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Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method
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Original Article
Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method Ni Dai, Bin Zhang*, Yixue Chen School of Nuclear Science and Engineering, North China Electric Power University, Beijing, 102206, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 August 2019 Received in revised form 21 October 2019 Accepted 20 November 2019 Available online xxx
The discrete ordinates method (SN) is one of the major shielding calculation method, which is suitable for solving deep-penetration transport problems. Our objective is to explore the available quadrature sets and to improve the accuracy in shielding problems involving strong anisotropy. The linear discontinuous finite-element (LDFE) quadrature sets based on the icosahedron (in short, ICLDFE quadrature sets) are developed by defining projected points on the surfaces of the icosahedron. Weights are then introduced in the integration of the discontinuous finite-element basis functions in the relevant angular regions. The multivariate secant method is used to optimize the discrete directions and their corresponding weights. The numerical integration of polynomials in the direction cosines and the Kobayashi benchmark are used to analyze and verify the properties of these new quadrature sets. Results show that the ICLDFE quadrature sets can exactly integrate the zero-order and first-order of the spherical harmonic functions over one-twentieth of the spherical surface. As for the Kobayashi benchmark problem, the maximum relative error between the fifth-order ICLDFE quadrature sets and references is only 0.55%. The ICLDFE quadrature sets provide better integration precision of the spherical harmonic functions in local discrete angle domains and higher accuracy for simple shielding problems. © 2019 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Shielding calculation Discrete ordinates method Quadrature sets Discontinuous finite elements Icosahedron
1. Introduction Particle transport calculations play a vital role in many fields, such as shielding design for nuclear devices [1], radiation therapy [2], astrophysics [3] and so on. The key issue of the shielding calculation is the steady particle transport equation, which is a linear differential-integral equation containing six independent variables: energy, space, and angle. This equation is usually discretized in each variable for convenience when solving. The discrete ordinates (SN) method directly discretizes the direction variables into a finite number of discrete directions and employs quadrature sets to approximate integrals over the angular variables [4]. However, the angular distribution of flux is not smooth and may even be discontinuous for shielding problems with strong anisotropy. Insufficient integration accuracy may result in larger discrete errors, reducing the accuracy and reliability of shielding calculations. Over the last decades, many quadrature sets have been proposed and shown to be useful in particle transport calculations. Carlson
* Corresponding author. E-mail address: [email protected] (B. Zhang).
and Lee proposed level symmetric (LS) quadrature sets in 1961 [5]. The LS quadrature sets are used widely, but some of their weights may become negative when the quadrature order is larger than 20. In 1987, Walters proposed the Legendre-Chebyshev quadrature sets (PNTN), which have the advantage of high integral accuracy and numerous discrete directions [6]. In 1995, Thurgood developed the symmetric quadrature sets, denoted TN, which are more accurate than the LS quadrature sets and always have positive weights [7]. Longoni studied regional angular refinement (RAR) based on the PNTN quadrature sets in 2002 [8], and in 2010 Jarrell developed the linear discontinuous finite-element (LDFE) quadrature sets based on the octahedron, which can exactly integrate the zero-order and first-order spherical harmonic functions on one-eighth of the spherical surface [9]. It also can generate a large number of discrete directions and is easier to refine locally. Later, Ahrens proposed a rotationally invariant quadrature sets based on the icosahedron [10]. These quadrature sets are highly efficient in the integrations of spherical harmonics but their reflecting boundaries are difficult to treat. In 2012, K. Atkison and W. Han discussed quadrature formula over spherical triangles including the 4-sided tetrahedron, the 8sided octahedron and the 20-sided icosahedron. They have chosen the centroid of sub-triangles as quadrature points and their
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surfaces areas as corresponding weights. The icosahedral triangulation has much smaller integration error than other triangulations with a similar directions [11]. In 2015, Lau and Adams studied linear and quadratic discontinuous finite-element methods and their local refinement quadrature sets based on spherical quadrilaterals [12]. The linear discontinuous finite element quadrature sets based on the icosahedron (termed the ICLDFE quadrature sets) can produce a more uniform distribution and strictly positive weights because the icosahedron is the closest polyhedron of the Platonic solids to the sphere. In this paper, we focus on the construction of these quadrature sets and analyze their properties. The rest of the paper is organized as follows: Section 2 introduces the construction of ICLDFE quadrature sets including those in discrete directions and the corresponding weights. In Section 3 the results of spherical harmonic integration and benchmarks are presented. Section 4 states our conclusions.
Fig. 2. Coordinate Systems of the ICLDFE quadrature sets.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi xÞðh yÞ 10 þ 2 5ð1 ~
m ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 3þpffiffi5ffi hpffiffiffi i2ffi; pffiffiffi ð Þ
2. Construction of ICLDFE quadrature sets
4
2.1. Quadrature points of ICLDFE
52 5 ~ x2 þ 1 ,
2
ðh yÞ2 þ
51 yþh
(5) A quadrature set is a set of weights and directions defined with the ordinates using the direction cosines (m, h, x) of the polar angle q and the azimuth g,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ux ¼ m ¼ cos g sin q ¼ cos g 1 x2 ;
(1)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uy ¼ h ¼ sin g sin q ¼ sin g 1 x2 ;
(2)
Uz ¼ x ¼ cos q:
(3)
(6)
The construction of the ICLDFE quadrature sets begins with the projection of the regular icosahedron, which has 20 faces of regular triangles, onto the unit sphere (Fig. 1). Fig. 2 illustrates the coordinate system of the direction cosines, where the h axis is defined as one side of a triangle of the icosahedron and the x axis lies on the line from the center point of the icosahedron to one of its vertices. The points are defined on a flat triangle and projected onto the surface of the unit sphere to obtain the corresponding unique quadrature points. The coordinates (x, y) of the flat regular triangle are established through the geometric relationship between the spherical and flat triangles (Fig. 2), and more detailed deduction is similar with the octahedron [13].
~ x¼
x 2hx ; ¼ xmax bðh yÞ
Fig. 1. Projection of regular icosahedron onto the sphere.
i pffiffiffi pffiffiffih 52 5 ~ x þ 1 ðh yÞ 3þ 5 h ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 3þpffiffi5ffi hpffiffiffi i2ffi; pffiffiffi 2 ð Þ 52 5 ~ x þ 1 , 2 ðh yÞ2 þ 4 51 yþh
(4)
pffiffiffi 51 yþh x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 3þpffiffi5ffi hpffiffiffi h i2ffi; pffiffiffi 2 ð Þ 52 5 ~ x þ 1 , 2 ðh yÞ2 þ 51 yþh (7) where b and h are the base length and height of the flat triangle determined by the geometric properties of the icosahedron,
4 b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi; 10 þ 2 5 h¼
pffiffiffi pffiffiffi 2 3 3 b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi: 2 10 þ 2 5
(8)
(9)
Each triangle contains four quadrature points for the lowest order ICLDFE quadrature sets. The centroid of each sub-triangle is chosen as the initial quadrature point. The ICLDFE quadrature sets increase its discrete directions by refining the triangles (Fig. 3). The N-order ICLDFE quadrature sets on each flat triangle contain 4N
Fig. 3. Projection of points from the icosahedron to the unit sphere and subsequent refinement: (a) first order and (b) second order.
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quadrature points. This refinement process ensures the directions grow exponentially with the order of the quadrature sets making it easy to construct a large number of discrete directions. Moreover, the ICLDFE quadrature sets can be refined locally in each triangle, which means every triangle may have different orders. The icosahedron does not have a 90 rotation symmetry. Nevertheless, the points on the remaining 19 triangles of the icosahedron can be attained by rotation and symmetry. In Fig. 4, the directions on Triangle 1 labeled as P1 (g1,q1) are rotated successively
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ð3þpffiffi5ffiÞ pffiffiffi 52 5 , 5h h y 2 jJj ¼ 8
3
ð ð weight ¼
ð ð bðq; gÞdqdg ¼
Dq Dg
jJjbðx; yÞdxdy;
(13)
Dx Dy
where the Jacobian of the icosahedron is
!
: ! #2 93=2 !2 " = i 3þpffiffi5ffi p ffiffiffi pffiffiffi 2 ð Þ þ 51 yþh 52 5 ~ x þ1 , 2 hy ;
by 72 azimuthally about a five-fold axis to obtain the points on Triangles 2 to 5.
PN ðgN ; qN Þ ¼ P1
2 g1 þ ðN 1Þp; q1 ; N ¼ 2; 3; 4; 5 5
(10)
Triangles 6 and 1 are symmetric about the plane through the common side and the origin. Triangles 19 and 1 are symmetric about the origin of the coordinate system (Fig. 2).
2.2. Quadrature weights of ICLDFE The discontinuous finite-element basis function is defined on each triangle and is a spherical harmonic expansion of zero- and first-order,
! b i ¼ a0;i þ a1;i m þ a2;i h þ a3;i x; i ¼ 1; 2; 3; 4:
(11)
The constants of the basis function are determined by the initial direction cosines, which has a value of one at its corresponding discrete direction and a value of zero in the remaining three directions. Then the weights are obtained by integrating each basis function in its corresponding discrete angular region,
ð weighti ¼
! ! b id U :
(14)
2.3. Optimization of ICLDFE The weights obtained from the initial discrete directions have large differences with the corresponding surface area of the spherical triangle (Table 1). The weights in some triangles may have a negative value as refinement continues. The L-methods [9] use the secant method to optimize the directions and corresponding weights to ensure that the surface area of the center triangle is equal to its weight. To reduce the degrees of freedom in adjusting the directions, each point is constrained to move only on the line connecting the center and the vertex, and does not exceed its located sub-triangle. A ratio L, defined as the value of the length from the center point to the initial point dci divided by the length from the center point to the corner point dcc, that is,
L¼
dci ; dcc
(15)
is introduced (Fig. 5). The secant method is used to approximate the derivative of the relative error between the surface area SAc and the weight weightc of the central triangle to the ratio,
½n vd d½n d½n1 z ½n ; vL L L½n1
(16)
SAc weightc ; SAc
(17)
(12)
Utriangle
Because the boundaries of spherical triangles are curved, it is difficult to integrate numerically. The Jacobian is used to map the integration from the surface of the sphere to the flat triangle,
d¼
d½n1
L½n ¼ L½n1 ½n1 :
(18)
vd vL
By iterating, a reasonable ratio is obtained to guarantee the
Table 1 Initial and optimized weights of ICLDFE-S1 quadrature sets.
