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ANGULAR DISCRETIZATION METHODS

Following from: Discrete ordinates and finite volume methods

In the article Mathematical formulation, we have seen that the angular discretization of the radiative transfer equation (RTE) requires the selection of a finite number of directions of propagation of radiation intensity and the associated quadrature weights in the discrete ordinates method (DOM), and the selection of discrete solid angles, also referred to as control angles, in the finite volume method (FVM). In general, any angular discretization method employed in the FVM may also be applied in the DOM, since the value of a solid angle defined in the FVM may be regarded as a weight in the DOM, and the center of that solid angle may be taken as the direction of propagation of radiation. The reverse is not true. In fact, although the weight of a quadrature in the DOM may be thought of as a solid angle, its boundaries are not always defined geometrically, preventing in such a case its direct application in the FVM. The angular discretization is largely arbitrary, but there are some recommended guidelines and a few common alternatives available, which are the subject of the present article.

Guidelines for the Choice of Discrete Ordinates Directions in the DOM

The following guidelines are generally recommended in the selection of the discrete ordinates directions of the DOM (Carlson and Lathrop, 1965; Fiveland, 1991; Koch et al., 1995; Koch and Becker, 2004):

  1. The set of directions and weights should be invariant to any rotation of 90 deg about any one of the coordinate axis.
  2. The quadrature weights ωi should be positive.
  3. The zeroth-, first-, and second-order moments of the direction cosines should be exactly satisfied:
    (1a)
    (1b)
    (1c)
    where δ stands for the unit tensor, s = ξi + ηj + μk is the unit vector along the direction of propagation of radiation intensity, and the summations extend over all the discrete directions, M. The first two equations allow an accurate calculation of the incident radiation and radiative heat flux, respectively. Equation (1c) allows accurate results when the diffusion approximation is satisfactory, i.e., in optically thick media. These equations are necessary, but not sufficient, to yield good numerical approximations.
  4. The first moment over a half range of 2π should be exactly satisfied as
    (2)
    where n is the unit vector normal to the wall and pointing to the interior of the medium. This equation allows an accurate calculation of the incident heat flux normal to walls aligned with the Cartesian axes.

It is possible to select different discrete ordinates sets that satisfy all or most of the above guidelines. In the case of anisotropically scattering media, Carlson and Lathrop (1965) and Fiveland (1991) recommend the choice of quadratures that satisfy as many moments of the direction cosines as possible, aiming at the accurate integration of anisotropic scattering phase functions approximated by a finite series of Legendre polynomials. In addition, Koch and Becker (2004) recommend rotational invariance with respect to the group of regular polyhedrons and to the infinite group of cyclic rotations by an angle 2π/n, with n = 1, 2,… around any axis through the center of the unit sphere. This last requirement cannot be satisfied by any quadrature with a finite number of directions.

Polar/Azimuthal Discretization

The simplest angular discretization method consists of the division of the angular domain into a finite number of discrete, nonoverlapping, solid angles defined by the intersection of lines of constant latitude and lines of constant longitude. This choice is typical of the FVM, but it may also be employed in the DOM.

In its simplest form, the polar angle domain of π and the azimuthal angle domain of 2π are divided into a prescribed number of equally spaced angles, Nθ and Nϕ, respectively, whose amplitude is evaluated as Δθ = Nθ/π and Δϕ = Nϕ/2π [see Fig. 1(a)]. In this case, the solid angles closer to θ = 0 or θ = π have smaller values than those closer to θ = π/2. Hence, an alternative option is to maintain the same equally spaced azimuthal angle discretization while selecting the amplitude of the polar angles to enforce that the discrete solid angles have the same amplitude [Fig. 1(b)]. Both options are sometimes referred to as piecewise constant angular discretization. However, the discretization may be more general, and discontinuous solid angles may be used, as exemplified in Fig. 1(c). The angular discretization may also be adjusted to the physics of the problem under consideration, as illustrated in Fig. 1(d) for the case of a collimated beam.

