Iteratively-improved Robin boundary conditions for the finite element solution of scattering problems in unbounded domains

An iterative procedure is described for the finite-element solution of scalar scattering problems in unbounded domains. The scattering objects may have multiple connectivity, may be of different materials or with different boundary conditions. A fictitious boundary enclosing all the objects involved is introduced. An appropriate Robin (mixed) condition is initially guessed on this boundary and is iteratively improved making use of Green's formula. It will be seen that the best choice for the Robin boundary condition is an absorbing-like one. A theorem about the theoretical convergence of the procedure is demonstrated. An analytical study of the special case of a circular cylindrical scatterer is made. Comparisons are made with other methods. Some numerical examples are provided in order to illustrate and validate the procedure and to show its applicability whatever the frequency of the incident wave. Although particular emphasis is laid in the paper on electromagnetic problems, the procedure is fully applicable to other kinds of physical phenomena such as acoustic ones. © 1998 John Wiley & Sons, Ltd.     

Diffusion Calculation of Change of Back-scattered Light Beam Intensity from Turbid Medium Owing to the Existence of an Inhomogeneity

Abstract

A solution to the searchlight problem on back-scattering of a narrow pencil light from a turbid medium with an inhomogeneity is presented for semi-infinite geometry in the framework of the diffusion approximation. The inhomogeneity may be an object with a given reflection coefficient or a variation of mean scattering and absorption properties of a turbid medium. A solution to the diffusion equation for mean diffuse radiant intensity with given boundary conditions is obtained by a perturbation theory method relative to the inhomogeneity effect. The spatially limited inhomogeneity contribution to the back-scattering light intensity is expressed in terms of an inhomogeneity diffuse scattering amplitude and the probability density distribution of photon paths in depth. The theoretical results obtained are applied to the interpretation of the Cui, Kumar and Chance experiment on the change of the back-scattered light beam intensity from turbid biological tissue phantom owing to the existence of small absorbers.     

Scattering of Thermal Waves and Non-steady Effective Thermal Conductivity of Unidirectional Fibrous Composites with Functionally Graded Interface

In this paper, a thermal wave method is applied to investigate the non-steady effective thermal conductivity of unidirectional fibrous composites with a functionally graded interface, and the analytical solution of the problem is obtained. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattering and refraction of thermal waves by a cylindrical fiber with an inhomogeneous interface layer in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material layers are expressed by using the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions of the layers. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and the non-steady effective thermal conductivity of composites. As an example, the effects of a graded interface on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In the region of intermediate and high frequencies, the effect of the properties of the interface on the effective thermal conductivity is greater. Comparisons with the steady thermal conductivity obtained from other methods are also presented.     

Asymptotic solutions for optical properties of large particles with strong absorption

The transverse wave condition is not applicable to the refracted electromagnetic wave within the context of geometrical optics when absorption is involved. Either the TM or the TE wave condition can be assumed for the wave to locally satisfy the electromagnetic boundary condition in a ray-tracing calculation. The assumed wave mode affects both the reflection and the refraction coefficients. As a result, nonunique solutions for these coefficients are inevitable. In this study the appropriate solutions for the Fresnel reflection-refraction coefficients are identified in light-scattering calculations based on the ray-tracing technique. In particular, a 3 x 2 refraction or transmission matrix is derived to account for the inhomogeneity of the refracted wave in an absorbing medium. An asymptotic solution that completely includes the effect of medium absorption on Fresnel coefficients is obtained for the scattering properties of a general polyhedral particle. Numerical results are presented for hexagonal plates and columns with both preferred and random orientations.     

Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method

This work presents a novel finite-element solution to the problem of scattering from a finite and an infinite array of cylindrical objects with arbitrary shapes and materials over perfectly conducting ground planes. The formulation is based on using the surface integral equation with Green's function of the first or second kind as a boundary constraint. The solution region is divided into interior regions containing the cylindrical objects and the region exterior to all the objects. The finite-element formulation is applied inside the interior regions to derive a linear system of equations associated with nodal field values. Using two-boundary formulation, the surface integral equation is then applied at the truncation boundary as a boundary constraint to connect nodes on the boundaries to interior nodes. The technique presented here is highly efficient in terms of computing resources, versatile, and accurate in comparison with previously published methods. The near and far fields are generated for a finite and an infinite array of objects. While the surface integral equation in combination with the finite-element method was applied before to the problem of scattering from objects in free space, the application of the method to the important problem of scattering from objects above infinite flat ground planes is presented here for the first time, to our knowledge.     