Fig. 4. (a) Rotation and (b) mirror symmetry strategy for the ICLDFE quadrature sets.
Initial weights
Surface areas
Optimized weights
1.8096E01 1.8096E01 1.8096E01 8.5449E02
1.4949E01 1.4949E01 1.4949E01 1.7986E01
1.4949E01 1.4949E01 1.4949E01 1.7986E01
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To explore the capability of the ICLDFE quadrature sets to integrate the spherical harmonic functions over a local angular domain, the numerical integrations of polynomials in the direction cosines for each quadrature set in the hemispherical space and onetwentieth spherical surface were calculated using
ð
!
ma hb xc d U z
Ihemisphere ¼ Uhemisphere
ð Ionetwentieth ¼
M=2 X
surface area of the center triangle equals its weight and strictly non-negative. Fig. 6 shows the optimized ICLDFE quadrature sets from the first-order to the fifth-order. The number of directions correlates according to the order by 20 4N. (i.e., in case of ICLDFE eS1, 20 4N ¼ 20 4 ¼ 80).
3. Analysis and results 3.1. Numerical integration of polynomials in the direction cosines The quadrature sets are required to integrate the spherical harmonic function as exactly as possible over the full angular domain to guarantee particle conservation. However, for some realistic shielding problems, the angular flux in the local angular domain usually changes sharply or is even discontinuous.
(19)
m¼1
!
ma hb xc d U z
Uonetewentieth Fig. 5. Optimized Strategy of the L-Methods in one triangle.
um mam hbm xcm ;
M=20 X
um mam hbm xcm ;
(20)
m¼1
where a, b, and c are non-negative integers, and u denotes a weight of the quadrature set. The relative errors in the hemispherical space between the quadrature sets used in the calculations and the results obtained numerically are listed in Table 2. Table 3 lists the relative errors in the one-twentieth spherical space between the results using quadrature sets and the calculations using GausseLegendre integral formula. The LS-SN sets are the N-order LS quadrature sets, PNTN-SN are the Norder Legendre-Chebyshev quadrature sets, and ICLDFE-SN are the N-order LDFE quadrature sets based on the icosahedron. In the hemispherical domain, the relative errors of the LS and the PNTN quadrature sets are more than those of the ICLDFE quadrature sets for the same discrete direction cosines (Table 2). Note that the relative errors of (2,0,0) for listed quadrature sets are all equal 0.00%. The maximum relative errors of the ICLDFE-S4 and ICLDFE-S5 quadrature sets are only 0.05% and 0.01%, respectively. Table 3 shows that the relative errors of (0,0,0) and (1,0,0) are all equal 0.00%; the ICLDFE-S4 and ICLDFE-S5 quadrature sets both have higher integration accuracy for the polynomials up to fifth-
Fig. 6. Distributions of different-order ICLDFE quadrature sets: (a) S1, (b) S2, (c) S3, (d) S4, (e) S5.
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Table 2 Relative errors for the polynomials of direction cosines integration over the hemisphere for the ICLDFE quadrature sets. Quadrature
ICLDFE-S1 LS-S8 PNTN-S16 ICLDFE-S2 PNTN eS32 ICLDFE-S3 PNTN eS70 ICLDFE-S4 ICLDFE-S5
Directions
Relative Errors/%
80 80 288 320 1088 1280 5040 5120 20480
(1,0,0)
(2,0,0)
(3,0,0)
(1,2,0)
(5,0,0)
(3,2,0)
(1,2,2)
1.29 1.70 0.47 0.25 0.13 0.07 0.03 0.02 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.21 0.14 0.01 0.00 0.00 0.00 0.00 0.00 0.00
2.68 3.54 0.89 0.62 0.24 0.17 0.05 0.04 0.01
0.44 0.03 0.00 0.02 0.00 0.00 0.00 0.00 0.00
0.16 0.47 0.03 0.01 0.00 0.00 0.00 0.00 0.00
2.26 5.61 1.64 1.20 0.43 0.19 0.09 0.05 0.01
Table 3 Relative errors for the spherical harmonic function integration in one-twentieth of the spherical surface for the ICLDFE quadrature sets. ICLDFE
Directions
Relative Errors/% (0,0,0)
(1,0,0)
(2,0,0)
(3,0,0)
(1,2,0)
(5,0,0)
(3,2,0)
(1,2,2)
S1 S2 S3 S4 S5
80 320 1280 5120 20480
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.07 0.00 0.00 0.00 0.00
4.82 0.06 0.00 0.00 0.00
4.24 0.19 0.01 0.00 0.00
14.41 0.90 0.02 0.00 0.00
5.95 0.64 0.01 0.00 0.00
2.21 0.70 0.01 0.00 0.00
order in the one-twentieth spherical surface. The numerical results show that compared with the traditional LS quadrature sets and the PNTN quadrature sets, the accuracy of the numerical integration in the hemispherical surface of the ICLDFE quadrature sets has obviously improved, and can exactly integrate the zero-order and firstorder spherical harmonic functions in the one-twentieth spherical surface. As the discrete directions of the ICLDFE quadrature sets increase, the accuracy of the numerical integration gradually improves. However, because PNTN quadrature sets was developed based on Legendre and Chebyshev polynomial, it can exactly integrates any polynomial of a given order of the directions cosines globally on the sphere, that is, PNTN quadrature sets perform better than ICLDFE quadrature sets in a full spherical surface.
3.2. Kobayashi Benchmark problems 3.2.1. Problem 1 The Kobayashi Benchmark [14] was proposed by the OECD Nuclear Energy Agency to test the reliability of neutron transport programs. The one-fourth geometry for problem 1 of the Kobayashi
benchmark (Fig. 7) models a large void region that may cause strong ray effects when discrete directions are not enough. Because the ICLDFE quadrature sets are difficult to treat on the reflecting boundaries, all the vacuum boundaries used for the full geometry model are configured in the calculation. The geometry mesh size is divided into 2 cm 2 cm 2 cm. The spatial discrete method chooses the diamond differencing schemes (DZ) [15], and the iterative convergence criterion is 104. Table 4 lists the details about source strengths and cross sections in each region. The PNTN and ICLDFE quadrature sets are used to calculate the neutron fluxes using the ARES program [16] at key points along the X-axis, Y-axis, and diagonal. Fig. 8 plots the root mean square (RMS) of the relative errors between calculations and reference values versus the number of directions in the full angular domain. Fig. 9 shows the neutron flux distribution of the ICLDFE-S4 quadrature sets. The RMS of relative errors of the PNTN-S8 and the ICLDFE-S1 quadrature sets with 80 directions are 1.20 100 and 5.05 101 (Fig. 8). The maximum in the relative error of PNTN-S8 is up to 354.84% resulting in a larger discrete error and serious ray effects. The ICLDFE quadrature sets except for the second-order all have relative errors less than those of the PNTN quadrature sets with a similar number of directions.
3.2.2. Problem 2 Problem 2 of the Kobayashi Benchmark involves a typical deep penetration with a straight duct and poses a challenge to traditional transport calculations. The one-fourth geometry of this problem is shown in Fig. 10. Similarly, all the vacuum boundaries of the full
Table 4 Cross sections and sources associated with the Kobayashi Benchmark.
Fig. 7. One-fourth geometry for Problem 1 of the Kobayashi Benchmark.
Regions
Source /n$cm3 s1
Total Cross Sections/cm1
Scattering Cross Sections/cm1
1 2 3
1 0 0
0.1 10e4 0.1
0.05 0.5 104 0.05
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Fig. 10. One-fourth geometry of problem 2 of the Kobayashi benchmark. Fig. 8. RMS of the relative errors at key points for Problem 1 of the Kobayashi Benchmark for the quadrature sets.
Fig. 11. RMS of the relative errors at key points for Problem 2 of the Kobayashi Benchmark for quadrature sets.
Fig. 9. Neutron Flux (n$cm2 s1) distribution of the ICLDFE-S4 quadrature sets at X ¼ 100 cm.
geometry model are used and configured for the calculation using the ARES program. Other information such as source strength and cross section are the same as in problem 1. The PNTN and the ICLDFE quadrature sets are again used to calculate the neutron fluxes at key points along the duct. Fig. 11 shows the RMS of the relative errors between the calculations and reference values versus the number of directions in the full angular domain. The RMS of the relative errors for the PNTN-S20 quadrature sets with 440 directions is 1.95 101, and that for ICLDFE-S2 with only 320 directions is 1.11 101 (see Fig. 11). Similarly, PNTN-S32 and ICLDFE-S3 quadrature sets have approximately 1000 directions yielding a RMS for the relative errors of 6.78 102 and 1.65 102, respectively. All the RMS values of the ICLDFE quadrature sets are less than those of PNTN, indicating that the
computational accuracy of the former is higher than that of the latter when the discrete directions are the same or similar. Moreover, the RMS for the relative errors gradually diminish as the numbers of directions of the quadrature sets increase. It is noticed that the duct has oriented along Z axis..Actually, the performance of quadrature sets for this problem can depend on the orientation of duct, and that they have found different errors for different orientation.We have tested and analyzed different orientation cases and ICLDFE quadrature sets have shown certain advantages compared to PNTN quadrature sets. We performed an analysis to assess the capability of the local refinement for the ICLDFE quadrature sets. The local refined ICLDFE quadrature sets with a total 3584 directions for problem 2 were developed (Fig. 12). With a focus on the duct, the triangles along the axis where the duct is located received the highest order ICLDFE-S5 quadrature sets, whereas other triangles were ascribed lower-order quadrature sets. The neutron flux and relative errors of the local refined quadrature sets are listed in Table 5. The relative errors of
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Fig. 13. One-fourth geometry for Problem 3 of the Kobayashi Benchmark.