(a) (b)
(c) (d)

Figure 1. Polar/azimuthal angular discretization: (a) equal polar and azimuthal subdivision; (b) equal solid angle subdivision; (c) arbitrary discretization; (d) discretization suitable for a collimated beam.

Li et al. (1998) proposed a spherical rings arithmetic progression quadrature (SRAPN) that is a particular case of a discontinuous polar/azimuthal discretization. In this method, a hemisphere is divided into N spherical rings, starting from the top of the sphere, where the spherical ring degenerates on a crown. The spherical rings are divided into a different number of identical solid angles, which increases in arithmetic progression from the top of the hemisphere to the bottom, as shown in Fig. 2(a). The centers of the solid angles obtained in this way define the discrete directions. The area of each solid angle, i.e., the quadrature weight, is the same for all discrete directions. The total number of directions for SRAPN is equal to 8[2 + 3 +..+ (N+ 1)]. The zeroeth and second moments are exactly satisfied by this quadrature. According to Li et al. (1998), the accuracy of the SRAPN is comparable to the accuracy of the TN quadrature, which will also be addressed in this article.

(a) (b)

Figure 2. Discontinuous polar/azimuthal discretizations: (a) SRAPN quadrature; (b) FTn FVM quadrature.

A similar quadrature, referred to as FTn FVM [see Fig. 2(b)], was developed by Kim and Huh (2000) in the framework of the FVM. There are only two minor differences between FTn FVM and SRAPN. The polar angle is uniformly divided in the FTn FVM, while the division is nonuniform in SRAPN, and the azimuthal angle for the first octant is not subdivided for the spherical ring at the top, while it is uniformly divided into two angles in the SRAPN.

The application of these quadratures to the DOM is straightforward. Once a discrete solid angle is defined, limited by [θmin, θmax] and [ϕmin, ϕmax], the discrete directions for the DOM may be defined by the center of the solid angle: θm = (θmin + θmax)/2, ϕm = (ϕmin + ϕmax)/2 and the quadrature weight is evaluated as

(3)

SN Quadratures

There are several different SN quadratures, the most common being the level symmetric quadratures (Carlson and Lathrop, 1965). In these quadratures, the points whose coordinates are the direction cosines of the discrete directions (ξm, ηm, μm) are arranged on N/2 levels (lines of constant latitude) relatively to each vertex of the first octant of a unit sphere centered at the origin of the reference frame, as illustrated in Fig. 3. The number of points at the ith level relative to a vertex is equal to N/2 – i + 1, where i ranges from 1 at the level most distant from the vertex up to N/2 at the level closest to the vertex. The order of an SN quadrature, N, represents the number of different direction cosines for every axis. In the SN quadrature, the total number of directions per octant is N (N + 2)/8, while the total number of directions is M = N (N + 2). Once the directions and weights for the first octant have been chosen, the directions and weights for the other octants become automatically defined by the requirement of invariance under a rotation of 90 deg about any coordinate axis. Due to symmetry, the RTE needs to be solved only for M/2 and M/4 directions in the case of 2D and 1D problems, respectively.

Figure 3. Level symmetric S6 quadrature.

Let the coordinates of a discrete direction be denoted by (si , sj , sk ), where the subscripts identify the level. The level symmetric quadratures use the same set of direction cosines for every axis, i.e., ξi = ηi = μi = si . The discrete directions correspond to all the combinations of subscripts that satisfy the relation i + j + k = 2 + N/2. For example, in the case of S8, these subscripts are (1,1,4), (1,2,3), (1,3,2), (1,4,1), (2,1,3), (2,2,2), (2,3,1), (3,1,2), (3,2,1), and (4,1,1). Points (si , sj , sk ) whose indices i, j, and k are permutations of each other have identical weights. As an example, in the S8 quadrature, the directions associated to indices (1,1,4), (1,4,1), and (4,1,1) have the same weight, as well as the directions associated with indices (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). The direction cosines must satisfy the relation si2 + sj2 + sk2 = 1, which may be rewritten as (Carlson and Lathrop, 1965)

