anderson acceleration for tiamat - icerm · alex toth tiamat overview anderson acceleration...
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Anderson Acceleration for Tiamat
A. Toth1, C.T. Kelley1, R. Pawlowski2
1North Carolina State University
2Sandia National Laboratories
ICERM Workshop on Numerical Methods for Large-ScaleNonlinear Problems and Their Applications
September 3, 2015
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Outline of Topics
1 Tiamat Overview
2 Anderson Acceleration Integration
3 Numerical TestsSingle Fuel RodSingle Assembly
4 Conclusions
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Tiamat
Tiamat is a tool being developed in CASL forpellet-cladding interaction (PCI) analysis
PCI is controlled by the complex interplay ofthe mechanical, thermal and chemicalbehavior of a fuel rod during operation
Tiamat couples the single rod fuelperformance code Bison-CASL with othertools in VERA which provide a whole corerepresentation of fission density and coolantconditions in order to compute quantities ofinterest for identifying PCI failure.
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
VERA Code Suite
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Components of Tiamat
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Bison-CASL
Fuel performance code which modelsthe thermo-mechanical behaviorbehavior of a single fuel rod
Built on INL MOOSE framework, usesfinite element geometric representationand JFNK to solve the governingsystems of PDEs
Used to compute key figures of merit(ex. max hoop stress) for identifyingfuel rods requiring further detailed PCIcalculations
Figure: Bison-CASL fuelrod hoop stress
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
COBRA-TF (CTF)
Subchannel thermal hydraulics code maintained byPenn State University
Utilizes a two-fluid, three-field representation oftwo-phase flow. Solves equations for:
Continuous vapor (mass, momentum andenergy)
Continuous liquid (mass, momentum andenergy)
Entrained liquid drops (mass and momentum)
Non-condensable gas mixture (mass)
Only parallel to the assembly level
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
MPACT
Primary neutronics code in VERA, co-developed by ORNL andUniversity of Michigan
Includes several methods for solving the neutron transportequation, workhorse method is 2D/1D solver with coarse-meshfinite-difference acceleration
Utilizes the subgroup method and embedded self-shieldingmethod for cross section evaluation
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Data Transfer Kit (DTK)
Software designed to provide parallel services for scalablemesh/geometry searching and data transfer developed at ORNL
Determines mapping for moving data between source and targetarrays using the rendezvous algorithm
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
PIKE
New Trilinos package for black box multiphysics coupling
Provides interfaces for:
single-physics model evaluatorsdata transfersobserversparallel distribution managementlocal/global status tests
Currently only includes Picard-based solvers
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Solution Process
1 Estimate hot full-power(HFP) state with CTF andMPACT
2 Model transition from coldzero-power (CZP) to hotzero-power (HZP) inBison-CASL
3 Model transition from HZPto HFP in Bison-CASL
4 Model reactor state at HFPconditions for one or moretime step
Figure: Bison-CASL ramp to HFP
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Data Transfers
In the coupled HFP solve phase, Tiamat utilizes 5 data transfers
Bison to MPACT: Fuel temperatures (Tf ,B → Tf ,M)MPACT to Bison-CASL: Fission heat generation (qM → qB)Bison-CASL to CTF: Heat flux (q′′B → q′′C)CTF to Bison-CASL: Clad surface temperatures (Tc,C → Tc,B)CTF to MPACT: Coolant temperature and densities(Tw,C → Tw,M, ρw,C → ρw,M)
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Picard Iteration - Block Gauss-Seidel Map
Algorithm 1: Gauss-Seidel Nonlinear Solve for Tiamat
Given q0M,T
0c,C,T
0w,C, ρ
0w,C,T
0f ,B, q
′′B,0.
