Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions

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1 Journal of Nuclear Science and Technology ISSN: (Print) (Online) Journal homepage: Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions Tomohiro ENDO & Akio YAMAMOTO To cite this article: Tomohiro ENDO & Akio YAMAMOTO (27) Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions, Journal of Nuclear Science and Technology, 44:1, To link to this article: Published online: 5 Jan 212. Submit your article to this journal Article views: 257 View related articles Citing articles: 7 View citing articles Full Terms & Conditions of access and use can be found at

2 Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 44, No. 1, p (27) ARTICLE Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions Tomohiro ENDO and Akio YAMAMOTO Department of Materials, Physics and Energy Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya , Japan (Received October 17, 26 and accepted in revised form May 2, 27) In this paper, we propose new solid angle quadrature sets named EO N for the S N method. The EO N quadrature sets are developed for the -D xyz geometry, and satisfy not only even-moment conditions but also odd-moment conditions for direction cosines over an octant of the sphere. We can accurately calculate the numerical integration of the polynomial of direction cosines using EO N. We verify the effectiveness of EO N through the 2-D bulk shield problem and the -D neutron transport benchmark problems. The verification results indicate that the angular discretization error of EO N is much smaller than that of the conventional level symmetric quadrature set. KEYWORDS: solid angle, quadrature set, S N, octant, moment condition, even moment, odd moment, level symmetric I. Introduction In order to obtain a numerical solution of the neutron transport equation, deterministic transport methods are often used in the field of reactor physics. The S N method is one of the deterministic transport methods. In the S N method, angular neutron flux is evaluated in a number of discrete directions, instead of using the spherical harmonics expansion. In order to reduce the discretization error of angular directions, a sufficient number of directions are necessary. The computational time is approximately in proportion to the number of discrete directions, so that an excessive number of directions result in a long calculation time. Thus, we need to use an effective solid angle quadrature set, which consists of pairs of discrete direction points and their s, in order to accurately evaluate the angular distribution of neutron flux using as few number of directions as possible. As a solid angle quadrature set in the S N method, the level symmetric quadrature set (LQ N ) 1) is widely used. The discrete directions of LQ N are chosen to be fully symmetric with respect to x-, y-, and z-axes. Their corresponding to each discrete direction is determined to satisfy the evenmoment conditions for direction cosines. Here, the even-moment condition means that the numerical integration of the nth power of direction cosine with respect to the solid angle is equal to the theoretical value when n is an even number. However, the S N -order of LQ N is limited to S 2, because some of the s become negative over S 2. 1) Thus, there is a possibility that sufficient accuracy would not be obtained Corresponding author, a-yamamoto@nucl.nagoya-u.ac.jp ÓAtomic Energy Society of Japan with LQ N up to S 2. In this case, we need to use an alternative quadrature set of which S N -order is not limited, e.g., P N - T N quadrature set. 2) In this paper, we propose new solid angle quadrature sets for the S N method. The new quadrature sets are developed for the -D xyz geometry, and satisfy not only even-moment conditions but also odd-moment conditions over an octant of the sphere, i.e., the numerical integration of the nth power of direction cosine over the octant is equal to the theoretical value even if n is an odd number. Thus, we name the new quadrature sets EO N from the initial letters of Even and Odd. Furthermore, the EO N quadrature sets also satisfy the moment conditions of cross terms like and, where,, and represent the direction cosines with respect to x-, y-, and z-axes, respectively. In section II, we describe the concept and details of EO N. In section III, we verify EO N through the 2-D bulk shield problem proposed by Gelbard and Crawford ) and -D neutron transport benchmark problems proposed by Takeda and Ikeda. 