Spherical Harmonics Method Research Articles

Method for Calculating the Optical Properties of Composites Based on a Transparent Matrix with Residual Porosity and Metal Nanoparticles

A method is proposed for calculating the optical characteristics of composites based on a transparent matrix with residual porosity and metal nanoparticles by solving the transport equation for monochromatic radiation employing a spherical harmonic method and Fresnel boundary conditions. The method is tested by modeling the transport of monochromatic radiation at four wavelengths of practical importance in cyclotrimethylenetrinitramine–aluminum nanoparticle composites with a Rayleigh distribution of pores with respect to radius. It is shown that in the case of small-radius pores the reflectivity increases as the mass fraction of nanoparticles is increased. In the case of pores with large radii, a kink corresponding to complete filling of the pores appears on the dependence of the transmission on the nanoparticle mass fraction. The application of these results to inverse problems in the spectroscopy of lightscattering system is discussed.

Three-dimensional CFD modelling and thermal performance analysis of porous volumetric receivers coupled to solar concentration systems

Porous volumetric receivers is a promising technology to improve the thermal performance of a new generation of concentrated solar power (CSP) plants. In this sense, this work addresses the Computational Fluid Dynamics (CFD) modelling and thermal performance analysis of porous volumetric receivers coupled to solar concentration systems. A cylindrical receiver element made of open-cell SiC ceramic foam was considered. The fluid flow and heat transfer processes in the porous media are modelled through volume averaged mass, momentum and energy conservation equations, considering the local thermal non-equilibrium (LTNE) approach, while the thermal radiation transfer is described by the P1 spherical harmonics method, using an open source software (OpenFOAM). An in-house algorithm based on the Monte Carlo Ray Tracing (MCRT) method was developed and coupled to the CFD mesh to model the propagation and absorption of solar radiation. The modelling of the receiver boundary conditions were improved, and a detailed analysis of a reference configuration of the receiver was conducted using a parabolic dish with a concentration ratio of 500 to generate the concentrated solar radiation field and a receiver element with diameter 5 cm, height 5 cm, pore size 3 mm and porosity 0.9. The thermal power output, thermal efficiency, mean fluid temperature at the outlet and pressure drop of this reference configuration are 628.92 W, 85.46%,474.22 K and 103.10 Pa, respectively. The use of receivers with high porosity and pores size increases the thermal efficiency slightly and decreases the pressure drop substantially. The convergent incidence of solar rays at the inlet of the receiver leads to high peaks of temperature in the porous structure and fluid, and a way to decrease these peaks is to design the concentration system or place the receiver in such way to obtain lower incidence angles at the inlet.

Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM)

Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.

Comparison of the PPMXL and UCAC5 Catalogs with the Gaia DR2

Abstract Astrometric ground-based catalogs usually suffer from varied systematic errors. These systematic errors were hard to detect because there was no independent reference catalog complete to very faint limiting magnitudes (∼20 mag). This situation has changed since the second data release of the Gaia mission (Gaia DR2). We aim to investigate positions and the proper-motion (PM) system of two ground-based catalogs, the UCAC5 and PPMXL, referring to the Gaia DR2. The individual position in the Gaia DR2 is transferred by its PM to the epoch of other catalogs for comparison. Systematic errors that depend on the magnitude, color, and sky regions in the UCAC5 and PPMXL could be clearly seen. A different behavior between the northern and southern sky is found in the PPMXL, which is possibly inherited from the imperfect calibration of the PM system. Besides, we perform a quantitative analysis of global differences for positions and PMs by the vector spherical harmonics method in terms of 3 rotation angles, 3 glide parameters, and 10 quadrupole parameters. We find a large glide component of ∼8 mas along Z-axis and a rotation angle of ∼5 mas about Z-axis for positional offsets between the PPMXL and Gaia DR2. These terms are found to be insignificant between the UCAC5 and Gaia DR2. We show that the position and PM system of the UCAC5, a new reduction of ground-based observations in the frame of the Gaia reference system, has been largely improved. This indicates that systematic errors in positions and PMs obtained from ground-based observations are mostly impacted by a relatively poor reference catalog. But these observations can be reconstructed in the frame of a space-based reference catalog. In this sense, our results justify the tradition of space-calibrated ground-based astrometric catalogs.

