Progress Review
Development of a Hexagonal Solution Module
for the PARCS Code
May, 2000
1
POWER-KAERI
Project Overview
 Objective
 Implement an Efficient Hexagonal Neutronic Solver in the PARCS code
 Work Scope
 Develop Hexagonal Solution Methods for Spatial Kinetics Calculation Satisfying:
- Fastness for Coupled 3D Kinetics/System T-H Calculations
- Accuracy for Solution Fidelity
- Versatility for Wide Range of Applications (MultiGroup, MultiRegion within a Hexagon)
 Implement a Hexagonal Solution Module in the PARCS Code
- Keep both Rectangular and Hexagonal Solvers in one Code (DMM Essential)
- Enable Coupled Calculation with System T-H Codes
 Verify Performance for
- Hexagonal Eigenvalue Benchmark Problems
- Transient Benchmark Problems Involving VVER Reactors
 Schedule
 Nov. 99 – April 00
 May 00
 June 00
 July 00 – Aug. 00
2
: Investigate Various Hexagonal Solvers and Select two (Based on
EVP Solution Performance)
: Implement the two Solvers into PARCS
: Perform VVER1000 Rod Ejection Transient Benchmarks with
RELAP/PARCS
: Refine the Solvers and Prepare Documentation
POWER-KAERI
Investigated Hexagonal Solvers
 Conformal Mapping
 Employs Prebuilt Mapping Functions to Transform a Hexagon to a Renangle
 Accurate for Practical Applications
 Vulnerable to Large Errors under Strong Flux Gradient Conditions
 Analytic Function Expansion Nodal (AFEN) Method
 Two-Dimensional Expansion using 12 Trigonometric and Exponential Functions
 Most Accurate  No Transverse Integration, Analytic Solution Basis
 Hard to Expand to Multigroup
 Local Fine-Mesh Finite Difference Method (LFMFDM)
 Nodal Coupling Resolved by Fine Mesh FDM Solution to Two-Node Problems in the
framework of CMFD
 Fast and Accurate (Accuracy Adjustable)
 Evolved to One-Node Formulation
 Higher Order Polynomial Expansion Nodal (HOPEN) Method
 Expansion using 2D Polynomials on a Triangle Basis
 Sufficiently Accurate with 6 Triangles per Hexagon, Further Mesh Refinement Possible
 No Limitations on Energy Groups and Allows Multiple regions within a Hexagon
 Evolved to Triangular-Z Polynomial Expansion Method (TPEN)
3
POWER-KAERI
PARCS Hexagonal Solver Overview
 CMFD Formulation
 Keep the Same Solution Methods as the Rectangular Solver
-
Eigenvalue Calculation by Wielandt Shift Method
Transient FSP Formulated by Theta Method and Analytic Precursor Integration
 Linear System Solver
-
Currently, Krylov Solver for Hexagonal Geometry
SOR or CCSI solver might replace the Krylov Solver for Flexibility in Symmetry Handling
 Dual Nodal Solvers
 Fine-Mesh FDM Solver
-
Transverse-Integrated  1D in Character
2nd Order Transverse Current Approximation along the Surfaces of the Hexagon
Surface Current Source Method Employed at the External Boundaries
Currently, Two-Node FDM
One-Node FDM will Replace the Two-Node Solver for Speed
 TPEN Solver
4
Separate Radial and Axial Directions
No Transverse Integration in the Radial Solution  Direct 2D Solution
Axial Direction Solution Resolved by NEM
POWER-KAERI
Two-Node FDM Solver
 Neutron Balance Equation for a Trapezoid
Two Node Geometry
J iT
J iR hi 1  J iL hi  iVi  ( J iT  J iB ) li
y
 g ,i 1   g ,i
   f 1   f 2 
R
   r1
,
J


2
D
g ,i
g
 r 2 
xi 1  xi
  12
Vi
1
i 1  i i 1
hi
hi 1
N
2 N
x
xi
 Constraints on Node-Average Fluxes
li
J iB
1
VH
N
1
iVi  L ; V
i 1
H
2N
iVi  R
i  N 1
 Resulting Linear System (LHS only)
5

 2   2 
d2 3
  
  
 3



  n 
 d n  n 1



 n 1 d n 1 
n 1 



A 
.

