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Acceleration of within group iteration for pin-by-pin calculations
Annals of Nuclear Energy 112 (2018) 225–235
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Acceleration of within group iteration for pin-by-pin calculations Tengfei Zhang a,⇑, E.E. Lewis b, M.A. Smith c, W.S. Yang d, Yongping Wang e, Hongchun Wu e a
Shanghai Jiao Tong University, Shanghai, China Department of Mechanical Engineering, Northwestern University, Evanston, IL, United States c Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL, United States d School of Nuclear Engineering, Purdue University, West Lafayette, IN, United States e Xi’an Jiaotong University, Xi’an, Shaanxi, China b
a r t i c l e
i n f o
Article history: Received 16 July 2017 Received in revised form 29 September 2017 Accepted 2 October 2017
Keywords: Heterogeneous variational nodal method Coarse nodes acceleration Acceleration of within-group iteration
a b s t r a c t In thermal reactor physics fine mesh neutron diffusion calculations in which homogenization is only at the pin cell level are of considerable interest. However, the slow convergence of iterations on finemesh spatial grids limits their applicability to practical problems. We present here an efficient way to accelerate the within-group iterations for pin-by-pin calculations based on a two-dimensional heterogeneous variational nodal method (VNM). Response matrix (RM) equations are formulated that incorporate multiple pins within each node. Within the nodes, finite elements in the x-y plane are employed to describe the piecewise constant heterogeneous geometry. On the nodal interfaces orthogonal polynomials are employed to approximate current distributions. The RM equations are solved using the Red-Black Gauss-Seidel (RBGS) iteration. Investigations on the coarse nodes acceleration (CNA) by combining homogenized pin cell nodes into larger heterogeneous nodes are performed. A series of meshing schemes are examined with a 2D small modular reactor core problem. With sufficient interface expansion order, CNA do not cause significant errors to either eigenvalues or fission rate distributions compared with fine mesh calculations. It is demonstrated that CNA accelerate the RBGS iteration significantly and achieves favorable accuracy-efficiency trade-off. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The traditional two-step method has been utilized as the treatment to the neutronic analysis of light water-cooled reactors (LWR) for decades (Roy, 2010). The first step is the performance of multi-group neutron transport calculations for each fuel assembly, with fine-grained detail of the two-dimensional assembly. The resulting flux spectrum is used to generate the few-group constants by spatial homogenization and energy group condensation. The second step is the execution of whole-core calculations, usually with the few-group coarse-mesh diffusion approximation. Due to adequate cost-accuracy trade-off, the two-step method has prevailed in industrial use. However, in whole core calculations, pin power information is not directly obtainable as a result of the homogenization over each coarse assembly mesh. Therefore pin power reconstruction must be performed to approximate high resolution results. ⇑ Corresponding author. E-mail addresses: [email protected] (T. Zhang), [email protected] (E.E. Lewis), [email protected] (M.A. Smith), [email protected] (W.S. Yang), [email protected] (Y. Wang), [email protected] (H. Wu). https://doi.org/10.1016/j.anucene.2017.10.006 0306-4549/Ó 2017 Elsevier Ltd. All rights reserved.
With the advancement of computational resources over the past decade, considerable effort has been expended on the pursuit of more advanced methods for mitigating the errors inherent in the traditional two-step procedure. One prominent approach is the pin-by-pin method which promotes the homogenization to the pin-cell level (Tatsumi and Yamamoto, 2002; Kozlowski et al., 2011). By performing pin-cell level fine mesh whole-core calculations, heterogeneity inside a fuel assembly as well as spectral mismatch at interfaces between different types of fuel assemblies can be treated more precisely, compared to the conventional coarse mesh methods. Moreover, pin cell information is directly obtained in the resulting outputs whereby eliminating the errors caused by pin power reconstructions. For these reasons, such pin-by-pin methods are incorporated in the production codes SCOPE2 (Tatsumi and Yamamoto, 2003), DYN3D (Beckert and Grundmann, 2008; Rohde and et al., 2016), and EFEN (Yang et al., 2014). However, power reactors consist of tens or hundreds of assemblies, resulting in millions of fine meshes. In addition, with pin-wise homogenization, the material heterogeneity is much more severe than that with the assembly-wise homogenization (Lee et al., 2014). In this case, every within-group sweep consumes substantial amount of time, and the optically thin meshes
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T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
deteriorate the convergence rate of within-group iteration (Lewis et al., 2011). No matter which method is selected – nodal (Beckert and Grundmann, 2008), finite difference (Tatsumi and Yamamoto, 2002; Tatsumi and Yamamoto, 2003) or response matrix (RM) based (Lewis et al., 2011) – the computational efficiency proves a crucial challenge limiting the application of the pin-by-pin calculations. Although parallel computing has achieved substantial speedups, pin-by-pin calculations remain burdensome. A typical 3D PWR problem with the nodal SP3 method, for example, requires more than 24 h on a single processor, and around 20 min on 100 processors (Beckert and Grundmann, 2008). To improve efficiency for a RM based method, investigators have introduced the orthogonal matrix aggregation (Lewis et al., 2011) (OMA) method. By combining N N fine mesh RMs with one degree of freedom (DOF) per interface into a coarse mesh RM with N orthogonal DOF per interface, favorable acceleration was achieved. However, the study was restricted to the piecewise constant discretization on interfaces and the cost-accuracy tradeoff was not surveyed thoroughly. In this paper we propose a heterogeneous variational nodal method (VNM) which is capable of accelerating the within-group iteration for pin-by-pin calculations by coarsening nodes. The method is realized by combining homogenized pin cell nodes into coarser nodes with piecewise constant cross sections (XSs) over each pin cell. The VNM has been applied predominantly to fast reactor problems (Lewis et al., 1996; Smith et al., 2014), and can be extended to describe unstructured geometry by utilizing finite element submeshes within each node (Zhang et al., 2017a,b). Generally in VNM the within-group iteration is performed using the standard Red-Black Gauss-Seidel (RBGS) iteration algorithm. This procedure expends the bulk of CPU time in VNM computation since it is the innermost iteration where the RM equations are solved iteratively (Lewis et al., 1996). Therefore here we focus our attention on improving within-group solution algorithms, employing the 2D diffusion approximation as a testbed. We first formulate the 2D diffusion heterogeneous VNM which serves as the basis for this research. Within each node, finite element trial functions in x-y are employed to describe the heterogeneous structures, while orthogonal polynomials are used to approximate the neutron current distributions along nodal interfaces. Continuity conditions are imposed on the neutron current moments across the nodal interfaces. Standard fission source iteration and RBGS iteration are performed to solve the resulting RM equations. To improve efficiency in VNM with unstructured geometry capability, the flat source region (FSR) acceleration (Zhang et al., 2017b) is used by evaluating the averaged scalar flux on the finite elements in place of the finite element nodes (i.e., vertices). The FSR is combined with heterogeneous coarse nodes to accelerate the RBGS iteration algorithm as for pin-by-pin calculations. We organize the rest of the paper as follows. In Section 2, we formulate the RM equations along with their spatial matrix components. In addition, FSR acceleration and coarse node acceleration (CNA) is elaborated to reduce further the computational costs. In Section 3, CNA is tested on an idealized 2D small modular reactor (SMR) core problem with pin-wise homogenized XSs. The orthogonal polynomial order on the interfaces is examined, convergence studies are performed, and the accuracy-efficiency trade-offs are compared. In Section 4, conclusions from this study are drawn and future directions discussed. For clarity, the nomenclatures present in the paper are listed in Appendix A (Table A). 2. Theory model In the diffusion approximation, the even-parity within-group equation is given as:
rð3rt Þ1 r/ð~ rÞ þ rr /ð~ rÞ ¼ qð~ rÞ;
ð1Þ
where rt and rr are the total and removal XSs (cm ), respectively. Other notations are conventional. The variational functional of the problem domain with respect to the scalar flux / and neutron currents j can be written as the superposition of the contributions from individual nodes and node surfaces: 1
F½/ð~ rÞ; jð~ rÞ ¼
X F m ½/ð~ rÞ; jð~ rÞ:
ð2Þ
m
The within-group nodal functional of each node can be written as:
F½/ð~ rÞ; jð~ rÞ ¼
Z
n o 2 2 dV ð3rt Þ1 ½r/ð~ rÞ þ rr /ð~ rÞ 2/ð~ rÞqð~ rÞ V Z rÞjð~ rÞ þ 2 dC/ð~ C
where
qðrÞ ¼
X
rsgg0 ð~rÞ/g0 ð~rÞ þ k1 vg
g 0 –g
X
mrfg0 ð~rÞ/g0 ð~rÞ;
ð3Þ
ð4Þ
g0
and C is the outer boundary. Requiring this functional to be stationary with respect to variations in d/ within V yields Eq. (1) as the Euler-Lagrange equation. Across the lateral and axial interfaces, j is defined to be continuous, respectively. 2.1. Ritz discretization In the standard VNM approach, the flux distribution is discretized with orthogonal polynomials both within nodes and on nodal interfaces. Here we limit the discussion on twodimensional (2D) problems for simplicity, but it can be extended to three-dimensional problems easily. Denoting a vector by an underline, the flux within a 2D node can be expanded as
/ð~ rÞ ¼ g T ðx; yÞ/;
ð5Þ
where gðx; yÞ is a vector of orthogonal, iso-parametric finite element trial functions and / is the vector of the scalar flux moments. For example, quadratic triangular finite elements are used for the numerical calculations presented in Section 3. Using the orthogonal polynomials f g defined in terms of local coordinate g as trial functions, the current j1 at a radial interface sented as
j1 ðgÞ ¼ f Tg ðgÞj1 :
1 ¼ 12 D1 can be repre-
1 2
1 ¼ D1; 1; g ¼ x; y;
where D1 is the node width in the
ð6Þ
1 direction.