Modeling of Optoacoustic Signals in a System with Elastic Scattering and Light Absorption

We propose a method for simulating optoacoustic signals initiated by a laser pulse in an absorbing medium with elastic radiation scattering and carry out corresponding calculations. It is shown that the use of the method of spherical harmonics allows obtaining an analytical expression for the optoacoustic signal. The deviation of the absorption profile from the Bouguer type near the sample surface leads to a strong influence of the medium parameters on the position of the signal maximum. The growth of the albedo of onefold scattering leads to a decrease in the effective signal growth constant and its amplitude.

Analysis of Combustion of Coal-Water Fuel in Low-Power Hot-Water Boiler via Numerical Modeling and Experiments

The paper presents an experimental and computational study of various regimes of firing coal-water fuel in a low-power hot-water boiler that enable both dry and liquid slag removal. The experimental studies were carried out on a pilot industrial boiler adapted for firing coal-water fuel. The fuel was prepared using flotation products of beneficiation of K grade hard coal. The experiments were accompanied by numerical modeling of combustion processes. The mathematical model includes description of the carrier phase motion (based on the RANS approach with the two-parameter Menter SST k-ω turbulence model), radiation transfer (based on the P1 approximation of the spherical harmonics method for a two-phase two-temperature gray medium), particle motion (based on the Lagrange approach), and gas phase combustion (based on a combination of the kinetic model of combustion of gas components with the vortex break model). The physico-mathematical model was tested on the problem of combustion of coal-water fuel (CWF). One of the goals of the work was verification of a complex mathematical model. A comparative analysis of the results of the numerical modeling and experimental data showed that the model reliably described the process of burning in a combustion chamber. An advanced design of hot-water boiler enabling regimes of both dry and liquid slag removal was investigated. The chamber was shown to provide the necessary conditions for firing CWF in terms of reliability and economy.

Construction of regional geoid using a virtual spherical harmonics model

Abstract The spherical cap harmonics (SCH) method can be used in regional geoid modeling. The core of this approach is the computation of its associated Legendre functions (ALF) with non-integer degree. However, it is unlikely to obtain a large number of zero-root values for the non-integer ALF. To overcome this problem, a new approach called virtual spherical harmonics (VSH) is proposed in this paper to transform the cap range into the whole sphere so that unlimited numbers of zero-root values can be obtained. The new approach was tested using four cap ranges with the radii of 30 ∘ {30^{\circ }} , 15 ∘ {15^{\circ }} , 10 ∘ {10^{\circ }} and 5 ∘ {5^{\circ }} , and geoid undulations for each of the regions are calculated from EGM2008. For each of the regions, the geoid undulations were used to construct three models with three different degrees of 20, 30 and 40. Numerical results showed that with the increase in the degree of the VSH model, the value of the maximum error decreases; and the maximum error of the model was less than 1 mm while the maximum degree is 40.

Bidirectional transmittance and reflectance models for soil signature analysis.

The propagation of radiation in an absorbing-scattering soil with constant or spatial variation of the refractive index is investigated. The soil consists of a plane parallel with Fresnel reflection at the boundaries and is exposed at one boundary to a diffuse or collimated incident radiation. The discrete spherical harmonics method using Marshak boundary conditions is introduced to approximate the directional hemispherical reflectance and transmittance as well as the bidirectional reflectance. The effect in spatial variation of the refractive index on the reflectance and transmittance predictions is examined. A comparison of the directional transmittance and reflectance with the literature results demonstrates that the present method gives accurate results for optically thin and thick soil with a maximum relative error in all cases less than 1%. The bidirectional radiance for variable refractive index soils also shows excellent agreement as compared to the literature results. The results demonstrated that the anisotropic soil interfaces cause a significant decrease of energy reflected and transmitted as well as the bidirectional reflectance.