(4)

where s12 ≤ 1/3. Once s1 is found, all other sj may be evaluated from this equation. The value of s1 and the point weights are determined by requesting a prescribed set of N/2 + 1 moments of the direction cosines to be satisfied, according to the following equation (Carlson and Lathrop, 1965, Fiveland, 1991):

(5)

where the summation extends over all levelslevels, k denotes the order of the moment to be satisfied, and pi is the ith level weight, which is equal to the sum of the weights of all points at that level.

The point weights may be found from Table 1, which lists the level indices i, j, k and the point weight index i of all discrete directions (the numbering of the directions is arbitrary) for several quadratures. The level coordinates si and all distinct point weights Wi obtained from Eqs. (4) and (5) are listed in Table 2 for level symmetric quadratures satisfying sequential even (LSE), odd (LSO), and hybrid (LSH) moments. The direction cosines and the weights of the LSE quadratures in Table 2, which satisfy moments of order 0, 2, 4,…, N, have been taken from Fiveland (1991), except S10, whose values were computed by us. Data for S16 and S20 may be found in Carlson and Lathrop (1965). Negative point weights appear for N > 22. The data for the LSO quadratures, which satisfy moments of order 0, 1, 2, 3,…, N/2, were taken from the same references. Negative point weights appear for N ≥ 10. The values for the LSH quadratures were taken from Balsara (2001). The direction cosines and the weights for any discrete direction of S4, S6, S8, S10, and S12 may be found from Tables 1 and 2. As an example, the direction cosines of the seventh direction of the S8 LSE quadrature are (s1 , s1, s4) and the point weight is W1, as listed in Table 1. Then, from Table 2 it can be concluded that (ξm, ηm, μm) = (0.2182179, 0.2182179, 0.9511897) and ωm = W1 = 0.1900470 for m = 7.

Table 1. Level and point weight indices for level symmetric SN quadratures

S4 S10 S12
Direction m Level index i Level index j Level index k Point weight index i Direction m Level index i Level index j Level index k Point weight index i Direction m Level index i Level index j Level index k Point weight index i
12111 15111 16111
21121 24122 25122
31211 34212 35212
S6 43133 44133
53224 54224
13111 63313 64313
22122 72142 73143
32212 82234 83235
41131 92324 93325
51222 102412 103413
61311 111151 112152
S8 121242 122244
131333 132335
14111 141422 142424
23122 151511 152512
33212 161161
42132 171252
52223 181343
62312 191433
71141 201522
81232 211611
91322
101411

Table 2. Direction cosines and point weights for level symmetric SN quadratures

Quadrature Level i LSELSOLSH
siWisiWisiWi
S4 10.35002120.52359870.29587590.52359880.29587590.5235988
20.86889030.90824830.9082483
S6 10.26663520.27666810.18386710.16095180.19148580.1780147
20.68150780.24694240.69505140.36264690.69402200.3455841
30.92618110.96560120.9626351
S8 10.21821790.19004700.14225550.17123590.16912770.1461389
20.57735030.14253520.57735030.09922840.57735030.1598389
30.78679580.14544410.80400870.46171780.79878810.1733461
40.95118970.97955440.9709746
S10 10.1893213 0.14027710.11204320.03928490.13727190.0944412
20.50888180.11392850.50312860.23973760.50468890.1483961
30.69431890.07075460.7026544–0.1237580.70041290.0173702
40.83976000.08471010.85691770.12859670.85231770.1149972
50.96349100.98736650.9809754
S12 10.16721270.11115360.09358990.13835670.12816520.0802617
20.45954760.08777780.4511138–0.0610410.45450030.1082299
30.62801910.05864980.63106910.10629600.62985290.0451194
40.7600210.07898270.77006020.62255480.76606720.0713859
50.87227050.04060680.8875457–0.3278230.88147780.0652525
60.97163770.99120220.9834365

El Wakil and Sacadura (1992) proposed a level symmetric quadrature with the first direction satisfying the half-range moment and the weights determined by geometric considerations rather than moment-matching criteria. Other SN quadratures are available, such as the equal weight quadrature satisfying sequential even moments (Carlson, 1971) and the equal weight quadrature satisfying sequential odd moments (Fiveland, 1991).