2.2. Response matrix equations By substitutions of Eqs. (5) and (6) in Eq. (3), we can put the functional in a discretized form: Z Z Z h i 1 dxdyg g T þ dxdyrx g rx g T þ dxdyry g ry g T F m /; j ¼ /T r1 t 3 Z Z þ dxdyrr gg T / 2/q þ 2/T dy gjx f Ty jx þ 2/T Z dy gjy f Tx jy ð7Þ
where x and y are suppressed in the spatial integrals, and
Z
q¼
v
dVgðx; yÞqð~ rÞ
ð8Þ
For brevity we employ the matrix notations shown in Table 1. Thus Eq. (7) is rewritten as
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
F m ½/; j ¼ /T A// / 2/q þ 2/T M /j j:
ð9Þ
where the variables with double underlines represent matrices. Based on the variational principle, requiring the discretized functional in Eq. (9) to be stationary with respect to variation in /. yields: 1 / ¼ A1 // q A// M /j j;
ð10Þ
which represents the neutron balance relationship within each node. Requiring the functional to be stationary with respect to the interface current term j:, stipulates that the following even parity surface moments, u, must be continuous across the nodal interfaces:
u ¼ MT/j /:
ð11Þ
Inserting Eq. (10) in Eq. (11) we obtain
u ¼ u Gj
ð12Þ
where
G ¼ MT/j A1 // M /j
ð13Þ
u ¼ M T/j A1 // q:
ð14Þ
To cast Eq. (12) into a RM form, we utilize a linear transformation of variables based on the relationships of partial currents:
j ¼
1 1 u j 4 2
ð15Þ
where j and jþ correspond respectively to neutrons entering and leaving the node. Inserting Eq. (15) into Eq. (12) yields the RM equation
jþ ¼ Bu þ R j
ð16Þ
where the expression of B and R matrices are given as
B¼
1 1 1 GþI 2 2
R¼I
ð17Þ
1 1 GþI G: 2
ð18Þ
Eq. (16) shows the relationship between interface currents and neutron sources. Using the standard RBGS iteration scheme, Eq. (16) can be solved iteratively as
~r þ Rrb j jþðlþ1Þ ¼q r b
ðlÞ
ð19Þ
~r þ Rrb j jþðlþ1Þ ¼q r b
ðlþ1Þ
ð20Þ
In the RBGS iteration, Eqs. (19) and (20) are successively solved using the relations between outgoing and incoming currents from the adjoining ‘red’ and ‘black’ nodes. This procedure is the most time-consuming process in VNM because it is the innermost calculation that has to be performed iteratively during each outer iteration. The spatial matrices in Eqs. (9)–(18) are defined in Table 1 in terms of the trial functions f and g. The integrals are evaluated with standard numerical quadrature schemes (Lewis et al., 2011). 2.3. Flat source region and coarse node acceleration The computational cost to solve the within-group RM equation in Eq. (16) increases with the total number of unknowns, that is, the number of nodes times the number of unknowns per node. Therefore, if the number of nodes and the number of unknowns per node can be reduced without any significant loss of accuracy, the efficiency of a RM based method can be improved significantly. The number of unknowns per node can be reduced using the FSR acceleration scheme (Zhang et al., 2017b) which we proposed before without a detailed description. Likewise, the number of nodes can be reduced by employing the OMA method (Lee et al., 2014) in which N N fine mesh RMs with one DOF per interface into a coarse node RM with N orthogonal DOF per interface. In this study, the CNA is utilized to accelerate the within-group iteration for pin-by-pin calculations by combining homogenized pin cell nodes into coarser nodes with piecewise constant XSs over each pin cell. In the FSR acceleration method, the averaged scalar flux is evaluated on the finite elements in place of the finite element vertices. As an example, consider the finite element mesh for a homogenized pin cell node shown in Fig. 1. The node is meshed with 4 quadratic triangular finite elements consisted of 13 finite element vertices (namely nodes in FEM). The finite element meshes employed within the node require evaluation of group sources and group fluxes on each finite element node, corresponding to 13 DOF. However, the DOF is reduced to 4 if the averaged scalar flux is evaluated on the finite elements. We start FSR by forming a vector of the lateral averages over each of the finite elements: _
/ ¼ N1
Z
dxdyhðx; yÞg T ðx; yÞ/
Z
H¼
dxdygðx; yÞhT ðx; yÞ
we have
~ ¼ Bu q
/ ¼ N1 HT /:
_
Table 1 Spatial Matrices. A// ¼ 13 F þ 13 P xx þ P yy þ F a h i M /j ¼ M /jx M/jy R T P ii ¼ dxdyr1 i ¼ x; y t ri g ri g R R T F a ¼ dxdyrr gg T F ¼ dxdyr1 t gg R R T M /jy ¼ dyjx f y M /jy ¼ dyjy f Tx
ð22Þ
where hðx; yÞ is a piecewise constant trial function vector, the ith component of which is unity over the area of the ith finite element and zero elsewhere; N is a diagonal matrix of the finite element areas. Defining a matrix
where r and b represent respectively the ‘red’ and ‘black’ nodes, l is the iteration index and
ð21Þ
227
Fig. 1. The Finite Element Mesh for a Homogenized Pin Cell.
ð23Þ
ð24Þ
228
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Fig. 2. Five Different Node Meshing Schemes.
Fig. 3. The 2D Schematic View of the SMR Problem.
229
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235 Table 2 Results vs. Node Meshing Schemes for the Unrodded SMR problem. f g ðgÞ order
Eigenvalue
Error – pcm
RMS
MAX
17 17
0 1 2
1.28635 1.28648 1.28654
15 5 Reference
0.17 0.05
0.99 0.32
5 5a
0 1 2 3
1.28615 1.28643 1.28652 1.28652
30 9 2 2
1.29 0.44 0.08 0.07
8.27 2.04 0.41 0.30
5 5b
0 1 2 3
1.28527 1.28611 1.28643 1.28650
99 34 9 4
1.69 0.68 0.22 0.07
6.59 2.36 0.88 0.28
33
0 1 2 3 4 5
1.28609 1.28618 1.28624 1.28634 1.28642 1.28649
35 28 24 16 10 4
1.61 0.76 0.59 0.34 0.22 0.12
15.05 3.45 2.17 1.26 0.92 0.46
11
0 1 2 3 4 5 6
1.28745 1.28658 1.28652 1.28653 1.28654 1.28654 1.28654
70 3 2 1 1 0 0
2.64 0.65 0.51 0.38 0.19 0.10 0.04
32.10 9.42 7.52 5.66 2.82 1.23 0.41
Fission Rate Error -%
Table 3 Results vs. Node Meshing Schemes for the Rodded SMR Problem. g
f g order
Eigenvalue
Error – pcm
RMS
MAX
17 17
0 1 2
1.22467 1.22456 1.22451
13 4 Reference
0.34 0.11
2.20 0.50
5 5a
0 1 2 3
1.22489 1.22459 1.22453 1.22453
31 7 1 1
1.32 0.49 0.09 0.08
8.15 2.70 0.65 0.54
5 5b
0 1 2 3
1.22541 1.22481 1.22461 1.22455
73 24 8 3
2.41 0.95 0.29 0.10
13.12 5.88 1.87 0.59
33
0 1 2 3 4 5
1.22539 1.22469 1.22467 1.22462 1.22458 1.22453
72 14 13 9 6 2
1.68 0.96 0.75 0.45 0.33 0.13
13.78 5.56 4.31 2.46 1.25 0.61
11
0 1 2 3 4 5 6
1.22853 1.22523 1.22446 1.22448 1.22450 1.22451 1.22451
328 58 4 3 1 0 0
3.68 1.24 0.50 0.37 0.19 0.10 0.04
32.04 10.25 7.14 5.42 2.76 1.10 0.40
This treatment substantially reduce the number of flux values that needs to be stored, because the length of / is equal to the _
number of finite element vertices, while the length of / is equal to the number of finite elements. Substitution of Eq. (10) in Eq. (24), we obtain _
1 T 1 / ¼ N1 HT A1 // q N H A// M /j j
ð25Þ
Pin Fission Rate Error -%
1 T 1 in which N1 HT A1 // and N H A// M /j are evaluated when forming the response matrices. We next evaluate qð~ rÞ of Eq. (4) by using the condition that within a pin-cell node, the XSs are piecewise constant in x and y with a distinct value over each finite element. Thus we may write reaction rates as
_
rðx; yÞ/ðx; yÞ ¼ hT ðx; yÞR /
ð26Þ
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T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
where R is a diagonal matrix containing the XS for each finite element. In other words, the XSs are treated as piecewise constants within each node, allowing for an exact heterogeneous description of the nodal geometry. Including the appropriate group subscripts on the XSs and averaged fluxes, we may combine Eqs. (4) and (26) to obtain
qð~ rÞ ¼ hT ðx; yÞ
X g 0 –g
_
Rsgg0 /g0 þ k1 vg
X g0
_
!
mRfg0 /g0 :
ð27Þ
To find the discretized group source we insert this expression into Eq. (8) we have
q¼H
X g 0 –g
_
1
Rsgg0 /g0 þ k vg
X g0
_
!
