numerical methods for nuclear nonlinear coupled system

Numerical Methods for Nuclear Nonlinear Coupled

System

J. Gan, Y. XuT. J. Downar

School of Nuclear EngineeringPurdue University

October, 2002

Outline

Numerical methods for nonlinear system: Marching forward method Nested loop method Newton – Krylov method (intro at end)

Results analysis Conclusion

Nonlinear Coupled System

Neutronics

eg. PARCS

Thermal –

Hydraulics

eg. TRAC-MT, , , B

Q

Nonlinear subsystems already handled in each side,

How about the above global nonlinear system?

Marching Forward Method for Coupled System

Current marching forward method:

No convergence constrained !

Only once feedback happened in each time step,

Therefore time step size limited

TRAC-M

PARCS

step n step n+1

A. Original Marching Forward

Iterative Methods for Coupled System – Nested Loop

Alternative nested loop method:

Convergence criteria constrained, more accurate !

0)()( ,*1,, ninin xSxxM

n: time step index; i: nested loop index

TRAC-M

PARCS

step n step n+1

B. Gauss Seidel style Nested iteration

Converged?N

Y Converged?N

Y Converged?N

Y

Neutronics/Thermal-Hydraulics Coupled Problem

)()( xbxxA

It can be viewed as :

variablesNeutronicsofvectortheΦ

variablesHTofvectortheT

:

/:

SΦTM

ΦQTTN

)(

)()( Thermal-Hydraulic equation

Neutronics spatial kinetic equation

0)()()( xbxxAxF

Nonlinear System:

)()(:2

:11

0

iii xbxxAstep

xofguessinitialstep

SΦTMstep

ΦQTTNstep

ΦTofguessinitialstep

ii

iii

1

11

00

)(:3

)()(:2

,:1

)(\)()(1 iiii xbxAxgx

Iterate step 2 and 3 until converged

Iterate step 2 until converged

It can be also viewed as :Fix point iteration for The nonlinear system

Gauss Seidel style Iteration Parallel style Iteration

SΦTMstep

ΦQTTNstep

ΦTofguessinitialstep

ii

iii

11

11

00

)(:3

)()(:2

,:1

Nested Iteration

Independent Variables in Nonlinear System

)()()(

)()()(

ΦQTTNofsolutionisΦTwhere

SxbΦTMxAΦx

)()(

)(0

0)()(

ΦQ

Sxb

TN

TMxA

T

Φx

Gauss Seidel style Nested Iteration

Parallel style Nested Iteration

New Module in TRAC-M

nestedloop() plays the role of once BackupLoop in trans()

Convergence control passed from PARCS

Also changes in GlobalDatM.f90, NamlistInputM.f90, prep.f90, TDMRTimeCalcM.f90, TDMRVarDeclM.f90

nestedloop.F90

SetTimeStep(): called once in 1st iteration

RestoreOldTimeInfo()

TDMR(3): new power and convergence control from PARCS

prep()

hout()

post()

TDMR(2): new T/H feedback to PARCS

Test Case: OECD MSLB EX2

TRAC-M Core Nodalization

TMI-I nuclear power plant Transient Simulation: EOC, 177 FA, 2772 MW th, 24

axial layers

1 3

4

2

6 5

8

7 9

10

11

12 18

17

16

15

14

13

19

neutronic fuel assembly

reflector assembly

stuck rod location

PARCS and TRAC-M Thermal Hydraulic Channel Mapping

Consistency of Numerical Results

Power vs. Timewith delt_t = 0.1 sec

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

Pow

er (%

)

marching forward

nested loop

Total Mass Flow Comparison in Pipe 1with delt_t = 0.1 sec

700

720

740

760

0 5 10 15 20 25 30

time (sec)

tota

l mas

s-flo

w (k

g/s)

marching forward

nested loop

Various Time Scales

Fixed fine time step size (0.1 sec) applied during SCRAM phase: 5 sec – 10 sec

Coarse time step size applied in other transient phases

Thereafter different time step sizes are used for testing

Reactivity vs. Time

-7

-6

-5

-4

-3

-2

-1

0

1

0 5 10 15 20 25 30

time (sec)