TN Quadrature

In the TN quadrature (Thurgood et al., 1995), the equilateral triangle whose vertices are (1, 0, 0), (0, 1, 0), and (0, 0, 1), is mapped onto the first octant of a unit radius sphere using the relation s = r/|r|, where r stands for a point on the triangle and s denotes the position vector of a mapped point on the sphere. The triangle is tessellated into smaller identical triangles by dividing each side into N equally spaced segments and connecting the points that result from that division by lines parallel to the sides of the original triangle, as illustrated in Fig. 4(a) for N = 4. The projection of the centroids of every small triangle onto the surface of the sphere defines the direction cosines of the discrete directions, while the projection of the small triangles defines spherical triangles on the surface of the sphere, whose areas are the quadrature weights [see Fig. 4(b)]. There are N2 directions per octant and a total of 8N2 directions for the TN quadrature. This satisfies exactly the zeroeth- and second-order moments.

(a) (b)
(c)

Figure 4. TN and PQLA(1) quadratures: (a) planar triangles; (b) discrete directions of the T4 quadrature; (c) discrete directions of the PQLA(1) quadrature.

A disadvantage of the TN quadratures is that all discrete directions and weights are fixed by geometrical considerations, and no degrees of freedom are available to satisfy additional moment conditions (Koch and Becker, 2004). The SN and the TN quadratures are compared in Thurgood et al. (1995). They conclude that although the TN quadrature is less accurate than the level symmetric SN quadratures presented above, the weights of TN are always positive, so that there are no restrictions on the choice of N.

Piecewise Quasilinear Angular Quadratures

Rukolaine and Yuferev (2001) proposed two different piecewise quasilinear angular (PQLA) quadratures that share some features of the TN quadratures. The equilateral triangle whose vertices are (1, 0, 0), (0, 1, 0), and (0, 0, 1) is considered again. In quadratures of the first type, PQLA(1), the triangle is tessellated exactly as in the TN quadrature, but the discrete directions are defined by the vertices of the spherical triangles after projection onto the sphere, as shown in Fig. 4(c). The Nth-order quadrature of this type comprises 4N2 + 2 discrete directions. In quadratures of the second type, PQLA(2), the triangle is tessellated by lines normal to the sides of the original triangle, as illustrated in Fig. 5(a), and the discrete directions are defined again by the vertices of the spherical triangles after projection onto the sphere, as shown in Fig. 5(b). In both cases, the quadrature weights are determined from integration over the sphere of shape functions defined according to the finite element method. The zeroeth- and second-order moments are exactly satisfied by this quadrature.

(a) (b)

Figure 5. PQLA(2) quadratures: (a) planar triangles; (b) discrete directions of the PQLA(2) quadrature.

The analysis reported in Rukolaine and Yuferev (2001) suggests that the accuracy of PQLA is not as good as that of SN and TN quadratures with a similar number of discrete directions. The main advantage of this quadrature is that it allows solving radiation problems with specular reflective boundaries using the DOM, since the angular dependence of the radiation intensity may be expressed analytically.

Double Cyclic Triangle Quadratures

The double cyclic triangle (DCT) quadratures (Koch et al., 1995) are a generalization of the level symmetric and equal weight SN quadratures. They relax the arrangement of the discrete directions in levels employed in SN level symmetric quadratures, and take the superposition of DCT as the principle to generate the quadratures. A DCT is a set of six discrete directions on one octant that is invariant to permutations of the direction cosines: (ξ, η, μ), (ξ, μ, η), (η, ξ, μ), (η, μ, ξ), (μ, ξ, η), and (μ, η, ξ). The weights of the discrete directions of a DCT are equal. There are two degenerated forms of DCT. If two of the direction cosines are equal (level of degeneration 1), then the discrete directions form a spherical triangle. If the direction cosines are all equal (level of degeneration 2), then there is only one discrete direction. In all cases, the discrete directions of a DCT are located concentrically around the directions (±1, ±1, ±1), as illustrated in Fig. 6. The discrete directions and the weights are determined by requiring a few moments to be satisfied.