mRfg0 /g0
ð28Þ
thus completing the discretization process. Eq. (28) is used to construct the source vectors in Eqs. (19), (20) and (25). In within-group iteration, Eqs. (19) and (20) are solved with the standard RBGS iter_
ation as illustrated in Section 2.2, then / is updated by Eq. (25). _
Obviously with FSR the length of the solution vector / decreases, hence the matrix operation in Eq. (25) are accelerated due to the 1 T 1 reduced size of N1 HT A1 // and N H A// M /j . In the heterogeneous VNM, each node is constructed to include one or more homogenized pin cells. Therefore, by utilizing finite element meshes and the piece-wise constant description of XSs, coarse nodes are employed without altering the original heterogeneous configuration. As an example, Fig. 2 shows five different coarse node meshes for a 17 17 fuel assembly. In each meshing scheme, coarse nodes are outlined by red lines. The 17 17 mesh consists of single pin-cell nodes, while the other meshes are composed of heterogeneous nodes containing several pin cells. In order to retain solution accuracy with the coarse nodes, the order of orthogonal polynomial f g ðgÞ in Eq. (6) is increased to represent more accurately the spatial variation of neutron on each coarse nodal interface. 3. Numerical results The foregoing heterogeneous VNM with the diffusion approximation is implemented in the Fortran 90 code PANX (Purdue – Argonne – Northwestern – Xi’an) as a branch option to the pin-resolved transport solver. The code is based on NODAL
(Smith et al., 2014), a diffusion code being developed at Argonne National Laboratory. The within-group RBGS iteration scheme and the standard fission source iteration scheme without acceleration are embedded in PANX. The performance of the proposed acceleration schemes for the within-group iterations of a RM based method for pin-by-pin LWR calculations were examined using a 2D Small Modular Reactor (SMR) problem with pin-wise homogenized XSs, which are based on the released NuScale design (Haugh and Mohamed, 2012). The problem incorporates two different configurations: rodded and unrodded cases. Fig. 3 shows the schematic view of the core and assembly configurations. The SMR problem utilizes a 17 17 UO2 pin-cell assembly with a pin pitch of 1.26 cm. Since the main interest of this work is to study the effect of CNA on acceleration of within-group iteration, we assumed a uniform configuration of UO2 assemblies in the unrodded case. In the rodded case, the guide tubes of the control rod assembly are replaced with control rod materials. Vacuum boundary conditions are applied to all of the four boundaries. 7-group pin-cell-homogenized XSs for standard PWR pin cell models are employed, which are included in Appendix B (Tables B1–B6). Every pin cell was meshed with 4 quadratic triangular finite elements as shown in Fig. 1, and five different nodal meshes shown in Fig. 2 were considered. The total number of finite element meshes remains the same in all the five node meshing schemes. A maximum number RBGS iteration of 5 was used to guarantee a relatively tight within-group convergence that ensures the convergence of the calculation. The convergence criteria for eigenvalue and fission source were set to 1 106 and 1 105 , respectively, in L2 norm. The calculations were performed on a single 2.60 GHz Intel i7-U6600U processor without acceleration on the outer iterations. To compare the computational efficiency of five different meshing schemes on a consistent basis, the solution accuracy obtained with the finest mesh was used as a reference. As the total number of finite elements within nodes is fixed in all the five meshing schemes, the order of interface polynomial f g ðgÞ in Eq. (6) becomes the dominating factor that influences the solution accuracy. Intuitively, when the nodes are coarsened, higher order f g ðgÞ should be needed to represent the spatial distribution of interface currents. Therefore, the order of interface polynomial f g ðgÞ required to achieve the reference accuracy obtained with the finest mesh was first examined for each nodal mesh. Subsequently, the compu-
Fig. 4. Fission Rate Error Distribution of 5 5a F3.
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Fig. 5. Fission Rate Error Distribution of 5 5b F3.
Fig. 6. Fission Rate Error Distribution of 3 3 F5.
Fig. 7. Fission Rate Error Distribution of 1 1 F5.
231
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T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Fig. 8. Convergence Rates of the RBGS Iteration.
3.1. Accuracy
Table 4 Performance Gain in the RBGS Iteration by CNA. Heterogeneous Nodes
17 17 F2 5 5a F3 5 5b F3 3 3 F5 1 1 F6
Gain Unrodded
Rodded
1.0 6.3 6.3 7.9 54.5
1.0 6.5 6.5 8.2 56.6
Using each of the five nodal meshes in Fig. 2, the 2D SMR problems were solved by varying the order of interface polynomial f g ðgÞ in Eq. (6). Tables 2 and 3 show the root mean square (RMS) and maximum (MAX) errors of the fission rates in individual pin cells as well as the eigenvalue and its error for the unrodded and rodded configurations, respectively. The fission rate ðFRÞi of the ith pin cell is defined by the volumetric average of the fission rate over the finite elements e in pin cell i:
ðFRÞi ¼ tational efficiencies of RBGS iteration achieved with different nodal meshes were compared, and the trade-off between accuracy and efficiency was investigated.
G X X
, g;e f /e V e
r
g¼1 e2i
X Ve
ð29Þ
e2i
where Ve is the volume of the finite element e, and /eg represents the group g scalar flux of the finite element e, which is directly obtained using Eq. (25).
Fig. 9. Eigenvalue Error vs. Number of FLOPs for the Rodded Case.
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
When the finest 17 17 mesh is used, the increase in the interface polynomial from linear to quadratic yield only marginal improvement in eigenvalue (5 pcm for the unrodded case and 4 pcm for the rodded case) and in RMS fission rate (0.05% for the unrodded case and 0.11% for the rodded case). These results indicate that for the 17 17 mesh, the quadratic interface polynomial yields an asymptotically converged solution. Therefore, the solution of the 17 17 F2 case was taken as the reference solution. Here we introduce a brief notation n n Fm with the number of nodes on each side of the assembly n and the interface polynomial order m. For a fixed nodal mesh, the solution accuracy is improved monotonically with increasing order of interface polynomial for both the rodded and unrodded cases. In addition, the order of interface polynomial required to achieve the same solution accuracy increases as the node becomes coarser. More importantly, the results in Tables 2 and 3 show that each of the coarse mesh yields a solution comparable to the reference solution obtained with 17 17 F2 by using a higher order of interface polynomial. The solutions of 5 5a F3, 5 5b F3, 3 3 F5, and 1 1 F6 show comparable accuracy to the reference solution. Fig. 4 through Fig. 7 show the quarter-core distributions of fission rate errors for 5 5a F3, 5 5b F3, 3 3 F5 and 1 1 F6. The nodes are outlined by white grids, and the center line of the core is described with black dashed lines. Note that the positions near inserted control rod exhibit slightly larger errors than others. Also observe that the pin fission rates inside each node agree well with the reference solution. Although minor errors are observed along nodal interfaces, the overall error is negligible with the maximum error of less than 0.6% among all the cases.
3.2. Computational efficiency Although CNA requires higher order interface polynomials to obtain solutions that closely approximate the reference fine node solution, the RBGS iteration converges substantially faster. To measure the relative efficiency of different nodal meshing schemes in accelerating the RBGS iteration, we employ the number of floating point operations (FLOPs) to measure computational effort. The number of DOFs (N) at each interface are 3, 4, 4, 6, and 7 for 17 17 F2, 5 5a F3, 5 5b F3, 3 3 F5 and 1 1 F6, respectively. Each R matrix multiplication in Eqs. (19) and (20) requires ð4NÞ2 FLOPs. Thus the total number of FLOPs per RBGS sweep becomes (4N)2 times the number of nodes in the problem. By recording the number of RBGS sweeps performed during the outer iterations, the number of FLOPs was evaluated during each calculation. Plots for the residual L2 norm of fission source vs. FLOPs expended are shown in Fig. 8. Observe that the convergence rates do not depend noticeably on the interface polynomial order, thus only the highest polynomial orders are plotted in Fig. 8. For both rodded and unrodded cases, the fission source converges more rapidly with increasing size of heterogeneous nodes. Moreover, the 5 5a F3 and 5 5b F3 configurations show nearly identical convergence rates because they have the same number of coarse nodes. Table 4 compares the performance gains for the RBGS iteration achieved by applying heterogeneous coarse nodes in term of the reduced number of FLOPs relative to the reference case 17 17 F2. Clearly the performance gain increases monotonically with increasing coarse node size. In particular, for both the unrodded and rodded cases, the use of 1 1 F6 reduces the number of FLOPs required for a converged solution by a factor of more than 50.
233
3.3. Accuracy-Efficiency Trade-off Fig. 9 compares the eigenvalue error versus the number of FLOPs of the five node meshing schemes for the rodded case. The results for different meshing schemes are distinguished by different marks, and the corresponding orders of interface polynomials are attached by number labels. The 1 1 F0 results are excluded in Fig. 9 because an eigenvalue error of 328 pcm was too large to be included in a compact plot. Fig. 9 clearly indicates that with the same order of interface polynomial, coarser nodes require fewer FLOPs compared to the finer meshes. With increasing order of interface polynomials, the eigenvalue error decreases monotonically while computing costs increase accordingly. The coarser nodes clearly show superior trade-offs between accuracy and efficiency. 4. Discussion This paper presents a preliminary study on accelerating within-group iteration of a RM based method for pin-by-pin diffusion calculations. By incorporating finite element meshes in the nodal framework, a 2D heterogeneous VNM is formulated with the diffusion approximation. The FSR acceleration method is derived focused on forming the RM equations with reduced DOFs. By combining the FSR acceleration method with coarse heterogeneous nodes, coarse-mesh RM equations are formed to preserve the original problem geometry. Using a SMR core problem, the effect of CNA on the convergence of RBGS iteration has been investigated. Rodded and unrodded problems of the 2D SMR with pin-cellhomogenized XSs were solved using five different nodal meshes for a 17 17 fuel pin assembly. Results show that coarse heterogeneous nodes can reproduce the fine mesh solution as the order of interface polynomials is increased. Although the interface polynomial order is increased, the total DOFs per assembly decreases with increasing size of coarse mesh; thus the number of FLOPs required for converged solutions is reduced significantly. Coarser nodes also show superior trade-offs between accuracy and efficiency. It should be noted that employing heterogeneous coarse nodes shifts the computational burden to forming response matrices: the times required to form the response matrices range from approximately 20% of the solution time for the 5 5 configurations to substantially larger than the solution times for the coarser grids. However, it will not be a significant roadblock to carrying out calculations compared with speeding up the RBGS iterations. By transforming the response matrices into block-diagonal forms (Yang et al., 2001), the computational costs of forming response matrices can be reduced significantly. More importantly, the formations are performed only once instead of at each iteration, and are completely un-coupled, thus the processes can be easily threaded with high efficiency. With parallelization they should consume only a small fraction of the solution times obtained in this paper. Therefore it can be expected that large gains in computing efficiency can be achieved with heterogeneous coarse nodes. These results suggest a possible direction for accelerating the within-group iteration for pin-by-pin calculations. Moreover, the proposed method can be extended to a 3D diffusion formulation by including orthogonal z polynomials (Zhang et al., 2016), as well as to a transport capability by employing spherical harmonics expansions as employed in VARIANT (Lewis et al., 1996; Zhang et al., 2016). Works on reducing the response matrix formation time and employing 3D CNA are underway.
