Multidimensional Media Research Articles

Inverse problem in nonstationary multidimensional medium

The problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered. The boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method. The integral-differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method. The order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary. It was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium. The cause and character of the multidimensional inverse problem overdefiniteness are considered. The application of the proposed procedure for a statistical problem is discussed.

Stability of zero solution of mathematical model of multidimensional barotropic viscoelastic medium

Stability of zero solution of mathematical model of multidimensional barotropic viscoelastic medium

Statistical characteristics of a point-source field in a randomly stratified medium in the absence of regular refraction

The problem of a point-source field in a multidimensional ramdomly stratified medium is considered. The inhomogeneities are described by a delta-correlated Gaussian process with a zero mean. The behavior of the wave-field intensity at the boundary of the half-space is analyzed under given fluctuations of the medium. Analytical results are compared with the results of statistical simulation.

The source approximation method for multidimensional media using the surface-integral form of the radiative transfer equation

A radiation source approximation of linear and quadratic interpolation over an optical length is constructed to simplify surface-integral equations of radiative transfer for an absorbing, emitting and linear anisotropic scattering medium. The expressions formulated here for incident radiation and radiative heat flux can be applied easily to various coordinate systems. Test cases of radiation in participating media with two-dimensional cylindrical and three-dimensional rectangular geometries are solved by the new technique. The numerical solutions are compared with other results obtained by the product-integration method (PIM). For all the cases considered the radiative transfer results predicted by the quadratic source approximation is in good agreement with the benchmark PIM solution and is more accurate than those obtained with the linear approximation. Since only integration at the bounding surface is involved in the equations of radiative transfer, the solution method is computationally efficient.

Multidimensional autoresonant mode conversion

It is shown that the autoresonance effect is characteristic of wave interactions in slowly varying weakly nonlinear multidimensional media. The theory of the phenomenon is presented for the mode conversion case and illustrated in two dimensions. It is demonstrated that multidimensional autoresonance is stable with respect to weak damping and transverse nonuniformity of the incident wave.

The source approximation method for multidimensional media using the surface-integral form of the radiative transfer equation

A radiation source approximation of linear and quadratic interpolation over an optical length is constructed to simplify surface-integral equations of radiative transfer for an absorbing, emitting and linear anisotropic scattering medium. The expressions formulated here for incident radiation and radiative heat flux can be applied easily to various coordinate systems. Test cases of radiation in participating media with two-dimensional cylindrical and three-dimensional rectangular geometries are solved by the new technique. The numerical solutions are compared with other results obtained by the product-integration method (PIM). For all the cases considered the radiative transfer results predicted by the quadratic source approximation is in good agreement with the benchmark PIM solution and is more accurate than those obtained with the linear approximation. Since only integration at the bounding surface is involved in the equations of radiative transfer, the solution method is computationally efficient.

MULTIDIMENSIONAL, NONHOMOGENEOUS, DENSE SPRAY CONTRIBUTIONS TO DIFFRACTION-BASED PARTICLE SIZE DISTRIBUTION MEASUREMENTS

The effect of dense sprays on particle size measurement by laser-diffraction-based techniques is examined for log-normal distributions of particles in multidimensional media. A Monte Carlo modeling technique is used to simulate scattering of light by particles, and general scattering phenomena are included by using Mie theory. The measurement errors in particle size distribution parameters due to multiple scattering are analyzed and compared to corrections previously presented for homogeneous planar media. The errors for measurements in homogeneous cylindrical media equal those for planar media with the same optical depth, but the cylindrical geometry requires the use of a scaled obscuration. Specifically, the obscuration is dependent on the size of the laser beam relative to the cylinder. Nonhomogeneities are examined in planar media and found to have no effect on the measured size distribution or obscuration. In nonhomogeneous cylindrical media, measurement errors in particle size distribution parameters due to multiple scattering are shown to be predicted by planar corrections for line-of-sight measurements in sprays. Simulations done with wide beams result in measured distributions that are within acceptable accuracy limits of equivalent homogeneous results, and multiple scattering biases can be corrected as if the medium were homogeneous.

Zero White Noise Limit through Dirichlet forms, with application to diffusions in a random medium

We study the Zero White Noise Limit for diffusions in a continuous multidimensional medium: given a continuous function on ℝ n ,W, we consider diffusions whose drift term is the gradient ofW and whose diffusion coefficient is constant equal to e. We describe the asymptotics of the exit time from a domain and of the law of the process when e tends to zero. By applying these results to a random self-similar mediumW we prove limit theorems for a diffusion in a random medium. Our theorems agree with results usually proved through the large deviation principle, although, in our setup, this last tool is not available. We extend to the multidimensional case properties of diffusions in a random medium already known in one dimension.

