The Pomraning-Eddington approximation to diffusion of light in turbid materials

The source-free diffusion problem of light in turbid media with generalized boundary conditions is considered. The intensity of light is considered as a sum of collimated and diffused radiance. In this way the problem is transformed to a source problem with a collimated source (problem 1). This problem is solved in terms of the corresponding source-free problem of simple boundary conditions (problem 2). The Pomraning-Eddington method is used to solve problem 2. Two coupled first-order differential equations are obtained involving the energy density and the radiation net flux. Weight functions are introduced in order to force the boundary conditions to be fulfilled. Numerical results are given and compared with previous calculations. The calculations show that the accuracy depends on the choice of the weight function.     

Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering

Exact integral equations are derived describing the source function and radiative flux in a two-dimensional, radially infinite cylindrical medium which scatters anisotropically. The problem is two-dimensional and cylindrical because of axisymmetric loading. Radially varying collimated radiation is incident normal to the upper surface while the lower boundary has no radiation incident upon it. The scattering phase function is represented by a spike in the forward direction plus a series of Legendre polynomials. The two-dimensional integral equations are reduced to a one-dimensional form by separating variables for the case when the radial variation of the incident radiation is a Bessel function. The one-dimensional form consists of a system of linear, singular Fredholm integral equations of second kind. Other more complex boundary conditions are shown to be solvable by a superposition of this basic Bessel function case. Diffusely incident radiation is also considered.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

On the flexoelectric deformations of finite size bodies

Exact equations describing flexoelectric deformation in solids, derived previously within the framework of a continuum media theory, are partial differential equations of the fourth order. They are too complex to be used in the cases interesting for applications. In this paper, using the fact of smallness of the elastic moduli of a higher order, simplified equations are proposed. Solution of the exact equations is approximately represented as a sum of two parts: the first part obeys one-dimensional differential equations and exponentially decays near surface and the second part satisfies the equations of classical theory of elasticity. The first part can be constructed in an explicit form. For the second part, boundary conditions are obtained. They have a form of the classical boundary conditions for the body under external forces on surface.     

Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity

Stress-strain relation in Eringen's nonlocal elasticity theory was originally formulated within the framework of an integral model. Due to difficulty of working with that integral model, the differential model of nonlocal constitutive equation is widely used for nanostructures. However, paradoxical results may be obtained by the differential model for some boundary and loading conditions. Presented in this article is a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory without employing any simplification in the model. The entire procedure of deriving equations of motion is carried out in the matrix form of representation, and hence, they can be easily used in the finite element analysis. For comparison purpose, the differential counterparts of equations are also derived. To study the outcome of analysis based on the integral and differential models, some case studies are presented in which the influences of boundary conditions, nonlocal length scale parameter and loading factor are analyzed. It is concluded that, in contrast to the differential model, there is no paradox in the numerical results of developed integral model of nonlocal continuum theory for different situations of problem characteristics. So, resolving the mentioned paradoxes by means of a purely numerical approach based on the original integral form of nonlocal elasticity theory is the major contribution of present study.     

Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

Abstract

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Pomraning–Eddington approximation for radiative transfer in a homogeneous solid cylinder

Abstract

The radiative transfer in a solid cylinder containing a homogeneous turbid medium with anisotropic scattering is considered. The medium has a diffuse and specular reflecting boundary illuminated by an external incidence and contains an internal energy source. This general problem can be solved in terms of the solution of the corresponding source-free problem with a specular reflecting boundary and isotropic external incidence. The Pomraning–Eddington approximation is used to solve the source-free problem. Three different weight functions are used to verify the boundary condition to find the constants of the solution. The partial flux, the irradiance and the net flux at the boundary for the general problem are calculated.     

Two-dimensional isotropic scattering in a semi-infinite cylindrical medium

Beginning with the integral equation for the source function, the solutions for the source function, flux and intensity at the boundary of a two-dimensional, isotropically scattering cylindrical medium are found. The incident radiation is collimated and normal to the surface of the medium and depends only on the radial coordinate. For a Bessel function boundary condition, separation of variables is used to reduce the source function integral equation to a one-dimensional equation. The resulting integral equation is shown to be the same as that for the two-dimensional planar case. Solutions for other boundary conditions are then shown to be superpositions of the Bessel function solution. Numerical results are presented for a Gaussian distribution of incident radiation which closely models a laser beam. These multiple scattering results are compared to the single scattering approximation. Also, the solution for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function is expressed in terms of the isotropic results.     

Relation between multidimensional radiative transfer in cylindrical and rectangular coordinates with anisotropic scattering

An exact integral equation is derived for the source function in a three-dimensional rectangular medium which scatters anisotropically. The upper boundary of the finite medium is exposed to collimated radiation, while the lower boundary has no radiation incident on it. The problem is multidimensional because the incident radiation varies spatially. The scattering phase function is represented by a series of Legendre polynomials. A double Fourier transform reduces the problem to a one-dimensional integral equation for the source function. The transformed equation is compared with the integral equation for a two-dimensional cylindrical medium which scatters anisotropically and is exposed to Bessel-varying collimated radiation. A simple relation is found between the two source functions which will greatly reduce the number of computations required for the three-dimensional case. The relation also illustrates the wide utility of the generalized one-dimensional source function. Simplification of the two-dimensional rectangular case to the generalized source function is also presented. The results are extended to problems with a strong anisotropic phase function which has a diffraction spike in the forward direction.     