234
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Appendix A Table A Nomenclatures. Abbreviations
Meanings
VNM RM RBGS OMA CNA DOF FSR FEM FLOP n n Fm
Variational Nodal Method Response Matrix Red-Black Gauss-Seidel Orthogonal Matrix Aggregation Coarse Node Acceleration Degree Of Freedom Flat Source Region Finite Element Method FLOating-Point operation The CNA meshing scheme using n nodes on each side of the assembly and up to mth polynomial order on nodal interfaces
Appendix B
Table B1 7-Group XSs for the Homogenized UO2 Pin Cell (cm1). Group g t g f
r r mrfg vg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1
2
3
4
5
6
7
1.70270E1 4.25641E3
3.64310E1 4.79383E4
5.27240E1 3.70264E3
5.67400E1 1.04931E2
4.85130E1 1.02095E2
7.70080E1 4.68075E2
1.51810E+0 1.17245E1
1.18390E2
1.18620E3
9.01160E3
2.55381E2
2.48479E2
1.13920E1
2.85351E1
5.87910E1 9.34960E2
4.11760E1 7.14850E2
3.39060E4 3.02070E4
1.17610E7 1.54010E6
0.00000E+0 2.17970E8
0.00000E+0 0.00000E+0
0.00000E+0 0.00000E+0
0.00000E+0
3.06980E1
5.48680E2
2.58660E4
1.99160E5
3.09050E6
4.33790E7
0.00000E+0
0.00000E+0
4.05890E1
9.72560E2
7.24550E3
1.12710E3
2.14580E4
0.00000E+0
0.00000E+0
0.00000E+0
2.95370E1
1.83810E1
2.77100E2
5.27800E3
0.00000E+0
0.00000E+0
0.00000E+0
1.02290E4
2.14960E1
2.24270E1
2.61260E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.69770E3
4.55210E1
2.43910E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
6.51920E2
1.28240E+0
Table B2 7-Group XSs for the Homogenized Guide Tube Pin Cell (cm1). Group
1
2
3
4
5
6
7
rtg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1.40020E1 5.70180E2
3.43760E1 8.19860E2
4.13750E1 4.68980E4
4.09290E1 2.42710E6
4.96530E1 3.43680E8
8.57900E1 0.00000E+0
1.78880E+0 0.00000E+0
0.00000E+0
2.58100E1
8.51700E2
4.07650E4
3.13870E5
4.87170E6
6.84990E7
0.00000E+0
0.00000E+0
2.51830E1
1.48330E1
1.11990E2
1.74210E3
3.31670E4
0.00000E+0
0.00000E+0
0.00000E+0
8.40020E2
2.73920E1
4.18910E2
7.97900E3
0.00000E+0
0.00000E+0
0.00000E+0
5.17450E5
1.16390E1
3.35690E1
4.00090E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.46820E3
4.79330E1
3.65440E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
8.42680E2
1.67540E+0
Table B3 7-Group XSs of the Homogenized Fission Chamber Pin Cell (cm1). Group
1
2
3
4
5
6
7
rtg rfg mrfg vg r1g s r2g s r3g s r4g s r5g s
1.40010E1 2.77136E9
3.43750E1 3.36592E9
4.13750E1 2.67516E7
4.09330E1 3.02584E6
4.96580E1 8.39346E8
8.58060E1 4.11455E7
1.79060E+0 1.21000E6
7.65680E9
8.28821E9
6.51080E7
7.36429E6
2.04280E7
1.00140E6
2.94490E6
5.87910E1 5.70260E2
4.11760E1 8.19660E2
3.39060E4 4.68820E4
1.17610E7 2.42630E6
0.00000E+0 3.43560E8
0.00000E+0 0.00000E+0
0.00000E+0 0.00000E+0
0.00000E+0
2.58090E1
8.51590E2
4.07600E4
3.13830E5
4.87110E6
6.84910E7
0.00000E+0
0.00000E+0
2.51900E1
1.48260E1
1.11940E2
1.74130E3
3.31520E4
0.00000E+0
0.00000E+0
0.00000E+0
8.41320E2
2.73840E1
4.18800E2
7.97680E3
0.00000E+0
0.00000E+0
0.00000E+0
5.17530E5
1.16400E1
3.35730E1
4.00130E2
235
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235 Table B3 (continued) Group 6g s 7g s
r r
1
2
3
4
5
6
7
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.46860E3
4.79430E1
3.65510E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
8.43720E2
1.67710E+0
Table B4 7-Group XSs for the Homogenized Moderator (cm1). Group
1
2
3
4
5
6
7
rtg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1.59206E1 4.44777E2
4.12970E1 1.13400E1
5.90310E1 7.23470E4
5.84350E1 3.74990E6
7.18000E1 5.31840E8
1.25445E+0 0.00000E+0
2.65038E+0 0.00000E+0
0.00000E+0
2.82334E1
1.29940E1
6.23400E4
4.80020E5
7.44860E6
1.04550E6
0.00000E+0
0.00000E+0
3.45256E1
2.24570E1
1.69990E2
2.64430E3
5.03440E4
0.00000E+0
0.00000E+0
0.00000E+0
9.10284E2
4.15510E1
6.37320E2
1.21390E2
0.00000E+0
0.00000E+0
0.00000E+0
7.14370E5
1.39138E1
5.11820E1
6.12290E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
2.21570E3
6.99913E1
5.37320E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.32440E1
2.48070E+0
Table B5 7-Group XSs for the Control Rod (cm1). Group
1
2
3
4
5
6
7
rtg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1.92550E1 1.17520E1
4.51960E1 7.34290E2
7.58420E1 3.61350E4
7.90490E1 1.65160E6
8.16430E1 2.23750E8
1.19770E+0 0.00000E+0
2.28680E+0 0.00000E+0
0.00000E+0
3.91940E1
5.48690E2
2.61330E4
2.01230E5
3.12250E6
4.38270E7
0.00000E+0
0.00000E+0
6.04530E1
9.74610E2
7.34650E3
1.14280E3
2.17570E4
0.00000E+0
0.00000E+0
0.00000E+0
3.47440E1
1.94200E1
2.96670E2
5.65070E3
0.00000E+0
0.00000E+0
0.00000E+0
6.84300E5
1.74270E1
2.52050E1
2.99190E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.63590E3
4.57850E1
2.78170E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
7.45640E2
1.66260E+0
References Beckert, C., Grundmann, U., 2008. Development and verification of a nodal approach for solving the multigroup SP3 equations. Ann. Nucl. Energy 35, 75–86. Haugh, B., Mohamed, A., 2012. Comparison of a NuScale SMR conceptual core design using CASMO5/simulate5 and MCNP5. PHYSOR 2012, Knoxville, Tennessee, USA. April 15–20, 2012. Kozlowski, T., Xu, Y., Downar, T.J., Lee, D.J., 2011. Cell homogenization method for pin-by-pin neutron transport calculations. Nucl. Sci. Eng. 169, 1–18. Lee, D.J., Downar, T.J., Kim, Y.H., Wu, H., Cao, L., Li, Y., 2014. A nodal and finite pinby-pin heterogeneous three-dimensional light water reactor diffusion calculations. Nucl. Sci. Eng. 146, 319–339. Lewis, E.E., Carrico, C.B., Palmiotti, G., 1996. Variational nodal formulation for the spherical harmonics equations. Nucl. Sci. Eng. 122, 194. Lewis, E.E., Smith, M.A., Yang, W.S., Wollaber, A., 2011. Preconditioned Krylov And Gauss-Seidel Solutions of Response Matrix Equations. Proc. M&C 2011, Rio, Brazil, May 8–12 (2011). Rohde, U. et al., 2016. The reactor dynamics code DYN3D – models, validation and applications. Prog. Nucl. Energy 89, 170–190. Roy, R., 2010. Reactor core methods. In: Azmy, Y., Sartores, E. (Eds.), Nuclear Computational Science, a Century in Review. Springer, New York. Smith, M.A., Lewis, E.E., Shemon, E.R., 2014. DIF3D-VARIANT 11.0: A Decade of Updates, ANL/NE-14/1. Argonne National Laboratory (Jan. 2014).
