Theory and methods used in Serpent 2 for spatial
homogenization
New: homogeneous diffusion flux solver
Maria Pusa
October 21th, 2015
1
Outline
• Nodal calculations
• Lattice calculations
– Homogenization
– Discontinuity and peaking factors
– Boundary conditions
• Multi-group diffusion equation in homogenized node
– Numerical solution
• Example: MIT Beavrs benchmark
2
Nodal Calculations
• Computationally expensive to solve transport equation in the true
heterogeneous geometry of the reactor core, i.e. to compute global
heterogeneous flux
⇒ Nodal calculations
– System divided into nodes that have homogeneous properties
– Solution based on multi-group diffusion theory
−D △ϕ
g
g
g g
(r)+Σt ϕ (r)
=
∑
h→g h
Σs ϕ (r)+
h
1 g∑ h h
χ
νΣf ϕ (r) ,
keff
h
)
(
−D△ϕ + Σt − Σs ϕ =
1
Fϕ
keff
• Solution called global homogeneous flux
• Lattice calculations ⇒ homogenized constants for each node
3
g = 1, . . . , G
• Key idea:
Lattice calculations
– Assume global heterogeneous ϕglob
het flux to be known
– Preserve nodal reaction rates:
∫
∫ Eg−1
glob
dr
dE
ϕ
het (r, E)Σx (r, E)
V
Eg
Σx,g =
∫
∫ Eg−1
glob
dE
ϕ
dr
het (r, E)
V
Eg
g
ϕhet
∫
∫
=
dr
V
g
Σx,g ϕhet
∫
Eg−1
dr
V
dE ϕglob
het (r, E)
Eg
∫
=
Eg−1
dE ϕglob
het (r, E)Σx (r, E)
Eg
• Practise: Solve transport equation for a set of smaller sub-problems ⇒ local
heterogeneous flux.
0
• Approximate global heterogeneous flux ϕglob
het by local heterogeneous flux ϕhet
when computing the homogenized constants
4
Continuity conditions for nodal model
• Nodes are coupled together using discontinuity factors obtained from
lattice calculations.
• Idea:
– Global heterogeneous flux is known to be continuous across node
boundaries
– There is no no physical requirement for the global homogeneous
flux to be continuous
– Discontinuity factors (DFs) couple local homogeneous flux to local
heterogeneous flux
• Approximation: DFs are used to couple global homogeneous flux to
global heterogeneous flux
⇒ Continuity conditions for the nodal model
5
Discontinuity Factors
Si−j
• Definition:
∫
g
Fi→j
g
dSϕ
het (r, E)
Si−j
= ∫
g
dS
ϕ
i (r, E)
Si−j
φi
φj
Node i
Node j
g
• Idea: On the boundary, ϕgi multiplied by the discontinuity factor Fi→j
equals ϕghet on average.
• Corner
DF s
can be defined similarly as
6
DF s
for the sides
Pin-Power Reconstruction
• Due to homogenization, nodal solution does not provide detailed
pin-by-pin flux
• It is still necessary to have an estimate for the power distribution inside
the nodes
⇒ Peaking factors (PFs) couple local homogeneous flux to pin-powers
corresponding to the local heterogeneous solution
– Approximation: Same coupling is used for global homogeneous
flux.
7
Case 1: Local solutions based on reflective boundary
conditions
• Typical practice: consider each node separately with reflective
boundary conditions ("sub-problem = node")
⇒ Homogenized constants for each assembly can be computed
separately.
⇒ Net currents over assembly boundaries are zero.
⇒ Diffusion equation has constant solution ϕ =
assembly.
⇒
DF s
0
ϕhet
inside the
and peaking factors can be computed based on ϕ0het alone.
⇒ No need for a diffusion solver
8
Case 2: Reflective BCs + leakage correction
B1 critical spectrum calculation:
• Homogenized reaction cross sections are calculated using an
intermediate micro-group structure (by default WIMS 69-group
structure)
• B1 -equations are formed and solved by critical buckling iteration
• The result is a leakage-corrected micro-group spectrum, which is used
for collapsing the cross sections into group constants using the final
macro-group structure (by default 2 energy groups)
9
Case 3: Local solutions based on colorset calculations
• In some cases, reflective boundary conditions cannot be used or the
approximation is poor:
• Reflectors
• Strong absorbers
• Assembly positioning
⇒ Node must be modeled with some surroundings ("sub-problem = colorset").