Spherical Harmonic Expansions for the Gravitational Field of a Polyhedral Body with Polynomial Density Contrast

We provide the spherical harmonic solutions to evaluate the external gravitational field of a general polyhedral body with arbitrary polynomial density contrast, including the gravitational potential and its arbitrary-order derivatives. The linear recursive algorithm for computation of the spherical harmonic coefficients of the potential is derived by using the Gauss divergence theorem and the Stokes theorem, and the computations are performed on the basis of the division of general polygonal pyramid of the polyhedron instead of the division of tetrahedron. The algorithm of this paper can handle the density contrast in both horizontal and vertical directions and the polynomial function of the density at arbitrary degree. Both the conversion relations of the density function and the arbitrary-order derivatives of the spherical harmonic potential between the initial and rotated reference frames are given in tensor product forms, which assist the calculations. The space-domain method for evaluating the gravity field of the polyhedral body may leads to numerical problems at a remote observation point. However, the spherical harmonic method is numerically stable at arbitrary observation points outside the smallest enclosed sphere. The numerical experiments for three actual and synthetic polyhedral models including a right rectangular prism with cubic density varying with depth, the asteroid 433 EROS with cubic polynomial density and a right rectangular prism with quartic polynomial density, are implemented to test the accuracy, convergence, and numerical stability of the spherical harmonic algorithm.

Spherical harmonics method for computing the image stress due to a spherical void

We develop an efficient numerical method for calculating the image stress field induced by spherical voids in materials, and applied the method to dislocation-void interactions. The method is constructed based on a complete set of basis functions for the displacement potential of the elastic boundary value problem for a spherical hole, as well as the corresponding displacement, stress, and traction fields, all in terms of linear combinations of spherical harmonics. Using the fast transformation between the real and spherical-harmonics spaces provided by the SHTOOLS package, the method is more efficient than other image stress solvers such as the finite-element method. This method can be readily extended for solving elasticity problems involving inclusions and inhomogeneities, as well as contact between spheres. The tools developed here can also be useful for fast solution of differential equations with spherical boundaries beyond elasticity.

Solution of the finite slab criticality problem using an alternative phase function with the second kind of Chebyshev polynomials

The critical size of a finite homogenous slab is investigated for one-speed neutrons using the alternative phase function (AG, Anli–Gungor) in place of the scattering function of the transport equation. First of all, the neutron angular flux expanded in terms of the Chebyshev polynomials of second kind (UN approximation) together with the AG phase function is applied to the transport equation to obtain a criticality condition for the system. Then, using various values of the scattering parameters, the numerical results for the critical half-thickness of the slab are calculated and they are tabulated in the tables together with the ones obtained from the conventional spherical harmonic (PN) method for comparison. They can be said to be in good accordance with each other.

Characteristic of the Disturbed Days Ionospheric Current System in the 180-Degree Magnetic Meridian

The behavior of the ionospheric current during disturbed days using 10 geomagnetic observatories along 180-degree magnetic meridian was analysed. Separation of external and internal current during geomagnetic storm on 12 April until 15 April 2012 data was carried out using spherical harmonic analysis method. The contour of day to day variation of total current, external current and internal current were distinguished based on geomagnetic storm phase. These current then were correlate with the solar wind parameter, IMF Bz, and geomagnetic indices in order to identify their relationship. The results reveal that these current intensities were found higher during the initial phase. This due to the enhancement of the magnetic field during the initial phase caused by the increment of solar wind dynamic pressure and southward IMF Bz. Hence, the induced current identified in the internal current. This study demonstrated that the electric current induction can be obtained from the weak geomagnetic storm.

Моделирование оптоакустических сигналов в системе с упругим рассеянием и поглощением света

AbstractWe propose a method for simulating optoacoustic signals initiated by a laser pulse in an absorbing medium with elastic radiation scattering and carry out corresponding calculations. It is shown that the use of the method of spherical harmonics allows obtaining an analytical expression for the optoacoustic signal. The deviation of the absorption profile from the Bouguer type near the sample surface leads to a strong influence of the medium parameters on the position of the signal maximum. The growth of the albedo of onefold scattering leads to a decrease in the effective signal growth constant and its amplitude.