Fig. 12. Distribution of the ICLDFE-LocalRefine quadrature sets.
the ICLDFE-S5 quadrature sets are within 0.55%. The local refined quadrature sets with 3584 directions have similar relative errors to those for the ICLDFE-S5 quadrature sets, which have 20480 directions. 3.2.3. Problem 3 The Kobayashi Benchmark Problem 3 (Fig. 13) has a bend in the ducts to provide a more anisotropic angular flux than problem 2. Other information such as the source strength and cross section are the same as for problem 1. The geometry mesh size is divided into 2 cm 2 cm 2 cm and 1 cm 1 cm 1 cm cells. The spatial discrete method with the diamond differencing schemes (DZ) were adopted. The PNTN quadrature sets and the ICLDFE quadrature sets are used to calculate the neutron fluxes along the duct using the ARES program. The RMS of the relative errors versus the number of directions were plotted (Fig. 14) and showed the same trends as for problem 2. All the RMS values of the ICLDFE quadrature sets are less than the PNTN quadrature sets. For example, the RMSs of the relative errors for the PNTN-S20 quadrature sets with 440 directions and the ICLDFE-S2 quadrature sets with 320 directions are 1.13 101 and 6.15 102, respectively, when the spatial mesh size is 2 cm. Note
Table 5 The relative errors at key points for Problem 2 of Kobayashi Benchmark using the ICLDFE-S5 and local refined quadrature sets. Key Points
References
ICLDFE-S5 (20480)
ICLDFE-LocalRefine (3584)
Calculations
Relative Errors
Calculations
Relative Errors
/cm
/n$cm2 s1
/n$cm2 s1
/%
/n$cm2 s1
/%
(5,5,5) (5,15,5) (5,25,5) (5,35,5) (5,45,5) (5,55,5) (5,65,5) (5,75,5) (5,85,5) (5,95,5)
8.61696Eþ00 2.16123Eþ00 8.93437E01 4.77452E01 2.88719E01 1.88959E01 1.31026E01 9.49890E02 7.12403E02 5.44807E02
8.59055Eþ00 2.15303Eþ00 8.89937E01 4.75680E01 2.87127E01 1.88018E01 1.30815E01 9.51621E02 7.13797E02 5.45151E02
0.31 0.38 0.39 0.37 0.55 0.50 0.16 0.18 0.20 0.06
8.58523Eþ00 2.14985Eþ00 8.88368E01 4.75656E01 2.87184E01 1.87802E01 1.30808E01 9.51288E02 7.13555E02 5.45086E02
0.37 0.53 0.57 0.38 0.53 0.61 0.17 0.15 0.16 0.05
Fig. 14. RMS of the relative errors at key points for Problem 3 of the Kobayashi Benchmark for quadrature sets.
that the relative errors do not gradually diminish as the spatial mesh size is refined, indicating the mesh size has little impact on the discrete error. Further, the discrete error consists of spatial discrete errors and angular discrete errors that may mutually compensate to reduce the error when the spatial mesh size is larger. 4. Conclusions The ICLDFE quadrature sets with icosahedral symmetry can generate a large number of discrete directions, be easier to refine locally and have strictly positive weights. The numerical results indicate that compared with the LS quadrature sets and the PNTN quadrature sets, the numerical integration accuracy in the local angular domain of the ICLDFE quadrature sets are obviously improved and can exactly integrate the zero-order and first-order spherical harmonic functions over the one-twentieth spherical surface and not only in the full angular region. For the Kobayashi benchmark problems, the ICLDFE quadrature sets outperform other quadrature sets in general. The local refined quadrature sets may
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have better computational accuracy and be more efficient. However, because of the geometric properties of the icosahedron, only vacuum boundaries can be used in the ICLDFE quadrature sets. The adaptability of the ICLDFE quadrature sets for much more realistic problems needs further research. Exploring the quadratic and cubic discontinuous finite-element quadrature sets based on the icosahedron remains an open problem as does the p-adaptive method in the angular domain.
financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Declaration of competing interests
Appendix
Acknowledgements This work was supported by the National Natural Science Foundation of China (11975097) and the Fundamental Research Funds for the Central Universities (2019MS038).
The authors declare that they have no known competing
Table 6 ICLDFE-S1 quadrature sets data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
m
h
x
weight
1.456751E-01 6.938161E-01 1.456751E-01 3.568221E-01 1.456751E-01 1.456752E-01 6.938161E-01 3.568221E-01 2.357073E-01 7.838483E-01 5.744771E-01 5.773502E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 2.357073E-01 5.744771E-01 7.838483E-01 5.773502E-01 5.744771E-01 7.838483E-01 2.357073E-01 5.773503E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 5.744771E-01 2.357074E-01 7.838483E-01 5.773503E-01 9.295234E-01 7.201522E-01 9.295234E-01 9.341724E-01 9.295234E-01 9.295234E-01 7.201522E-01 9.341724E-01
2.005046E-01 3.786064E-01 7.768541E-01 4.911234E-01 2.005046E-01 7.768541E-01 3.786064E-01 4.911235E-01 7.658595E-02 1.015158E-01 5.428625E-01 1.875925E-01 2.478373E-01 7.141139E-01 7.141139E-01 6.070620E-01 7.658595E-02 5.428625E-01 1.015158E-01 1.875925E-01 7.906999E-01 5.025251E-01 9.007728E-01 7.946545E-01 9.773588E-01 8.672858E-01 8.672858E-01 9.822470E-01 7.906999E-01 9.007728E-01 5.025251E-01 7.946545E-01 3.020205E-01 5.901952E-01 5.418310E-02 3.035310E-01 3.020205E-01 5.418309E-02 5.901952E-01 3.035310E-01
9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01
1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
m
h
x
weight
5.744771E-01 7.838483E-01 2.357073E-01 5.773503E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 5.744771E-01 2.357074E-01 7.838483E-01 5.773503E-01 9.295234E-01 7.201522E-01 9.295234E-01 9.341724E-01 9.295234E-01 9.295234E-01 7.201522E-01 9.341724E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 2.357073E-01 5.744771E-01 7.838483E-01 5.773502E-01 1.456751E-01 6.938161E-01 1.456751E-01 3.568221E-01 1.456751E-01 1.456752E-01 6.938161E-01 3.568221E-01 2.357073E-01 7.838483E-01 5.744771E-01 5.773502E-01
7.906999E-01 5.025251E-01 9.007728E-01 7.946545E-01 9.773588E-01 8.672858E-01 8.672858E-01 9.822470E-01 7.906999E-01 9.007728E-01 5.025251E-01 7.946545E-01 3.020205E-01 5.901952E-01 5.418310E-02 3.035310E-01 3.020205E-01 5.418309E-02 5.901952E-01 3.035310E-01 2.478373E-01 7.141139E-01 7.141139E-01 6.070620E-01 7.658595E-02 5.428625E-01 1.015158E-01 1.875925E-01 2.005046E-01 3.786064E-01 7.768541E-01 4.911234E-01 2.005046E-01 7.768541E-01 3.786064E-01 4.911235E-01 7.658595E-02 1.015158E-01 5.428625E-01 1.875925E-01
2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01
1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02
Table 7 ICLDFE-S2 quadrature sets data
1 2 3 4 5 6 7 8 9 10 11
m
h
x
weight
6.549282E-02 3.629794E-01 7.250052E-02 1.666666E-01 5.804812E-01 7.867750E-01 5.804812E-01 6.666667E-01 7.250062E-02 3.629792E-01 6.549292E-02
9.014313E-02 1.941707E-01 4.052160E-01 2.293970E-01 2.648416E-01 3.245022E-01 5.702684E-01 3.918568E-01 6.339108E-01 7.282930E-01 8.485443E-01
9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812360E-01 5.250554E-01
2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03
161 162 163 164 165 166 167 168 169 170 171
m
h
x
weight
5.517467E-01 7.217364E-01 4.312575E-01 5.786893E-01 8.048146E-01 8.272520E-01 6.252892E-01 7.696723E-01 2.968341E-01 4.077871E-01 1.059698E-01
7.594142E-01 6.879574E-01 8.990026E-01 7.964975E-01 5.736100E-01 3.802139E-01 6.319413E-01 5.336320E-01 9.426794E-01 7.899657E-01 9.042559E-01
3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01
2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
N. Dai et al. / Nuclear Engineering and Technology xxx (xxxx) xxx
9
Table 7 (continued )
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85
m
h
x
weight
1.666667E-01 4.480860E-01 1.497516E-01 4.480860E-01 3.568221E-01 6.549282E-02 7.250053E-02 3.629794E-01 1.666666E-01 7.250078E-02 6.549308E-02 3.629790E-01 1.666667E-01 5.804811E-01 5.804811E-01 7.867752E-01 6.666666E-01 4.480860E-01 4.480860E-01 1.497516E-01 3.568221E-01 1.059696E-01 4.077872E-01 2.968341E-01 2.696723E-01 6.252892E-01 8.272520E-01 8.048146E-01 7.696723E-01 4.312577E-01 7.217363E-01 5.517467E-01 5.786893E-01 7.250184E-01 4.266840E-01 5.406376E-01 5.773502E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01 4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 1.059696E-01 2.968341E-01 4.077872E-01 2.696723E-01 4.312579E-01 5.517468E-01 7.217361E-01 5.786893E-01 6.252889E-01 8.048146E-01 8.272520E-01 7.696723E-01 7.250184E-01 5.406376E-01 4.266840E-01 5.773502E-01 5.517467E-01 7.217364E-01 4.312575E-01 5.786893E-01 8.048146E-01
