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):
- The set of directions and weights should be invariant to any rotation of 90 deg about any one of the coordinate axis.
- The quadrature weights ω_{i} should be positive.
- The zeroth-, first-, and second-order moments of the direction cosines should be exactly satisfied:
(1a) (1b) (1c) - The first moment over a half range of 2π should be exactly satisfied as
(2)
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, N_{θ} and N_{ϕ}, respectively, whose amplitude is evaluated as Δθ = N_{θ}/π and Δϕ = N_{ϕ}/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 (SRAP_{N}) 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 SRAP_{N} 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 SRAP_{N} is comparable to the accuracy of the T_{N} quadrature, which will also be addressed in this article.
(a) | (b) |
Figure 2. Discontinuous polar/azimuthal discretizations: (a) SRAP_{N} quadrature; (b) FT_{n} FVM quadrature.
A similar quadrature, referred to as FT_{n} 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 FT_{n} FVM and SRAP_{N}. The polar angle is uniformly divided in the FT_{n} FVM, while the division is nonuniform in SRAP_{N}, 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 SRAP_{N}.
The application of these quadratures to the DOM is straightforward. Once a discrete solid angle is defined, limited by [θ_{min}, θ_{max}] and [ϕ_{min}, ϕ_{max}], the discrete directions for the DOM may be defined by the center of the solid angle: θ^{m} = (θ_{min} + θ_{max})/2, ϕ^{m} = (ϕ_{min} + ϕ_{max})/2 and the quadrature weight is evaluated as
(3) |
S_{N }Quadratures
There are several different S_{N} 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 (ξ^{m}, η^{m}, μ^{m}) 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 S_{N } quadrature, N, represents the number of different direction cosines for every axis. In the S_{N } 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 S_{6} quadrature.
Let the coordinates of a discrete direction be denoted by (s_{i }, s_{j }, s_{k }), where the subscripts identify the level. The level symmetric quadratures use the same set of direction cosines for every axis, i.e., ξ_{i } = η_{i } = μ_{i } = s_{i }. 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 S_{8}, 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 (s_{i }, s_{j }, s_{k }) whose indices i, j, and k are permutations of each other have identical weights. As an example, in the S_{8} 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 s_{i}^{2} + s_{j}^{2} + s_{k}^{2} = 1, which may be rewritten as (Carlson and Lathrop, 1965)
(4) |
where s_{1}^{2} ≤ 1/3. Once s_{1} is found, all other s_{j} may be evaluated from this equation. The value of s_{1} 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 p_{i} 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 s_{i} and all distinct point weights W_{i} 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 S_{10}, whose values were computed by us. Data for S_{16} and S_{20} 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 S_{4}, S_{6}, S_{8}, S_{10}, and S_{12} may be found from Tables 1 and 2. As an example, the direction cosines of the seventh direction of the S_{8} LSE quadrature are (s_{1} , s_{1}, s_{4}) and the point weight is W_{1}, as listed in Table 1. Then, from Table 2 it can be concluded that (ξ^{m}, η^{m}, μ^{m}) = (0.2182179, 0.2182179, 0.9511897) and ω_{m} = W_{1} = 0.1900470 for m = 7.
Table 1. Level and point weight indices for level symmetric S_{N} quadratures
S_{4} | S_{10} | S_{12} | ||||||||||||
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 |
1 | 2 | 1 | 1 | 1 | 1 | 5 | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 1 |
2 | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 2 | 2 | 2 | 5 | 1 | 2 | 2 |
3 | 1 | 2 | 1 | 1 | 3 | 4 | 2 | 1 | 2 | 3 | 5 | 2 | 1 | 2 |
S_{6} | 4 | 3 | 1 | 3 | 3 | 4 | 4 | 1 | 3 | 3 | ||||
5 | 3 | 2 | 2 | 4 | 5 | 4 | 2 | 2 | 4 | |||||
1 | 3 | 1 | 1 | 1 | 6 | 3 | 3 | 1 | 3 | 6 | 4 | 3 | 1 | 3 |
2 | 2 | 1 | 2 | 2 | 7 | 2 | 1 | 4 | 2 | 7 | 3 | 1 | 4 | 3 |
3 | 2 | 2 | 1 | 2 | 8 | 2 | 2 | 3 | 4 | 8 | 3 | 2 | 3 | 5 |
4 | 1 | 1 | 3 | 1 | 9 | 2 | 3 | 2 | 4 | 9 | 3 | 3 | 2 | 5 |
5 | 1 | 2 | 2 | 2 | 10 | 2 | 4 | 1 | 2 | 10 | 3 | 4 | 1 | 3 |
6 | 1 | 3 | 1 | 1 | 11 | 1 | 1 | 5 | 1 | 11 | 2 | 1 | 5 | 2 |
S_{8} | 12 | 1 | 2 | 4 | 2 | 12 | 2 | 2 | 4 | 4 | ||||