  

  2 n 1



 2 n 1 d 2 n 1  2 n

 2 n 1 

  2 n 
 n 1   2 n 1  2 n



1   1 
 2   n
POWER-KAERI
Transverse Current Approximation
 Quadratic Representation of Transverse Currents
1

J ( )  J  ( J R  J L )  ( J R  J L  2 J ) 3 2  
4

 Three Vector Addition Scheme at Corner
 1 


J c  J x  J u  J v 
3

Ju

Jv

Jx
 Use only at the interior surfaces of the hexagon
6
POWER-KAERI
Surface Current Source Method
 To Determine the Current Profile at the External Surface
 Utilizes Precalculated Response of Corner Current to the Unit Current Source
Placed a Segment of the other Surfaces
J L  RL J S
JL
J L  [ J1L , J 2L ]T ; J s  [ J11 , J 21 , J12 , J 22 ,, J115 , J 215 ]T
1
1
15
 11

 21
 112  212    1115  21
RL   1
1
2
2
15
15 








 12
22
12
22
12
22 
J12
 Use Fine Mesh FDM to Obtain the Response for
the Boundary Composition - Needed only Once for a Core
7
POWER-KAERI
TPEN Solver Development Overview
 One-Node Hexagon Formulation
 To use TPEN within the Framework of CMFD
 Partial Incoming Currents and Hexagon Corner Point Fluxes are Used as Constraint for
the One-Node TPEN Solver
 CMFD vs. CMR Comparison
 CMFD turned out to be more efficient in terms of the number of nodal updates
 CMFD Solver
 Point and Line-SOR  Convenience in Handling Various Symmetries
 Wielandt Shift Method for Accelerating Eigenvalue Convergence
 Global Iteration Logic Refinement
 Symmetric Gauss-Seidel Sweep (both ways) in the One-Node Nodal Calculation
 Use of Node Average Flux Ratios (Post-CMFD Flux/Post-Nodal Flux) to Update the BC
for the One-Node Nodal Calculation
- J_in, Phi_corner, Flux Moments
8
POWER-KAERI
Triangular PEN Formulation
 Unknowns Selected for a Triangle (9 in total per Group)
p
x
u
p
~ 
x ~
y
u
x
1
 ( x, y ) dydx
V 
d   (rd )

Nodal Volume Average Flux,  

Fluxes at three Corners,

Surface average fluxes at three surfaces
x 
x

1
 ( x, y ) dy
h
Moments
2 3 1
x ( x, y ) dydx
3h V 
2 1
~
y 
y ( x, y ) dydx
h V 
~
x 
u
p
 Flux Expansion for a Triangle
 ( x, y)  c0  ax x  ay y  bx x 2  bu u 2  bp p 2  cx x3  cu u 3  c p p3
9
POWER-KAERI
Constraints Used in TPEN
 Nodal Balance for the Triangle
1
D
16 D
 D

 x  u   p 
 80 2   r    f   32 2  x  u   p  
k
h
3 h2
 h

 Two 1-st Order Weighted Residual Balance (x and y directions)
~ 8 D
 D
~ 1
2 x  u   p   8 D2 2 x  u   p 
 80 2   r    f  
2
k
3h
9h
 h

D
8 D
~
 D
~ 1
u   p 
 80 2   r    f   8 2 u   p  
2
h
k
h
3
h


 Surface Average Current Continuity
Jx 

D
~
2 x  u   p  24 x  20  120 x
3h

 Corner Point Balance (CPB)
Lx 
10

4 3D
~
  x   x  15 x
h

POWER-KAERI
Hexagonal TPEN Formulation
 Boundary Conditions Given only at the Hexagon Boundary
- Incoming Partial Current : ji1 , ji2 , ji3 , ji4 , ji5 , ji6
- Corner Point Flux :  1 ,  2 ,  3 ,  4 ,  5 ,  6
2
3
x  y
j 1
2
 3 jo3