7.551281E-01 6.167375E-01 5.198028E-01 3.030501E-01 4.911234E-01 9.014313E-02 4.052160E-01 1.941707E-01 2.293970E-01 6.339110E-01 8.485442E-01 7.282931E-01 7.551281E-01 2.648414E-01 5.702687E-01 3.245021E-01 3.918568E-01 6.167375E-01 3.030501E-01 5.198028E-01 4.911235E-01 3.443161E-02 5.626652E-02 2.852119E-01 8.762184E-02 1.269369E-01 1.999270E-01 1.201586E-01 7.483798E-02 4.702298E-01 3.758476E-01 6.479910E-01 5.129472E-01 2.355728E-01 3.325075E-01 1.820568E-02 1.875925E-01 1.114230E-01 3.704414E-01 3.704413E-01 2.835502E-01 5.554597E-01 7.249826E-01 8.025552E-01 7.088757E-01 5.554594E-01 8.025553E-01 7.249825E-01 7.088757E-01 7.623295E-01 5.085510E-01 5.085510E-01 6.070620E-01 3.443161E-02 2.852119E-01 5.626653E-02 8.762184E-02 4.702299E-01 6.479908E-01 3.758478E-01 5.129472E-01 1.269370E-01 1.201588E-01 1.999271E-01 7.483799E-02 2.355728E-01 1.820568E-02 3.325075E-01 1.875925E-01 7.594142E-01 6.879574E-01 8.990026E-01 7.964975E-01 5.736100E-01
6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01
3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03
172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245
m
h
x
weight
2.696723E-01 5.406376E-01 4.266840E-01 7.250184E-01 5.773503E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01 4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 5.517467E-01 4.312575E-01 7.217364E-01 5.786893E-01 2.968341E-01 1.059700E-01 4.077869E-01 2.696724E-01 8.048146E-01 6.252889E-01 8.272521E-01 7.696723E-01 5.406376E-01 7.250184E-01 4.266840E-01 5.773503E-01 8.927449E-01 8.773151E-01 9.882682E-01 9.363390E-01 7.942369E-01 6.172399E-01 7.942369E-01 7.453560E-01 9.882682E-01 8.773152E-01 8.927450E-01 9.363390E-01 8.747700E-01 9.887237E-01 8.747700E-01 9.341724E-01 8.927449E-01 9.882682E-01 8.773151E-01 9.363390E-01 9.882681E-01 8.927450E-01 8.773153E-01 9.363390E-01 7.942369E-01 7.942369E-01 6.172397E-01 7.453560E-01 8.747700E-01 8.747700E-01 9.887237E-01 9.341724E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01
8.969032E-01 7.441238E-01 9.009675E-01 6.842149E-01 7.946545E-01 9.386875E-01 9.807952E-01 9.807952E-01 9.845251E-01 9.371183E-01 7.938460E-01 8.787870E-01 8.841194E-01 9.371184E-01 8.787870E-01 7.938459E-01 8.841194E-01 9.197876E-01 9.796966E-01 9.796966E-01 9.822470E-01 7.594142E-01 8.990026E-01 6.879574E-01 7.964975E-01 9.426793E-01 9.042559E-01 7.899659E-01 8.969032E-01 5.736100E-01 6.319414E-01 3.802137E-01 5.336320E-01 7.441238E-01 6.842149E-01 9.009675E-01 7.946545E-01 2.900704E-01 4.738216E-01 1.323432E-01 3.042350E-01 5.881689E-01 6.692708E-01 3.994047E-01 5.671005E-01 8.997907E-03 1.437157E-01 1.786472E-01 2.068470E-02 2.842300E-01 1.273863E-01 4.780995E-01 3.035310E-01 2.900704E-01 1.323432E-01 4.738216E-01 3.042350E-01 8.997919E-03 1.786470E-01 1.437155E-01 2.068469E-02 5.881689E-01 3.994045E-01 6.692710E-01 5.671005E-01 2.842300E-01 4.780995E-01 1.273863E-01 3.035310E-01 1.114230E-01 3.704414E-01 3.704413E-01 2.835502E-01 5.554597E-01
3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01
3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03
(continued on next page)
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
10
N. Dai et al. / Nuclear Engineering and Technology xxx (xxxx) xxx
Table 7 (continued )
86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
m
h
x
weight
8.272520E-01 6.252892E-01 7.696723E-01 2.968341E-01 4.077871E-01 1.059698E-01 2.696723E-01 5.406376E-01 4.266840E-01 7.250184E-01 5.773503E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01 4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 5.517467E-01 4.312575E-01 7.217364E-01 5.786893E-01 2.968341E-01 1.059700E-01 4.077869E-01 2.696724E-01 8.048146E-01 6.252889E-01 8.272521E-01 7.696723E-01 5.406376E-01 7.250184E-01 4.266840E-01 5.773503E-01 8.927449E-01 8.773151E-01 9.882682E-01 9.363390E-01 7.942369E-01 6.172399E-01 7.942369E-01 7.453560E-01 9.882682E-01 8.773152E-01 8.927450E-01 9.363390E-01 8.747700E-01 9.887237E-01 8.747700E-01 9.341724E-01 8.927449E-01 9.882682E-01 8.773151E-01 9.363390E-01 9.882681E-01 8.927450E-01 8.773153E-01 9.363390E-01 7.942369E-01 7.942369E-01 6.172397E-01 7.453560E-01 8.747700E-01 8.747700E-01 9.887237E-01 9.341724E-01
3.802139E-01 6.319413E-01 5.336320E-01 9.426794E-01 7.899657E-01 9.042559E-01 8.969032E-01 7.441238E-01 9.009675E-01 6.842149E-01 7.946545E-01 9.386875E-01 9.807952E-01 9.807952E-01 9.845251E-01 9.371183E-01 7.938460E-01 8.787870E-01 8.841194E-01 9.371184E-01 8.787870E-01 7.938459E-01 8.841194E-01 9.197876E-01 9.796966E-01 9.796966E-01 9.822470E-01 7.594142E-01 8.990026E-01 6.879574E-01 7.964975E-01 9.426793E-01 9.042559E-01 7.899659E-01 8.969032E-01 5.736100E-01 6.319414E-01 3.802137E-01 5.336320E-01 7.441238E-01 6.842149E-01 9.009675E-01 7.946545E-01 2.900704E-01 4.738216E-01 1.323432E-01 3.042350E-01 5.881689E-01 6.692708E-01 3.994047E-01 5.671005E-01 8.997907E-03 1.437157E-01 1.786472E-01 2.068470E-02 2.842300E-01 1.273863E-01 4.780995E-01 3.035310E-01 2.900704E-01 1.323432E-01 4.738216E-01 3.042350E-01 8.997919E-03 1.786470E-01 1.437155E-01 2.068469E-02 5.881689E-01 3.994045E-01 6.692710E-01 5.671005E-01 2.842300E-01 4.780995E-01 1.273863E-01 3.035310E-01
4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01
2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03
246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
m
h
x
weight
4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 1.059696E-01 2.968341E-01 4.077872E-01 2.696723E-01 4.312579E-01 5.517468E-01 7.217361E-01 5.786893E-01 6.252889E-01 8.048146E-01 8.272520E-01 7.696723E-01 7.250184E-01 5.406376E-01 4.266840E-01 5.773502E-01 6.549282E-02 3.629794E-01 7.250052E-02 1.666666E-01 5.804812E-01 7.867750E-01 5.804812E-01 6.666667E-01 7.250062E-02 3.629792E-01 6.549292E-02 1.666667E-01 4.480860E-01 1.497516E-01 4.480860E-01 3.568221E-01 6.549282E-02 7.250053E-02 3.629794E-01 1.666666E-01 7.250078E-02 6.549308E-02 3.629790E-01 1.666667E-01 5.804811E-01 5.804811E-01 7.867752E-01 6.666666E-01 4.480860E-01 4.480860E-01 1.497516E-01 3.568221E-01 1.059696E-01 4.077872E-01 2.968341E-01 2.696723E-01 6.252892E-01 8.272520E-01 8.048146E-01 7.696723E-01 4.312577E-01 7.217363E-01 5.517467E-01 5.786893E-01 7.250184E-01 4.266840E-01 5.406376E-01 5.773502E-01
7.249826E-01 8.025552E-01 7.088757E-01 5.554594E-01 8.025553E-01 7.249825E-01 7.088757E-01 7.623295E-01 5.085510E-01 5.085510E-01 6.070620E-01 3.443161E-02 2.852119E-01 5.626653E-02 8.762184E-02 4.702299E-01 6.479908E-01 3.758478E-01 5.129472E-01 1.269370E-01 1.201588E-01 1.999271E-01 7.483799E-02 2.355728E-01 1.820568E-02 3.325075E-01 1.875925E-01 9.014313E-02 1.941707E-01 4.052160E-01 2.293970E-01 2.648416E-01 3.245022E-01 5.702684E-01 3.918568E-01 6.339108E-01 7.282930E-01 8.485443E-01 7.551281E-01 6.167375E-01 5.198028E-01 3.030501E-01 4.911234E-01 9.014313E-02 4.052160E-01 1.941707E-01 2.293970E-01 6.339110E-01 8.485442E-01 7.282931E-01 7.551281E-01 2.648414E-01 5.702687E-01 3.245021E-01 3.918568E-01 6.167375E-01 3.030501E-01 5.198028E-01 4.911235E-01 3.443161E-02 5.626652E-02 2.852119E-01 8.762184E-02 1.269369E-01 1.999270E-01 1.201586E-01 7.483798E-02 4.702298E-01 3.758476E-01 6.479910E-01 5.129472E-01 2.355728E-01 3.325075E-01 1.820568E-02 1.875925E-01
5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812360E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01
2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
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Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, 2011. [10] C.D. Ahrens, Highly efficient, exact quadratures for three-dimensional discrete ordinates transport calculations, Nucl. Sci. Eng. 170 (1) (2012) 98e101. [11] K.E. Atkinson, W. Han, I. Ebrary, Spherical Harmonics and Approximations on the Unit Sphere : an Introduction, Springer Verlag, 2012. [12] C.Y. Lau, M.L. Adams, Discrete Ordinates Quadratures Based on Linear and Quadratic Discontinuous Finite Elements over Spherical Quadrilaterals, LaGrange Park: 2015 ANS Joint International Conference on Mathematics and Computation, M&C 2015), 2015. [13] J. Jarrell, An Adaptive Angular Discretization Method for Neutral-Particle Transport in Three-Dimensional Geometries, Dissertations & Theses-Gradworks, 2012. [14] K. Kobayashi, N. Sugimura, Y. Nagaya, 3D radiation transport benchmark problems and results for simple geometries with void region, Prog. Nucl. Energy 39 (2) (2001) 119e144. [15] E.W. Larsen, Spatial convergence properties of the diamond difference method in x, y geometry, Nucl. Sci. Eng. 80 (4) (1982) 710e713. [16] Y.X. Chen, B. Zhang, L. Zhang, et al., ARES: a parallel discrete ordinates transport code for radiation shielding applications and reactor physics analysis, Sci. Technol.Nucl. Install. 2017 (2017) 1e11.