The Improved Differential Approximation for Radiative Transfer in Multidimensional Media

One particularly popular approximate method for radiative transfer in participating media is the so-called differential approximation or P-1 approximation. Unfortunately, the differential approximation is accurate only in optically thick media, and may become very inaccurate in optically thin situations, in particular in two- and three-dimensional geometries. To increase the accuracy, a number of improvements have been proposed. Olfe (1967, 1968) separated wall emission from medium emission in what he called the modified differential approximation (MDA). Modest (1974, 1975) combined the P-1 approximation with an optically thin correction term in what he called the improved differential approximation (IDA). While very accurate, both models were limited to relatively simple situations. Recently, Modest (1989) demonstrated that Olfe's MDA may be formulated for a general three-dimensional case with an absorbing/emitting, anisotropically scattering medium bounded by gray, diffusely emitting and reflecting walls. It is the purpose of the present note to show that the IDA may also be extended to the most general case, reducing to the correct optically thin and optically thick limits for all situations, resulting in accuracy similar to that of the MDA, but being computationally more efficient.

Multidimensional radiative transfer: A single integral representation of anisotropic scattering kernels

A complicated kernel arising in the study of anisotropic scattering in a multidimensional media is investigated. The scattering phase function is represented by a series of Legendre polynomials. The kernel is transformed with a symbolic manipulator to a single integral form suitable for use in Ambarzumian's method. The results for a five term phase function are obtained. Single integral representations of the generalized exponential integral and its derivatives are also developed.

On the convergence of the multigroup approximations for multidimensional media

In this paper, we investigate the stability, convergence, and consistency properties of steady-state multigroup models for submultiplying media with spatial dimensions greater than one. We define these concepts in a Banach space whose norm measures the collision density integrated over phase space. Stability and consistency occur under the conditions that the maximum fluctuations in the total cross section, and in the expected number of secondary particles arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding example and discussion deal with pathologies of the multigroup model in situations where these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of both Belleni-Morante and Busoni for isotropic slabs and of Yang Mingzhu and Zhu Guangtian for bounded, three-dimensional media. This work directly extends the steady-state results of Paul Nelson, Jr. and H. D. Victory, Jr. for slab media to steady-state transport in multidimensional media.

Evaluation of spectrally-integrated radiative fluxes of molecular gases in multi-dimensional media

A new multi-dimensional model has been developed which makes it possible to calculate the spectrally-integrated total radiative flux for a molecular-gas band based on the solution of two simple differential equations. The newmodel employs the exponential-wide-band model and makes it unnecessary to evaluate the spectral flux for a large number of wavenumbers with subsequent spectral integrations, thus considerably reducing the numerical effort required. Comparison with spectrally-integrated results from the differential ( P-1) approximation, on which the present method is based, and with some exact results, shows excellent agreement for all situations.

Generation of neutron transport benchmark problems using BEAPAC-3T

A method is described how a quantitative measure can be obtained for the accuracy of numerical methods for the solution of neutron transport problems in an arbitrary homogeneous multidimensional medium. An expansion for the combining coefficients of singular eigenfunctions is developed that yields exact elementary plane-wave solutions that can be evaluated very accurately in terms of exponential integral functions. Linear combinations of these solutions are automatically generated which approximate, in the least squares sense, a user specified incident flux distribution on the boundary of a multidimensional benchmark cell domain. The generated solution is exact for a new benchmark problem which has incident flux given precisely by the generated solution. The quantitative error analysis method has been implemented in the code BEAPAC-3T and is illustrated by application to the TWOTRAN-II code for a square domain problem.

The Physical Effects of Radiative Transfer in Multidimensional Media Including Models of the Solar Atmosphere

view Abstract Citations (17) References (50) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Physical Effects of Radiative Transfer in Multidimensional Media Including Models of the Solar Atmosphere Jones, H. P. ; Skumanich, A. Abstract We review the astrophysical literature concerning radiative transfer in multidimensional media where one requires the solution of the transfer equation under scattering conditions for a medium in which some combination of boundary configuration, external illumination, and internal thermodynamic structure causes the radiation field to vary with more than one spatial dimension. In constant opacity atmospheres, the radiation field is shown to scale systematically with to a characteristic geometric scale for a wide variety of configurations and types of scattering. Some effects of radiative exchange between different regions of multidimensional media are reviewed, and the constraining influence of an exponential vertical variation of opacity is discussed. Particular emphasis is given to recent applications of multidimensional transfer to the interpretation of the fine spatial structure on the Sun. Publication: The Astrophysical Journal Supplement Series Pub Date: February 1980 DOI: 10.1086/190649 Bibcode: 1980ApJS...42..221J full text sources ADS |