Features of laser diagnostics of laminar spherical and annular inhomogeneities

The direct problem of refraction by a laser plane in a medium with nonmonotone radial profile of the index of refraction is solved. The possibility of solving the inverse problem on the basis of the Abel integral is analyzed. A computer simulation of the direct problem for models of hot and cold layers in a liquid in the presence, or absence, of an inhomogeneity core, is conducted. A technique for diagnosis of transparent spherical and cylindrical inhomogeneities based on a combined laser-computer refraction method is proposed. __________ Translated from Izmeritel’naya Tekhnika, No. 1, pp. 34–37, January, 2007.     

Diffuse light propagation in a turbid medium with varying refractive index: Monte Carlo modeling in a spherically symmetrical geometry

We report on the development of Monte Carlo software that can model media with spatially varying scattering coefficient, absorption, and refractive index. The varying refractive index is implemented by calculating curved photon paths in the medium. The results of the numerical simulations are compared with analytical solutions obtained using the diffusion approximation. The model under investigation is a scattering medium that contains a spherically symmetrical inclusion (inhomogeneity) created by variation in optical properties and having no sharp boundaries. The following steady-state cases are considered: (a) a nonabsorbing medium with a spherically symmetrical varying refractive index, (b) an inclusion with varying absorption and scattering coefficients and constant refractive index, and (c) an inclusion with varying absorption, scattering, and refractive index. In the latter case it is shown that the interplay between the absorption coefficient and the refractive index may create the effect of a hidden inclusion.     

A pseudopotential approach to scattering

Scalar wave scattering is discussed using pseudopotentials. These are singular, wavefunction-dependent source terms subsuming the scattering properties of the object. The latter can be an impenetrable body or an inhomogeneity. In this approach, the boundary condition on the scatterer does not enter into computations explicitly, but the scattering amplitude in an unbounded medium appears in the Helmholtz equation. The pseudopotential gives the correct solution of the scattering problem outside the smallest sphere circumscribing the scatterer. If the scattering medium is spatially limited by a boundary (e.g., a waveguide or an enclosure), then the pseudopotentials decouple the scatterer boundary condition from that on the limiting surface. This simplifies the problem in that only the Green’s function for the limiting boundary needs to be considered. The formulation is entirely in terms of the free-space Green’s function, and hence is independent of any particular assumption or specific expression of the incident field. Numerical results are presented for a spherical inhomogeneity in a spherical Dirichlet enclosure. The pseudopotential formulation is anticipated to be useful in situations involving multiple scattering centers, and extraneous bounding surfaces.     

Two-dimensional scattering from an impenetrable cylindrical obstacle in an acoustic quarterspace

The problem of sound scattering by an infinitely long hard or soft circular cylindrical obstacle suspended near a rigid corner is investigated. The separation of variables technique, the appropriate wave field expansions and the method of images along with the translational addition theorem for cylindrical wave functions are used to derive a closed-form analytical solution in form of infinite series. The analytical results are illustrated with a numerical example in which the cylindrical obstacle is positioned near the rigid boundary of a water-filled acoustic quarter-space. The backscattering form function amplitude and spatial distribution of the total acoustic pressure are evaluated and discussed for representative values of the parameters characterizing the system. The effects of incident wave frequency, angle of incidence and proximity of the cylinder to the rigid boundary are examined. Limiting case involving an infinite cylinder in an acoustic halfspace is considered and fair agreement with a well-known solution is established.     