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Original Article
Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method Ni Dai, Bin Zhang*, Yixue Chen School of Nuclear Science and Engineering, North China Electric Power University, Beijing, 102206, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 August 2019 Received in revised form 21 October 2019 Accepted 20 November 2019 Available online xxx
The discrete ordinates method (SN) is one of the major shielding calculation method, which is suitable for solving deep-penetration transport problems. Our objective is to explore the available quadrature sets and to improve the accuracy in shielding problems involving strong anisotropy. The linear discontinuous finite-element (LDFE) quadrature sets based on the icosahedron (in short, ICLDFE quadrature sets) are developed by defining projected points on the surfaces of the icosahedron. Weights are then introduced in the integration of the discontinuous finite-element basis functions in the relevant angular regions. The multivariate secant method is used to optimize the discrete directions and their corresponding weights. The numerical integration of polynomials in the direction cosines and the Kobayashi benchmark are used to analyze and verify the properties of these new quadrature sets. Results show that the ICLDFE quadrature sets can exactly integrate the zero-order and first-order of the spherical harmonic functions over one-twentieth of the spherical surface. As for the Kobayashi benchmark problem, the maximum relative error between the fifth-order ICLDFE quadrature sets and references is only 0.55%. The ICLDFE quadrature sets provide better integration precision of the spherical harmonic functions in local discrete angle domains and higher accuracy for simple shielding problems. © 2019 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Shielding calculation Discrete ordinates method Quadrature sets Discontinuous finite elements Icosahedron
1. Introduction Particle transport calculations play a vital role in many fields, such as shielding design for nuclear devices [1], radiation therapy [2], astrophysics [3] and so on. The key issue of the shielding calculation is the steady particle transport equation, which is a linear differential-integral equation containing six independent variables: energy, space, and angle. This equation is usually discretized in each variable for convenience when solving. The discrete ordinates (SN) method directly discretizes the direction variables into a finite number of discrete directions and employs quadrature sets to approximate integrals over the angular variables [4]. However, the angular distribution of flux is not smooth and may even be discontinuous for shielding problems with strong anisotropy. Insufficient integration accuracy may result in larger discrete errors, reducing the accuracy and reliability of shielding calculations. Over the last decades, many quadrature sets have been proposed and shown to be useful in particle transport calculations. Carlson
* Corresponding author. E-mail address: [email protected] (B. Zhang).
and Lee proposed level symmetric (LS) quadrature sets in 1961 [5]. The LS quadrature sets are used widely, but some of their weights may become negative when the quadrature order is larger than 20. In 1987, Walters proposed the Legendre-Chebyshev quadrature sets (PNTN), which have the advantage of high integral accuracy and numerous discrete directions [6]. In 1995, Thurgood developed the symmetric quadrature sets, denoted TN, which are more accurate than the LS quadrature sets and always have positive weights [7]. Longoni studied regional angular refinement (RAR) based on the PNTN quadrature sets in 2002 [8], and in 2010 Jarrell developed the linear discontinuous finite-element (LDFE) quadrature sets based on the octahedron, which can exactly integrate the zero-order and first-order spherical harmonic functions on one-eighth of the spherical surface [9]. It also can generate a large number of discrete directions and is easier to refine locally. Later, Ahrens proposed a rotationally invariant quadrature sets based on the icosahedron [10]. These quadrature sets are highly efficient in the integrations of spherical harmonics but their reflecting boundaries are difficult to treat. In 2012, K. Atkison and W. Han discussed quadrature formula over spherical triangles including the 4-sided tetrahedron, the 8sided octahedron and the 20-sided icosahedron. They have chosen the centroid of sub-triangles as quadrature points and their
https://doi.org/10.1016/j.net.2019.11.025 1738-5733/© 2019 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
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N. Dai et al. / Nuclear Engineering and Technology xxx (xxxx) xxx
surfaces areas as corresponding weights. The icosahedral triangulation has much smaller integration error than other triangulations with a similar directions [11]. In 2015, Lau and Adams studied linear and quadratic discontinuous finite-element methods and their local refinement quadrature sets based on spherical quadrilaterals [12]. The linear discontinuous finite element quadrature sets based on the icosahedron (termed the ICLDFE quadrature sets) can produce a more uniform distribution and strictly positive weights because the icosahedron is the closest polyhedron of the Platonic solids to the sphere. In this paper, we focus on the construction of these quadrature sets and analyze their properties. The rest of the paper is organized as follows: Section 2 introduces the construction of ICLDFE quadrature sets including those in discrete directions and the corresponding weights. In Section 3 the results of spherical harmonic integration and benchmarks are presented. Section 4 states our conclusions.
Fig. 2. Coordinate Systems of the ICLDFE quadrature sets.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi xÞðh yÞ 10 þ 2 5ð1 ~
m ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 3þpffiffi5ffi hpffiffiffi i2ffi; pffiffiffi ð Þ
2. Construction of ICLDFE quadrature sets
4
2.1. Quadrature points of ICLDFE
52 5 ~ x2 þ 1 ,
2
ðh yÞ2 þ
51 yþh
(5) A quadrature set is a set of weights and directions defined with the ordinates using the direction cosines (m, h, x) of the polar angle q and the azimuth g,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ux ¼ m ¼ cos g sin q ¼ cos g 1 x2 ;
(1)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uy ¼ h ¼ sin g sin q ¼ sin g 1 x2 ;
(2)
Uz ¼ x ¼ cos q:
(3)
(6)
The construction of the ICLDFE quadrature sets begins with the projection of the regular icosahedron, which has 20 faces of regular triangles, onto the unit sphere (Fig. 1). Fig. 2 illustrates the coordinate system of the direction cosines, where the h axis is defined as one side of a triangle of the icosahedron and the x axis lies on the line from the center point of the icosahedron to one of its vertices. The points are defined on a flat triangle and projected onto the surface of the unit sphere to obtain the corresponding unique quadrature points. The coordinates (x, y) of the flat regular triangle are established through the geometric relationship between the spherical and flat triangles (Fig. 2), and more detailed deduction is similar with the octahedron [13].
~ x¼
x 2hx ; ¼ xmax bðh yÞ
Fig. 1. Projection of regular icosahedron onto the sphere.
i pffiffiffi pffiffiffih 52 5 ~ x þ 1 ðh yÞ 3þ 5 h ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 3þpffiffi5ffi hpffiffiffi i2ffi; pffiffiffi 2 ð Þ 52 5 ~ x þ 1 , 2 ðh yÞ2 þ 4 51 yþh
(4)
pffiffiffi 51 yþh x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 3þpffiffi5ffi hpffiffiffi h i2ffi; pffiffiffi 2 ð Þ 52 5 ~ x þ 1 , 2 ðh yÞ2 þ 51 yþh (7) where b and h are the base length and height of the flat triangle determined by the geometric properties of the icosahedron,
4 b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi; 10 þ 2 5 h¼
pffiffiffi pffiffiffi 2 3 3 b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi: 2 10 þ 2 5
(8)
(9)
Each triangle contains four quadrature points for the lowest order ICLDFE quadrature sets. The centroid of each sub-triangle is chosen as the initial quadrature point. The ICLDFE quadrature sets increase its discrete directions by refining the triangles (Fig. 3). The N-order ICLDFE quadrature sets on each flat triangle contain 4N
Fig. 3. Projection of points from the icosahedron to the unit sphere and subsequent refinement: (a) first order and (b) second order.
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quadrature points. This refinement process ensures the directions grow exponentially with the order of the quadrature sets making it easy to construct a large number of discrete directions. Moreover, the ICLDFE quadrature sets can be refined locally in each triangle, which means every triangle may have different orders. The icosahedron does not have a 90 rotation symmetry. Nevertheless, the points on the remaining 19 triangles of the icosahedron can be attained by rotation and symmetry. In Fig. 4, the directions on Triangle 1 labeled as P1 (g1,q1) are rotated successively
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ð3þpffiffi5ffiÞ pffiffiffi 52 5 , 5h h y 2 jJj ¼ 8
3
ð ð weight ¼
ð ð bðq; gÞdqdg ¼
Dq Dg
jJjbðx; yÞdxdy;
(13)
Dx Dy
where the Jacobian of the icosahedron is
!
: ! #2 93=2 !2 " = i 3þpffiffi5ffi p ffiffiffi pffiffiffi 2 ð Þ þ 51 yþh 52 5 ~ x þ1 , 2 hy ;
by 72 azimuthally about a five-fold axis to obtain the points on Triangles 2 to 5.
PN ðgN ; qN Þ ¼ P1
2 g1 þ ðN 1Þp; q1 ; N ¼ 2; 3; 4; 5 5
(10)
Triangles 6 and 1 are symmetric about the plane through the common side and the origin. Triangles 19 and 1 are symmetric about the origin of the coordinate system (Fig. 2).
2.2. Quadrature weights of ICLDFE The discontinuous finite-element basis function is defined on each triangle and is a spherical harmonic expansion of zero- and first-order,
! b i ¼ a0;i þ a1;i m þ a2;i h þ a3;i x; i ¼ 1; 2; 3; 4:
(11)
The constants of the basis function are determined by the initial direction cosines, which has a value of one at its corresponding discrete direction and a value of zero in the remaining three directions. Then the weights are obtained by integrating each basis function in its corresponding discrete angular region,
ð weighti ¼
! ! b id U :
(14)
2.3. Optimization of ICLDFE The weights obtained from the initial discrete directions have large differences with the corresponding surface area of the spherical triangle (Table 1). The weights in some triangles may have a negative value as refinement continues. The L-methods [9] use the secant method to optimize the directions and corresponding weights to ensure that the surface area of the center triangle is equal to its weight. To reduce the degrees of freedom in adjusting the directions, each point is constrained to move only on the line connecting the center and the vertex, and does not exceed its located sub-triangle. A ratio L, defined as the value of the length from the center point to the initial point dci divided by the length from the center point to the corner point dcc, that is,
L¼
dci ; dcc
(15)
is introduced (Fig. 5). The secant method is used to approximate the derivative of the relative error between the surface area SAc and the weight weightc of the central triangle to the ratio,
½n vd d½n d½n1 z ½n ; vL L L½n1
(16)
SAc weightc ; SAc
(17)
(12)
Utriangle
Because the boundaries of spherical triangles are curved, it is difficult to integrate numerically. The Jacobian is used to map the integration from the surface of the sphere to the flat triangle,
d¼
d½n1
L½n ¼ L½n1 ½n1 :
(18)
vd vL
By iterating, a reasonable ratio is obtained to guarantee the
Table 1 Initial and optimized weights of ICLDFE-S1 quadrature sets.