4) II. New Solid Angle Quadrature Sets (EO N ) 1. Concept In the S N method, the accuracy of numerical integration of angular flux ð ~Þ with respect to the solid angle, i.e., total neutron flux, is very important. An accurate total neutron flux leads to an accurate reaction rate which is essential to the accurate evaluation of the neutron multiplication factor k eff and the power distribution. Here, the total neutron flux is calculated by taking the ed sum of m which is the angular flux corresponding to the discrete directions ~ m ð m ; m ; m Þ, i.e., 1249

3 125 T. ENDO and A. YAMAMOTO Z ð ~Þd 4 XM w m m ; ð1þ 4 where M is the total number of discrete directions; w m is the solid angle quadrature corresponding to ~ m and normalized as the total sum equal to unity: X M m w m 1: If we choose the appropriate set of the discrete directions and their s, we can evaluate total neutron flux more accurately using fewer numbers of directions. ð2þ In order to consider the appropriate solid angle quadrature set, we assume that the angular neutron flux ð ~Þ can be expanded by the polynomial of direction ~ ð; ; Þ: ð ~Þ X X X f ijk i j k ; ðþ i j k where i, j, and k are the exponents of,, and, respectively; f ijk is the coefficient corresponding to i j k. Let us assume that we can choose the appropriate discrete directions and their s to reproduce the theoretical value of the solid angle integral i j k : 4 XM w m i m j m k m Z 4 Z Z i j k d Z 2 d ðsin cos Þ i ðsin sin Þ j ðcos Þ k Z 2 ðsin Þ iþjþ1 ðcos Þ k d ðcos Þ i ðsin Þ j d ; where and are the polar and azimuthal angles, respectively. Then, the solid angle integral i j k can be preserved exactly using these discrete directions and their s. If the polynomial expressed by Eq. () holds to a fairly good approximation of the actual angular flux, and the discrete directions and their s satisfy Eq. (4) as higher order moments as much as possible, we expect that we can accurately evaluate the total neutron flux using this solid angle quadrature set. Although the angular flux is expanded by Eq. (), i.e., polynomials of direction cosines, there is ambiguity on the expansion method. For example, the angular flux is commonly expanded by the spherical harmonic functions. Therefore, another expansion method may give a more efficient solid angular quadrature set, if the expansion can capture angular flux distribution with fewer expansion terms. Such investigation is interesting and is considered as an open problem. It is noted that the angular flux is not always continuous for all directions. The solid angle dependency of angular flux might be discontinuous at the interface between different materials. Discontinuous points of angular flux degrade the results of numerical integration as in the case of Gauss-Legendre quadrature. In this paper, we suppose the -D xyz geometry, and assume that angular flux is discontinuous at,, or, but continuous otherwise. Thus, we determined the discrete directions and their s to satisfy the following moment conditions over an octant of the sphere ( =2, =2): 4 XM=8 w m i m j m k m d i j k ðsin Þ iþjþ1 ðcos Þ k d ðcos Þ i ðsin Þ j d : These discrete directions over the octant are expanded over the total sphere by exchanging the positive and negative signs of m, m, and m. Namely, these discrete directions satisfy mirror symmetry with respect to the yz-, zx-, and xy-planes. Here, we