13 | 1 | 3 | 3 | 3 | 13 | 2 | 3 | 3 | 5 | |||||
1 | 4 | 1 | 1 | 1 | 14 | 1 | 4 | 2 | 2 | 14 | 2 | 4 | 2 | 4 |
2 | 3 | 1 | 2 | 2 | 15 | 1 | 5 | 1 | 1 | 15 | 2 | 5 | 1 | 2 |
3 | 3 | 2 | 1 | 2 | 16 | 1 | 1 | 6 | 1 | |||||
4 | 2 | 1 | 3 | 2 | 17 | 1 | 2 | 5 | 2 | |||||
5 | 2 | 2 | 2 | 3 | 18 | 1 | 3 | 4 | 3 | |||||
6 | 2 | 3 | 1 | 2 | 19 | 1 | 4 | 3 | 3 | |||||
7 | 1 | 1 | 4 | 1 | 20 | 1 | 5 | 2 | 2 | |||||
8 | 1 | 2 | 3 | 2 | 21 | 1 | 6 | 1 | 1 | |||||
9 | 1 | 3 | 2 | 2 | ||||||||||
10 | 1 | 4 | 1 | 1 |
Table 2. Direction cosines and point weights for level symmetric S_{N} quadratures
Quadrature | Level i | LSE | LSO | LSH | ||||
s_{i} | W_{i} | s_{i} | W_{i} | s_{i} | W_{i} | |||
S_{4} | 1 | 0.3500212 | 0.5235987 | 0.2958759 | 0.5235988 | 0.2958759 | 0.5235988 | |
2 | 0.8688903 | 0.9082483 | 0.9082483 | |||||
S_{6} | 1 | 0.2666352 | 0.2766681 | 0.1838671 | 0.1609518 | 0.1914858 | 0.1780147 | |
2 | 0.6815078 | 0.2469424 | 0.6950514 | 0.3626469 | 0.6940220 | 0.3455841 | ||
3 | 0.9261811 | 0.9656012 | 0.9626351 | |||||
S_{8} | 1 | 0.2182179 | 0.1900470 | 0.1422555 | 0.1712359 | 0.1691277 | 0.1461389 | |
2 | 0.5773503 | 0.1425352 | 0.5773503 | 0.0992284 | 0.5773503 | 0.1598389 | ||
3 | 0.7867958 | 0.1454441 | 0.8040087 | 0.4617178 | 0.7987881 | 0.1733461 | ||
4 | 0.9511897 | 0.9795544 | 0.9709746 | |||||
S_{10} | 1 | 0.1893213 | 0.1402771 | 0.1120432 | 0.0392849 | 0.1372719 | 0.0944412 | |
2 | 0.5088818 | 0.1139285 | 0.5031286 | 0.2397376 | 0.5046889 | 0.1483961 | ||
3 | 0.6943189 | 0.0707546 | 0.7026544 | –0.123758 | 0.7004129 | 0.0173702 | ||
4 | 0.8397600 | 0.0847101 | 0.8569177 | 0.1285967 | 0.8523177 | 0.1149972 | ||
5 | 0.9634910 | 0.9873665 | 0.9809754 | |||||
S_{12} | 1 | 0.1672127 | 0.1111536 | 0.0935899 | 0.1383567 | 0.1281652 | 0.0802617 | |
2 | 0.4595476 | 0.0877778 | 0.4511138 | –0.061041 | 0.4545003 | 0.1082299 | ||
3 | 0.6280191 | 0.0586498 | 0.6310691 | 0.1062960 | 0.6298529 | 0.0451194 | ||
4 | 0.760021 | 0.0789827 | 0.7700602 | 0.6225548 | 0.7660672 | 0.0713859 | ||
5 | 0.8722705 | 0.0406068 | 0.8875457 | –0.327823 | 0.8814778 | 0.0652525 | ||
6 | 0.9716377 | 0.9912022 | 0.9834365 |
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 S_{N} 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).
T_{N} Quadrature
In the T_{N} 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 N^{2} directions per octant and a total of 8N^{2} directions for the T_{N} quadrature. This satisfies exactly the zeroeth- and second-order moments.
(a) | (b) |
A disadvantage of the T_{N} 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 S_{N} and the T_{N} quadratures are compared in Thurgood et al. (1995). They conclude that although the T_{N} quadrature is less accurate than the level symmetric S_{N} quadratures presented above, the weights of T_{N} 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 T_{N} 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 T_{N} 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 4N^{2} + 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 S_{N} and T_{N} 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 S_{N} quadratures. They relax the arrangement of the discrete directions in levels employed in S_{N} 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 S_{8} 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 | 1 | 0.24154201 | 0.24154201 | 0.93984834 | 0.243753132 | |
1 | 0.26524016 | 0.68177989 | 0.68177989 | 0.279845644 | ||
DCT111-24681012 | 0 | 0.20417467 | 0.53562198 | 0.81940331 | 0.16427082 | |
1 | 0.18046478 | 0.18046478 | 0.96688413 | 0.13527009 | ||
2 | 0.57735027 | 0.57735027 | 0.57735027 | 0.17936111 |
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 (LM_{N}), which integrate exactly the spherical harmonics of order N = 9, 11, 13, 15, 17, 19, and 23, and the quadratures of the Chebyshev type (LC_{N}), 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 S_{N}, DCT, T_{N}, 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 LC_{11}, 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 S_{N} quadratures. Li et al. (2002) proposed two spherical symmetrical equal dividing (SSD_{N}) 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 S_{N} 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 T_{N} 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
- 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.
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