Tatsumi, M., Yamamoto, A., 2002. Object-oriented three-dimensional fine-mesh transport calculation on parallel/distributed environments for advanced reactor core analyses. Nucl. Sci. Eng. 141, 190. Tatsumi, M., Yamamoto, A., 2003. Advanced PWR core calculation based on multigroup nodal-transport method in three-dimensional pin-by-pin geometry. J. Nucl. Sci. Technol. 40 (6), 376–387. Yang, W.S., Palmiotti, G., Lewis, E.E., 2001. Numerical optimization of computing algorithms of the variational nodal method based on transformation of variables. Nucl. Sci. Eng. 139 (2), 174–185. Yang, W., Zheng, Y., Wu, H., Cao, L., Li, Y., 2014. High-performance whole core pinby-pin calculation based on EFEN-SP3 method. Nucl. Power Eng. (in Chinese) 35, 164–167. Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., 2016. A variational nodal approach to 2D/ 1D pin resolved neutron transport: I diffusion theory. Trans. Am. Nucl. Soc. 114. Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., 2016. A variational nodal approach to 2D/1D pin resolved neutron transport: II spherical harmonics. Trans. Am. Nucl. Soc. 114. Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., Wu, H., 2017. Assessment of 2DTransport/1D-Diffusion Approximations in a Pin Resolved Variational Nodal Method for PWR Calculations. Proc. M&C 2017, Jeju, Korea. April 16–20 (2017). Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., Wu, H., 2017b. A variational nodal approach to 2D/1D pin resolved neutron transport for pressurized water reactors. Nucl. Sci. Eng. 186, 120–133.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Acceleration of within group iteration for pin-by-pin calculations Tengfei Zhang a,⇑, E.E. Lewis b, M.A. Smith c, W.S. Yang d, Yongping Wang e, Hongchun Wu e a
Shanghai Jiao Tong University, Shanghai, China Department of Mechanical Engineering, Northwestern University, Evanston, IL, United States c Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL, United States d School of Nuclear Engineering, Purdue University, West Lafayette, IN, United States e Xi’an Jiaotong University, Xi’an, Shaanxi, China b
a r t i c l e
i n f o
Article history: Received 16 July 2017 Received in revised form 29 September 2017 Accepted 2 October 2017
Keywords: Heterogeneous variational nodal method Coarse nodes acceleration Acceleration of within-group iteration
a b s t r a c t In thermal reactor physics fine mesh neutron diffusion calculations in which homogenization is only at the pin cell level are of considerable interest. However, the slow convergence of iterations on finemesh spatial grids limits their applicability to practical problems. We present here an efficient way to accelerate the within-group iterations for pin-by-pin calculations based on a two-dimensional heterogeneous variational nodal method (VNM). Response matrix (RM) equations are formulated that incorporate multiple pins within each node. Within the nodes, finite elements in the x-y plane are employed to describe the piecewise constant heterogeneous geometry. On the nodal interfaces orthogonal polynomials are employed to approximate current distributions. The RM equations are solved using the Red-Black Gauss-Seidel (RBGS) iteration. Investigations on the coarse nodes acceleration (CNA) by combining homogenized pin cell nodes into larger heterogeneous nodes are performed. A series of meshing schemes are examined with a 2D small modular reactor core problem. With sufficient interface expansion order, CNA do not cause significant errors to either eigenvalues or fission rate distributions compared with fine mesh calculations. It is demonstrated that CNA accelerate the RBGS iteration significantly and achieves favorable accuracy-efficiency trade-off. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The traditional two-step method has been utilized as the treatment to the neutronic analysis of light water-cooled reactors (LWR) for decades (Roy, 2010). The first step is the performance of multi-group neutron transport calculations for each fuel assembly, with fine-grained detail of the two-dimensional assembly. The resulting flux spectrum is used to generate the few-group constants by spatial homogenization and energy group condensation. The second step is the execution of whole-core calculations, usually with the few-group coarse-mesh diffusion approximation. Due to adequate cost-accuracy trade-off, the two-step method has prevailed in industrial use. However, in whole core calculations, pin power information is not directly obtainable as a result of the homogenization over each coarse assembly mesh. Therefore pin power reconstruction must be performed to approximate high resolution results. ⇑ Corresponding author. E-mail addresses: [email protected] (T. Zhang), [email protected] (E.E. Lewis), [email protected] (M.A. Smith), [email protected] (W.S. Yang), [email protected] (Y. Wang), [email protected] (H. Wu). https://doi.org/10.1016/j.anucene.2017.10.006 0306-4549/Ó 2017 Elsevier Ltd. All rights reserved.
With the advancement of computational resources over the past decade, considerable effort has been expended on the pursuit of more advanced methods for mitigating the errors inherent in the traditional two-step procedure. One prominent approach is the pin-by-pin method which promotes the homogenization to the pin-cell level (Tatsumi and Yamamoto, 2002; Kozlowski et al., 2011). By performing pin-cell level fine mesh whole-core calculations, heterogeneity inside a fuel assembly as well as spectral mismatch at interfaces between different types of fuel assemblies can be treated more precisely, compared to the conventional coarse mesh methods. Moreover, pin cell information is directly obtained in the resulting outputs whereby eliminating the errors caused by pin power reconstructions. For these reasons, such pin-by-pin methods are incorporated in the production codes SCOPE2 (Tatsumi and Yamamoto, 2003), DYN3D (Beckert and Grundmann, 2008; Rohde and et al., 2016), and EFEN (Yang et al., 2014). However, power reactors consist of tens or hundreds of assemblies, resulting in millions of fine meshes. In addition, with pin-wise homogenization, the material heterogeneity is much more severe than that with the assembly-wise homogenization (Lee et al., 2014). In this case, every within-group sweep consumes substantial amount of time, and the optically thin meshes
226
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
deteriorate the convergence rate of within-group iteration (Lewis et al., 2011). No matter which method is selected – nodal (Beckert and Grundmann, 2008), finite difference (Tatsumi and Yamamoto, 2002; Tatsumi and Yamamoto, 2003) or response matrix (RM) based (Lewis et al., 2011) – the computational efficiency proves a crucial challenge limiting the application of the pin-by-pin calculations. Although parallel computing has achieved substantial speedups, pin-by-pin calculations remain burdensome. A typical 3D PWR problem with the nodal SP3 method, for example, requires more than 24 h on a single processor, and around 20 min on 100 processors (Beckert and Grundmann, 2008). To improve efficiency for a RM based method, investigators have introduced the orthogonal matrix aggregation (Lewis et al., 2011) (OMA) method. By combining N N fine mesh RMs with one degree of freedom (DOF) per interface into a coarse mesh RM with N orthogonal DOF per interface, favorable acceleration was achieved. However, the study was restricted to the piecewise constant discretization on interfaces and the cost-accuracy tradeoff was not surveyed thoroughly. In this paper we propose a heterogeneous variational nodal method (VNM) which is capable of accelerating the within-group iteration for pin-by-pin calculations by coarsening nodes. The method is realized by combining homogenized pin cell nodes into coarser nodes with piecewise constant cross sections (XSs) over each pin cell. The VNM has been applied predominantly to fast reactor problems (Lewis et al., 1996; Smith et al., 2014), and can be extended to describe unstructured geometry by utilizing finite element submeshes within each node (Zhang et al., 2017a,b). Generally in VNM the within-group iteration is performed using the standard Red-Black Gauss-Seidel (RBGS) iteration algorithm. This procedure expends the bulk of CPU time in VNM computation since it is the innermost iteration where the RM equations are solved iteratively (Lewis et al., 1996). Therefore here we focus our attention on improving within-group solution algorithms, employing the 2D diffusion approximation as a testbed. We first formulate the 2D diffusion heterogeneous VNM which serves as the basis for this research. Within each node, finite element trial functions in x-y are employed to describe the heterogeneous structures, while orthogonal polynomials are used to approximate the neutron current distributions along nodal interfaces. Continuity conditions are imposed on the neutron current moments across the nodal interfaces. Standard fission source iteration and RBGS iteration are performed to solve the resulting RM equations. To improve efficiency in VNM with unstructured geometry capability, the flat source region (FSR) acceleration (Zhang et al., 2017b) is used by evaluating the averaged scalar flux on the finite elements in place of the finite element nodes (i.e., vertices). The FSR is combined with heterogeneous coarse nodes to accelerate the RBGS iteration algorithm as for pin-by-pin calculations. We organize the rest of the paper as follows. In Section 2, we formulate the RM equations along with their spatial matrix components. In addition, FSR acceleration and coarse node acceleration (CNA) is elaborated to reduce further the computational costs. In Section 3, CNA is tested on an idealized 2D small modular reactor (SMR) core problem with pin-wise homogenized XSs. The orthogonal polynomial order on the interfaces is examined, convergence studies are performed, and the accuracy-efficiency trade-offs are compared. In Section 4, conclusions from this study are drawn and future directions discussed. For clarity, the nomenclatures present in the paper are listed in Appendix A (Table A). 2. Theory model In the diffusion approximation, the even-parity within-group equation is given as:
rð3rt Þ1 r/ð~ rÞ þ rr /ð~ rÞ ¼ qð~ rÞ;
ð1Þ
where rt and rr are the total and removal XSs (cm ), respectively. Other notations are conventional. The variational functional of the problem domain with respect to the scalar flux / and neutron currents j can be written as the superposition of the contributions from individual nodes and node surfaces: 1
F½/ð~ rÞ; jð~ rÞ ¼
X F m ½/ð~ rÞ; jð~ rÞ:
ð2Þ
m
The within-group nodal functional of each node can be written as:
F½/ð~ rÞ; jð~ rÞ ¼
Z
n o 2 2 dV ð3rt Þ1 ½r/ð~ rÞ þ rr /ð~ rÞ 2/ð~ rÞqð~ rÞ V Z rÞjð~ rÞ þ 2 dC/ð~ C
where
qðrÞ ¼
X
rsgg0 ð~rÞ/g0 ð~rÞ þ k1 vg
g 0 –g
X
mrfg0 ð~rÞ/g0 ð~rÞ;
ð3Þ
ð4Þ
g0
and C is the outer boundary. Requiring this functional to be stationary with respect to variations in d/ within V yields Eq. (1) as the Euler-Lagrange equation. Across the lateral and axial interfaces, j is defined to be continuous, respectively. 2.1. Ritz discretization In the standard VNM approach, the flux distribution is discretized with orthogonal polynomials both within nodes and on nodal interfaces. Here we limit the discussion on twodimensional (2D) problems for simplicity, but it can be extended to three-dimensional problems easily. Denoting a vector by an underline, the flux within a 2D node can be expanded as
/ð~ rÞ ¼ g T ðx; yÞ/;
ð5Þ
where gðx; yÞ is a vector of orthogonal, iso-parametric finite element trial functions and / is the vector of the scalar flux moments. For example, quadratic triangular finite elements are used for the numerical calculations presented in Section 3. Using the orthogonal polynomials f g defined in terms of local coordinate g as trial functions, the current j1 at a radial interface sented as
j1 ðgÞ ¼ f Tg ðgÞj1 :
1 ¼ 12 D1 can be repre-
1 2
1 ¼ D1; 1; g ¼ x; y;
where D1 is the node width in the
ð6Þ
1 direction.