for k = 0, 1, . . . until converged doTransfer Bison-CASL to MPACT, Tk
f ,B → Tkf ,M
Transfer CTF to MPACT, Tkw,C → Tk
w,M and ρkw,C → ρk
w,M
Using Tkf ,M, Tk
w,M, and ρkw,M, solve MPACT and obtain qk+1
M
Transfer MPACT to Bison-CASL, qk+1M → qk+1
BTransfer CTF to Bison-CASL, Tk
c,C → Tkc,B
Using Tkc,B and qk+1
B , solve Bison-CASL and obtain Tk+1f ,B and
q′′B,k+1
Transfer Bison-CASL to CTF, q′′B,k+1 → q′′C,k+1
Using q′′C,k+1, solve CTF and obtain Tk+1c,C ,Tk+1
w,C , and ρk+1w,C
end
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Block Gauss-Seidel Map with Damping
Algorithm 2: Damped Gauss-Seidel Nonlinear Solve for TiamatGiven q0
M,T0c,C,T
0w,C, ρ
0w,C,T
0f ,B, q
′′B,0.
for k = 0, 1, . . . until converged doTransfer Bison-CASL to MPACT, Tk
f ,B → Tkf ,M
Transfer CTF to MPACT, Tkw,C → Tk
w,M and ρkw,C → ρk
w,M
Using Tkf ,M, Tk
w,M, and ρkw,M, solve MPACT and obtain qk+1
M
Transfer MPACT to Bison-CASL, qk+1M → qk+1
Bif k > 1 then
Damp the transferred power, qk+1B = (1− ω)qk
B + ωqk+1B
endTransfer CTF to Bison-CASL, Tk
c,C → Tkc,B
Using Tkc,B and qk+1
B , solve Bison-CASL and obtain Tk+1f ,B and q′′B,k+1
Transfer Bison-CASL to CTF, q′′B,k+1 → q′′C,k+1
Using q′′C,k+1, solve CTF and obtain Tk+1c,C ,Tk+1
w,C , and ρk+1w,C
end
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Tiamat Communication Layers
Issue: Applications live in independent processor space, sosignificant idle time for sequential solves
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Picard Iteration - Block Jacobi Map
Algorithm 3: Jacobi Nonlinear Solve for Tiamat
Given q0M,T
0c,C,T
0w,C, ρ
0w,C,T
0f ,B, q
′′B,0.
for k = 0, 1, . . . until converged doTransfer Bison-CASL to MPACT, Tk
f ,B → Tkf ,M
Transfer CTF to MPACT, Tkw,C → Tk
w,M and ρkw,C → ρk
w,M
Transfer MPACT to Bison-CASL, qkM → qk
BTransfer CTF to Bison-CASL, Tk
c,C → Tkc,B
Transfer Bison-CASL to CTF, q′′B,k → q′′C,kUsing Tk
f ,M, Tkw,M, and ρk
w,M, solve MPACT and obtain qk+1M
Using Tkc,B and qk
B, solve Bison-CASL and obtain Tk+1f ,B and
q′′B,k+1
Using q′′C,k, solve CTF and obtain Tk+1c,C ,Tk+1
w,C , and ρk+1w,C
end
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Convergence criteria
Global convergence of the coupled system is determined by the followingcriteria
Successful local convergence of each of the applications
Bison-CASL: the change in the maximum fuel temperature acrosseach of the fuel rods is less than some tolerance εT
CTF: the change in the maximum clad temperature and maximumcoolant temperature is less than εT
MPACT: the relative change (in the l2 norm) in the power distributionis less than a tolerance εP, and the change in the dominanteigenvalue keff is less than a tolerance εk
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Advantages/Drawbacks of Picard Iteration
AdvantagesSimple to implementFew requirements forapplication codes
DrawbacksRelatively slow convergencePoor robustness 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
20
Damping factor
Ite
ratio
ns
80% Power
100% Power
120% Power
Figure: Picard iteration dependenceon damping
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Can We Do Better?
Would like to utilize a method which converges more quickly thanPicard, or is at least less sensitive to ad hoc damping factors
Newton isn’t an option as the residual evaluation is too expensivefor JFNK (Roger’s talk), and we can’t get derivatives from theapplications
Anderson acceleration is an ideal candidate, as it requires nomore information to implement than Picard and only one functionevaluation per iteration.