A note on the integral formulation of Einstein’s equations induced on a braneworld

We revisit the integral formulation (or Greens function approach) of Einsteins equations in the context of braneworlds. The integral formulation has been proposed independently by several authors in the past, based on the assumption that it possible to give a reinterpretation of the local metric field in curved spacetimes as an integral expression involving sources and boundary conditions. This allows one to separate source-generated and source-free contributions to the metric field. As a consequence, an exact meaning to Machs Principle can be achieved in the sense that only source-generated (matter fields) contributions to the metric are allowed for; universes which do not obey this condition would be non-Machian. In this paper, we revisit this idea concentrating on a Randall–Sundrum-type model with a non-trivial cosmology on the brane. We argue that the role of the surface term (the source-free contribution) in the braneworld scenario may be quite subtler than in the 4D formulation. This may pose, for instance, an interesting issue to the cosmological constant problem.     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

Stochastic radiative transfer in a finite plane-parallel medium with general boundary conditions

The time-independent radiative transfer problem in a scattering and absorbing planar random medium with general boundary conditions and internal energy source is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The problem is solved in terms of the solution of the corresponding free-source problem with simple boundary conditions which is solved using Pomraning-Eddington approximation in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average partial heat fluxes are calculated in terms of the albedoes of the source-free problem. Results are obtained for isotropic and anisotropic scattering for specular and diffused reflecting boundaries.     

The albedo problem with Fresnel reflection

The classic albedo problem for a half-space is generalised to include the effect of refraction at the boundary by inclusion of the Fresnel boundary conditions. The problem is solved using the Wiener-Hopf technique with both specular and diffuse reflection. The non-singular Fredholm integral equations that arise for the surface angular distribution are solved numerically and the solutions are illustrated by a number of results in graphical and tabular form. The significant effect of refraction on the albedo and the associated angular distributions is highlighted.     

Scattering from a rough layer of a random medium

This paper deals with scattering from a random-medium layer with rough boundaries. The fluctuations of the surface heights and medium permittivity are assumed to be small and smooth. All random quantities are assumed to be stationary and independent of each other. After the introduction of approximate boundary conditions, the system of partial differential equations is transformed into an integral equation where the fluctuations of the problem are represented as a zero-mean random operator. Employing smoothing, integral equations for the coherent fields are obtained. Use of the Helmholtz operator leads to solution for the coherent propagation constant while the boundary operators lead to coherent Fresnel coefficients. The characteristics of the results are illustrated by considering several examples.     

Three-dimensional radiative transfer in an isotropically scattering, plane-parallel medium: generalized X- and Y-functions

The topic of this work is the generalized X- and Y-functions of multidimensional radiative transfer. The physical problem considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, plane-parallel medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is used to derive coupled, integro-differential equations for the source functions at the boundaries of the medium. The resulting equations are said to be in double-integral form because the integration is over both angular variables. Numerical results are presented to illustrate the computational characteristics of the formulation.     

Method of frustrated total reflection in spectroscopy of surface polaritons in almost perfectly conducting metals

The standard technique for surface polaritons excitation by a prism coupling in Otto configuration is applied for investigation of almost perfectly conducting (pec) metals like tantalum irradiated by a collimated He-Ne laser radiation (λ ₀ = 632.8 nm). In pec metals the imaginary part of the relative dielectric permittivity (ɛ″) is quite larger than the modulus of the real part of the same quantity (ɛ′ < 0, ɛ″ ≫ | ɛ′ |). Under this condition the single Lorentz dip of the reflectivity coefficient is proven to exist and is given in an analytical form, which follows from simplification of the usual multilayer Fresnel formula. In the case of a deterministically curved metal surface an approximate solution to the reduced Rayleigh integral equation appropriate for the Otto configuration is also given. These two theoretical deductions are compared with experimental data that have been produced by us for the reflectivity into the prism region from a bulk tantalum sample.     

Three-dimensional radiative transfer in an isotropically scattering, semi-infinite medium: generalized reflection function

The focus of this study is the generalized reflection function of multidimensional radiative transfer. The physical situation considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, semi-infinite medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The resulting equation is said to be in double-integral form because the integration is over both angular variables. Computational issues associated with this generalized reflection function formulation are investigated. The source function and reflection function formulations are compared, and the relative merits of the two approaches are discussed.     

On the uniqueness of the problem of acoustic diffraction by an infinite plate with local irregularities

The question concerning the uniqueness of the solution to the problem of the acoustic diffraction by an immersed and isolated thin infinite plate with a finite scatterer is studied. It is shown that, to provide the uniqueness of the solution, the conditions at the scatterer must lead to an energy inequality for a source-free field, which determines the absence of the energy-carrying field components at infinity. A formula that generalizes the Sommerfeld formula is obtained and is used to prove the uniqueness of the solution to the problem of diffraction by a plate immersed in an acoustic medium. For the problem of diffraction of a flexural wave by an irregularity of the plate, the uniqueness theorem is proved only for the case of a fixed or hinged edge. When boundary conditions of a general form are imposed on the scatterer in an isolated plate, the uniqueness of the solution is generally lost, which is also corroborated by an example.     

Covariant jump conditions in electromagnetism

A generally covariant four-dimensional representation of Maxwell’s electrodynamics in a generic material medium can be achieved straightforwardly in the metric-free formulation of electromagnetism. In this setup, the electromagnetic phenomena are described by two tensor fields, which satisfy Maxwell’s equations. A generic tensorial constitutive relation between these fields is an independent ingredient of the theory. By use of different constitutive relations (local and non-local, linear and non-linear, etc.), a wide area of applications can be covered. In the current paper, we present the jump conditions for the fields and for the energy–momentum tensor on an arbitrarily moving surface between two media. From the differential and integral Maxwell equations, we derive the covariant boundary conditions, which are independent of any metric and connection. These conditions include the covariantly defined surface current and are applicable to an arbitrarily moving smooth curved boundary surface. As an application of the presented jump formulas, we derive a Lorentzian type metric as a condition for existence of the wave front in isotropic media. This result holds for ordinary materials as well as for metamaterials with negative material constants.