2.2. Response matrix equations By substitutions of Eqs. (5) and (6) in Eq. (3), we can put the functional in a discretized form: Z Z Z h i 1 dxdyg g T þ dxdyrx g rx g T þ dxdyry g ry g T F m /; j ¼ /T r1 t 3 Z Z þ dxdyrr gg T / 2/q þ 2/T dy gjx f Ty jx þ 2/T Z dy gjy f Tx jy ð7Þ
where x and y are suppressed in the spatial integrals, and
Z
q¼
v
dVgðx; yÞqð~ rÞ
ð8Þ
For brevity we employ the matrix notations shown in Table 1. Thus Eq. (7) is rewritten as
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
F m ½/; j ¼ /T A// / 2/q þ 2/T M /j j:
ð9Þ
where the variables with double underlines represent matrices. Based on the variational principle, requiring the discretized functional in Eq. (9) to be stationary with respect to variation in /. yields: 1 / ¼ A1 // q A// M /j j;
ð10Þ
which represents the neutron balance relationship within each node. Requiring the functional to be stationary with respect to the interface current term j:, stipulates that the following even parity surface moments, u, must be continuous across the nodal interfaces:
u ¼ MT/j /:
ð11Þ
Inserting Eq. (10) in Eq. (11) we obtain
u ¼ u Gj
ð12Þ
where
G ¼ MT/j A1 // M /j
ð13Þ
u ¼ M T/j A1 // q:
ð14Þ
To cast Eq. (12) into a RM form, we utilize a linear transformation of variables based on the relationships of partial currents:
j ¼
1 1 u j 4 2
ð15Þ
where j and jþ correspond respectively to neutrons entering and leaving the node. Inserting Eq. (15) into Eq. (12) yields the RM equation
jþ ¼ Bu þ R j
ð16Þ
where the expression of B and R matrices are given as
B¼
1 1 1 GþI 2 2
R¼I
ð17Þ
1 1 GþI G: 2
ð18Þ
Eq. (16) shows the relationship between interface currents and neutron sources. Using the standard RBGS iteration scheme, Eq. (16) can be solved iteratively as
~r þ Rrb j jþðlþ1Þ ¼q r b
ðlÞ
ð19Þ
~r þ Rrb j jþðlþ1Þ ¼q r b
ðlþ1Þ
ð20Þ
In the RBGS iteration, Eqs. (19) and (20) are successively solved using the relations between outgoing and incoming currents from the adjoining ‘red’ and ‘black’ nodes. This procedure is the most time-consuming process in VNM because it is the innermost calculation that has to be performed iteratively during each outer iteration. The spatial matrices in Eqs. (9)–(18) are defined in Table 1 in terms of the trial functions f and g. The integrals are evaluated with standard numerical quadrature schemes (Lewis et al., 2011). 2.3. Flat source region and coarse node acceleration The computational cost to solve the within-group RM equation in Eq. (16) increases with the total number of unknowns, that is, the number of nodes times the number of unknowns per node. Therefore, if the number of nodes and the number of unknowns per node can be reduced without any significant loss of accuracy, the efficiency of a RM based method can be improved significantly. The number of unknowns per node can be reduced using the FSR acceleration scheme (Zhang et al., 2017b) which we proposed before without a detailed description. Likewise, the number of nodes can be reduced by employing the OMA method (Lee et al., 2014) in which N N fine mesh RMs with one DOF per interface into a coarse node RM with N orthogonal DOF per interface. In this study, the CNA is utilized to accelerate the within-group iteration for pin-by-pin calculations by combining homogenized pin cell nodes into coarser nodes with piecewise constant XSs over each pin cell. In the FSR acceleration method, the averaged scalar flux is evaluated on the finite elements in place of the finite element vertices. As an example, consider the finite element mesh for a homogenized pin cell node shown in Fig. 1. The node is meshed with 4 quadratic triangular finite elements consisted of 13 finite element vertices (namely nodes in FEM). The finite element meshes employed within the node require evaluation of group sources and group fluxes on each finite element node, corresponding to 13 DOF. However, the DOF is reduced to 4 if the averaged scalar flux is evaluated on the finite elements. We start FSR by forming a vector of the lateral averages over each of the finite elements: _
/ ¼ N1
Z
dxdyhðx; yÞg T ðx; yÞ/
Z
H¼
dxdygðx; yÞhT ðx; yÞ
we have
~ ¼ Bu q
/ ¼ N1 HT /:
_
Table 1 Spatial Matrices. A// ¼ 13 F þ 13 P xx þ P yy þ F a h i M /j ¼ M /jx M/jy R T P ii ¼ dxdyr1 i ¼ x; y t ri g ri g R R T F a ¼ dxdyrr gg T F ¼ dxdyr1 t gg R R T M /jy ¼ dyjx f y M /jy ¼ dyjy f Tx
ð22Þ
where hðx; yÞ is a piecewise constant trial function vector, the ith component of which is unity over the area of the ith finite element and zero elsewhere; N is a diagonal matrix of the finite element areas. Defining a matrix
where r and b represent respectively the ‘red’ and ‘black’ nodes, l is the iteration index and
ð21Þ
227
Fig. 1. The Finite Element Mesh for a Homogenized Pin Cell.
ð23Þ
ð24Þ
228
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Fig. 2. Five Different Node Meshing Schemes.
Fig. 3. The 2D Schematic View of the SMR Problem.
229
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235 Table 2 Results vs. Node Meshing Schemes for the Unrodded SMR problem. f g ðgÞ order
Eigenvalue
Error – pcm
RMS
MAX
17 17
0 1 2
1.28635 1.28648 1.28654
15 5 Reference
0.17 0.05
0.99 0.32
5 5a
0 1 2 3
1.28615 1.28643 1.28652 1.28652
30 9 2 2
1.29 0.44 0.08 0.07
8.27 2.04 0.41 0.30
5 5b
0 1 2 3
1.28527 1.28611 1.28643 1.28650
99 34 9 4
1.69 0.68 0.22 0.07
6.59 2.36 0.88 0.28
33
0 1 2 3 4 5
1.28609 1.28618 1.28624 1.28634 1.28642 1.28649
35 28 24 16 10 4
1.61 0.76 0.59 0.34 0.22 0.12
15.05 3.45 2.17 1.26 0.92 0.46
11
0 1 2 3 4 5 6
1.28745 1.28658 1.28652 1.28653 1.28654 1.28654 1.28654
70 3 2 1 1 0 0
2.64 0.65 0.51 0.38 0.19 0.10 0.04
32.10 9.42 7.52 5.66 2.82 1.23 0.41
Fission Rate Error -%
Table 3 Results vs. Node Meshing Schemes for the Rodded SMR Problem. g
f g order
Eigenvalue
Error – pcm
RMS
MAX
17 17
0 1 2
1.22467 1.22456 1.22451
13 4 Reference
0.34 0.11
2.20 0.50
5 5a
0 1 2 3
1.22489 1.22459 1.22453 1.22453
31 7 1 1
1.32 0.49 0.09 0.08
8.15 2.70 0.65 0.54
5 5b
0 1 2 3
1.22541 1.22481 1.22461 1.22455
73 24 8 3
2.41 0.95 0.29 0.10
13.12 5.88 1.87 0.59
33
0 1 2 3 4 5
1.22539 1.22469 1.22467 1.22462 1.22458 1.22453
72 14 13 9 6 2
1.68 0.96 0.75 0.45 0.33 0.13
13.78 5.56 4.31 2.46 1.25 0.61
11
0 1 2 3 4 5 6
1.22853 1.22523 1.22446 1.22448 1.22450 1.22451 1.22451
328 58 4 3 1 0 0
3.68 1.24 0.50 0.37 0.19 0.10 0.04
32.04 10.25 7.14 5.42 2.76 1.10 0.40
This treatment substantially reduce the number of flux values that needs to be stored, because the length of / is equal to the _
number of finite element vertices, while the length of / is equal to the number of finite elements. Substitution of Eq. (10) in Eq. (24), we obtain _
1 T 1 / ¼ N1 HT A1 // q N H A// M /j j
ð25Þ
Pin Fission Rate Error -%
1 T 1 in which N1 HT A1 // and N H A// M /j are evaluated when forming the response matrices. We next evaluate qð~ rÞ of Eq. (4) by using the condition that within a pin-cell node, the XSs are piecewise constant in x and y with a distinct value over each finite element. Thus we may write reaction rates as
_
rðx; yÞ/ðx; yÞ ¼ hT ðx; yÞR /
ð26Þ
230
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
where R is a diagonal matrix containing the XS for each finite element. In other words, the XSs are treated as piecewise constants within each node, allowing for an exact heterogeneous description of the nodal geometry. Including the appropriate group subscripts on the XSs and averaged fluxes, we may combine Eqs. (4) and (26) to obtain
qð~ rÞ ¼ hT ðx; yÞ
X g 0 –g
_
Rsgg0 /g0 þ k1 vg
X g0
_
!
mRfg0 /g0 :
ð27Þ
To find the discretized group source we insert this expression into Eq. (8) we have
q¼H
X g 0 –g
_
1
Rsgg0 /g0 þ k vg
X g0
_
!