~1 ~ 1 2
3 ~3 ~3



x y s
s x  y
1
o
 Unknowns in the Interior (31 in total)
- Node Average Flux :  1 ,  2 ,  3 ,  4 ,  5 ,  6
~ ~ ~ ~ ~ ~
- x - momentum :  x1 ,  x2 ,  x3 ,  x4 ,  x5 ,  x6
~ ~ ~ ~ ~ ~
- y - momentum :  y1 ,  y2 ,  y3 ,  y4 ,  y5 ,  y6
- inner - surface flux : s1 , s2 , s3 , s4 , s5 , s6
- out - going partial currents :
jo1 , jo2 , jo3 , jo4 , jo5 , jo6
jo2
~2 ~2
1
4
 s1  c  s4
~4 ~4
~6 ~ 6 6

5 x y
x  y s

4
4
5 s
6
6

j

o
jo 
~5 ~ 5
x  y
5
5
6
jo
- Center Point Flux :  c
11
POWER-KAERI
Hexagonal TPEN Formulation
 Constraints to Determine the 31 Unknowns





6 Nodal Balance Equations for 6 Node Average Flux
12 WRM Equations for 12 Moments
6 Net Current Continuity Conditions for 6 Inner Surface Flux
6 Incoming Current Conditions for 6 Outgoing Currents
1 Net Leakage Balance Equation for 1 Center Flux
Caa
 0

 0
Csa
C
 ja
 0
12
0
C xx
0
Csx
C jx
Cpx
0
0
C yy
Csy
0
0
Cas
C xs
C ys
Css
0
0
Caj Cap   φ   q  S ji 
~  ~ ~ 
C xj C xp  φ
q S
 ~ x   x ~ xj i 
0 C yp  φ y   q y 

0 Csp  φ s   0 
C jj C jp   jo   S jo ji 

  
φ
Cpj Cpp   p   Spji 
POWER-KAERI
Hexagon TPEN Linear System
  1     q 1  S 1  
  2     2

  3    q  S 2  
 a1 0 0 0 0 0 

a
a
0
0
0
0
a
a
0
0
0
0
0







2 3
5
3
3
4
  
0 0 0 0 0 0 
0 0 0 0 0 0 
 0 a 0 0 0 0
 0 a2 a3 0 0 0   0 a4 0 0 0 0  a5     4     q  S  
0 0 0 0 0 0 
0 0 0 0 0 0 
 4

  0 01 a 0 0 0 







q  S 4 
0 0 0 0 0 0 
0 0 0 0 0 0 
0 0 a2 a3 0 0
a  5  
0 0 a4 0 0 0
1


 

  5     6     5
5




000000
000000
  0 0 0 a1 0 0 
 0 0 0 a2 a3 0   0 0 0 a4 0 0  a5         q  S  
0 0 0 0 0 0 
0 0 0 0 0 0 
  0 0 0 0 a1 0 
 0 0 0 0 a2 a3   0 0 0 0 a4 0  a5     ~1     q 6  S 6  




  0 0 0 0 0 a1 
 a3 0 0 0 0 a2   0 0 0 0 0 a4  a5    ~x2     ~1 ~1  
0 0 0 0 0 0 
0 0 0 0 0 0 
q  S 

 
 x2 x3 0 0 0 0   x4 0 0 0 0 0   x5     ~x3     ~ x2 ~x2  
 x1 0 0 0 0 0 
0 0 0 0 0 0 
 0 0 0 0 0 0 





 qx  S x 
 0 x2 x3 0 0 0   0 x4 0 0 0 0   x5   x
 0 x1 0 0 0 0 
0 0 0 0 0 0 
 0 0 0 0 0 0 
 ~ 4    ~ 3 ~ 3  




0 0 x 0 0 0


 qx  S x 



 0 0 0 0 0 0 

0 0 x2 x3 0 0
x
0 0 x4 0 0 0
000000
1

 


  5     ~x5     ~ 4 ~ 4  
0 0 0 0 0 0 
 0 0 0 0 0 0 
0
0
0
x
0
0
0
0
0
x
x
0
0
0
0
x
0
0
x



1
2 3
4

 