Reac

tivity

($)

marching forward 0.1s

marching forward 0.2s

Analysis of Time Step Size

Power vs. Timewith marching forward method

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

Pow

er (%

) 0.1sec

0.2 sec

0.5 sec

1.0 sec

Power vs. Timewith nested loop method

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

Pow

er (%

) 0.1 sec

0.2 sec

0.5 sec

1.0 sec

Marching Forward Method vs. Nested Loop Method

Cost ComparisonTotal Cost Summary for Various Numerical Solvers

time step size: 0.1 sec, time steps: 250

CPU-time consumed ratios compared with marching forward method are listed in the grey cells

The ratios of PARCS are little smaller than those of TRAC-M: nested outer iteration is the intrinsic solver for sub-nonlinear system of PARCS; however SETS (like predictor-corrector) is applied in TRAC-M, whose cost is more linear to the number of nested loop iterations

Numerical SolverDiff. Power

(%)

CPU-Time (sec)

Nested IterationsTRAC-M PARCS

Marching Forward 0.153 69.3 1.0 298.6 1.0 250

Nested Loop Method - 253.7 3.7 582.7 2.0 971

Excluding the SCRAM time phase

Cost Comparison (cont.)

Total Cost Summary for Various Numerical Solverstime step size: 0.5 sec, time steps: 50

CPU-time consumed ratios still compared with marching forward method with 0.1 sec step size

The cost of PARCS for nested loop method is cheaper than that for marching forward method with 0.1 sec time step size

Excluding the SCRAM time phase

Numerical SolverDiff. Power

(%)

CPU-Time (sec)Nested

IterationsTRAC-M PARCS

Marching Forward 2.97 15.9 0.2 90.5 0.3 50

Nested Loop Method 1.26 61.1 0.9 165.96 0.6 223

Test Case: OECD MSLB EX3

Full plant system: Task 1

Task 2

Transient Simulation

Total Mass Flow in Pipe 1 (hot leg b)

7400

7600

7800

8000

8200

8400

8600

8800

9000

0 10 20 30 40 50 60 70 80 90 100

time (sec)

mas

s flo

w (k

g/se

c)

Reactor Power

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 10 20 30 40 50 60 70 80 90 100

time (sec)

reac

tor p

ower

(%fp

)

Reactor Power

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 10 20 30 40 50 60 70 80 90 100

time (sec)

reac

tor p

ower

(%fp

)

Time Scale Selection

Only remove SCRAM effect for time step size testing

Fine time step size (10-6-10-1 sec) applied during SCRAM phase: 0 sec – 10 sec

Coarse time step size applied in other transient phases

Thereafter different time step sizes are used for testing

Results for Various Methods

Total Mass Flow in Pipe 1 (hot leg b)

8600

8650

8700

8750

8800

8850

0 5 10 15 20 25 30

time (sec)

mas

s flo

w (k

g/se

c)

marching forward method

nested loop method

Reactor Power

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

reac

tor p

ower

(%fp

)

marching forward method

nested loop method

Power curve is much more stable for two methods with same time step size, 0.1 second Signal trips are delayed in the result with nested loop method

Effects of Various Time Step Size Testing cases for larger step sizes are failed around 30-40 second caused by large perturbation Fewer oscillations captured with larger time step size Later study will test large step size after ECC signal triggered Furthermore, adaptive time step size and CS-trigger time step backup for nested loop iteration

Total Mass Flow in Pipe 1 (hot leg b)with marching forward method

8600

8650

8700

8750

8800

8850

0 5 10 15 20 25 30

time (sec)

mas

s flo

w (k

g/se

c)

0.1 sec

0.2 sec

Total Mass Flow in Pipe 1 (hot leg b)with nested loop method

8600

8650

8700

8750

8800

8850

0 5 10 15 20 25 30

time (sec)

mas

s flo

w (k

g/se

c)

0.1 sec

0.2 sec

0.3 sec

Current Results of Larger Step Sizes

Large step size after ECC signal triggered was tested for marching forward method till 0.4 second

Test failed for nested loop method since heat structure backup problem in BackupHS()