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Anderson Acceleration Algorithm
Algorithm 4: Anderson accelerationGiven initial iterate u0, storage depth parameter m ∈ N, andmixing parameter βSet u1 = (1− β)u0 + βG(u0)for k = 1, 2, . . . until converged do
Set mk = min{m, k}Determine α(k) which solves:
minα∈Rmk+1
∥∥∥∥∥∥mk∑j=0
αjF(uk−mk+j)
∥∥∥∥∥∥ ,such that
∑mkj=0 αj = 1, where F(u) = G(u)− u
Set uk+1 = (1− β)∑mk
j=0 α(k)j uk−mk+j + β
∑mkj=0 α
(k)j G(uk−mk+j)
end
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Implementation
What do we choose for u? How do we define G?
Let u be comprised of some subset of the transferred dataDerive G from Picard iteration, i.e. take specified input u,perform one Picard iteration, and define the correspondingoutput as G.In order to leverage existing data transfer objects, we letapply Anderson acceleration by intercepting and overwritingtransfer target arrays
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Block Gauss-Seidel Map
Given Tf ,M,Tw,M, ρw,M,Tc,B
1 Using Tf ,M, Tw,M, and ρw,M, solve MPACT and obtain q̂M
2 Transfer MPACT to Bison-CASL, q̂M → q̂B
3 Using Tc,B and q̂B, solve Bison-CASL and obtain T̂f ,B and q̂′′B4 Transfer Bison-CASL to MPACT, T̂f ,B → T̂f ,M
5 Transfer Bison-CASL to CTF, q̂′′B → q̂′′C6 Using q̂′′C, solve CTF and obtain T̂c,C,T̂w,C, and ρ̂w,C
7 Transfer CTF to MPACT, T̂w,C → T̂w,M and ρ̂w,C → ρ̂w,M
8 Transfer CTF to Bison-CASL, T̂c,C → T̂c,B
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Block Gauss-Seidel Map
Define:
GGS
Tf ,MTw,Mρw,MTc,B
=
T̂f ,M
T̂w,Mρ̂w,M
T̂c,B
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Block Jacobi Map
Given Tf ,M,Tw,M, ρw,M,Tc,B, qB, q′′C1 Using Tf ,M, Tw,M, and ρw,M, solve MPACT and obtain q̂M
2 Using Tc,B and qB, solve Bison-CASL and obtain T̂f ,B and q̂′′B3 Using q′′C, solve CTF and obtain T̂c,C,T̂w,C, and ρ̂w,C
4 Transfer Bison-CASL to MPACT, T̂f ,B → T̂f ,M
5 Transfer CTF to MPACT, T̂w,C → T̂w,M and ρ̂w,C → ρ̂w,M
6 Transfer MPACT to Bison-CASL, q̂M → q̂B
7 Transfer CTF to Bison-CASL, T̂c,C → T̂c,B
8 Transfer Bison-CASL to CTF, q̂′′B → q̂′′C
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Block Jacobi Map
Define:
GJAC
Tf ,MTw,Mρw,MTc,BqBq′′C
=
T̂f ,M
T̂w,Mρ̂w,M
T̂c,B(1− ω)qB + ωq̂B
q̂′′C
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Scaling of Variables
Issue: The different fields comprising u may exist on vastly differentscales, and large magnitude fields may dominate the least-squaresproblem in Anderson acceleration
We introduce scaled variables v = Mu, where M is a diagonal scalingmatrix, and instead of u = G(u) apply Anderson to the scaled fixed-pointproblem
v = MG(M−1v) ≡ H(v)
For temperature and density unknowns we let Mi,i = (u0)i. For powerunknowns, we scale by the the initial average power in the fuel rod. Heatflux unknowns are scaled by the global average initial heat flux.