Fig. 4. (a) Rotation and (b) mirror symmetry strategy for the ICLDFE quadrature sets.
Initial weights
Surface areas
Optimized weights
1.8096E01 1.8096E01 1.8096E01 8.5449E02
1.4949E01 1.4949E01 1.4949E01 1.7986E01
1.4949E01 1.4949E01 1.4949E01 1.7986E01
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To explore the capability of the ICLDFE quadrature sets to integrate the spherical harmonic functions over a local angular domain, the numerical integrations of polynomials in the direction cosines for each quadrature set in the hemispherical space and onetwentieth spherical surface were calculated using
ð
!
ma hb xc d U z
Ihemisphere ¼ Uhemisphere
ð Ionetwentieth ¼
M=2 X
surface area of the center triangle equals its weight and strictly non-negative. Fig. 6 shows the optimized ICLDFE quadrature sets from the first-order to the fifth-order. The number of directions correlates according to the order by 20 4N. (i.e., in case of ICLDFE eS1, 20 4N ¼ 20 4 ¼ 80).
3. Analysis and results 3.1. Numerical integration of polynomials in the direction cosines The quadrature sets are required to integrate the spherical harmonic function as exactly as possible over the full angular domain to guarantee particle conservation. However, for some realistic shielding problems, the angular flux in the local angular domain usually changes sharply or is even discontinuous.
(19)
m¼1
!
ma hb xc d U z
Uonetewentieth Fig. 5. Optimized Strategy of the L-Methods in one triangle.
um mam hbm xcm ;
M=20 X
um mam hbm xcm ;
(20)
m¼1
where a, b, and c are non-negative integers, and u denotes a weight of the quadrature set. The relative errors in the hemispherical space between the quadrature sets used in the calculations and the results obtained numerically are listed in Table 2. Table 3 lists the relative errors in the one-twentieth spherical space between the results using quadrature sets and the calculations using GausseLegendre integral formula. The LS-SN sets are the N-order LS quadrature sets, PNTN-SN are the Norder Legendre-Chebyshev quadrature sets, and ICLDFE-SN are the N-order LDFE quadrature sets based on the icosahedron. In the hemispherical domain, the relative errors of the LS and the PNTN quadrature sets are more than those of the ICLDFE quadrature sets for the same discrete direction cosines (Table 2). Note that the relative errors of (2,0,0) for listed quadrature sets are all equal 0.00%. The maximum relative errors of the ICLDFE-S4 and ICLDFE-S5 quadrature sets are only 0.05% and 0.01%, respectively. Table 3 shows that the relative errors of (0,0,0) and (1,0,0) are all equal 0.00%; the ICLDFE-S4 and ICLDFE-S5 quadrature sets both have higher integration accuracy for the polynomials up to fifth-
Fig. 6. Distributions of different-order ICLDFE quadrature sets: (a) S1, (b) S2, (c) S3, (d) S4, (e) S5.
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Table 2 Relative errors for the polynomials of direction cosines integration over the hemisphere for the ICLDFE quadrature sets. Quadrature
ICLDFE-S1 LS-S8 PNTN-S16 ICLDFE-S2 PNTN eS32 ICLDFE-S3 PNTN eS70 ICLDFE-S4 ICLDFE-S5
Directions
Relative Errors/%
80 80 288 320 1088 1280 5040 5120 20480
(1,0,0)
(2,0,0)
(3,0,0)
(1,2,0)
(5,0,0)
(3,2,0)
(1,2,2)
1.29 1.70 0.47 0.25 0.13 0.07 0.03 0.02 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.21 0.14 0.01 0.00 0.00 0.00 0.00 0.00 0.00
2.68 3.54 0.89 0.62 0.24 0.17 0.05 0.04 0.01
0.44 0.03 0.00 0.02 0.00 0.00 0.00 0.00 0.00
0.16 0.47 0.03 0.01 0.00 0.00 0.00 0.00 0.00
2.26 5.61 1.64 1.20 0.43 0.19 0.09 0.05 0.01
Table 3 Relative errors for the spherical harmonic function integration in one-twentieth of the spherical surface for the ICLDFE quadrature sets. ICLDFE
Directions
Relative Errors/% (0,0,0)
(1,0,0)
(2,0,0)
(3,0,0)
(1,2,0)
(5,0,0)
(3,2,0)
(1,2,2)
S1 S2 S3 S4 S5
80 320 1280 5120 20480
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.07 0.00 0.00 0.00 0.00
4.82 0.06 0.00 0.00 0.00
4.24 0.19 0.01 0.00 0.00
14.41 0.90 0.02 0.00 0.00
5.95 0.64 0.01 0.00 0.00
2.21 0.70 0.01 0.00 0.00
order in the one-twentieth spherical surface. The numerical results show that compared with the traditional LS quadrature sets and the PNTN quadrature sets, the accuracy of the numerical integration in the hemispherical surface of the ICLDFE quadrature sets has obviously improved, and can exactly integrate the zero-order and firstorder spherical harmonic functions in the one-twentieth spherical surface. As the discrete directions of the ICLDFE quadrature sets increase, the accuracy of the numerical integration gradually improves. However, because PNTN quadrature sets was developed based on Legendre and Chebyshev polynomial, it can exactly integrates any polynomial of a given order of the directions cosines globally on the sphere, that is, PNTN quadrature sets perform better than ICLDFE quadrature sets in a full spherical surface.
3.2. Kobayashi Benchmark problems 3.2.1. Problem 1 The Kobayashi Benchmark [14] was proposed by the OECD Nuclear Energy Agency to test the reliability of neutron transport programs. The one-fourth geometry for problem 1 of the Kobayashi
benchmark (Fig. 7) models a large void region that may cause strong ray effects when discrete directions are not enough. Because the ICLDFE quadrature sets are difficult to treat on the reflecting boundaries, all the vacuum boundaries used for the full geometry model are configured in the calculation. The geometry mesh size is divided into 2 cm 2 cm 2 cm. The spatial discrete method chooses the diamond differencing schemes (DZ) [15], and the iterative convergence criterion is 104. Table 4 lists the details about source strengths and cross sections in each region. The PNTN and ICLDFE quadrature sets are used to calculate the neutron fluxes using the ARES program [16] at key points along the X-axis, Y-axis, and diagonal. Fig. 8 plots the root mean square (RMS) of the relative errors between calculations and reference values versus the number of directions in the full angular domain. Fig. 9 shows the neutron flux distribution of the ICLDFE-S4 quadrature sets. The RMS of relative errors of the PNTN-S8 and the ICLDFE-S1 quadrature sets with 80 directions are 1.20 100 and 5.05 101 (Fig. 8). The maximum in the relative error of PNTN-S8 is up to 354.84% resulting in a larger discrete error and serious ray effects. The ICLDFE quadrature sets except for the second-order all have relative errors less than those of the PNTN quadrature sets with a similar number of directions.
3.2.2. Problem 2 Problem 2 of the Kobayashi Benchmark involves a typical deep penetration with a straight duct and poses a challenge to traditional transport calculations. The one-fourth geometry of this problem is shown in Fig. 10. Similarly, all the vacuum boundaries of the full
Table 4 Cross sections and sources associated with the Kobayashi Benchmark.
Fig. 7. One-fourth geometry for Problem 1 of the Kobayashi Benchmark.
Regions
Source /n$cm3 s1
Total Cross Sections/cm1
Scattering Cross Sections/cm1
1 2 3
1 0 0
0.1 10e4 0.1
0.05 0.5 104 0.05
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Fig. 10. One-fourth geometry of problem 2 of the Kobayashi benchmark. Fig. 8. RMS of the relative errors at key points for Problem 1 of the Kobayashi Benchmark for the quadrature sets.
Fig. 11. RMS of the relative errors at key points for Problem 2 of the Kobayashi Benchmark for quadrature sets.
Fig. 9. Neutron Flux (n$cm2 s1) distribution of the ICLDFE-S4 quadrature sets at X ¼ 100 cm.
geometry model are used and configured for the calculation using the ARES program. Other information such as source strength and cross section are the same as in problem 1. The PNTN and the ICLDFE quadrature sets are again used to calculate the neutron fluxes at key points along the duct. Fig. 11 shows the RMS of the relative errors between the calculations and reference values versus the number of directions in the full angular domain. The RMS of the relative errors for the PNTN-S20 quadrature sets with 440 directions is 1.95 101, and that for ICLDFE-S2 with only 320 directions is 1.11 101 (see Fig. 11). Similarly, PNTN-S32 and ICLDFE-S3 quadrature sets have approximately 1000 directions yielding a RMS for the relative errors of 6.78 102 and 1.65 102, respectively. All the RMS values of the ICLDFE quadrature sets are less than those of PNTN, indicating that the
computational accuracy of the former is higher than that of the latter when the discrete directions are the same or similar. Moreover, the RMS for the relative errors gradually diminish as the numbers of directions of the quadrature sets increase. It is noticed that the duct has oriented along Z axis..Actually, the performance of quadrature sets for this problem can depend on the orientation of duct, and that they have found different errors for different orientation.We have tested and analyzed different orientation cases and ICLDFE quadrature sets have shown certain advantages compared to PNTN quadrature sets. We performed an analysis to assess the capability of the local refinement for the ICLDFE quadrature sets. The local refined ICLDFE quadrature sets with a total 3584 directions for problem 2 were developed (Fig. 12). With a focus on the duct, the triangles along the axis where the duct is located received the highest order ICLDFE-S5 quadrature sets, whereas other triangles were ascribed lower-order quadrature sets. The neutron flux and relative errors of the local refined quadrature sets are listed in Table 5. The relative errors of
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Fig. 13. One-fourth geometry for Problem 3 of the Kobayashi Benchmark.