半空间饱和土中圆形壳结构对平面P波的散射

在Biot波动理论的基础上,引入更符合工程实际的半透水边界条件,给出了半空间均质饱和土中半渗透圆柱形壳结构对平面P波散射的级数解。具体做法是利用波函数展开法,将入射波和散射波的势函数展开成Fourier-Bessel函数的无穷级数的形式,采用大半径凸圆弧来近似半空间表面,根据问题的连续性条件和边界条件,确定波函数展开式中的未知系数,进而得到问题的求解。计算结果表明:半空间饱和土和圆柱形壳界面处的径向应力与壳和土体的相对渗透性、入射角等有关,壳和饱和土体的相对渗透性越小,饱和土体和壳结构底部界面处径向应力所受影响越大。     

Inverse problem in optical diffusion tomography. II. Role of boundary conditions

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. The role of boundary conditions in the derivation of Fourier-Laplace inversion formulas is considered. Boundary conditions of a general mixed type are discussed, with purely absorbing and purely reflecting boundaries obtained as limiting cases. Four different geometries are considered with boundary conditions imposed on a single plane and on two parallel planes and on a cylindrical and on a spherical surface.     

Finite-difference time-domain and steady-state analysis with novel boundary conditions of optical transport in a three-dimensional scattering medium illuminated by an isotropic point source

We evaluate the numerical accuracy of finite-difference time-domain (FDTD) analysis of optical transport in a three-dimensional scattering medium illuminated by an isotropic point source. This analysis employs novel boundary conditions for the diffusion equation. The power radiated from an isotropic point source located at a depth equal to the reciprocal of the reduced scattering coefficient (1/μ'(s)) below the surface at the irradiated position is introduced to the integral form of the diffusion equation. Finite-difference approximations of the diffusion equation for a surface cell are derived by utilizing new boundary conditions that include an isotropic source even in a surface cell. Steady-state and time-resolved reflectances are calculated by FDTD analysis for a semi-infinite uniform scattering medium illuminated by an isotropic point source. The numerical results agree reasonably with the analytical solutions for μ'(s)=1-3 mm(-1) without resizing the mesh elements.     

The interaction between a penny-shaped crack and a spherical inhomogeneity in an infinite solid under uniaxial tension

Summary This paper presents an approximate three-dimensional analytical solution to the elastic stress field of a penny-shaped crack and a spherical inhomogeneity embedded in an infinite and isotropic matrix. The body is subjected to an uniaxial tension applied at infinity. The inhomogeneity is also isotropic but has different elastic moduli from the matrix. The interaction between the crack and the inhomogeneity is treated by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. The stress intensity factor at the boundary of the penny-shaped crack and the stress field inside the inhomogeneity are evaluated in the form of a series which involves the ratio of the radii of the spherical inhomogeneity and the crack to the distance between the centers of inhomogeneity and crack. Numerical calculations are carried out and show the variation of the stress intensity factor with the configuration and the elastic properties of the matrix and the inhomogeneity.     

Response of acoustic imaging systems using convergent leaky waves to cylindrical flaws

A theoretical study of imaging systems utilizing focused leaky surface acoustic waves (SAWs), and their response to certain kind of defects is presented. In particular, circular cylindrical inhomogeneities with axes perpendicular to the surface are considered. The scattering of the SAW from this cylinder is formulated with some approximations. The surface wave incident on the inhomogeneity is initially found as an angular spectrum of plane waves. However, to apply the boundary conditions at the cylindrical surface, the incident field has to be transformed into a superposition of cylindrical waves. Similarly, the scattered field, which is found in the form of outgoing cylindrical SAWs, is converted back to a plane wave spectrum. A formula is obtained for the transducer output voltage in terms of the position and the radius of the cylinder, and it is suitable for computer evaluation. By considering various locations for the cylinder, the sensitivity of the system around the focal point is studied. By comparing the output voltages for cylinders of different radii, the sensitivity of the system to the size of the inhomogeneity is examined. The numerical results are in agreement with the experimental observations.     

Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media

The reconstruction of the location and optical properties of objects in turbid media requires the solution of the inverse problem. Iterative solutions to this problem can require large amounts of computing time and may not converge to a unique solution. Instead, we propose a fast, simple method for approximately solving this problem in which calculated effective absorption and reduced scattering coefficients are backprojected to create an image of the objects. We reconstructed images of objects with centimeter dimensions embedded in a diffusive medium with optical characteristics similar to those of human tissue. Data were collected by a frequency-domain spectrometer operating at 120 MHz with a laser diode light source emitting at 793 nm. Intensity and phase of the incident photon density wave were collected from linear scans at different projection angles. Although the positions of the objects are correctly identified by the reconstructed images, the optical parameters of the objects are recovered only qualitatively.     

Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities

We develop a scattering matrix approach for the numerical calculation of resonant states and Q values of a nonideal optical disk cavity with an arbitrary shape and with an arbitrary varying refraction index. The developed method is applied to study the effect of surface roughness and inhomogeneity of the refraction index on Q values of microdisk cavities for lasing applications. We demonstrate that even small surface roughness (deltar < or approximately equal to lambda/50) can lead to a drastic degradation of high-Q cavity modes by many orders of magnitude. The results of the numerical simulation are analyzed and explained in terms of wave reflection at a curved dielectric interface, combined with an examination of Poincaré surfaces of section and of Husimi distributions.     

Use of constant,linear and quadratic boundary elements in 3D wave diffraction analysis

The performance of the Boundary Element Method (BEM) depends on the size of the elements and the interpolation function used. However, improvements in accuracy and efficiency obtained with both expansion and grid refinement increases demand on the computational effort. This paper evaluates the performance of constant, linear and quadratic elements in the analysis of the three-dimensional scattering caused by a cylindrical cavity buried in an infinite homogeneous elastic medium subjected to a point load. A circular cylindrical cavity for which analytical solutions are known is used in the simulation analysis. First, the dominant BEM errors are identified in the frequency domain and related to the natural vibration modes of the inclusion. Comparisons of BEM errors are then made for different types of boundary elements, maintaining similar computational costs. Finally, the accuracy of the BEM solution is evaluated when the nodal points are moved inside linear and quadratic discontinuous elements.     

Modal vibrations of an infinite cylinder in an acoustic halfspace

Acoustic radiation from an infinite cylindrical surface vibrating with an arbitrary, time-harmonic surface velocity distribution while positioned near the rigid/compliant boundary of a semi-infinite ideal compressible fluid medium is determined in an exact fashion using the classical method of separation of variables. The formulation utilizes the appropriate wave-field expansions and the method of images along with the pertinent translational addition theorem to develop a closed-form solution in form of infinite series. The analytical results are illustrated with numerical examples in which the cylindrical source, vibrating in the monopole and dipole-like modes, is positioned near the rigid/compliant boundary of a water-filled acoustic halfspace. Subsequently, the basic acoustic field quantities such as the modal acoustic radiation impedance load, radiated far-field pressure and the radiation intensity distribution are evaluated for representative values of the parameters characterizing the system.     

Flow Birefringence Studies: Scattering of a light wave by a turbulent flow showing optical anisotropy

The Mueller matrix of a scattering volume is calculated in two cases. In the first case, the medium is subjected to thermal fluctuations and the index of refraction is a random function. In the second case, the medium is a turbulent boundary layer showing flow birefringence. For both cases, the turbulence is considered as a random homogeneous isotropic process and the spectrum correlation functions of the index of the refraction are assumed to be gaussian. The results are compared in the case of a wave scattered perpendicular to the wave direction.     

Penetration depth limits of in vivo confocal reflectance imaging

We present experiments to predict the maximum penetration depth atwhich typical biological structures in amelanotic tissue can bedetected with confocal microscopy. The detected signal is examinedas the signal source strength (index of refraction mismatch), thesource depth, and the medium scattering coefficient are varied. Thedetected background produced by scattering outside the focal volume isexamined as the medium scattering coefficient, the depth in the medium, the dimensionless pinhole radius, nu(p), and theshape of the scattering phase function are varied. When the systemapproaches ideal confocal performance (nu(p) ? 3), the penetration depth is limited by the signal-to-noiseratio to approximately 3-4 optical depths (OD's) for a 0.05 indexmismatch. As nu(p) increases to 8, thepenetration depth is limited by the signal-to-background ratio and isdependent on the scattering coefficient. At mu(s) = 100 cm(-1) (l(s) = 100 mum) and an index mismatch of 0.05, the maximum penetrationdepth is approximately 2 OD.