Reactor Powerwith marching forward method

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 10 20 30 40 50 60 70 80 90 100

time (sec)

reac

tor

pow

er (%

fp)

0.1 sec

0.2 sec

0.4 sec

Total Mass Flow in Pipe 1 (hot leg b)with marching forward method

7400

7600

7800

8000

8200

8400

8600

8800

9000

0 20 40 60 80 100

time (sec)

mas

s flo

w (k

g/se

c)

0.1 sec

0.2 sec

0.4 sec

Some “bug” in BackupHS() Nested loop method crushed by an infinite

logarithm operand in the reactor power period signal evaluation at svProcessHS()

In each nested loop iteration, old time information is reinstalled, and new power from PARCS is used to get new T/H feedback in TRAC-M

The associated variables of SV55, “rpowrn” and “rpowro”, will be backup in BackupHS()

“rpowrn” is changed with whole power, but “rpowro” is always increased about one order in each backup

“rpowro” is kept as initialization value if no backup when original marching forward method applied

*idsv isvn ilcn icn1 icn2

055 19 999 0 0 * Reactor Power Period

rpowrn rpowro

New and old time reactor core power in fuel rod 999

BackupHS()

temp = rodTab(cco)%rpowrnrodTab(cco)%rpowrn = rodTab(cco)%rpowro- (rodTab(cco)%rpowr-rodTab(cco)%rpowro)/ min(dto/delt,2.0d0)rodTab(cco)%rpowro = rodTab(cco)%rpowro- (temp-rodTab(cco)%rpowro)*dto/delt

Newton’s Method

)( :iteration sNewton’ ) F(xx –J xx ii-1i1 i

ii1i

iiii

iii

ii

Δxx

RΔJΔ

)F(x )J(xJ

)F(xR

4)

from solve 3)

2)

1)

Solve: F(x)=0

Inexact Newton method:solve

by iterative solver, such as GMRES, BICG,…

iii RΔJ

Generalized Minimum Residual Method (GMRES … thanks Prof

Saad!)To solve: J Δ =R drop the superscript i

Methodology: find ΔmKm(P-1J, P-1R)=span{ P-1R, P-1JP-1R,…(P-1J)mP-1R}

by minimizing ||rm=R-JΔm||2 increase m, until ||rm||2<tolerance

Algorithm: Δ0=0, r0=P-1R , β :=||r0||2, and v1:= r0 /β

Define the (m+1)×m matrix Hm={hi,j}1≤i≤m+1,1≤j≤m. set Hm=0

For j=1,2,…m Do:Compute wj:=P-1JvjFor I=1,2,…j Do

h i,j :=(wj,vi)

wj :=wj - h i,jvi

Enddohj+1,j=||wj||2. If hj+1,j=0, set m :=j, and break

v j+1=wj/ h j+1,j

EnddoCompute ym the minizer of || βe1-Hmy||2, and Δm= Δ0+Vmym

Matrix Free Inexact Newton MethodWith the following approximation of Jivj,

)()( i

ji

ji xFvxFvJ

vjKm(P-1Ji, P-1Ri), a combination of residuals

Ji does not need to be Constructed Matrix Free

Only need evaluate the residuals: F(xi), and F (xi +vj)

Question: Can we implement MF Newton’s method with a good preconditioner P, without constructing P?

And even do not evaluate the residuals F(xi), and F (xi +vj)?

And implement this in “legacy” codes like TRACM and PARCS

Perhaps … and do it based on codes using the nested method!

Derivation of MF Newton’s MethodBased on Nested Method

)()()( xbxxAxF

)()()( ji

ji

ji

ji vxgvxvxAvxF

)()()()()(

)()()()()()(

)()()()(

1

1)(1

jii

j

iij

ij

i

iiij

ij

ij

ii

ij

ii

jnn

vxgxgv

xgxvxgvx

xgxxAvxgvxvxAxA

xFvxFxAvJxA

No F(xi) and F (xi +vj), Don’t construct Preconditioner

Need g(xi) and g(xi +vj), can be obtained by nested method

)()(

)()()()()()()()(iii

iiiiiiiiii

xgxxA

xbxgxAxbxxAxbxxAxF

Recall: )(\)()( xbxAxg and

Preconditioning

A(xi) is the most convenient preconditioner we can get,Fortunately, it is often a good preconditioner, as:

J(xi)= F(xi)= A(xi)+ A(xi) xi - b(xi)

Implementation in TRACM/PARCS

finish

No

Yes

MATRIX - FREE PRECONDITIONING GMRESIteration j:

estimatewj: = P-1 J vj ≈ [P-1 F(xi+εvj) - r0] / ε

Newton_perturb (xi + εvj )

GMRES_MFP

Back to GMRES:P-1 F(xi+εvj) = (xi + εvj ) -f (xi + εvj )

GMRES Convergence?

GM

RE

S Iteration

PARCS

TRAC-M

'Nx

'/ HTx

Once Nested Coupling Loop

Consistency of Numerical Results

Total Mass Flow Comparison in Pipe 1with delt_t = 0.1 sec

700

720

740

760

0 5 10 15 20 25 30

time (sec)

tota

l mas

s-flo

w (k

g/s)

marching forward

nested loop

newton - krylov

Power vs. Timewith delt_t = 0.1 sec

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

Pow

er (%

) marching forward

nested loop

newton - krylov

Power vs. Timewith newton - krylov method

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

Pow

er (%

) 0.1 sec

0.2 sec

0.5 sec

Analysis of Time Step Size (cont.)

Marching Forward Method vs. Newton – Krylov Method

Power vs. Timewith marching forward method

0.00E+00

2.00E+01

4.00E+01

6.00E+01

8.00E+01

1.00E+02

1.20E+02

1.40E+02

0 5 10 15 20 25 30

time (sec)

Pow

er (%

) 0.1sec

0.2 sec

0.5 sec

Cost ComparisonTotal Cost Summary for Various Numerical Solvers

time step size: 0.1 sec, time steps: 250

CPU-time consumed ratios compared with marching forward method are listed in the grey cells

The ratios of PARCS are little smaller than those of TRAC-M: nested outer iteration is the intrinsic solver for sub-nonlinear system of PARCS; however SETS (like predictor-corrector) is applied in TRAC-M, whose cost is more linear to the number of nested loop iterations

Numerical SolverDiff. Power

(%)

CPU-Time (sec) Iterations

TRAC-M PARCS NESTED NEWTON GMRES

Marching Forward 0.153 69.3 1.0 298.6 1.0 250 - -

Nested Loop Method - 253.7 3.7 582.7 2.0 971 - -

Newton Method 0.0511 602.6 8.7 1686.7 6.6 2351 693 774Excluding the SCRAM time phase

Cost Comparison (cont.)Total Cost Summary for Various Numerical Solvers

time step size: 0.5 sec, time steps: 50

CPU-time consumed ratios still compared with marching forward method with 0.1 sec step size

The cost of PARCS for nested loop method is cheaper than that for marching forward method with 0.1 sec time step size

Perturbed system in GMRES with large time step size is difficult to achieve convergence for PARCS intrinsic nested loop solver

Excluding the SCRAM time phase

Numerical SolverDiff.

Power (%)

CPU-Time (sec) Iterations

TRAC-M PARCS NESTED NEWTON GMRES

Marching Forward 2.97 15.9 0.2 90.5 0.3 50 - -

Nested Loop Method

1.26 61.1 0.9 165.96 0.6 223 - -

Newton Method 1.35 334.9 4.8 1564.6 5.2 1251 192 480

Conclusions

The Nested and Newton methods have been implemented in the U.S. NRC code system

w/o construction of Jacobi matrix, nonlinear residuals, and w/ innovative preconditioner

Nested and Newton methods improve the accuracy of nonlinear coupled system in nuclear fields for the same time step size, but require a large increase in CPU time

The time step size with Nested and Newton methods can be increased to reduce the CPU requirements

Continuing Work

Work is continuing on optimizing the numerical behavior of the Nested and Newton methods Convergence study for outer Newton & inner GMRES

of Matrix Free Newton-Krylov Alterative approximation study

The tradeoff in accuracy vrs cpu time will then be examined for a wider range of coupled transient systems with more complicated T/H model, like OECD

MSLB EX3, PBTT EX3