point out the difference between LQ N and our developed quadrature sets. In the case of LQ N, the discrete directions only satisfy the even-moment conditions over the total sphere, 1) i.e., X M w m n m XM w m n m XM w m n m 1 ; if n is an even number: n þ 1 ð6þ ð4þ ð5þ On the other hand, as shown in Eq. (5), our developed quadrature sets satisfy not only even-moment conditions but also odd-moment conditions over an octant of the sphere. Thus, we name the new quadrature sets EO N from the initial letters of Even and Odd. It is noted that the EO N quadrature sets also satisfy the moment conditions of cross terms, e.g.,,, and so on. A quadrature set that satisfies the moment conditions of cross terms for even moments has been developed. 5) From this viewpoint, the present EO N quadrature set can be considered as the extension of the previous one. When odd moments and cross terms (including even and odd moments) are not essential to represent the angular flux distribution, the accuracy of EO N will be comparable to that of LQ N. However, as discussed later, EO N gives better results than LQ N. This suggests that odd moments and cross terms (with even and odd moments) are important for angular flux representation. This is the essential cause for the superiority of EO N. JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

4 Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions Development Procedure In this subsection, we describe the development procedure of the new solid angle quadrature sets, EO N. The total number of discrete directions of EO N is NðN þ 2Þ as in the case of LQ N. The EO N quadrature sets satisfy the -fold symmetry around ð; ; Þ ðp 1 ffiffi ; p 1ffiffi ; 1ffiffi p Þ, i.e., there are combinations of symmetrical discrete directions by changing the order of m, m, and m. These combinations are divided into three cases: Case 1: Triangle type, e.g., ða; a; bþ, ða; b; aþ, and ðb; a; aþ. Case 2: Hexagon type, e.g., ða; b; cþ, ðb; a; cþ, ða; c; bþ, ðc; a; bþ, ðb; c; aþ, and ðc; b; aþ. Case : Only itself, i.e., ðp 1 ffiffi ; p 1ffiffi ; 1ffiffi p Þ. Here, the is different for each combination. Thus, we determine the discrete directions and s for each combination to satisfy moment conditions, i.e., Eq. (5). The number of moment conditions should be equal to the degrees of freedom in discrete directions, i.e., the directional cosines and s for all combinations. In the following part, the choice of moment conditions is discussed. At first, the following recurrence equation offers a hint for this choice: iþ2 j k þ i jþ2 k þ i j kþ2 i j k : Equation (7) can be derived from the fact that ð; ; Þ is on the unit sphere, i.e., 2 þ 2 þ 2 1. Integrating Eq. (7) with respect to an octant part of the sphere, we obtain d ð iþ2 j k þ i jþ2 k þ i j kþ2 Þ d i j k : As shown in Eq. (8), if the quadrature set satisfies the moment condition of i j k, we can integrate ð iþ2 j k þ i jþ2 k þ i j kþ2 Þ exactly using this quadrature set. For example, when i j k, Eq. (8) becomes d ð 2 þ 2 þ 2 Þ ð7þ ð8þ d 2 : ð9þ In R Eq. (9), the th-order moment condition, i.e., =2 R =2 d is already satisfied by the normalization condition of s, Eq. (2). The discrete directions satisfy the -fold symmetry, so that d 2 d 2 d 2 : ð1þ As shown in Eqs. (9) and (1), it is found that the moment conditions of 2, 2, and 2 are automatically satisfied. For another example, when i 1 and j k, Eq. (8) becomes d ð þ 2 þ 2 Þ d : From the -fold symmetry, we obtain d 2 ð11þ d 2 : ð12þ From Eqs. (11) and (12), it is also found that, if the quadrature set satisfies the moment conditions of and, the moment conditions of 2 and 2 are also automatically satisfied. Namely, if the lower order moment conditions are satisfied, some parts of the higher moment conditions are automatically satisfied. Therefore, Eq. (8) indicates that the moment conditions should be satisfied from lower to higher order ones to efficiently reduce the discretization