mRfg0 /g0
ð28Þ
thus completing the discretization process. Eq. (28) is used to construct the source vectors in Eqs. (19), (20) and (25). In within-group iteration, Eqs. (19) and (20) are solved with the standard RBGS iter_
ation as illustrated in Section 2.2, then / is updated by Eq. (25). _
Obviously with FSR the length of the solution vector / decreases, hence the matrix operation in Eq. (25) are accelerated due to the 1 T 1 reduced size of N1 HT A1 // and N H A// M /j . In the heterogeneous VNM, each node is constructed to include one or more homogenized pin cells. Therefore, by utilizing finite element meshes and the piece-wise constant description of XSs, coarse nodes are employed without altering the original heterogeneous configuration. As an example, Fig. 2 shows five different coarse node meshes for a 17 17 fuel assembly. In each meshing scheme, coarse nodes are outlined by red lines. The 17 17 mesh consists of single pin-cell nodes, while the other meshes are composed of heterogeneous nodes containing several pin cells. In order to retain solution accuracy with the coarse nodes, the order of orthogonal polynomial f g ðgÞ in Eq. (6) is increased to represent more accurately the spatial variation of neutron on each coarse nodal interface. 3. Numerical results The foregoing heterogeneous VNM with the diffusion approximation is implemented in the Fortran 90 code PANX (Purdue – Argonne – Northwestern – Xi’an) as a branch option to the pin-resolved transport solver. The code is based on NODAL
(Smith et al., 2014), a diffusion code being developed at Argonne National Laboratory. The within-group RBGS iteration scheme and the standard fission source iteration scheme without acceleration are embedded in PANX. The performance of the proposed acceleration schemes for the within-group iterations of a RM based method for pin-by-pin LWR calculations were examined using a 2D Small Modular Reactor (SMR) problem with pin-wise homogenized XSs, which are based on the released NuScale design (Haugh and Mohamed, 2012). The problem incorporates two different configurations: rodded and unrodded cases. Fig. 3 shows the schematic view of the core and assembly configurations. The SMR problem utilizes a 17 17 UO2 pin-cell assembly with a pin pitch of 1.26 cm. Since the main interest of this work is to study the effect of CNA on acceleration of within-group iteration, we assumed a uniform configuration of UO2 assemblies in the unrodded case. In the rodded case, the guide tubes of the control rod assembly are replaced with control rod materials. Vacuum boundary conditions are applied to all of the four boundaries. 7-group pin-cell-homogenized XSs for standard PWR pin cell models are employed, which are included in Appendix B (Tables B1–B6). Every pin cell was meshed with 4 quadratic triangular finite elements as shown in Fig. 1, and five different nodal meshes shown in Fig. 2 were considered. The total number of finite element meshes remains the same in all the five node meshing schemes. A maximum number RBGS iteration of 5 was used to guarantee a relatively tight within-group convergence that ensures the convergence of the calculation. The convergence criteria for eigenvalue and fission source were set to 1 106 and 1 105 , respectively, in L2 norm. The calculations were performed on a single 2.60 GHz Intel i7-U6600U processor without acceleration on the outer iterations. To compare the computational efficiency of five different meshing schemes on a consistent basis, the solution accuracy obtained with the finest mesh was used as a reference. As the total number of finite elements within nodes is fixed in all the five meshing schemes, the order of interface polynomial f g ðgÞ in Eq. (6) becomes the dominating factor that influences the solution accuracy. Intuitively, when the nodes are coarsened, higher order f g ðgÞ should be needed to represent the spatial distribution of interface currents. Therefore, the order of interface polynomial f g ðgÞ required to achieve the reference accuracy obtained with the finest mesh was first examined for each nodal mesh. Subsequently, the compu-
Fig. 4. Fission Rate Error Distribution of 5 5a F3.
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Fig. 5. Fission Rate Error Distribution of 5 5b F3.
Fig. 6. Fission Rate Error Distribution of 3 3 F5.
Fig. 7. Fission Rate Error Distribution of 1 1 F5.
231
232
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Fig. 8. Convergence Rates of the RBGS Iteration.
3.1. Accuracy
Table 4 Performance Gain in the RBGS Iteration by CNA. Heterogeneous Nodes
17 17 F2 5 5a F3 5 5b F3 3 3 F5 1 1 F6
Gain Unrodded
Rodded
1.0 6.3 6.3 7.9 54.5
1.0 6.5 6.5 8.2 56.6
Using each of the five nodal meshes in Fig. 2, the 2D SMR problems were solved by varying the order of interface polynomial f g ðgÞ in Eq. (6). Tables 2 and 3 show the root mean square (RMS) and maximum (MAX) errors of the fission rates in individual pin cells as well as the eigenvalue and its error for the unrodded and rodded configurations, respectively. The fission rate ðFRÞi of the ith pin cell is defined by the volumetric average of the fission rate over the finite elements e in pin cell i:
ðFRÞi ¼ tational efficiencies of RBGS iteration achieved with different nodal meshes were compared, and the trade-off between accuracy and efficiency was investigated.
G X X
, g;e f /e V e
r
g¼1 e2i
X Ve
ð29Þ
e2i
where Ve is the volume of the finite element e, and /eg represents the group g scalar flux of the finite element e, which is directly obtained using Eq. (25).
Fig. 9. Eigenvalue Error vs. Number of FLOPs for the Rodded Case.
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
When the finest 17 17 mesh is used, the increase in the interface polynomial from linear to quadratic yield only marginal improvement in eigenvalue (5 pcm for the unrodded case and 4 pcm for the rodded case) and in RMS fission rate (0.05% for the unrodded case and 0.11% for the rodded case). These results indicate that for the 17 17 mesh, the quadratic interface polynomial yields an asymptotically converged solution. Therefore, the solution of the 17 17 F2 case was taken as the reference solution. Here we introduce a brief notation n n Fm with the number of nodes on each side of the assembly n and the interface polynomial order m. For a fixed nodal mesh, the solution accuracy is improved monotonically with increasing order of interface polynomial for both the rodded and unrodded cases. In addition, the order of interface polynomial required to achieve the same solution accuracy increases as the node becomes coarser. More importantly, the results in Tables 2 and 3 show that each of the coarse mesh yields a solution comparable to the reference solution obtained with 17 17 F2 by using a higher order of interface polynomial. The solutions of 5 5a F3, 5 5b F3, 3 3 F5, and 1 1 F6 show comparable accuracy to the reference solution. Fig. 4 through Fig. 7 show the quarter-core distributions of fission rate errors for 5 5a F3, 5 5b F3, 3 3 F5 and 1 1 F6. The nodes are outlined by white grids, and the center line of the core is described with black dashed lines. Note that the positions near inserted control rod exhibit slightly larger errors than others. Also observe that the pin fission rates inside each node agree well with the reference solution. Although minor errors are observed along nodal interfaces, the overall error is negligible with the maximum error of less than 0.6% among all the cases.
3.2. Computational efficiency Although CNA requires higher order interface polynomials to obtain solutions that closely approximate the reference fine node solution, the RBGS iteration converges substantially faster. To measure the relative efficiency of different nodal meshing schemes in accelerating the RBGS iteration, we employ the number of floating point operations (FLOPs) to measure computational effort. The number of DOFs (N) at each interface are 3, 4, 4, 6, and 7 for 17 17 F2, 5 5a F3, 5 5b F3, 3 3 F5 and 1 1 F6, respectively. Each R matrix multiplication in Eqs. (19) and (20) requires ð4NÞ2 FLOPs. Thus the total number of FLOPs per RBGS sweep becomes (4N)2 times the number of nodes in the problem. By recording the number of RBGS sweeps performed during the outer iterations, the number of FLOPs was evaluated during each calculation. Plots for the residual L2 norm of fission source vs. FLOPs expended are shown in Fig. 8. Observe that the convergence rates do not depend noticeably on the interface polynomial order, thus only the highest polynomial orders are plotted in Fig. 8. For both rodded and unrodded cases, the fission source converges more rapidly with increasing size of heterogeneous nodes. Moreover, the 5 5a F3 and 5 5b F3 configurations show nearly identical convergence rates because they have the same number of coarse nodes. Table 4 compares the performance gains for the RBGS iteration achieved by applying heterogeneous coarse nodes in term of the reduced number of FLOPs relative to the reference case 17 17 F2. Clearly the performance gain increases monotonically with increasing coarse node size. In particular, for both the unrodded and rodded cases, the use of 1 1 F6 reduces the number of FLOPs required for a converged solution by a factor of more than 50.
233
3.3. Accuracy-Efficiency Trade-off Fig. 9 compares the eigenvalue error versus the number of FLOPs of the five node meshing schemes for the rodded case. The results for different meshing schemes are distinguished by different marks, and the corresponding orders of interface polynomials are attached by number labels. The 1 1 F0 results are excluded in Fig. 9 because an eigenvalue error of 328 pcm was too large to be included in a compact plot. Fig. 9 clearly indicates that with the same order of interface polynomial, coarser nodes require fewer FLOPs compared to the finer meshes. With increasing order of interface polynomials, the eigenvalue error decreases monotonically while computing costs increase accordingly. The coarser nodes clearly show superior trade-offs between accuracy and efficiency. 4. Discussion This paper presents a preliminary study on accelerating within-group iteration of a RM based method for pin-by-pin diffusion calculations. By incorporating finite element meshes in the nodal framework, a 2D heterogeneous VNM is formulated with the diffusion approximation. The FSR acceleration method is derived focused on forming the RM equations with reduced DOFs. By combining the FSR acceleration method with coarse heterogeneous nodes, coarse-mesh RM equations are formed to preserve the original problem geometry. Using a SMR core problem, the effect of CNA on the convergence of RBGS iteration has been investigated. Rodded and unrodded problems of the 2D SMR with pin-cellhomogenized XSs were solved using five different nodal meshes for a 17 17 fuel pin assembly. Results show that coarse heterogeneous nodes can reproduce the fine mesh solution as the order of interface polynomials is increased. Although the interface polynomial order is increased, the total DOFs per assembly decreases with increasing size of coarse mesh; thus the number of FLOPs required for converged solutions is reduced significantly. Coarser nodes also show superior trade-offs between accuracy and efficiency. It should be noted that employing heterogeneous coarse nodes shifts the computational burden to forming response matrices: the times required to form the response matrices range from approximately 20% of the solution time for the 5 5 configurations to substantially larger than the solution times for the coarser grids. However, it will not be a significant roadblock to carrying out calculations compared with speeding up the RBGS iterations. By transforming the response matrices into block-diagonal forms (Yang et al., 2001), the computational costs of forming response matrices can be reduced significantly. More importantly, the formations are performed only once instead of at each iteration, and are completely un-coupled, thus the processes can be easily threaded with high efficiency. With parallelization they should consume only a small fraction of the solution times obtained in this paper. Therefore it can be expected that large gains in computing efficiency can be achieved with heterogeneous coarse nodes. These results suggest a possible direction for accelerating the within-group iteration for pin-by-pin calculations. Moreover, the proposed method can be extended to a 3D diffusion formulation by including orthogonal z polynomials (Zhang et al., 2016), as well as to a transport capability by employing spherical harmonics expansions as employed in VARIANT (Lewis et al., 1996; Zhang et al., 2016). Works on reducing the response matrix formation time and employing 3D CNA are underway.