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Single Fuel Rod
First consider simulation of a single fuel rod at HFP
Tests run with 12 processors, 10 allocated to MPACT, 1 forBison-CASL, 1 for CTF
We use convergence tolerancesεP = 1e− 4, εT = 0.1◦C, εk = 1e− 5
8-group test cross sections are used in MPACT
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Sensitivity to Damping/Storage Depth
0.2 0.4 0.6 0.8 1300
350
400
450
500
550
600
Damping Factor
Solu
tion T
ime (
s)
Picard
Anderson−1
Anderson−2
Anderson−3
Figure: Run times for Gauss-Seidel map, varying storage depthparameter and damping level
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Convergence of the Fixed-Point Residual
0 2 4 6 8 1010
−5
10−4
10−3
10−2
10−1
100
Iteration
Re
lative
fix
ed
−p
oin
t re
dis
ua
l
Gauss−Seidel map, damping = 0.5
Picard
Anderson−1
Anderson−2
Anderson−3
0 5 10 15 20 25 3010
−5
10−4
10−3
10−2
10−1
100
IterationR
ela
tive
fix
ed
−p
oin
t re
dis
ua
l
Jacobi map, damping = 0.5
Picard
Anderson−1
Anderson−2
Anderson−3
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Agreement With Picard Solution
0 100 200 300 400−5
−4
−3
−2
−1
0
1
2
3
4x 10
−4
Height (cm)
Re
lative
diffe
ren
ce
Fuel Temperature
Gauss−Seidel
Jacobi
0 100 200 300 400−1.5
−1
−0.5
0
0.5
1
1.5x 10
−4
Height (cm)
Re
lative
diffe
ren
ce
Clad Temperature
Gauss−Seidel
Jacobi
Figure: Relative difference of Anderson-2 solutions from Picard solutions
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Agreement With Picard Solution
0 100 200 300 400−2.5
−2
−1.5
−1
−0.5
0
0.5
1x 10
−3
Height (cm)
Re
lative
diffe
ren
ce
Fission Rate
Gauss−Seidel
Jacobi
0 100 200 300 400−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1x 10
−3
Height (cm)
Re
lative
diffe
ren
ce
Heat Flux
Gauss−Seidel
Jacobi
Figure: Relative difference of Anderson-2 solutions from Picard solutions
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Sensitivity to Power Variation - GS Map
0.2 0.4 0.6 0.8 1300
350
400
450
500
550
600
Damping Factor
Solu
tion T
ime (
s)
80% Power
Picard
Anderson−2
0.2 0.4 0.6 0.8 1300
350
400
450
500
550
600
Damping Factor
Solu
tion T
ime (
s)
100% Power
Picard
Anderson−2
0.2 0.4 0.6 0.8 1300
350
400
450
500
550
600
Damping Factor
Solu
tion T
ime (
s)
120% Power
Picard
Anderson−2
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Sensitivity to Power Variation - Jacobi Map
0.2 0.4 0.6 0.8 1300
350
400
450
500
550
600
650
700
750
Damping Factor
Solu
tion T
ime (
s)
80% Power
100% Power
120% Power
Figure: Anderson-2 run times
At each power level, Picard only converges in fewer than 30 iterations atone tested damping level (Anderson iterations take between 10-20)
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Scaled vs Unscaled Fixed-Point Problem
0 0.2 0.4 0.6 0.8 10
5
10
15
Mixing parameter
Ite
ratio
ns
Gauss−Seidel Map
Scaled
Unscaled
(a) Block Gauss-Seidel map
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
Mixing parameter
Ite
ratio
ns
Jacobi Map
Scaled
Unscaled
(b) Block Jacobi map
Figure: Iteration counts from applying Anderson-2 to the unscaledproblem u = G(u) and the scaled problem v = MG(M−1v)
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
CASL Progression Problem P6a
Progression problem P6asimulates a single 17x17 PWRassembly at HFP
Tests run with 64 processors, 32allocated to MPACT, 31 toBison-CASL, 1 for CTF
We use convergence tolerancesεP = 1e− 4, εT = 1◦C, εk = 1e− 5
Except when noted otherwise,results use 8-group test crosssections in MPACT
Figure: 17x17 lattice, fuel in blue,guide tubes in white, instrument
tube in yellow
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Single Assembly Results
Iteration Count Time(s)Picard (Gauss-Siedel) 8 6111
Anderson-2 (Gauss-Seidel) 6 5237Picard (Jacobi) 17 5675
Anderson-2 (Jacobi) 12 4919
Table: Single assembly test results, damping factor 0.5
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Application timing breakdown