  5    ~x   q x  S~x  
0 0 0 0 0 0 
 0 0 0 0 0 0 
0
0
0
0
x
0
0
0
0
0
x
0
0
0
0
0
x
x





  x5     x6     q~x5  S x5  
1
4
2
3







0
0
0
0
0
0
0
0
0
0
0
0





0
0
0
0
0
x
0
0
0
0
0
x
x
0
0
0
0
x




1
4

  x5    ~1     q~x6  S~x6  
2 

 3


y
 0 0 0 0 0 0 
 y1 0 0 0 0 0   y2 y3 0 0 0 0 
 y4     ~ 2     q~1  
0 0 0 0 0 0 
0 0 0 0 0 0 
y

 0 0 0 0 0 0 

y










0
y
y
0
0
0
0 y1 0 0 0 0
y4
0 0 0 0 0 0 
0 0 0 0 0 0 
  2 
2
3
 
0 0 y 0 0 0 0 0 y y 0 0 
 y    ~y3    q~y  

0 0 0 0 0 0 
0 0 0 0 0 0 
2
3
1
4 
 0 0 0 0 0 0 










~4
0 0 0 0 0 0 
0 0 0 0 0 0 
  q~ 3  
 0 0 0 0 0 0 
 0 0 0 y1 0 0   0 0 0 y2 y3 0 
 y4     y      ~y4  




 0 0 0 0 0 0 
 0 0 0 0 y1 0   0 0 0 0 y2 y3 
 y4    ~ 5    q y  
0 0 0 0 0 0 
0 0 0 0 0 0 
 0 0 0 0 0 0 
 0 0 0 0 0 y1   y3 0 0 0 0 y2 
 y4     ~y6     q~ 5  
0 0 0 0 0 0 
0 0 0 0 0 0 
y

 
s
0
0
0
0
s
s
0
0
0
0
s
s
0
0
0
0
0
s
0
0
0
0
s
 3
 5

 s8     y    q~ 6  
4
6  7
2
0 0 0 0 0 0 
 1

1 
y 



 s8    s 
0 0 0 0 0 0 
  s2 s1 0 0 0 0   s4 s3 0 0 0 0   s6 s5 0 0 0 0   0 s7 0 0 0 0 
 s     2    0 
0 0 0 0 0 0 
  0 s2 s1 0 0 0   0 s4 s3 0 0 0   0 s6 s5 0 0 0   0 0 s7 0 0 0 
 8     s3    0 
0 0 0 0 0 0 
 0 0 s s 0 0 0 0 s s 0 0 0 0 s s 0 0 0 0 0 s 0 0
2 1
4 3
6 5
7
 
 

 s8     s    0 
 
0 0 0 0 0 0 

 s8     s4    0 


  0 0 0 s2 s1 0   0 0 0 s4 s3 0   0 0 0 s6 s5 0   0 0 0 0 s7 0 
 s8     5    0 
0 0 0 0 0 0 
  0 0 0 0 s2 s1   0 0 0 0 s4 s3   0 0 0 0 s6 s5   0 0 0 0 0 s7 
 s      
  0 0 0 0 0   0 0 0 0 0 
 3 0 0 0 0 0   4     6    0 
0 0 0 0 0 0
0 0 0 0 0 0 
  01  0 0 0 0   02  0 0 0 0  

 0  3 0 0 0 0   4    s1     1  
0 0 0 0 0 0 
0 0 0 0 0 0 
1
2

 0 0  0 0 0       jo     S s2  



0 0 0 0 0 0 
3
  0 0  1 0 0 0   0 0  2 0 0 0  0 0 0 0 0 0 
S

  4   2  
0 0 0 0 0 0 
0 0 0  3 0 0   4     jo3     s3  
  0 0 0  1 0 0   0 0 0  2 0 0  0 0 0 0 0 0 





S



j
  0 0 0 0 1 0   0 0 0 0  2 0 
000000
 0 0 0 0  3 0   4     o4     s4  
0 0 0 0 0 0 
S
  0 0 0 0 0    0 0 0 0 0   0 0 0 0 0 0