⇒ Computation of DFs and PFs requires solving the local homogeneous flux
inside the node.
⇒ Separate deterministic calculation is required to solve the multi-group diffusion
equation inside the homogenized node
– This capability has been implemented in Serpent 2
10
Solutions of Homogeneous Diffusion Equation (1)
• Diffusion equation (DE) in the homogenized node
(
)
(
)
1
D ϕxx + ϕyy
=
Σt − Σs −
F ϕ = Aϕ
keff
⇔
ϕxx + ϕyy
= Mϕ ,
M = D −1 A .
• Trial function
ψ(x, y) = eB 1 x+B 2 y c = eB 1 x eB 2 y
• Substitute to
DE
ψ xx + ψ yy =
• Function ψ satisfies
DE
B 21 ψ
+
B 22 ψ
( 2
)
2
= B1 + B2 ψ
if
B 21 + B 22 = M = D −1 A
11
Solutions of Homogeneous Diffusion Equation (2)
• Functions of the form
ψ(x, y) = eB 1 x+B 2 y ,
satisfy
DE .
B 21 + B 22 = M ,
These functions are called basis functions.
• Matrix square root: If C 2 = A , C is a matrix square root of A
• Examples of basis functions: e
√
Mx
,e
√
− My
√
and e
M (x+y)
2
• General solution of DE is a linear combination of all basis functions.
Boundary conditions determine the coefficients of the basis functions.
• When constructing a solution to DE, the number of boundary
conditions must equal the number of basis functions.
• Local homogeneous solution should be consistent with the nodal code.
12
Constant Current on Every Boundary
∂
• Boundary condition: −D ∂n
Φ(x, y) = J S
net /S = const., when (x, y) ∈ S.
– For example, ϕx (x, y) = −D −1 J W
net /a when x = −a/2
N
Jnet
(− a2 , a2 )
W
Jnet
( a2 , a2 )
E
Jnet
(− a2 , − a2 )
S
Jnet
( a2 , − a2 )
• Solution must be of the form:
√
ϕ(x, y) = e
Ax
√
− Ax
c1 + e
√
c2 + e
Ay
c3 + e
√
− Ay
c4
• Unknown coefficient vectors c1 , . . . , c4 are solved from 4 boundary conditions
• In this case, the solution is unique.
13
Constant Current on Every Boundary
• Forcing the homogeneous current to a constant value on each boundary can
lead to overestimation or underestimation of homogeneous flux near the
corners
• Some nodal codes use corner ADFs in addition to surface ADFs
Thermal heterogeneous flux
x 10
Thermal homogeneous flux
−5
−4
x 10
12
6.5
11
6
10
9
5.5
8
5
7
10
10
5
5
10
5
0
10
5
0
0
−5
0
−5
−5
−10
−5
−10
−10
14
−10
Method implemented in Serpent
• Boundary conditions:
∫
– S J hom (r) dS = J net,S for each boundary surface
∫
– Γ J hom (r) dS = J net,Γ for each corner
• Rectangular geometry:
– 8 boundary conditions and 8 basis functions
– Basis functions:
±
f±
=
e
x
√
Ax
, f±
y = e
√
± Ay
ΓNW
±
, f±
=
e
x+y
SN
A (x+y)
2
ΓNE
SW
ΓSW
√
SE
SS
ΓSE
• Similar approach for hexagonal geometry
15
, f±
x−y = e
√
(x−y)
± A
2
Numerical solution of homogeneous flux in Serpent
• Starting point: Homogenized constants and boundary conditions
• Form matrix M :
(
M = D −1 Σt − Σs −
1
F
keff
)
• Compute complex Schur form M = U T U ∗
– Hessenberg reduction and QR decomposition based on Householder