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T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235
Appendix A Table A Nomenclatures. Abbreviations
Meanings
VNM RM RBGS OMA CNA DOF FSR FEM FLOP n n Fm
Variational Nodal Method Response Matrix Red-Black Gauss-Seidel Orthogonal Matrix Aggregation Coarse Node Acceleration Degree Of Freedom Flat Source Region Finite Element Method FLOating-Point operation The CNA meshing scheme using n nodes on each side of the assembly and up to mth polynomial order on nodal interfaces
Appendix B
Table B1 7-Group XSs for the Homogenized UO2 Pin Cell (cm1). Group g t g f
r r mrfg vg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1
2
3
4
5
6
7
1.70270E1 4.25641E3
3.64310E1 4.79383E4
5.27240E1 3.70264E3
5.67400E1 1.04931E2
4.85130E1 1.02095E2
7.70080E1 4.68075E2
1.51810E+0 1.17245E1
1.18390E2
1.18620E3
9.01160E3
2.55381E2
2.48479E2
1.13920E1
2.85351E1
5.87910E1 9.34960E2
4.11760E1 7.14850E2
3.39060E4 3.02070E4
1.17610E7 1.54010E6
0.00000E+0 2.17970E8
0.00000E+0 0.00000E+0
0.00000E+0 0.00000E+0
0.00000E+0
3.06980E1
5.48680E2
2.58660E4
1.99160E5
3.09050E6
4.33790E7
0.00000E+0
0.00000E+0
4.05890E1
9.72560E2
7.24550E3
1.12710E3
2.14580E4
0.00000E+0
0.00000E+0
0.00000E+0
2.95370E1
1.83810E1
2.77100E2
5.27800E3
0.00000E+0
0.00000E+0
0.00000E+0
1.02290E4
2.14960E1
2.24270E1
2.61260E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.69770E3
4.55210E1
2.43910E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
6.51920E2
1.28240E+0
Table B2 7-Group XSs for the Homogenized Guide Tube Pin Cell (cm1). Group
1
2
3
4
5
6
7
rtg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1.40020E1 5.70180E2
3.43760E1 8.19860E2
4.13750E1 4.68980E4
4.09290E1 2.42710E6
4.96530E1 3.43680E8
8.57900E1 0.00000E+0
1.78880E+0 0.00000E+0
0.00000E+0
2.58100E1
8.51700E2
4.07650E4
3.13870E5
4.87170E6
6.84990E7
0.00000E+0
0.00000E+0
2.51830E1
1.48330E1
1.11990E2
1.74210E3
3.31670E4
0.00000E+0
0.00000E+0
0.00000E+0
8.40020E2
2.73920E1
4.18910E2
7.97900E3
0.00000E+0
0.00000E+0
0.00000E+0
5.17450E5
1.16390E1
3.35690E1
4.00090E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.46820E3
4.79330E1
3.65440E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
8.42680E2
1.67540E+0
Table B3 7-Group XSs of the Homogenized Fission Chamber Pin Cell (cm1). Group
1
2
3
4
5
6
7
rtg rfg mrfg vg r1g s r2g s r3g s r4g s r5g s
1.40010E1 2.77136E9
3.43750E1 3.36592E9
4.13750E1 2.67516E7
4.09330E1 3.02584E6
4.96580E1 8.39346E8
8.58060E1 4.11455E7
1.79060E+0 1.21000E6
7.65680E9
8.28821E9
6.51080E7
7.36429E6
2.04280E7
1.00140E6
2.94490E6
5.87910E1 5.70260E2
4.11760E1 8.19660E2
3.39060E4 4.68820E4
1.17610E7 2.42630E6
0.00000E+0 3.43560E8
0.00000E+0 0.00000E+0
0.00000E+0 0.00000E+0
0.00000E+0
2.58090E1
8.51590E2
4.07600E4
3.13830E5
4.87110E6
6.84910E7
0.00000E+0
0.00000E+0
2.51900E1
1.48260E1
1.11940E2
1.74130E3
3.31520E4
0.00000E+0
0.00000E+0
0.00000E+0
8.41320E2
2.73840E1
4.18800E2
7.97680E3
0.00000E+0
0.00000E+0
0.00000E+0
5.17530E5
1.16400E1
3.35730E1
4.00130E2
235
T. Zhang et al. / Annals of Nuclear Energy 112 (2018) 225–235 Table B3 (continued) Group 6g s 7g s
r r
1
2
3
4
5
6
7
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.46860E3
4.79430E1
3.65510E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
8.43720E2
1.67710E+0
Table B4 7-Group XSs for the Homogenized Moderator (cm1). Group
1
2
3
4
5
6
7
rtg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1.59206E1 4.44777E2
4.12970E1 1.13400E1
5.90310E1 7.23470E4
5.84350E1 3.74990E6
7.18000E1 5.31840E8
1.25445E+0 0.00000E+0
2.65038E+0 0.00000E+0
0.00000E+0
2.82334E1
1.29940E1
6.23400E4
4.80020E5
7.44860E6
1.04550E6
0.00000E+0
0.00000E+0
3.45256E1
2.24570E1
1.69990E2
2.64430E3
5.03440E4
0.00000E+0
0.00000E+0
0.00000E+0
9.10284E2
4.15510E1
6.37320E2
1.21390E2
0.00000E+0
0.00000E+0
0.00000E+0
7.14370E5
1.39138E1
5.11820E1
6.12290E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
2.21570E3
6.99913E1
5.37320E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.32440E1
2.48070E+0
Table B5 7-Group XSs for the Control Rod (cm1). Group
1
2
3
4
5
6
7
rtg r1g s r2g s r3g s r4g s r5g s r6g s r7g s
1.92550E1 1.17520E1
4.51960E1 7.34290E2
7.58420E1 3.61350E4
7.90490E1 1.65160E6
8.16430E1 2.23750E8
1.19770E+0 0.00000E+0
2.28680E+0 0.00000E+0
0.00000E+0
3.91940E1
5.48690E2
2.61330E4
2.01230E5
3.12250E6
4.38270E7
0.00000E+0
0.00000E+0
6.04530E1
9.74610E2
7.34650E3
1.14280E3
2.17570E4
0.00000E+0
0.00000E+0
0.00000E+0
3.47440E1
1.94200E1
2.96670E2
5.65070E3
0.00000E+0
0.00000E+0
0.00000E+0
6.84300E5
1.74270E1
2.52050E1
2.99190E2
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
1.63590E3
4.57850E1
2.78170E1
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
0.00000E+0
7.45640E2
1.66260E+0
References Beckert, C., Grundmann, U., 2008. Development and verification of a nodal approach for solving the multigroup SP3 equations. Ann. Nucl. Energy 35, 75–86. Haugh, B., Mohamed, A., 2012. Comparison of a NuScale SMR conceptual core design using CASMO5/simulate5 and MCNP5. PHYSOR 2012, Knoxville, Tennessee, USA. April 15–20, 2012. Kozlowski, T., Xu, Y., Downar, T.J., Lee, D.J., 2011. Cell homogenization method for pin-by-pin neutron transport calculations. Nucl. Sci. Eng. 169, 1–18. Lee, D.J., Downar, T.J., Kim, Y.H., Wu, H., Cao, L., Li, Y., 2014. A nodal and finite pinby-pin heterogeneous three-dimensional light water reactor diffusion calculations. Nucl. Sci. Eng. 146, 319–339. Lewis, E.E., Carrico, C.B., Palmiotti, G., 1996. Variational nodal formulation for the spherical harmonics equations. Nucl. Sci. Eng. 122, 194. Lewis, E.E., Smith, M.A., Yang, W.S., Wollaber, A., 2011. Preconditioned Krylov And Gauss-Seidel Solutions of Response Matrix Equations. Proc. M&C 2011, Rio, Brazil, May 8–12 (2011). Rohde, U. et al., 2016. The reactor dynamics code DYN3D – models, validation and applications. Prog. Nucl. Energy 89, 170–190. Roy, R., 2010. Reactor core methods. In: Azmy, Y., Sartores, E. (Eds.), Nuclear Computational Science, a Century in Review. Springer, New York. Smith, M.A., Lewis, E.E., Shemon, E.R., 2014. DIF3D-VARIANT 11.0: A Decade of Updates, ANL/NE-14/1. Argonne National Laboratory (Jan. 2014).
Tatsumi, M., Yamamoto, A., 2002. Object-oriented three-dimensional fine-mesh transport calculation on parallel/distributed environments for advanced reactor core analyses. Nucl. Sci. Eng. 141, 190. Tatsumi, M., Yamamoto, A., 2003. Advanced PWR core calculation based on multigroup nodal-transport method in three-dimensional pin-by-pin geometry. J. Nucl. Sci. Technol. 40 (6), 376–387. Yang, W.S., Palmiotti, G., Lewis, E.E., 2001. Numerical optimization of computing algorithms of the variational nodal method based on transformation of variables. Nucl. Sci. Eng. 139 (2), 174–185. Yang, W., Zheng, Y., Wu, H., Cao, L., Li, Y., 2014. High-performance whole core pinby-pin calculation based on EFEN-SP3 method. Nucl. Power Eng. (in Chinese) 35, 164–167. Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., 2016. A variational nodal approach to 2D/ 1D pin resolved neutron transport: I diffusion theory. Trans. Am. Nucl. Soc. 114. Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., 2016. A variational nodal approach to 2D/1D pin resolved neutron transport: II spherical harmonics. Trans. Am. Nucl. Soc. 114. Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., Wu, H., 2017. Assessment of 2DTransport/1D-Diffusion Approximations in a Pin Resolved Variational Nodal Method for PWR Calculations. Proc. M&C 2017, Jeju, Korea. April 16–20 (2017). Zhang, T., Lewis, E.E., Smith, M.A., Yang, W.S., Wu, H., 2017b. A variational nodal approach to 2D/1D pin resolved neutron transport for pressurized water reactors. Nucl. Sci. Eng. 186, 120–133.