Evolutionary isotropic radiation

A directionally dependent source distribution activates within a multi-dimensional isotropic dispersive medium. It then develops into a purely pulsatory state. The radiation problem is approached via a convolution principle. Certain basic postulates are imposed to help secure integral convergence at various stages. The Sommerfeld radiation principle holds through an applied initial condition. One field quantity results from stationary phase approximations and represents a superposition of slowly transient spherical wavemodes over a continuously evolving set of wavenumbers and frequencies that must avoid the source frequency. Their radial group velocities vary coincidentally with a positive reception velocity parameter; the associated dispersion is basically a function of the medium, but admits amplitudes with some dependence on source frequency. Another field quantity accumulates relevant residues which contribute to a superposition of quasisteady spherical wavemodes over an intermittently growing set. This depends not only on the medium, but also on the source frequency, imparted to all such wavemodes, as well as an observation criterion, namely that any specific wavemode is observed after its enclosing energy front crosses the observer, in particular, with an invariant group velocity exceeding that common to all slowly transient wavemodes; amplitudes quickly lose their source-induced time dependence. It is this last quantity that survives in the long run and progresses into a non-trivial purely pulsatory steady state consistent with Lighthill’s radiation principle. Its ultimate survival is accomplished through a permanent flow of source-generated non-transient energy, permanency of the energy supply being guaranteed by an indefinitely sustained source amplitude; moreover, both medium and source never conspire to cancel the supply. On an energy basis, the fading of slowly transient modes may be due to the decreasing group velocity of their energy arrival. Spherically symmetric and axisymmetric cases are briefly examined. Finally, arguments and results are applied, with some modifications, to (i) an unsteadily vibrating elastic plate problem, and (ii) radiation of certain internal gravity waves in a Boussinesq fluid.

Isotropic wave dispersion

An isotropic multidimensional medium is propagating dispersive radiation from a suddenly triggered immersed point source. The latter’s subsequent action is postulated as being transient. A contour integration technique accounts for a part time zero condition plus other supplementary hypotheses (e.g. that of stability), leading via the stationary phase technique to an asymptotic solution valid for large time. One set of results holds at comparably far range. In the ensuing dispersion, concentrically expanding trains of Kelvin approximated waves get emitted. These wave trains are generally bounded by spherical advancing frontal and/or rear edges near which Airy-type approximations can be made. Several such edges may coincide or almost coincide. Another set of results, involving Hankel or Bessel functions, holds at any given finite range; it indicates slow wave packets and implies, consistent with a transient source, the ultimate attainment of a steady state of silence, possibly within a ‘slowest’ spherical edge. Applications are illustrated for elastic plate deflexions and Klein-Gordon governed motions.

The Formation of Resonance Lines in Multidimensional Media. III. Interpolation Functions, Accuracy, and Stability

The accuracy and stability for several specific representations of the integral-operator technique presented in Paper II (Jones and Skumanich 1973) are discussed. Solutions are tested against independently calculated results for an effectively thin "embedded slab." It is found that the cardinal interpolation functions for approximating the source function along characteristics must be "local," must give accurate representation of a variety of functions and their second derivatives on coarse, irregular grids, and must be compatible with interpolation functions used to map functions of spatial position to functions of path length. Cubic splines appear to meet these requirements and give good overall results. Subject headings: line formation - radiative transfer

The Formation of Resonance Lines in Multidimensional Media. II. Radiation Operators and Their Numerical Representation

The Formation of Resonance Lines in Multidimensional Media. II. Radiation Operators and Their Numerical Representation

The Formation of Resonance Lines in Multidimensional Media. I. Scaling Properties in Two Dimensions

Resonance line formation in multidimensional media, applying to non-LTE line transfer for two dimensional temperature variations

Line Formation in Multi-Dimensional Media

AbstractThe flux divergence technique of Athay and Skumanich (1967) is generalized for application to media whose properties vary in more than one spatial dimension. In this method, the flux divergence is viewed as an integro-differential functional of the source function. The source function is then expanded in terms of basis functions along characteristic paths, and, with the help of various interpolations, the flux divergence is converted to an approximate linear algebraic operator on a discrete spatial grid. A large but finite set of linear, inhomogeneous, simultaneous algebraic equations with known matrix coefficients is thus generated and is solved by direct matrix inversion for the source function at each point of the spatial grid.Some aspects of the accuracy, stability, and computational convenience of the technique are discussed. Sample solutions for depth dependent, axially symmetric variations of temperature are shown.