Figure 6. Discrete directions of a DCT quadrature.

The DCT quadratures are obtained by superposing several DCT arrangements. They are identified as DCTxyz-abcd..., where x, y, and z denote how many DCT arrangements of degeneration level 0, 1, and 2 are used, respectively, and a, b, c, d identify the moments satisfied by the quadrature. Koch et al. (1995) recommend the use of the DCT111–24681012 quadrature for general radiative transfer problems, as a good compromise between accuracy and computational requirements. This quadrature, comprising 10 discrete directions per octant, provides improved accuracy in comparison with S8 quadratures, which have the same number of discrete directions. Koch and Becker (2004) recommend the DCT020–2468 quadrature in the case of quadratures with approximately 50 discrete directions. The discrete directions and the weights of these quadratures are given in Table 3.

Table 3. Direction cosines and point weights for DCT020–2468 and DCT111–24681012 quadratures

Quadrature Degeneration level ξ η μ ω
DCT020-2468 10.241542010.241542010.939848340.243753132
10.265240160.681779890.681779890.279845644
DCT111-24681012 00.204174670.535621980.819403310.16427082
10.180464780.180464780.966884130.13527009
20.577350270.577350270.577350270.17936111

Lebedev Quadratures

The Lebedev quadratures(Lebedev, 1975, 1976) are rotationally invariant to the group of regular polyhedrons and exactly integrate the spherical harmonics functions on the unit sphere up to a certain order. Two types of quadrature are available, namely, the quadratures of the Markov type (LMN), which integrate exactly the spherical harmonics of order N = 9, 11, 13, 15, 17, 19, and 23, and the quadratures of the Chebyshev type (LCN), which integrate exactly the spherical harmonics of order N = 11 and 15. Quadratures of this kind were compared with DCT by Rukolaine and Yuferev (2001) and with SN, DCT, TN, and PQLA quadratures by Koch and Becker (2004). They found that the best accuracy among the studied quadratures, with up to ~100 discrete directions, was achieved by LC11, which has 96 directions, and integrates exactly all moments up to order 11, except the first-order moment.

Other Quadratures

Standard Gauss quadratures were used by Fiveland (1987) to solve 1D radiative transfer problems in anisotropically scattering media. Their accuracy was found to be lower than that of SN quadratures. Li et al. (2002) proposed two spherical symmetrical equal dividing (SSDN) quadratures for the DOM. They are both based on geometric considerations, with equal weights for all directions, and preserve symmetry to any rotation of 90 deg about the coordinate axes. The construction method is somewhat awkward, and the number of discrete directions is limited to 96. The accuracy is reported to be similar to that of the SN quadratures.

Adaptive Quadratures

All the quadratures mentioned above are fixed prior to the solution of the RTE and remain unchanged during the calculation. Adaptive quadratures have been described by Cumber (1999, 2000) and Verteeg et al. (2003) and applied to the discrete transfer method. The calculation starts with a standard polar/azimuthal discretization, and during the course of the calculation, based on a refinement criterion, a few solid angles are refined. Additional refinements may be performed during the calculation, as illustrated in Fig. 7, until the refinement criterion is satisfied. There is no theoretical difficulty in the application of such an adaptive strategy to the FVM and DOM.

Figure 7. Polar/azimuthal discretization during the course of an adaptive solution procedure.

REFERENCES

Carlson, B. G., Tables of Equal Weight Quadrature Over the Unit Sphere, Los Alamos Scientific Laboratory, Report LA–4737, 1971.