Bison-CASL CTF MPACTPicard (Gauss-Siedel) 145 131 180
Anderson-2 (Gauss-Seidel) 155 126 193Picard (Jacobi) 148 127 172
Anderson-2 (Jacobi) 149 137 190
Table: Average application solve times
As a result of good balance in the solve times, each Jacobiiteration takes on average approximately 40% of the solve time ofa Gauss-Seidel iteration
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Agreement With Picard Solution
0 50 100 150 200 250 300 350−1.5
−1
−0.5
0
0.5
1
1.5x 10
−4
Height (cm)
Re
lative
diffe
ren
ce
Assembly Averaged Fuel Temperature
Gauss−Seidel
Jacobi
0 50 100 150 200 250 300 350−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−5
Height (cm)
Re
lative
diffe
ren
ce
Assembly Averaged Clad Temperature
Gauss−Seidel
Jacobi
Figure: Relative difference of assembly averaged Anderson-2 solutionsfrom Picard solutions
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Agreement With Picard Solution
0 50 100 150 200 250 300 350
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−4
Height (cm)
Re
lative
diffe
ren
ce
Assembly Averaged Fission Rate
Gauss−Seidel
Jacobi
0 50 100 150 200 250 300 350
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−4
Height (cm)
Re
lative
diffe
ren
ce
Assembly Averaged Heat Flux
Gauss−Seidel
Jacobi
Figure: Relative difference of assembly averaged Anderson-2 solutionsfrom Picard solutions
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Anderson-2 Mixing Parameter Variation
Mixing Parameter Iteration Count Time(s)0.25 11 71720.5 6 5237
0.75 8 61751.0 7 6029
Table: Results for Anderson-2 with Gauss-Seidel map,varying mixing parameter
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Varying Cross Section Libraries
Iteration Count Time(s) keffPicard (8-group) 8 6111 1.165224
Anderson-2 (8-group) 6 5237 1.165224Picard (47-group) 8 13930 1.164680
Anderson-2 (47-group) 7 13190 1.164681
Table: Comparison of Picard and Anderson-2 for Gauss-Seidel map,varying cross section libraries
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Varying Cross Section Libraries
Bison-CASL CTF MPACTPicard (8-group) 145 131 180
Anderson-2 (8-group) 155 126 193Picard (47-group) 147 139 961
Anderson-2 (47-group) 166 118 1042
Table: Average application solve times
With higher fidelity cross sections, MPACT takes roughly 75% ofthe Gauss-Seidel iteration time, so little gain from solvingapplications concurrently
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Conclusions
Anderson acceleration displays improved robustness overPicard iteration in that its performance is generally lesssensitive to variation in parameters that need to be tuned forPicardIn general, Anderson at worst converges as quickly as Picardwith optimally chosen damping, generally provides marginalimprovementAnderson with block Jacobi map can outperformGauss-Seidel map, but good balance in application solvetimes is criticalScaling of the unknowns has been seen to be important inrelated calculations, and this merits further investigation inthis context
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Acknowledgements
This research was supported by the Consortium for AdvancedSimulation of Light Water Reactors (http://www.casl.gov), anEnergy Innovation Hub (http://www.energy.gov/hubs) for Modelingand Simulation of Nuclear Reactors under U.S. Department ofEnergy Contract No. DE-AC05-00OR22725
This research used resources of the Oak Ridge LeadershipComputing Facility at the Oak Ridge National Laboratory, which issupported by the Office of Science of the U.S. Department ofEnergy under Contract No. DE-AC05-00OR22725.
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Andersonfor Tiamat
Alex Toth
Tiamat Overview
AndersonAccelerationIntegration
Numerical TestsSingle Fuel Rod
Single Assembly
Conclusions
Questions?Comments?Suggestions?
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