0
0
0
0
0
0
s
j







0
0
0
0
0



  5 
1 
2
3  4


 o
 p1 p1 p1 p1 p1 p1  0 0 0 0 0 0
0 0 0 0 0 0  p2 p2 p2 p2 p2 p2   p3     jo5     S s6  
 0 0 0 0 0 0
6
 S

  jo     sc  
  c   S

 
 
 The linear system was solved analytically by using Mathematica.
13
POWER-KAERI
Eigenvalue Benchmark Problems Examined
Dim
# of
Nodes
x (cm)
Reflector
2D
169
20.0
Yes
2D
127
20.0
No
VVER1000 2D
163
23.6
No
Large HWR 2D
919
17.78
Yes
0.125
2D
421
3D
5052
14.7
Yes
0.5
Tight Lattice
2D
LWR
199
22.4
Yes
0.5
Modified
IAEA
VVER440
14
Albedo
0.5
0.125
0.5
0.125
0.5
0.125
Remarks
Water Reflector
No Reflector
Zero Power Region
Present in Core
Zero Power Region
Present in Core
Rod Inserted Halfway
UO2 only or MOX
partially loaded
POWER-KAERI
TPEN Calculation Flow
ˆ 0
D
 p    s
IF1 :
l=1
Update D̂
l=l+1
n=1
no
F.S. Calculation
15
1
l 2
2
 1 or n  5
l n,m  l n,m1
IF2 : max
  2  or m  5
l n,m1
Backward Sweep ?
IF3 :
 ln  ln1

n
l 2
2
 3
TPEN Solution
Inner Iteration(SOR)
IF1 ?
yes
IF3 ?
yes
End
 
2
l
n=n+1
m=1
IF2 ?
yes
yes
 ln  ln1
m=m+1
NEM Axial Solution
no
CPB Solution
no
no
Calculation of
Multiplier, f
Update Triangular Flux,
Moments, Currents From f
POWER-KAERI
Comparison of Solution Accuracy
40
8
MASTER
TZPEN
7
2-Node FDM
20
6
10
5
0
4
-10
3
-20
2
-30
1
-40
0
2D IAEA w/o 2D IAEA w/o
R.(0.5)
R.(0.125)
16
2D IAEA w
R.(0.5)
2D IAEA w
R.(0.125)
2D VVER1000(0.5)
Max. Node Power Error,%
k_eff Error,pcm
30
2D VVER- 2D VVER-4403D VVER-10003D VVER-440
1000(0.125)
POWER-KAERI
Comparison of Calculation Performance
(VVER440 3D Problem Only)
Computation Time
Method/Code
CPU time(sec)*
CMR(AFEN-NEM)/MASTER
18.2
1-Node CMFD(TPEN)
9.8
2-Node CMFD(FDM)
8.2
* Pentinum III 500 MHz
Comparison of Iteration Characteristics of CMFD and CMR for Accelerating TPEN
17
Method/Code
CMR
1-Node CMFD(TPEN)
CPU time(sec)
10.4
9.8
No. of Outer Iterations/
No. of Nodal Updates
156/32
51/15
Tnodal/Ttotal(%)
64.0
66.0
POWER-KAERI
One-Node FDM
 To Reduce the Computational Burden of the Two-Node FDM Problems
 Incoming Partial Currents are Chosen as BC Instead of Node Avg. Flux
 Solves Three Directions Simultaneously
 FDM Formulation for a System of three second order 1-D Diffusion Equations
(Coupled through the transverse leakage terms appearing on the RHS)
 Balance Equation at each Mesh
y
J iT
J iR hi 1  J iLhi  iVi  ( J iT  J iB ) li   u
 Unknowns (3*N+4)
- 3*N Fine Mesh Flux
- 3 Adjusted Transverse Leakage Source (  u )
- Node (Hexagon) Averaged Flux
 Equations
- 3*N Mesh Balance Equation
- 3 Node Average Flux Constraints
- 1 Nodal Balance Equation
Vi
1
i 1  i i 1
N
x
hi
hi 1
xi
li
J iB
 Linear System can be Solved by Gaussian Elimination very Effectively
18
POWER-KAERI