Comparison with the exact solution is presented for surface impact loading. To cite this article: J. Kaplunov et al., C. R. Mecanique 332 (2004).     

应变损伤材料平面应力动态裂纹尖端场的渐近解

采用塑性动力学方程,对应变损伤材料的平面应力动态裂纹尖端场进行了渐近分析。假定损伤规律服从反比例关系,对平面应力问题,导出了本构方程,并给出了动态弹塑性场的渐近解,揭示了场的渐近特性。     

Elastodynamic wave scattering by an incompressible elastic inclusion

The problem of scattering of time-harmonic elastodynamic waves by an incompressible elastic inclusion is solved by means of the null field approach. The solution is obtained both directly and as a limit of the solution to the corresponding problem for a compressible inclusion. It is also demonstrated that the null field solution to the problem of scattering by a rigid movable scatterer can be obtained from a null field solution for the incompressible scatterer by taking the limit of infinite shear modulus. Some numerical results for spherical and spheroidal inclusions are given.     

有限散射信号下二维缺陷形状识别的罚函数方法

研究在有限照射角度和频带宽度下二维缺陷的形状识别问题。首先,通过引进介质参数扰动函数,建立介质参数扰动函数和弹性波散射场之间的非线性关系,并将所关心的缺陷的形状识别问题转化为关于扰动函数的反演;然后,利用变分技术和优化方法求解,为了弥补散射数据的不足,在总的目标函数中,采用附加度量函数作为罚函数;最后,对后场散射远场测量时有限照射角度和频带宽度下几种典型缺陷进行了模拟识别,表明了;表明了罚函数法的有效性。