errors. Furthermore, Eq. (8) is the recurrence equation between the ði þ j þ kþth- and ði þ j þ k þ 2Þth-order moments, so that the ði þ j þ kþth moment conditions are not directly reflected on the ði þ j þ k þ 1Þth ones. Based on the above mentioned observation, we make the Nth-order EO N quadrature set to satisfy Eq. (5) perfectly in the range of ði þ j þ kþ ðn=2 1Þ. Nevertheless, we have additional choices to satisfy other higher moments. These choices require an engineering judgment. In this paper, we satisfied the other N=2 and ðn=2 þ 1Þth moment conditions alternately like N=2, N=2þ1, N=2 1, N=2, and so on. Therefore, further improvement of EO N might be achieved by the proper choice of moment conditions. The EO N quadrature sets were developed using Mathematica, 6) i.e., the discrete directions and their s are numerically solved using the FindRoot function of Mathematica, which searches for numerical solutions to the simultaneous equations of Eq. (5). We show the input and output examples of EO 6 in Fig. 1. We cannot find the quadrature set over S 16, because the nonphysical solutions, e.g., negative or imaginary solutions, are numerically obtained over S 16. Therefore, our solid angle quadrature set is limited to S 16.. Numerical Solutions Tables 1 7 show the numerical solutions of EO 4 EO 16 quadrature sets. It is noted that one must expand their discrete directions listed in Tables 1 7 over the total sphere, with -fold and mirror symmetries in mind. In order to get a visual image in our mind, Figs. 2 8 show that the discrete directions point in ð; ; Þ space. In Figs. 2 8, their s are represented by the gradation from white to black. As shown in Tables 1 7 and Figs. 2 8, the surface density of discrete directions is high around 1, 1, and 1. This results in small s corresponding to the discrete directions around 1, 1, and 1. These characteristics are quite contrary to the characteristics of LQ N, see Fig. 9, which show the discrete directions of LQ 16 and their s. In order to ensure that the moment conditions are satisfied, Tables 8 1 show the relative errors of the numerical integration of i j k using EO 4 EO 16. As shown in Tables 8 1, we can confirm that Nth-order EO N preserve the numerical integration of i j k perfectly in the range of ði þ j þ kþ ðn=2 1Þ; furthermore, the relative errors of VOL. 44, NO. 1, OCTOBER 27

5 1252 T. ENDO and A. YAMAMOTO Fig. 1 Input and output examples of Mathematica for EO 6 Table 1 EO 4 quadrature set Table 2 EO 6 quadrature set Table EO 8 quadrature set Table 4 EO 1 quadrature set Table 5 EO 12 quadrature set JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

6 Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions 125 Table 6 EO 14 quadrature set Table 7 EO 16 quadrature set ξ.5.45 ξ Fig. 2 Discrete directions of EO 4 and their s Fig. 4 Discrete directions of EO 8 and their s ξ.5.21 ξ Fig. Discrete directions of EO 6 and their s Fig. 5 Discrete directions of EO 1 and their s VOL. 44, NO. 1, OCTOBER 27

7 1254 T. ENDO and A. YAMAMOTO ξ.5.7 ξ Fig. 6 Discrete directions of EO 12 and their s Fig. 8 Discrete directions of EO 16 and their s ξ.5.49 ξ Fig. 7 numerical integration become smaller as the S N -order increases. III. Verification Discrete directions of EO 14 and their s In this section, we verify EO N quadrature sets through the 2-D bulk shield problem proposed by Gelbard and Crawford, ) and the -D neutron transport benchmark problems proposed by Takeda and Ikeda. 4) In order to verify our quadrature set, we used a -D xyz S N code which is developed by the authors. Our S N code carries out computation with double precision. In our S N code, the diamond differencing scheme with negative flux fix up 1) and the exponential directional ed (EDW) differencing scheme 7) are available. Furthermore, coarse mesh finite-difference (CMFD) acceleration is utilized in order to accelerate the transport sweep. 