Fig. 12. Distribution of the ICLDFE-LocalRefine quadrature sets.
the ICLDFE-S5 quadrature sets are within 0.55%. The local refined quadrature sets with 3584 directions have similar relative errors to those for the ICLDFE-S5 quadrature sets, which have 20480 directions. 3.2.3. Problem 3 The Kobayashi Benchmark Problem 3 (Fig. 13) has a bend in the ducts to provide a more anisotropic angular flux than problem 2. Other information such as the source strength and cross section are the same as for problem 1. The geometry mesh size is divided into 2 cm 2 cm 2 cm and 1 cm 1 cm 1 cm cells. The spatial discrete method with the diamond differencing schemes (DZ) were adopted. The PNTN quadrature sets and the ICLDFE quadrature sets are used to calculate the neutron fluxes along the duct using the ARES program. The RMS of the relative errors versus the number of directions were plotted (Fig. 14) and showed the same trends as for problem 2. All the RMS values of the ICLDFE quadrature sets are less than the PNTN quadrature sets. For example, the RMSs of the relative errors for the PNTN-S20 quadrature sets with 440 directions and the ICLDFE-S2 quadrature sets with 320 directions are 1.13 101 and 6.15 102, respectively, when the spatial mesh size is 2 cm. Note
Table 5 The relative errors at key points for Problem 2 of Kobayashi Benchmark using the ICLDFE-S5 and local refined quadrature sets. Key Points
References
ICLDFE-S5 (20480)
ICLDFE-LocalRefine (3584)
Calculations
Relative Errors
Calculations
Relative Errors
/cm
/n$cm2 s1
/n$cm2 s1
/%
/n$cm2 s1
/%
(5,5,5) (5,15,5) (5,25,5) (5,35,5) (5,45,5) (5,55,5) (5,65,5) (5,75,5) (5,85,5) (5,95,5)
8.61696Eþ00 2.16123Eþ00 8.93437E01 4.77452E01 2.88719E01 1.88959E01 1.31026E01 9.49890E02 7.12403E02 5.44807E02
8.59055Eþ00 2.15303Eþ00 8.89937E01 4.75680E01 2.87127E01 1.88018E01 1.30815E01 9.51621E02 7.13797E02 5.45151E02
0.31 0.38 0.39 0.37 0.55 0.50 0.16 0.18 0.20 0.06
8.58523Eþ00 2.14985Eþ00 8.88368E01 4.75656E01 2.87184E01 1.87802E01 1.30808E01 9.51288E02 7.13555E02 5.45086E02
0.37 0.53 0.57 0.38 0.53 0.61 0.17 0.15 0.16 0.05
Fig. 14. RMS of the relative errors at key points for Problem 3 of the Kobayashi Benchmark for quadrature sets.
that the relative errors do not gradually diminish as the spatial mesh size is refined, indicating the mesh size has little impact on the discrete error. Further, the discrete error consists of spatial discrete errors and angular discrete errors that may mutually compensate to reduce the error when the spatial mesh size is larger. 4. Conclusions The ICLDFE quadrature sets with icosahedral symmetry can generate a large number of discrete directions, be easier to refine locally and have strictly positive weights. The numerical results indicate that compared with the LS quadrature sets and the PNTN quadrature sets, the numerical integration accuracy in the local angular domain of the ICLDFE quadrature sets are obviously improved and can exactly integrate the zero-order and first-order spherical harmonic functions over the one-twentieth spherical surface and not only in the full angular region. For the Kobayashi benchmark problems, the ICLDFE quadrature sets outperform other quadrature sets in general. The local refined quadrature sets may
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have better computational accuracy and be more efficient. However, because of the geometric properties of the icosahedron, only vacuum boundaries can be used in the ICLDFE quadrature sets. The adaptability of the ICLDFE quadrature sets for much more realistic problems needs further research. Exploring the quadratic and cubic discontinuous finite-element quadrature sets based on the icosahedron remains an open problem as does the p-adaptive method in the angular domain.
financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Declaration of competing interests
Appendix
Acknowledgements This work was supported by the National Natural Science Foundation of China (11975097) and the Fundamental Research Funds for the Central Universities (2019MS038).
The authors declare that they have no known competing
Table 6 ICLDFE-S1 quadrature sets data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
m
h
x
weight
1.456751E-01 6.938161E-01 1.456751E-01 3.568221E-01 1.456751E-01 1.456752E-01 6.938161E-01 3.568221E-01 2.357073E-01 7.838483E-01 5.744771E-01 5.773502E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 2.357073E-01 5.744771E-01 7.838483E-01 5.773502E-01 5.744771E-01 7.838483E-01 2.357073E-01 5.773503E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 5.744771E-01 2.357074E-01 7.838483E-01 5.773503E-01 9.295234E-01 7.201522E-01 9.295234E-01 9.341724E-01 9.295234E-01 9.295234E-01 7.201522E-01 9.341724E-01
2.005046E-01 3.786064E-01 7.768541E-01 4.911234E-01 2.005046E-01 7.768541E-01 3.786064E-01 4.911235E-01 7.658595E-02 1.015158E-01 5.428625E-01 1.875925E-01 2.478373E-01 7.141139E-01 7.141139E-01 6.070620E-01 7.658595E-02 5.428625E-01 1.015158E-01 1.875925E-01 7.906999E-01 5.025251E-01 9.007728E-01 7.946545E-01 9.773588E-01 8.672858E-01 8.672858E-01 9.822470E-01 7.906999E-01 9.007728E-01 5.025251E-01 7.946545E-01 3.020205E-01 5.901952E-01 5.418310E-02 3.035310E-01 3.020205E-01 5.418309E-02 5.901952E-01 3.035310E-01
9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01
1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
m
h
x
weight
5.744771E-01 7.838483E-01 2.357073E-01 5.773503E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 5.744771E-01 2.357074E-01 7.838483E-01 5.773503E-01 9.295234E-01 7.201522E-01 9.295234E-01 9.341724E-01 9.295234E-01 9.295234E-01 7.201522E-01 9.341724E-01 1.802645E-10 3.387697E-01 3.387697E-01 4.415465E-10 2.357073E-01 5.744771E-01 7.838483E-01 5.773502E-01 1.456751E-01 6.938161E-01 1.456751E-01 3.568221E-01 1.456751E-01 1.456752E-01 6.938161E-01 3.568221E-01 2.357073E-01 7.838483E-01 5.744771E-01 5.773502E-01
7.906999E-01 5.025251E-01 9.007728E-01 7.946545E-01 9.773588E-01 8.672858E-01 8.672858E-01 9.822470E-01 7.906999E-01 9.007728E-01 5.025251E-01 7.946545E-01 3.020205E-01 5.901952E-01 5.418310E-02 3.035310E-01 3.020205E-01 5.418309E-02 5.901952E-01 3.035310E-01 2.478373E-01 7.141139E-01 7.141139E-01 6.070620E-01 7.658595E-02 5.428625E-01 1.015158E-01 1.875925E-01 2.005046E-01 3.786064E-01 7.768541E-01 4.911234E-01 2.005046E-01 7.768541E-01 3.786064E-01 4.911235E-01 7.658595E-02 1.015158E-01 5.428625E-01 1.875925E-01
2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 2.115888E-01 3.647607E-01 3.647607E-01 1.875924E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01 9.688017E-01 6.125981E-01 6.125981E-01 7.946545E-01
1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02 1.189591E-02 1.189590E-02 1.189591E-02 1.431228E-02
Table 7 ICLDFE-S2 quadrature sets data
1 2 3 4 5 6 7 8 9 10 11
m
h
x
weight
6.549282E-02 3.629794E-01 7.250052E-02 1.666666E-01 5.804812E-01 7.867750E-01 5.804812E-01 6.666667E-01 7.250062E-02 3.629792E-01 6.549292E-02
9.014313E-02 1.941707E-01 4.052160E-01 2.293970E-01 2.648416E-01 3.245022E-01 5.702684E-01 3.918568E-01 6.339108E-01 7.282930E-01 8.485443E-01
9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812360E-01 5.250554E-01
2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03
161 162 163 164 165 166 167 168 169 170 171
m
h
x
weight
5.517467E-01 7.217364E-01 4.312575E-01 5.786893E-01 8.048146E-01 8.272520E-01 6.252892E-01 7.696723E-01 2.968341E-01 4.077871E-01 1.059698E-01
7.594142E-01 6.879574E-01 8.990026E-01 7.964975E-01 5.736100E-01 3.802139E-01 6.319413E-01 5.336320E-01 9.426794E-01 7.899657E-01 9.042559E-01
3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01
2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
N. Dai et al. / Nuclear Engineering and Technology xxx (xxxx) xxx
9
Table 7 (continued )
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85
m
h
x
weight
1.666667E-01 4.480860E-01 1.497516E-01 4.480860E-01 3.568221E-01 6.549282E-02 7.250053E-02 3.629794E-01 1.666666E-01 7.250078E-02 6.549308E-02 3.629790E-01 1.666667E-01 5.804811E-01 5.804811E-01 7.867752E-01 6.666666E-01 4.480860E-01 4.480860E-01 1.497516E-01 3.568221E-01 1.059696E-01 4.077872E-01 2.968341E-01 2.696723E-01 6.252892E-01 8.272520E-01 8.048146E-01 7.696723E-01 4.312577E-01 7.217363E-01 5.517467E-01 5.786893E-01 7.250184E-01 4.266840E-01 5.406376E-01 5.773502E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01 4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 1.059696E-01 2.968341E-01 4.077872E-01 2.696723E-01 4.312579E-01 5.517468E-01 7.217361E-01 5.786893E-01 6.252889E-01 8.048146E-01 8.272520E-01 7.696723E-01 7.250184E-01 5.406376E-01 4.266840E-01 5.773502E-01 5.517467E-01 7.217364E-01 4.312575E-01 5.786893E-01 8.048146E-01
