Spherical Harmonics Method Research Articles

A TWO-DIMENSIONAL INFRARED MAP OF THE EXTRASOLAR PLANET HD 189733b

We derive the first secondary eclipse map of an exoplanet, HD 189733b, based on Spitzer IRAC 8 micron data. We develop two complementary techniques for deriving the two dimensional planet intensity: regularized slice mapping and spherical harmonic mapping. Both techniques give similar derived intensity maps for the infrared day-side flux of the planet, while the spherical harmonic method can be extended to include phase variation data which better constrain the map. The longitudinal offset of the day-side hot spot is consistent with that found in prior studies, strengthening the claim of super-rotating winds, and eliminating the possibility of phase variations being caused by stellar variability. The latitude of the hot-spot is within 10.1 deg (68% confidence) of the planet's equator, confirming the predictions of general circulation models for hot Jupiters and indicative of a small planet obliquity.

Computation of hypersonic flow and radiation of viscous chemically reacting gas in a channel modeling a section of a scramjet

This work is devoted to a consideration of flow and combustion of a hydrogen-air mixture in a channel modeling a section of a supersonic combustion ramjet (scramjet). Fields of concentrations, pressure, and temperature are obtained. Based on them, the thermal radiation of gas within a scramjet combustor is computed. The density of the radiative heat flux to the chamber wall is computed by two methods, i.e., in a P1 approximation of the spherical harmonics method and in an approximation of the plane layer. It has been shown that the radiative heat flux contribute significantly to the total heating of the jet wall.

Arbitrary-Order Spherical Harmonics Method for Coupled Heat Transfer in Semitransparent Materials

An advanced numerical model and its application in thermal processing of nonlinearly anisotropic scattering semitransparent materials are presented in this paper. The decomposition of the radiative transfer equation is deduced using spherical harmonics (PN-approximation) method. Considering the transient problem, coupled radiative and conductive heat transfer model based on arbitrary-order PN-approximation method is established. With the coupled model, the calculation accuracy of high-order PN-approximation method is verified through comparison with theoretical solution and by the simulation results of other methods. Comparison shows that the arbitrary-order PN-approximation method reproduces results with a high accuracy for linear and non-linear anisotropic scattering, while it can easily treat the nonlinear anisotropic scattering phase functions with desired order. Keywords: semitransparent material, coupled heat transfer, numerical simulation, spherical harmonics method

Application of vector spherical harmonics for kinematic analysis of stars from zonal catalogues

We solve the problem on a kinematic analysis of the three-dimensional velocity field of stars from zonal catalogues, i.e., catalogues in which the stars are presented at all right ascensions in some declination zones. We have constructed a system of vector spherical harmonics with the properties of completeness and orthogonality for a chosen declination zone. We suggest a method that allows the Ogorodnikov-Milne model parameters in the Galactic coordinate system to be estimated by analyzing the proper motions and radial velocities of stars in the equatorial coordinate system. The vector spherical harmonics are shown to have the following advantages over the standard approach based on a direct leastsquares estimation of the parameters for a specific model. First, in contrast to the standard approach, the new method can reveal all systematic components of the velocity field irrespective of a particular model. Second, it allows one to get rid of the correlation between the sought-for parameters, which presents a serious problem for the conventional method in the case of zonal catalogues. Third, the method of vector spherical harmonics allows the kinematic parameters to be estimated at least by two techniques. Comparison of these two solutions makes it possible to test the standard kinematic model for compatibility with the observational data. The developed method has been tested on the basis of numerical experiments and applied for a kinematic analysis of the proper motions of Tycho-2 stars in the southern hemisphere for which the parallaxes can be estimated using data from the Tycho-2 Spectral Type Catalogue.