Application of diffusion approximation in the calculations of reactor with cavities

The importance of calculation of radiation fields inside in-reactor cavities is associated with the necessity to simulate the emergency modes in fast breeder reactors (FBR), as well as reactor states with different coolant levels in special dedicated channels of passive feedback devices in lead-cooled fast reactors (LFR) of BREST type or in sodium cavities in sodium-cooled fast reactors (SFR). The Last Flight (LF) method (Bell and Glesston 1974, Davison 1960, DOORS3.2 1988, Mynatt et. al. 1969, Rhoades and Childs 1988, Rhoades and Sipmson 1997, SCALE 2009, Voloschenko et. al. 2012), or the method of the unscattered component is widely known and is commonly used in computer codes based on the method of spherical harmonics for obtaining solution in a gas medium at a certain distance from the calculated volume domain (DORT (Rhoades and Childs 1988), TORT (Rhoades and Sipmson 1997) and others (SCALE 2009)). The practice of its application (DOORS3.2 1988) demonstrated that acceptable results are achievable at considerable distances from the surface separating dense and gas media (more than two meters). Obtaining high-quality solution is not guaranteed for cavities within the calculation area. In addition, it is desirable to implement the cavities calculation methodology within the framework of the approximations used in reactor calculations introducing certain specific features. In particular isotropy of the neutron flux density and the necessity of forced introduction of a “conditional” calculation cell on the boundary surface of the void cavity are assumed in the diffusion approximation. If the LF method is oriented on the connection of the source point with the detection point, then it is necessary to determine in the calculation of neutron field in the cavities the neutrons escaping the surface area of the source and neutrons reaching a certain surface area of the cavity. In order to solve the problem, the authors suggested using the approximate solution presented in the paper. Thus, an algorithm for calculation of in-reactor cavities using the diffusion approximation was developed and implemented by the authors.

Numerical investigation of the influence of operating conditions on the formation of nitrogen oxides in the combustion chamber of a low-power boiler during the combustion of coal-water fuel

The paper presents verification of the mathematical model and the results of a numerical research of the influence of operating conditions on the formation of nitrogen oxides during the combustion of coal-water fuel in a promising low-capacity boiler. For the numerical simulation of turbulent flow of an incompressible liquid, we used the approach of Reynolds-averaged Navier-Stokes equations (RANS) taking into account the interfacial interactions. To solve the equation of thermal radiation transfer, the P1 approximation of spherical harmonics method was employed. The optical properties of gases were described based on the sum of gray gases model. To describe the motion of coal particles we used the method of Lagrange multipliers. The combustion process of coal-water fuel is considered in terms of the following consecutive steps: evaporation of the water part of the droplet, evaporation of moisture from the fuel, devolatilization and the combustion of the volatile components, and the combustion of the coke residue. Comparative analysis has shown that the selection of the operating conditions of the boiler has a significant influence on the oxygen concentration in combustion chamber and the temperature of the flue gases. This leads to significant differences in the formation of nitrogen oxides during the combustion of coal-water fuel.

A stable algorithm for calculating one-group diffusion coefficient in a multi-zone cylindrical cell by the surface pseudo-sources method

In the surface harmonics method, designed to solve the neutron transport equation of in the core of a nuclear reactor, the distribution function is decomposed by trial functions. The first trial function is the absorption and division matrix. It is responsible for the main processes (absorption and fission) that occur inside the nuclear reactor cell. The second trial function is the diffusion coefficient matrix. It is associated with the processes of neutron flow from one cell to another. It is usually calculated in the P2 – and P4 – approximations of the spherical harmonics method and by the surface pseudo-sources method. In close lattices with high absorption of P2 – and P4 – approximations may not be enough, and the surface pseudo-sources method proves to be unstable in the calculation of multi-zone cylindrical cells. This work is dedicated to the development of stable one-group calculation of the diffusion coefficient by the surface pseudo-sources method first, by eliminating the divergent integrals, and secondly, the modification of the algorithm of matrix factorization for calculating one-group neutron transport equation. Developed program DICOF may use in the complexes of WIMS-SH and SVL for stable calculation of the diffusion coefficients matrix for calculation of the VVER and RBMK reactors.