7.551281E-01 6.167375E-01 5.198028E-01 3.030501E-01 4.911234E-01 9.014313E-02 4.052160E-01 1.941707E-01 2.293970E-01 6.339110E-01 8.485442E-01 7.282931E-01 7.551281E-01 2.648414E-01 5.702687E-01 3.245021E-01 3.918568E-01 6.167375E-01 3.030501E-01 5.198028E-01 4.911235E-01 3.443161E-02 5.626652E-02 2.852119E-01 8.762184E-02 1.269369E-01 1.999270E-01 1.201586E-01 7.483798E-02 4.702298E-01 3.758476E-01 6.479910E-01 5.129472E-01 2.355728E-01 3.325075E-01 1.820568E-02 1.875925E-01 1.114230E-01 3.704414E-01 3.704413E-01 2.835502E-01 5.554597E-01 7.249826E-01 8.025552E-01 7.088757E-01 5.554594E-01 8.025553E-01 7.249825E-01 7.088757E-01 7.623295E-01 5.085510E-01 5.085510E-01 6.070620E-01 3.443161E-02 2.852119E-01 5.626653E-02 8.762184E-02 4.702299E-01 6.479908E-01 3.758478E-01 5.129472E-01 1.269370E-01 1.201588E-01 1.999271E-01 7.483799E-02 2.355728E-01 1.820568E-02 3.325075E-01 1.875925E-01 7.594142E-01 6.879574E-01 8.990026E-01 7.964975E-01 5.736100E-01
6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01
3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03
172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245
m
h
x
weight
2.696723E-01 5.406376E-01 4.266840E-01 7.250184E-01 5.773503E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01 4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 5.517467E-01 4.312575E-01 7.217364E-01 5.786893E-01 2.968341E-01 1.059700E-01 4.077869E-01 2.696724E-01 8.048146E-01 6.252889E-01 8.272521E-01 7.696723E-01 5.406376E-01 7.250184E-01 4.266840E-01 5.773503E-01 8.927449E-01 8.773151E-01 9.882682E-01 9.363390E-01 7.942369E-01 6.172399E-01 7.942369E-01 7.453560E-01 9.882682E-01 8.773152E-01 8.927450E-01 9.363390E-01 8.747700E-01 9.887237E-01 8.747700E-01 9.341724E-01 8.927449E-01 9.882682E-01 8.773151E-01 9.363390E-01 9.882681E-01 8.927450E-01 8.773153E-01 9.363390E-01 7.942369E-01 7.942369E-01 6.172397E-01 7.453560E-01 8.747700E-01 8.747700E-01 9.887237E-01 9.341724E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01
8.969032E-01 7.441238E-01 9.009675E-01 6.842149E-01 7.946545E-01 9.386875E-01 9.807952E-01 9.807952E-01 9.845251E-01 9.371183E-01 7.938460E-01 8.787870E-01 8.841194E-01 9.371184E-01 8.787870E-01 7.938459E-01 8.841194E-01 9.197876E-01 9.796966E-01 9.796966E-01 9.822470E-01 7.594142E-01 8.990026E-01 6.879574E-01 7.964975E-01 9.426793E-01 9.042559E-01 7.899659E-01 8.969032E-01 5.736100E-01 6.319414E-01 3.802137E-01 5.336320E-01 7.441238E-01 6.842149E-01 9.009675E-01 7.946545E-01 2.900704E-01 4.738216E-01 1.323432E-01 3.042350E-01 5.881689E-01 6.692708E-01 3.994047E-01 5.671005E-01 8.997907E-03 1.437157E-01 1.786472E-01 2.068470E-02 2.842300E-01 1.273863E-01 4.780995E-01 3.035310E-01 2.900704E-01 1.323432E-01 4.738216E-01 3.042350E-01 8.997919E-03 1.786470E-01 1.437155E-01 2.068469E-02 5.881689E-01 3.994045E-01 6.692710E-01 5.671005E-01 2.842300E-01 4.780995E-01 1.273863E-01 3.035310E-01 1.114230E-01 3.704414E-01 3.704413E-01 2.835502E-01 5.554597E-01
3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01
3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03
(continued on next page)
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
10
N. Dai et al. / Nuclear Engineering and Technology xxx (xxxx) xxx
Table 7 (continued )
86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
m
h
x
weight
8.272520E-01 6.252892E-01 7.696723E-01 2.968341E-01 4.077871E-01 1.059698E-01 2.696723E-01 5.406376E-01 4.266840E-01 7.250184E-01 5.773503E-01 8.104344E-11 1.795258E-01 1.795258E-01 2.062397E-10 3.139492E-01 4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 5.517467E-01 4.312575E-01 7.217364E-01 5.786893E-01 2.968341E-01 1.059700E-01 4.077869E-01 2.696724E-01 8.048146E-01 6.252889E-01 8.272521E-01 7.696723E-01 5.406376E-01 7.250184E-01 4.266840E-01 5.773503E-01 8.927449E-01 8.773151E-01 9.882682E-01 9.363390E-01 7.942369E-01 6.172399E-01 7.942369E-01 7.453560E-01 9.882682E-01 8.773152E-01 8.927450E-01 9.363390E-01 8.747700E-01 9.887237E-01 8.747700E-01 9.341724E-01 8.927449E-01 9.882682E-01 8.773151E-01 9.363390E-01 9.882681E-01 8.927450E-01 8.773153E-01 9.363390E-01 7.942369E-01 7.942369E-01 6.172397E-01 7.453560E-01 8.747700E-01 8.747700E-01 9.887237E-01 9.341724E-01
3.802139E-01 6.319413E-01 5.336320E-01 9.426794E-01 7.899657E-01 9.042559E-01 8.969032E-01 7.441238E-01 9.009675E-01 6.842149E-01 7.946545E-01 9.386875E-01 9.807952E-01 9.807952E-01 9.845251E-01 9.371183E-01 7.938460E-01 8.787870E-01 8.841194E-01 9.371184E-01 8.787870E-01 7.938459E-01 8.841194E-01 9.197876E-01 9.796966E-01 9.796966E-01 9.822470E-01 7.594142E-01 8.990026E-01 6.879574E-01 7.964975E-01 9.426793E-01 9.042559E-01 7.899659E-01 8.969032E-01 5.736100E-01 6.319414E-01 3.802137E-01 5.336320E-01 7.441238E-01 6.842149E-01 9.009675E-01 7.946545E-01 2.900704E-01 4.738216E-01 1.323432E-01 3.042350E-01 5.881689E-01 6.692708E-01 3.994047E-01 5.671005E-01 8.997907E-03 1.437157E-01 1.786472E-01 2.068470E-02 2.842300E-01 1.273863E-01 4.780995E-01 3.035310E-01 2.900704E-01 1.323432E-01 4.738216E-01 3.042350E-01 8.997919E-03 1.786470E-01 1.437155E-01 2.068469E-02 5.881689E-01 3.994045E-01 6.692710E-01 5.671005E-01 2.842300E-01 4.780995E-01 1.273863E-01 3.035310E-01
4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01 3.447690E-01 7.623158E-02 7.623160E-02 1.752437E-01 1.524636E-01 4.136321E-01 4.578905E-01 3.504874E-01 1.524633E-01 4.578906E-01 4.136322E-01 3.504874E-01 3.924165E-01 7.872903E-02 7.872905E-02 1.875924E-01
2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03
246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
m
h
x
weight
4.457768E-01 1.344238E-01 3.090170E-01 3.139492E-01 1.344235E-01 4.457770E-01 3.090170E-01 5.544803E-10 1.843808E-01 1.843808E-01 4.415465E-10 1.059696E-01 2.968341E-01 4.077872E-01 2.696723E-01 4.312579E-01 5.517468E-01 7.217361E-01 5.786893E-01 6.252889E-01 8.048146E-01 8.272520E-01 7.696723E-01 7.250184E-01 5.406376E-01 4.266840E-01 5.773502E-01 6.549282E-02 3.629794E-01 7.250052E-02 1.666666E-01 5.804812E-01 7.867750E-01 5.804812E-01 6.666667E-01 7.250062E-02 3.629792E-01 6.549292E-02 1.666667E-01 4.480860E-01 1.497516E-01 4.480860E-01 3.568221E-01 6.549282E-02 7.250053E-02 3.629794E-01 1.666666E-01 7.250078E-02 6.549308E-02 3.629790E-01 1.666667E-01 5.804811E-01 5.804811E-01 7.867752E-01 6.666666E-01 4.480860E-01 4.480860E-01 1.497516E-01 3.568221E-01 1.059696E-01 4.077872E-01 2.968341E-01 2.696723E-01 6.252892E-01 8.272520E-01 8.048146E-01 7.696723E-01 4.312577E-01 7.217363E-01 5.517467E-01 5.786893E-01 7.250184E-01 4.266840E-01 5.406376E-01 5.773502E-01
7.249826E-01 8.025552E-01 7.088757E-01 5.554594E-01 8.025553E-01 7.249825E-01 7.088757E-01 7.623295E-01 5.085510E-01 5.085510E-01 6.070620E-01 3.443161E-02 2.852119E-01 5.626653E-02 8.762184E-02 4.702299E-01 6.479908E-01 3.758478E-01 5.129472E-01 1.269370E-01 1.201588E-01 1.999271E-01 7.483799E-02 2.355728E-01 1.820568E-02 3.325075E-01 1.875925E-01 9.014313E-02 1.941707E-01 4.052160E-01 2.293970E-01 2.648416E-01 3.245022E-01 5.702684E-01 3.918568E-01 6.339108E-01 7.282930E-01 8.485443E-01 7.551281E-01 6.167375E-01 5.198028E-01 3.030501E-01 4.911234E-01 9.014313E-02 4.052160E-01 1.941707E-01 2.293970E-01 6.339110E-01 8.485442E-01 7.282931E-01 7.551281E-01 2.648414E-01 5.702687E-01 3.245021E-01 3.918568E-01 6.167375E-01 3.030501E-01 5.198028E-01 4.911235E-01 3.443161E-02 5.626652E-02 2.852119E-01 8.762184E-02 1.269369E-01 1.999270E-01 1.201586E-01 7.483798E-02 4.702298E-01 3.758476E-01 6.479910E-01 5.129472E-01 2.355728E-01 3.325075E-01 1.820568E-02 1.875925E-01
5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812360E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01 9.937731E-01 9.113417E-01 9.113417E-01 9.589574E-01 7.700003E-01 5.250556E-01 5.812361E-01 6.340377E-01 7.700005E-01 5.812361E-01 5.250554E-01 6.340377E-01 6.471891E-01 8.410586E-01 8.410586E-01 7.946545E-01
2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03 2.352408E-03 3.247096E-03 3.247101E-03 3.049298E-03 3.247102E-03 2.352393E-03 3.247105E-03 3.049304E-03 3.247101E-03 3.247111E-03 2.352391E-03 3.049302E-03 3.530800E-03 3.530812E-03 3.530808E-03 3.719866E-03
Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025
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Please cite this article as: N. Dai et al., Discontinuous finite-element quadrature sets based on icosahedron for the discrete ordinates method, Nuclear Engineering and Technology, https://doi.org/10.1016/j.net.2019.11.025