Uncertainty reevaluation in determining the volume of a silicon sphere by spherical harmonics in an Avogadro project

To determine the Avogadro constant with a target relative uncertainty of 2 × 10−8, the uncertainty component of the silicon sphere's volume introduced by the spherical harmonics method, which is usually used in determining the sphere's volume, is reevaluated. By means of representing the shape of the silicon sphere by an ellipsoid with Gaussian white noise in its diameters, the uncertainty of the current mapping methods based on the spherical harmonics theory can be estimated theoretically. It is evidenced that the uncertainty component attributed to the current mapping method is underestimated. To eliminate this effect as much as possible, the number of mapping points should be increased to more than before. Moreover, a new mapping method is proposed to accomplish the equal-area mapping with large number points on the silicon sphere.

Cell Homogenization Method for Pin-by-Pin Neutron Transport Calculations

For practical reactor core applications, low-order transport approximations such as SP3 have been shown to provide sufficient accuracy for both static and transient calculations with considerably less computational expense than the discrete ordinate or the full spherical harmonics methods. These methods have been applied in several core simulators where homogenization was performed at the level of the pin cell. One of the principal problems has been to recover the error introduced by pin cell homogenization. One of the basic approaches to treat pin cell homogenization error is pin cell discontinuity factors (CDFs) based on well-established generalized equivalence theory to generate appropriate group constants. The method is able to treat all sources of error together, allowing even a few-group diffusion solution with one mesh per cell to reproduce a higher-order reference solution. However, a CDF has to be derived separately for each space-angle approximation. An additional difficulty is that in practice the CDFs have to be derived from a lattice calculation from which only the scalar flux and current are available, and therefore recovery of the exact SPN angular moment is not possible. This paper focuses on the pin cell scale homogenization. It demonstrates derivation of the CDF for the SP3 transport method with finite-difference spatial discretization with the limitation of only the scalar flux and interface current being available from the heterogeneous reference. The method is demonstrated using a sample benchmark application.

A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization

A simplified implementation of the analytical discrete ordinates (ADO) method in radiative transfer with polarization is presented in this work. The class of problems that can be solved with the simplified ADO approach consists of problems defined in plane-parallel geometry and driven by external illumination in the form of obliquely incident parallel rays. Numerical results of benchmark quality are tabulated for the albedo problem with polarization and Lambert reflection. The new results improve on a tabulation made available in a previous work by the authors that was based on the (less accurate) spherical harmonics method.

Gravity and geoid model in South Korea and its vicinity by spherical cap harmonic analysis

Long wavelength gravity anomalies derived from satellite data have limitations of spatial resolution and accuracy. To complement these weak points, combined datasets merged with shipborne/land gravity data and satellite data are generally compiled using global spherical harmonic, rectangular harmonic, or spherical cap harmonic analysis etc. However, each method also has a different resolution and accuracy. In this study, gravity anomalies and geoid undulations of South Korea and its vicinity are calculated using the spherical cap harmonic method, and the results are compared with those of GGM02C and EGM2008. Absolute and relative geoid height differences between GPS/leveling data and the results of various other methods show that the method used in this study has the lowest RMS error (0.271 m vs 0.344–0.416 m) and standard deviation (0.236 m vs 0.309–0.362 m). This implies that the spherical cap harmonic method has better resolution and accuracy than the other methods. The short wavelength gravity anomalies derived from the spherical cap harmonic coefficients are used to estimate the sediment thickness of the Ulleung Basin, which is poorly known. The sediment thickness of the Ulleung Basin is about 5 km at the center of the basin, and about 12 km at the southern margin.

A novel, customizable and optimizable parameter method using spherical harmonics for molecular shape similarity comparisons