transformations
– QR updates with Wilkinson shift
⇒ Matrix functions (matrix square root, matrix exponential) can be computed
efficiently with Parlett method
– Form boundary conditions
– Solve coefficients (Gaussian elimination with partial pivoting)
– Compute DFs and PFs based on local homogeneous solution
16
Example: BEAVRS benchmark
• The Benchmark for Evaluation and Validation of Reactor Simulations
(BEAVRS) [1]
R
1
• 1000 MW Westinghouse
PWR reactor core
• Fuel assemblies:
– Red: 1.6 wt.%
235
U
– Yellow: 2.4 wt.% 235 U
– Blue: 3.1 wt.%
235
U
P
N
M
L
K
J
H
G
F
E
D
III
III
III
III
III
III
III
III
III
I
I
I
I
I
I
II
III
III
III
III
III
II
II
II
2
III
III
3
III
II
15
II
16
6
16
16
6
20
16
6
20
16
16
III
III
5
III
II
6
III
I
7
III
I
8
III
I
9
III
I
10
III
I
11
III
I
12
III
III
II
16
13
III
II
15
14
III
III
II
III
III
II
II
III
III
III
II
I
I
I
I
I
III
III
III
III
III
III
III
15
16
6
16
16
20
6
12
16
6
12
12
20
12
12
16
12
12
16
16
16
16
6
III
III
15
II
III
16
II
III
III
I
III
I
III
I
III
I
III
I
III
I
III
II
III
III
16
20
20
6
20
16
16
6
16
16
II
III
15
II
III
II
III
III
III
16
6
6
16
16
A
16
16
16
6
II
12
12
20
III
12
12
16
III
12
12
12
II
12
12
12
II
12
16
12
III
16
12
16
III
16
12
16
III
16
12
16
12
16
12
12
12
12
16
12
B
16
4
16
C
II
II
III
I
III
III
III
III
III
N. Horelik and B. Herman, Benchmark for Evaluation and Validation of Reactor Simulations, MIT
Computational Reactor Physics Group, http://crpg.mit.edu/pub/beavrs/
17
[1]
Example: BEAVRS benchmark
Red assembly in the middle of the core: colorset calculation with 2.5 assembly widths of surroundings.
Fast homogeneous flux
Fast heterogeneous flux
x 10
−4
−3
x 10
3.4
2.1
3.2
2
3
1.9
2.8
1.8
2.6
10
10
5
5
10
5
0
10
5
0
0
−5
0
−5
−5
−10
−5
−10
−10
Thermal heterogeneous flux
−10
Thermal homogeneous flux
−5
−4
x 10
x 10
12
6.5
11
10
6
9
5.5
8
5
7
10
10
5
5
10
5
0
10
5
0
0
−5
0
−5
−5
−10
−5
−10
−10
18
−10
Example: BEAVRS benchmark
Top: surroundings included. Bottom: reflective boundary.
Fast heterogeneous flux
x 10
Thermal heterogeneous flux
−4
−5
x 10
12
3.4
11
3.2
10
3
9
2.8
8
2.6
10
10
7
5
5
10
5
0
10
5
0
0
−5
0
−5
−5
−10
−10
−10
−10
Thermal heterogeneous flux
Fast heterogeneous flux
x 10
−5
−4
−5
x 10
3.4
11.5
3.3
11
3.2
10.5
3.1
10
3
9.5
2.9
9
2.8
8.5
2.7
8
10
10
5
5
10
5
0
10
5
0
0
−5
0
−5
−5
−10
−5
−10
−10
19
−10
Example: BEAVRS benchmark
• Serpent–ARES code sequence
– ARES is LWR core simulator code developed at Finnish Radiation
and Nuclear Safety Authority (STUK)
– Homogenization calculations with Serpent
– Results compared to 3D reference calculation with Serpent
– Hot zero-power state of the initial core considered in:
Leppänen, J., Mattila, R., and Pusa, M. “Validation of the Serpent-ARES code sequence using
the MIT BEAVRS benchmark - Initial core at HZP conditions.” Ann. Nucl. Energy, 69 (2014)
212-225.