Carlson, B. G. and Lathrop, K. D., Discrete Ordinates Angular Quadrature of the Neutron Transport Equation, Los Alamos Scientific Laboratory, Report LA–3186, 1965.

Cumber, P. S., Application of Adaptive Quadrature to Fire Modeling, J. Heat Transfer, vol. 121, pp. 203–205, 1999..

Cumber, P. S., Ray Effect Mitigation in Jet Fire Radiation Modelling, Int. J. Heat Mass Transfer, vol. 43, pp. 935–943, 2000.

El Wakil, N. and Sacadura, J. F., Some Improvements of the Discrete Ordinates Method for the Solution of the Radiative Transport Equation in Multidimensional Anisotropically Scattering Media, Developments in Radiative Heat Transfer, S. T. Thynell, M. F. Modest, L. C. Burmeister, M. L. Hunt, T. W. Tong, R. D. Skocypec, W. W. Yuen, and W. A. Fiveland, Eds., ASME HTD-vol. 103, pp. 119–127, 1992.

Fiveland, W. A., Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically Scattering Media, J. Heat Transfer, vol. 109, pp. 809–812, 1987.

Fiveland, W. A., The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering, Fundamentals of Radiation Heat Transfer, W.A. Fiveland W. A. Fiveland, A. L. Crosbie, A. M. Smith and T. F. Smith, Eds., ASME HTD- vol. 160, pp. 89–96, 1991.

Kim, S. H. and Huh, K. Y., A New Angular Discretization Scheme of the Finite Volume Method for 3-D Radiative Heat Transfer in Absorbing, Emitting and Anisotropically Scattering Media, Int. J. Heat Mass Transfer, vol. 43, pp. 1233–1242, 2000.

Koch, R. and Becker, R., Evaluation of Quadrature Schemes for the Discrete Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 423–435, 2004.

Koch, R., Krebs, W., Wittig, S., and Viskanta, R., Discrete Ordinates Quadrature Schemes for Multidimensional Radiative Transfer, J. Quant. Spectrosc. Radiat. Transfer, vol. 53(4), pp. 353–372, 1995.

Lebedev, V. Values of the Nodes and Weights of Ninth to Seventeenth Order Gauss-Markov Quadrature Formulae Invariant under the Octahedron Group with Inversion, USSR Comput. Math. Math. Phys., vol. 15, pp. 44–51, 1975.

Lebedev, V., Quadratures on a Sphere, USSR Comput. Math. Math. Phys., vol. 16, pp. 10–24, 1976.

Li, B. W., Yao, Q, Cao, X.-Y., and Cen K.-F., A New Discrete Ordinates Quadrature Scheme for Three-Dimensional Radiative Heat Transfer, J. Heat Transfer, vol. 120, pp. 514–518, 1998.

Li, B. W., Chen, H.-G., Zhou, J.-H., Cao, X.-Y., and Cen K.-F., The Spherical Surface Symmetrical Equal Dividing Angular Quadrature Scheme for Discrete Ordinates Method, J. Heat Transfer, vol. 124, pp. 482–490, 2002.

Rukolaine, S. A. and Yuferev, V. S., Discrete Ordinates Quadrature Schemes Based on the Angular Interpolation of Radiation Intensity, J. Quant. Spectrosc. Radiat. Transfer, vol. 69, pp. 257–275, 2001.

Thurgood, C. P., Pollard, A., and Becker, H. A., The TN Quadrature Set for the Discrete Ordinates Method, J. Heat Transfer, vol. 117, pp. 1068–1070, 1995.

Verteeg, H. K., Henson, J. C., and Malalasekera, W., An Adaptive Angular Quadrature for the Discrete Transfer Method Based on Error Estimation, J. Heat Transfer, vol. 125, pp. 301–311, 2003.