A novel molecular shape similarity comparison method, namely SHeMS, derived from spherical harmonic (SH) expansion, is presented in this study. Through weight optimization using genetic algorithms for a customized reference set, the optimal combination of weights for the translationally and rotationally invariant (TRI) SH shape descriptor, which can specifically and effectively distinguish overall and detailed shape features according to the molecular surface, is obtained for each molecule. This method features two key aspects: firstly, the SH expansion coefficients from different bands are weighted to calculate similarity, leading to a distinct contribution of overall and detailed features to the final score, and thus can be better tailored for each specific system under consideration. Secondly, the reference set for optimization can be totally configured by the user, which produces great flexibility, allowing system-specific and customized comparisons. The directory of useful decoys (DUD) database was adopted to validate and test our method, and principal component analysis (PCA) reveals that SH descriptors for shape comparison preserve sufficient information to separate actives from decoys. The results of virtual screening indicate that the proposed method based on optimal SH descriptor weight combinations represents a great improvement in performance over original SH (OSH) and ultra-fast shape recognition (USR) methods, and is comparable to many other popular methods. Through combining efficient shape similarity comparison with SH expansion method, and other aspects such as chemical and pharmacophore features, SHeMS can play a significant role in this field and can be applied practically to virtual screening by means of similarity comparison with 3D shapes of known active compounds or the binding pockets of target proteins.

On simplified spherical harmonics equations for the radiative transfer equation

The radiative transfer equation (RTE) arises in a wide range of applications in sciences and engineering. Due to the high dimension and complicated form of the RTE, it is challenging to solve the equation directly. In the literature, several approximation methods were developed for the RTE. One approximation method, the simplified spherical harmonics (SP N ) method, provides an efficient way to generate good approximate solutions of the RTE with high absorption and small geometries. The main purpose of the paper is to study well-posedness of the simplified spherical harmonics system. The weak formulation used in the proof of the solution existence and uniqueness provides a starting point for developing the Galerkin finite element method to solve the simplified spherical harmonics system.

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x– y geometry. Part I: Modeling and verification for time-independent solution

This paper describes a novel method based on using cellular neural networks (CNN) coupled with spherical harmonics method ( P N ) to solve the time-independent neutron transport equation in x– y geometry. To achieve this, an equivalent electrical circuit based on second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. We use the CNN model to simulate spatial response of scalar flux distribution in the steady state condition for different order of spherical harmonics approximations. The accuracy, stability, and capabilities of CNN model are examined in 2D Cartesian geometry for fixed source and criticality problems.

Comparison of GPS strain rate computing methods and their reliability

Using modelled and simulated data for comparison of several methods to compute GPS strain rate fields in terms of their precision and robustness reveals that least-squares collocation is superior. Large scale (75°E–135°E and 20°N–50°N) analyses of 1° grid sampling data and decimated 50 per cent data by resampling (then erasing data in two 5°× 10° region) reveal that the Delaunay method has poor performance and that the other three methods show high accuracy. The correlation coefficients between theoretical results and calculated results obtained with different errors in input data show that the order in terms of robustness, from good to bad, is least-squares collocation, spherical harmonics, multisurface function and the Delaunay method. The influence of data sparseness on different methods shows that least-squares collocation is better than spherical harmonics and multisurface function when sample data are distributed from a 2° grid to a 1° grid. Analysis to medium scale (90°E–120°E, 25°N–40°N) in 1°–0.5° grid sampling data reveals that least-squares collocation is superior to other methods in terms of robustness and sensitivity to data sparseness, but their difference is slight. Strain rate results obtained for the Chinese mainland using GPS data from 1999 to 2004 show that the spherical harmonics method has edge effects and that its value and range increase concomitantly with increased sparseness. The multisurface function method shows non-steady-state characteristics; the errors of results increase concomitantly with increased sparseness. The least-squares collocation method shows steady characteristics. The errors of results show no significant increase even though 50 per cent of input data are decimated by resampling. The spherical harmonics and multisurface function methods are affected by the geometric distribution of input data, but the least-squares collocation method is not.

Spherical Cap Harmonic Analyses of Geomagnetic Field and Its Secular Variation in China for 2005.0

AbstractAs a basic technological product describing the geomagnetic spatial distribution, the national geomagnetic chart should choose an appropriate method to describe the geomagnetic field on standard epoch and its secular variations for the coming 5 years truly. We calculated the geomagnetic data at 1119 sites and 36 observatories in China by using the spherical cap harmonic (SCH) method, and obtained the space distribution of the normal and the anomalous geomagnetic fields in China for 2005.0 and the SCH model of geomagnetic secular variations in China for 2005~2010. The results showed that the errors of the geomagnetic field by IGRF could be ~ 5ʹ for the declination D and the inclination I and ~100 nT for the total intensity F. Because the geomagnetic survey data with new data and better resolution were used in the calculation, the geomagnetic chart for 2005.0 by the SCH method can better describe the space distribution of the geomagnetic field than the IGRF. This method can be stably used in compiling the China Geomagnetic Chart.