8) 1. Bulk Shield Problem In order to verify the ray effect of EO N, we calculate the 2-D xy bulk shield problem proposed by Gelbard and Crawford. ) The number of energy groups of this benchmark problem is two. The geometry is a homogeneous system (1 cm 14 cm) and we divided the geometry into 1 Fig. 9 Discrete directions of S 16 level symmetric quadrature set, LQ 16, and their s 14 spatial meshes with 1:cm 1:cm mesh widths. In this problem, we used the EDW differencing scheme in the transport sweep, 7) because the EDW differencing scheme can eliminate the small spatial oscillation of total neutron flux which arises in the case of the diamond difference scheme. The convergence criterion is 1 7 with respect to the total neutron flux at each mesh point. Figures 1 and 11 show the numerical results of the total neutron fluxes of fast and thermal energy groups at y 19:5 cm using EO 16 and LQ 16. As shown in Figs. 1 and 11, the total neutron fluxes of both quadrature sets have spatial fluctuations due to the ray effect, but the fluctuation of EO 16 is smaller than that of LQ 16. The EO N satisfies the moment conditions of not only the even moment but also the odd moment, and furthermore their cross term. Because of this characteristic of EO N, it is considered that EO N can mitigate the ray effect. 2. -D Neutron Transport Benchmark Problems In order to verify the discretization errors of EO N for the neutron multiplication factor k eff, we calculate -D neutron transport benchmark problems proposed by Takeda and Ikeda. 4) We briefly explain their models as follows: JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

8 Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions 1255 Table 8 Relative error of numerical integration R =2 R =2 d i j k in the range of ði þ j þ kþ 6 relative error [%] ði; j; kþ theoretical value EO 4 EO 6 EO 8 EO 1 EO 12 EO 14 EO 16 (,,) =2 (1,,) =4 (2,,) =6 (1,1,) 1= 1:825 (,,) = (2,1,) =16 6:84 (1,1,1) 1=8 :84 :21 (4,,) = :997 (,1,) 2=15 : (2,2,) = 2: (2,1,1) 1=15 6:29 1:82 (5,,) = :42.6 (4,1,) = : : 1 5 (,2,) =48 28: :12 (,1,1) 1=24 :84 :21 (2,2,1) =96 2:49 5: :6 1 4 (6,,) = :1.21 (5,1,) 8= : :17 (4,2,) =7 22: :524 (,,) 4=15 45: :2 1 4 (4,1,1) 1= :1 1 (,2,1) 2=15 2:615 :296 :791 :14 8:4 1 4 (2,2,2) =21 :62 18: Table 9 Relative error of numerical integration R =2 R =2 d i j k in the range of 7 ði þ j þ kþ 8 relative error [%] ði; j; kþ theoretical value EO 4 EO 6 EO 8 EO 1 EO 12 EO 14 EO 16 (7,,) = : (6,1,) 5= :9.256 :45 4:8 1 4 (5,2,) =96 1: :18 :21 (4,,) =128 49: :9 1 4 (5,1,1) 1= :11. :99.9 5: 1 6 (4,2,1) =256 14:216 :669 1: :9 1 4 (,,1) 1=96 9: :.99 :9 5: 1 6 (,2,2) =84 9:1 2: :168 :2 (8,,) = : (7,1,) 16= : :48 :5 (6,2,) = :47 2:8 :81 :9 (5,,) 16=945 45: : :2 1 :9 1 5 (6,1,1) 1= : :6. 2:5 1 4 (4,4,) =21 6: (5,2,1) 8= : :14.1 7:9 1 5 (4,,1) 2=15 41: : :9 :2 1 4 (4,2,2) =6 :62 18: (,,2) 4=945 51:124 25: :592 :6 : ) Model 1 Model 1 simulates the small LWR core of the Kyoto University Critical Assembly (KUCA). The number of energy groups is two. We calculated two cases: the control rod is withdrawn or inserted. In this calculation, we divided the geometry into spatial meshes of which the mesh widths are :5cm :5cm :5cm. 2) Model 2 Model 2 simulates the small FBR core. The number of energy groups is four. We calculated two cases: the control rod VOL. 44, NO. 1, OCTOBER 27

9 1256 T. ENDO and A. YAMAMOTO Table 1 Relative error of numerical integration R =2 R =2 d i j k in the range of i þ j þ k 9 relative error [%] ði; j; kþ theoretical value EO 4 EO 6 EO 8 EO 1 EO 12 EO 14 EO 16 (9,,) = : :2 1 (8,1,) 7= : :1. (7,2,) = :951 :18 :16 :5.5 (7,1,1) 1= : :62.64 :16 2:9 1 6 (6,,) =256 5: : :5 :19 1:5 1 5 (5,4,) =2 66: :2 1 5 (6,2,1) = : : :9 1 5 (5,,1) 1=24 : : : : 1 6 (4,4,1) =256 55: : :78 :6 6:5 1 5 (5,2,2) =96 18:975 16: :67 :6 1 5 (4,,2) =128 52:866 2: :52 : :4 1 5 (,,) 1=48 62:11 9: : :97 1: Total neutron flux [1 6 cm 2 s 1 ] EO16 LQ16 Total neutron flux [1 5 cm 2 s 1 ] EO16 LQ x [cm] Fig. 1 Total neutron flux of fast energy group at y 