Analysis of Small-Angle Neutron Scattering Spectra from Deformed Polymers with the Spherical Harmonic Expansion Method and a Network Model

We combine the recently proposed spherical harmonic expansion (SHE) method [Phys. Rev. X 2017, 7, 031003] with a modified network model to analyze the small-angle neutron scattering (SANS) spectra ...

On eigenvalues of the linearization of a free boundary problem modeling two-phase tumor growth

In this paper we study eigenvalues of the linearization of a free boundary problem modeling the growth of a tumor containing two species of cells: proliferating cells and quiescent cells. Such eigenvalues are potential bifurcation points from which nonradial solutions of the free boundary problem might bifurcate from the radial solution. A special feature of this problem is that it contains a singular ordinary differential equation which causes the main difficulty of this problem. By using the spherical harmonic expansion method combined with some techniques for solving singular differential integral equations developed in some previous literature, eigenvalues of the linearized problem are completely determined. Invertibility of some linear operators related to the linearized problem in suitable function spaces is also studied which might be useful in the analysis of the original free boundary problem.

The Multi-PN approximation to neutron transport equation

The Multi-PN (MPN) approximation to the first-order neutron transport equation in one- and two-dimensional geometries is presented using a variational approach. The directional variations of the angular flux have been modeled by separate spherical harmonics expansions in M subdomains, while finite elements have been utilized to represent the spatial dependence. A straightforward procedure has been proposed to handle any order of discretized polynomial expansions in both isotropic and anisotropic scattering medium. A key problem with much of the literature regarding the full-range spherical harmonic method is that it cannot exactly describe the discontinuous nature of the angular flux. Thus, the use of customary expansions leads to substantial defect at boundary and interface of two distinct media (e.g. interface of fuel elements and moderators). The major contribution of the MPN discretization is a theory that overcomes these difficulties. The work represents the generalization of the Double-PN (DPN) approximation previously applied to the neutron transport equation. Numerical results in one- and two-dimensional problems compare MPN and customary PN calculations to the reference transport calculations.

Effects of surface rates for the series reaction A → B → C on successive separated spherical sites

A steady - state series reaction A →B → C, consisting of a first reaction A →B on sphere 1 of radius a1 with first order kinetic rate constant kA and reactant diffusivity DA, followed by a second reaction B → C on sphere 2 of radius a2 with a first order kinetic rate constant kB and a B chemical species diffusivity DB, occurs on two discrete chemically active spheres a center - to – center distance d apart. The twin spherical harmonic expansion method with a Neumann iterative solution of the coefficient equations is used to generate a rigorous solution for the rate of series reaction in terms of an expansion in the inverse dimensionless center - to - center sphere separation d (=d⁄((a_1+a_2 ))) up to the inverse thirteenth power. The expansion coefficients depend on three dimensionless parameters (γ,λ_A,λ_B), where the dimensionless inverse surface kinetic constants are λ_A=D_A⁄a_1 kA on sphere 1, λ_B=D_B⁄(a_2 k_B ) on the sphere 2, while the geometry enters as the radius ratio γ=a_1⁄a_2 . Using these results an equivalent site kinetic change is imposed, first to λ_Aon site 1, then to λ_Bon site 2, to determine which site is more influential for the overall series reaction rate. Effects of γ geometry modification at several fixed kinetic surface site rates are examined. When either one of reactive spheres is diffusion controlled, various levels of λ kinetic controls at the other site or γ site geometries are generated to study the effects of site kinetics or geometry control on the series reaction rate. As the “other site” reaction rate also increases toward diffusion control, inflection points are observed to develop as precursors to a local maximum on the series reaction rate versus sphere separation curves. An application to the isomerization of n-pentane is presented.