Efficiency of reflection coefficients in the isotropic neutron transport equation and computation of the critical-half thicknesses

Abstract The critical slab problem of the reflected isotropic one-speed neutron transport equation is solved with the Chebyshev polynomial approximation. The efficiency of the reflection coefficient, R, in the neutron transport equation is obtained for different c values. For the solution, the Mark boundary condition is used. The values obtained from this approximation are compared with results obtained by the spherical harmonics method.

Deduction of a symmetric tensor formulation of the P N method for the linear transport equation

In the paper, a new formulation of the classic spherical harmonics method for the study of the linear transport equation is proposed. The method makes use of an expansion of the angular flux in terms of symmetric spherical harmonics and therefore it does not present privileged directions in its formulation. In the paper, the symmetric spherical harmonics are introduced in a simple way and their properties are presented in order to deduce the new equations for the moments of the flux.

Application of the modified differential approximation for radiative transfer to arbitrary geometry

The first-order spherical harmonics method (or P 1 approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P 1 approximation deteriorates as the optical thickness of the medium is decreased. The modified differential approximation (MDA) was originally proposed to remove the shortcomings of the P 1 approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry—in particular, geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results. The accuracy of MDA was found to be superior to the P 1 approximation for all optical thicknesses. Its accuracy, when compared with the discrete ordinates method (both S 6 and S 8), was found to be clearly superior in optically thin situations, but problem dependent in optically intermediate and thick situations. For 3D geometries, calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint. Several strategies to reduce both memory and computational time are discussed and demonstrated.

On the Relation Between Spherical Harmonics and Simplified Spherical Harmonics Methods

The purpose of the paper is, first, to recall the proof that the AN method and, therefore, the SP2N−1 method (of which AN was shown to be a variant) are equivalent to the odd order P2N−1, at least for a particular class of multi-region problems; namely the problems for which the total cross section has the same value for all the regions and the scattering is supposed to be isotropic. By virtue of the introduction of quadrature formulas representing first collision probabilities, this class is then enlarged in order to encompass the systems in which the regions may have different total cross sections. Some examples are reported to numerically validate the procedure.

Solution of the one velocity 2D and 3D source and criticality problems by the Boundary Element – Response Matrix (BERM) method in the A2-SP3

The paper illustrates some applications of a variant of the simplified spherical harmonics (SPN) method, called AN, in order to solve 2D and 3D source and criticality problems. The AN equations (here considered only for N = 2, which corresponds to the SP3 approximation) are solved by means of a Boundary Element-Response Matrix technique.

Study on photon transport problem based on the platform of molecular optical simulation environment.

As an important molecular imaging modality, optical imaging has attracted increasing attention in the recent years. Since the physical experiment is usually complicated and expensive, research methods based on simulation platforms have obtained extensive attention. We developed a simulation platform named Molecular Optical Simulation Environment (MOSE) to simulate photon transport in both biological tissues and free space for optical imaging based on noncontact measurement. In this platform, Monte Carlo (MC) method and the hybrid radiosity-radiance theorem are used to simulate photon transport in biological tissues and free space, respectively, so both contact and noncontact measurement modes of optical imaging can be simulated properly. In addition, a parallelization strategy for MC method is employed to improve the computational efficiency. In this paper, we study the photon transport problems in both biological tissues and free space using MOSE. The results are compared with Tracepro, simplified spherical harmonics method (S P n), and physical measurement to verify the performance of our study method on both accuracy and efficiency.

Finite-Volume Formulation and Solution of the P3 Equations of Radiative Transfer on Unstructured Meshes

Abstract The method of spherical harmonics (or PN) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the four P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the case without radiation.