19:5 cm using EO 16 and LQ x [cm] Fig. 11 Total neutron flux of thermal energy group at y 19:5 cm using EO 16 and LQ 16 is withdrawn or half-inserted. In this calculation, we divided the geometry into spatial meshes of which the mesh widths are 2:5cm 2:5cm 2:5cm. ) Model Model simulates the axially heterogeneous FBR core. The number of energy groups is four. We calculated two cases: the control rods are withdrawn or inserted. In this calculation, we divided the geometry into spatial meshes of which the mesh widths are 2:5cm 2:5cm 2:5cm. In these calculations, we used the diamond differencing scheme with negative flux fix up. The diamond differencing scheme is better suited for the evaluation of the neutron multiplication factor k eff. The convergence criteria are 1 7 with respect to the k eff and the total neutron flux at each mesh point. Tighter convergence criteria are used both for k eff and the total neutron flux to eliminate the effect of residual errors in comparison. Tables 11 1 show the numerical results of k eff calculated using LQ N and EO N. Here, the reference values are calculated using the double-p N 2 2 quadrature set of which the number of discrete directions is 124 per octant; the details are described in Appendix. It is noted that these relative errors are very small, so that the LQ N sets are sufficient to evaluate k eff for the present benchmark problems. As shown in Tables 11 1, the effectiveness of EO N depends on each Model. In Model 1 (Table 11), the relative errors of LQ N are smaller than those of EO N when the S N -orders are S 4 and S 6. In the rod-out case, it is observed that the k eff of LQ N converges as the S N -order increases, but the convergence value is slightly larger than reference value. On the other hand, the k eff of our S 16 quadrature set is almost equal to the reference value. In the rod-in case, the relative errors of EO N are much smaller than those of LQ N, when the S N - order is larger than S 6. In Model 2 (Table 12), the relative errors of EO N are smaller than those of LQ N, for all S N -orders. In Model (Table 1), the relative errors of EO N are much larger than those of LQ N when the S N -order is S 4. However, when the S N -order is larger than S 4,EO N can evaluate k eff more accurately than LQ N. These results imply that the priority of moment conditions depends on the calculation condition, e.g., the geometry and JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

10 Development of New Solid Angle Quadrature Sets to Satisfy Even- and Odd-Moment Conditions 1257 Table 11 Numerical results of k eff for Model 1 S Rod-out Rod-in LQ N EO N LQ N EO N ( :191) ( :577) ( :9) (.181) ( :115) (.167) (.48) (.171) ( :58) (.21) (.79) (.8) ( :29) (.15) (.7).9626 ( :1) ( :1) (.16) (.65) (.1) (.) (.1) (.56) (.) (.7).977 (.1) (.49) (.) (.12) (.4) (.14) (.8) Reference The relative error with respect to the reference value [%] The S N -order of EO N is limited to S 16 Table 12 Numerical results of k eff for Model 2 S Rod-out Rod-in LQ N EO N LQ N EO N ( :18).9765 (.1) ( :1) (.91) ( :57).9764 (.15) ( :7) (.15) ( :) (.) ( :4) (.4) ( :2).9762 (.) ( :28) (.1) ( :17).9762 (.) ( :2) (.) ( :14) ( :1) ( :16) (.) ( :11) ( :1) ( :1) (.) ( :1) ( :11) ( :8) ( :9) Reference The relative error with respect to the reference value [%] The S N -order of EO N is limited to S 16 Table 1 Numerical results of k eff for Model S Rod-out Rod-in LQ N EO N LQ N EO N ( :14) (.78) ( :22) (.874) ( :59) (.7) ( :91) (.9) ( :1) (.9) ( :48) (.12) ( :22) (.2) ( :4) (.2) ( :16) (.1) ( :24) (.1) ( :1) (.) ( :18) (.) ( :1) (.) ( :15) (.) ( :9) ( :12) ( :8).9714 ( :11) Reference The relative error with respect to the reference value [%] The S N -order of EO N is limited to S 16 number of energy groups. Generally speaking, in all Models, the relative error of LQ N decreases smoothly and slowly as the S N -order increases. On the other hand, the relative error of EO N decreases dramatically as the S N -order increases. The relative errors of EO 8 are of the same magnitude as, or much smaller than those of LQ 2. In view of the total number of discrete directions, the number of EO 8 is approximately one-fifth that of LQ 2, so that the computational time of the transport sweep is reduced by one-fifth. VOL. 44, NO. 1, OCTOBER 27