References

  1. Carlson, B. G., Tables of Equal Weight Quadrature Over the Unit Sphere, Los Alamos Scientific Laboratory, Report LA–4737, 1971.
  2. Carlson, B. G. and Lathrop, K. D., Discrete Ordinates Angular Quadrature of the Neutron Transport Equation, Los Alamos Scientific Laboratory, Report LA–3186, 1965.
  3. Cumber, P. S., Application of Adaptive Quadrature to Fire Modeling, J. Heat Transfer, vol. 121, pp. 203–205, 1999..
  4. Cumber, P. S., Ray Effect Mitigation in Jet Fire Radiation Modelling, Int. J. Heat Mass Transfer, vol. 43, pp. 935–943, 2000.
  5. El Wakil, N. and Sacadura, J. F., Some Improvements of the Discrete Ordinates Method for the Solution of the Radiative Transport Equation in Multidimensional Anisotropically Scattering Media, Developments in Radiative Heat Transfer, S. T. Thynell, M. F. Modest, L. C. Burmeister, M. L. Hunt, T. W. Tong, R. D. Skocypec, W. W. Yuen, and W. A. Fiveland, Eds., ASME HTD-vol. 103, pp. 119–127, 1992.
  6. Fiveland, W. A., Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically Scattering Media, J. Heat Transfer, vol. 109, pp. 809–812, 1987.
  7. Fiveland, W. A., The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering, Fundamentals of Radiation Heat Transfer, W.A. Fiveland W. A. Fiveland, A. L. Crosbie, A. M. Smith and T. F. Smith, Eds., ASME HTD- vol. 160, pp. 89–96, 1991.
  8. Kim, S. H. and Huh, K. Y., A New Angular Discretization Scheme of the Finite Volume Method for 3-D Radiative Heat Transfer in Absorbing, Emitting and Anisotropically Scattering Media, Int. J. Heat Mass Transfer, vol. 43, pp. 1233–1242, 2000.
  9. Koch, R. and Becker, R., Evaluation of Quadrature Schemes for the Discrete Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 423–435, 2004.
  10. Koch, R., Krebs, W., Wittig, S., and Viskanta, R., Discrete Ordinates Quadrature Schemes for Multidimensional Radiative Transfer, J. Quant. Spectrosc. Radiat. Transfer, vol. 53(4), pp. 353–372, 1995.
  11. Lebedev, V. Values of the Nodes and Weights of Ninth to Seventeenth Order Gauss-Markov Quadrature Formulae Invariant under the Octahedron Group with Inversion, USSR Comput. Math. Math. Phys., vol. 15, pp. 44–51, 1975.
  12. Lebedev, V., Quadratures on a Sphere, USSR Comput. Math. Math. Phys., vol. 16, pp. 10–24, 1976.
  13. Li, B. W., Yao, Q, Cao, X.-Y., and Cen K.-F., A New Discrete Ordinates Quadrature Scheme for Three-Dimensional Radiative Heat Transfer, J. Heat Transfer, vol. 120, pp. 514–518, 1998.
  14. Li, B. W., Chen, H.-G., Zhou, J.-H., Cao, X.-Y., and Cen K.-F., The Spherical Surface Symmetrical Equal Dividing Angular Quadrature Scheme for Discrete Ordinates Method, J. Heat Transfer, vol. 124, pp. 482–490, 2002.
  15. Rukolaine, S. A. and Yuferev, V. S., Discrete Ordinates Quadrature Schemes Based on the Angular Interpolation of Radiation Intensity, J. Quant. Spectrosc. Radiat. Transfer, vol. 69, pp. 257–275, 2001.
  16. Thurgood, C. P., Pollard, A., and Becker, H. A., The TN Quadrature Set for the Discrete Ordinates Method, J. Heat Transfer, vol. 117, pp. 1068–1070, 1995.
  17. Verteeg, H. K., Henson, J. C., and Malalasekera, W., An Adaptive Angular Quadrature for the Discrete Transfer Method Based on Error Estimation, J. Heat Transfer, vol. 125, pp. 301–311, 2003.
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