20
Example: BEAVRS benchmark
• Config. 1: Reflective boundary conditions
• Config. 2: Reflective boundary conditions + B1 leakage correction.
• Config. 4: Colorset with 0.5 assembly widths of surroundings.
• Config. 6: Colorset with 2.5 assembly widths of surroundings.
Table 1:
Effective multiplication factor calculated by ARES, maximum negative and positive differences
in assembly power between ARES and Serpent 3D, and error fractions and mean absolute errors in ARES
pin-power distribution.
Config.
keff
Differences
Error fractions
< 1%
< 2%
< 3%
1
0.99922
[-7.7 , 12.0]
13.5
29.0
44.8
2
0.99953
[-8.4 , 10.9]
16.8
31.5
43.8
4
0.99994
[-3.8 , 8.7]
27.9
58.8
80.8
6
0.99995
[-1.6 , 0.6]
82.4
97.7
99.7
21
Example: BEAVRS benchmark
• Assemblies homogenized with 2.5 assembly widths of surroundings
R
P
N
M
1
H
G
F
E
1.64
1.74
1.53
1.27
1.25
1.52
1.73
1.63
1.73
1.52
1.26
D
C
B
A
R
P
N
M
L
K
J
H
G
F
E
-1.1
-0.5
-0.1
-0.5
-0.7
-0.3
-1.1
D
C
B
A
3
1
2.08
1.96
1.56
1.89
1.57
1.89
1.57
1.97
2.08
1.53
2.06
1.94
1.56
1.88
1.57
1.88
1.57
1.96
2.07
1.51
1.52
2.05
1.99
1.72
1.70
1.52
1.63
1.52
1.70
1.72
1.99
2.06
1.52
1.51
2.05
1.99
1.71
1.69
1.51
1.63
1.52
1.70
1.72
1.99
2.05
1.50
2.07
2.07
1.98
1.99
2.41
2.41
1.85
1.85
1.55
1.55
1.72
1.70
1.49
1.48
1.72
1.71
1.56
1.55
1.86
1.85
2.42
2.41
2.00
1.99
2.09
2.06
1.26
1.95
1.71
1.85
1.62
1.75
1.50
1.68
1.50
1.76
1.63
1.86
1.72
1.97
1.26
1.96
1.72
1.85
1.62
1.74
1.48
1.66
1.48
1.74
1.62
1.85
1.71
1.94
1.25
1.52
1.56
1.69
1.55
1.75
1.49
1.62
1.39
1.63
1.50
1.76
1.56
1.71
1.57
1.54
1.52
1.57
1.70
1.55
1.74
1.48
1.61
1.39
1.61
1.48
1.74
1.55
1.69
1.56
1.52
1.73
1.88
1.51
1.71
1.49
1.62
1.33
1.40
1.34
1.62
1.50
1.72
1.52
1.89
1.74
1.73
1.88
1.52
1.71
1.48
1.61
1.33
1.39
1.33
1.61
1.48
1.70
1.51
1.88
1.73
1.62
1.56
1.62
1.48
1.66
1.39
1.39
1.23
1.40
1.39
1.68
1.49
1.63
1.57
1.64
4
7
J
1.73
1.50
3
6
K
1.53
1.52
2
5
L
1.27
2.5
1.27
2
-1.0
-0.6
-0.9
-0.4
-0.6
0.0
-0.6
-0.2
-0.5
-0.6
-1.2
3
-0.7
-0.4
-0.0
-0.1
-0.5
-0.5
-0.3
-0.4
-0.2
-0.4
-0.2
-0.9
-1.5
4
0.3
0.4
0.0
0.0
-0.4
-0.9
-0.7
-1.0
-0.5
-0.4
-0.5
-0.5
-1.0
2
2
5
-0.5
0.1
0.3
0.2
-0.1
-0.8
-0.9
-1.1
-0.9
-1.0
-0.6
-0.5
-0.5
-1.4
-1.6
6
0.3
0.4
0.4
0.0
-0.6
-0.6
-0.8
-0.5
-1.0
-1.0
-1.1
-0.7
-1.0
-0.9
-1.0
7
0.0
0.1
0.3
-0.2
-0.6
-0.7
-0.1
-0.2
-0.5
-0.8
-1.0
-1.1
-0.6
-1.0
-0.7
1
1.5
8
9
10
1.63
1.57
1.63
1.48
1.66
1.39
1.39
1.24
1.39
1.39
1.66
1.48
1.63
1.57
1.63
8
0.4
0.6
0.3
-0.2
-0.3
-0.3
0.3
0.3
-0.2
-0.5
-1.1
-0.7
-0.4
-0.1
-0.7
1.72
1.73
1.87
1.88
1.51
1.51
1.71
1.70
1.49
1.48
1.62
1.61
1.33
1.33
1.39
1.39
1.33
1.33
1.62
1.61
1.49
1.48
1.72
1.71
1.52
1.52
1.89
1.88
1.73
1.73
9
0.4
0.3
0.1
-0.3
-0.4
-0.5
-0.0
0.1
-0.1
-0.8
-0.7
-0.8
-0.3
-0.6
-0.5
1.52
1.56
1.69
1.55
1.74
1.49
1.61
1.39
1.62
1.49
1.75
1.55
1.70
1.57
1.53
1.52
1.56
1.69
1.55
1.74
1.48
1.61
1.39
1.61
1.48
1.74
1.55
1.70
1.57
1.52
10
0.1
0.1
-0.0
0.0
-0.2
-0.4
-0.3
-0.3
-0.6
-0.7
-0.8
-0.2
-0.3
-0.0
-0.4
1.26
1.95
1.71
1.84
1.61
1.75
1.49
1.66
1.49
1.75
1.62
1.85
1.72
1.96
1.26
1.25
1.94
1.71
1.85
1.62
1.74
1.48
1.66
1.48
1.74
1.62
1.85
1.72
1.96
1.26
11
-0.3
-0.4
0.3
0.4
0.1
-0.4
-0.4
-0.4
-0.6
-0.6
-0.1
0.1
0.0
-0.2
-0.6
2.07
1.98
2.41
1.85
1.55
1.71
1.48
1.71
1.55
1.85
2.41
1.99
2.07
2.06
1.99
2.41
1.85
1.55
1.71
1.48
1.70
1.55
1.85
2.41
1.99
2.07
12
-0.1
0.3
0.2
0.3
0.2
-0.3
-0.3
-0.3
-0.1
0.1
0.2
0.1
-0.1
1.51
2.04
1.98
1.71
1.69
1.51
1.62
1.52
1.70
1.71
1.98
2.05
1.52
1.50
2.05
1.99
1.72
1.70
1.52
1.63
1.51
1.69
1.71
1.99
2.05
1.51
13
-0.9
0.1
0.3
0.5
0.1
0.2
0.2
-0.0
-0.3
0.0
0.1
-0.4
-0.6
1.51
1.51
2.07
2.07
1.95
1.96
1.56
1.57
1.87
1.88
1.56
1.57
1.88
1.88
1.56
1.56
1.96
1.94
2.07
2.06
1.52
1.50
-0.3
0.2
0.1
0.4
0.3
0.4
-0.2
-0.1
-0.6
-0.2
-1.2
1.26
1.52
1.73
1.63
1.73
1.53
1.26
1.26
1.52
1.73
1.63
1.73
1.52
1.25
-0.2
0.1
-0.1
0.3
0.1
-0.2
-0.8
1
0
−1
11
12
13
14
15
0.5
14
15
0
−2
−3
Serpent and ARES results
Relative difference (%)
22
Summary
• In colorset calculations, the computation of discontinuity and peaking
factors requires solving diffusion equation for the homogenized node.
– Solution can be computed as a linear combination of matrix
functions
– Solution depends on boundary conditions
– Boundary conditions should be consistent with nodal code
• Method implemented in Serpent 2:
– 2D
– Rectangular and hexagonal geometry
– Boundary and corner net currents as boundary condition
• Future work:
– Need for other types of boundary conditions?
– Extend methodology to 3D.
23