11 1258 T. ENDO and A. YAMAMOTO IV. Conclusions In this paper, we developed new quadrature sets for the S N code, named EO N. The EO N quadrature sets are developed for the -D xyz geometry, and satisfy not only even-moment conditions but also odd-moment conditions over the octant. Specifically, the Nth-order EO N exactly satisfies the moment conditions of direction cosines i j k, i.e., Eq. (5), in the range of ðiþ j þ kþ ðn=2 1Þ. In other words, we can exactly calculate the numerical integration of i j k over an octant in the range of ðiþ j þ kþ ðn=2 1Þ using EO N. In order to verify the ray effect of EO N, we calculated the 2-D xy bulk shield benchmark problem. Through this benchmark problem, we confirm that the ray effect of EO N is smaller than that of the level symmetric quadrature set, LQ N. In order to verify the discretization error of EO N for the neutron multiplication factor k eff, we calculated the -D neutron transport benchmark problems. Through the -D neutron transport benchmark problems, we confirm that the discretization error of EO N decreases dramatically as the S N -order increases. The relative errors of EO 8 are of as the same magnitude as, or much smaller than those of the LQ 2. In the S N code, the computational time is approximately in proportion to the number of discrete directions, so that we can reduce the computational time drastically using EO N. However, it is noted that EO N has ambiguity in the choice of moment conditions. Therefore, further improvement of EO N might be achieved by the proper choice of moment conditions. The orthogonal polynomial defined over an octant, which is like the spherical harmonics, would be the aid for the choice of moment conditions. References 1) E. E. Lewis, W. F. Miller, Jr., Computational Methods of Neutron Transport, John Wiley & Sons, New York, (1984). 2) G. Longoni, A. Haghighat, Development of new quadrature sets with the ordinate splitting technique, The 21 ANS International Meeting on Mathematical Methods for Nuclear Applications, Salt Lake City, Utah, USA, Sept. 9 1, 21 (21). ) E. M. Gelbard, B. Crawford, Benchmark Problem Book, ANL- 7416, Supplement 1, Argonne National Laboratory (1972). 4) T. Takeda, H. Ikeda, -D neutron transport benchmarks, J. Nucl. Sci. Technol., 28[7], 656 (1991). 5) Y. Ronen ed., Handbook of Nuclear Reactors Calculations, CRC press, Boca Raton, Florida, Vol. 1, 9 (1986). 6) S. Wolfram, The Mathematica Book, 5th ed., Wolfram Media, Illinois (2). 7) G. E. Sjoden, A. Haghighat, The exponential directional ed (EDW) S N differencing scheme in -D Cartesian geometry, Proc. Joint Int. Conf. on Mathematical Methods and Supercomputing for Nuclear Applications, Saratoga Springs, New York, Oct. 5 9, 1997, Vol. 2, 1267 (1997). 8) K. S. Smith, J. D. Rhodes, CASMO-4 characteristics method for two-dimensional PWR and BWR core calculations, Trans. Am. Nucl. Soc., 8, (2). Appendix The double-p N 2 2 quadrature set for an octant of the sphere is defined as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 2 m cosð m Þ; ða1þ qffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 2 m sinð m Þ; ða2þ m 1 2 ðx L;i þ 1Þ; 1 i 2; ðaþ m 4 ðx L;j þ 1Þ; 1 j 2; ða4þ w m w L;iw L;j ; ða5þ 2 where x L;i is the ith root of 2nd-order Legendre Polynomial, 1 < x L;1 < x L;2 < < x L;2 < 1; w L;i is ith corresponding to x L;i and the total sum is two: X 2 i1 w L;i 2: ða6þ The total number of discrete directions of this quadrature set is per octant. Therefore, the discretization error of angular directions is negligible using the double- P N 2 2 quadrature set. JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY