1. Numerical Methods for Nuclear Nonlinear Coupled System J. Gan, Y. Xu T. J. Downar School of Nuclear Engineering Purdue University October, 2002

2. Outline Numerical methods for nonlinear system: Marching forward method Nested loop method Newton – Krylov method (intro at end) Results analysis Conclusion

3. Nonlinear Coupled System Neutronics eg. PARCS Thermal – Hydraulics eg. TRAC-M T, , , B Q Nonlinear subsystems already handled in each side, How about the above global nonlinear system?

4. 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

5. Iterative Methods for Coupled System – Nested Loop Alternative nested loop method: Convergence criteria constrained, more accurate ! 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 N Y N Y

6. Neutronics/Thermal-Hydraulics Coupled Problem It can be viewed as : Thermal-Hydraulic equation Neutronics spatial kinetic equation Nonlinear System:

7. 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 Nested Iteration

8. Independent Variables in Nonlinear System Gauss Seidel style Nested Iteration Parallel style Nested Iteration

9. 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 1 st iteration RestoreOldTimeInfo() TDMR(3): new power and convergence control from PARCS prep() hout() post() TDMR(2): new T/H feedback to PARCS

10. 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 PARCS and TRAC-M Thermal Hydraulic Channel Mapping

11. Consistency of Numerical Results

12. 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

13. Analysis of Time Step Size Marching Forward Method vs. Nested Loop Method

14. Cost Comparison Total 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 Solver Diff. Power (%) CPU-Time (sec) Nested Iterations TRAC-MPARCS Marching Forward0.15369.31.0298.61.0250 Nested Loop Method-253.73.7582.72.0971 Excluding the SCRAM time phase

15. 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 Excluding the SCRAM time phase Numerical Solver Diff. Power (%) CPU-Time (sec) Nested Iterations TRAC-MPARCS Marching Forward2.9715.90.290.50.350 Nested Loop Method1.2661.10.9165.960.6223

16. Test Case: OECD MSLB EX3 Full plant system:

17. Transient Simulation

18. 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

19. Results for Various Methods 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

20. 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

21. 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()

22. 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 rpowrnrpowro New and old time reactor core power in fuel rod 999 BackupHS() temp = rodTab(cco)%rpowrn rodTab(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

23. Newton’s Method Solve: F(x)=0 Inexact Newton method: solve by iterative solver, such as GMRES, BICG,…

24. Generalized Minimum Residual Method (GMRES … thanks Prof Saad!) To solve: J Δ =R drop the superscript i Methodology: find Δ m  K m (P -1 J, P -1 R)=span{ P -1 R, P -1 JP -1 R,…(P -1 J) m P -1 R} by minimizing ||r m =R-JΔ m || 2 increase m, until ||r m || 2 <tolerance Algorithm: Δ 0 =0, r 0 =P -1 R, β :=||r 0 ||2, and v 1 := r 0 /β Define the (m+1)×m matrix H m ={h i,j } 1≤i≤m+1,1≤j≤m. set H m =0 For j=1,2,…m Do: Compute w j :=P -1 Jv j For I=1,2,…j Do h i,j :=(w j,v i ) w j :=w j - h i,j v i Enddo h j+1,j =||w j || 2. If h j+1,j =0, set m :=j, and break v j+1 =w j / h j+1,j Enddo Compute y m the minizer of || βe 1 -H m y|| 2, and Δ m = Δ 0 +V m y m

25. Matrix Free Inexact Newton Method With the following approximation of J i v j, v j  K m (P -1 J i, P -1 R i ), a combination of residuals J i does not need to be Constructed  Matrix Free Only need evaluate the residuals: F(x i ), and F (x i +  v j )

26. Question: Can we implement MF Newton’s method with a good preconditioner P, without constructing P? And even do not evaluate the residuals F(x i ), and F (x i +  v j )? And implement this in “legacy” codes like TRACM and PARCS Perhaps … and do it based on codes using the nested method!

27. Derivation of MF Newton’s Method Based on Nested Method No F(x i ) and F (x i +  v j ), Don’t construct Preconditioner Need g(x i ) and g(x i +  v j ), can be obtained by nested method Recall:and

28. Preconditioning A(x i ) is the most convenient preconditioner we can get, Fortunately, it is often a good preconditioner, as: J(x i )=  F(x i )= A(x i )+  A(x i ) x i -  b(x i )

29. Implementation in TRACM/PARCS finish No Yes MATRIX - FREE PRECONDITIONING GMRES Iteration j: estimate w j : = P -1 J v j ≈ [P -1 F(x i +εv j ) - r 0 ] / ε Newton_perturb (x i + εv j ) GMRES_MFP Back to GMRES: P -1 F(x i +εv j ) = (x i + εv j ) -f (x i + εv j ) GMRES Convergence? GMRES Iteration PARCS TRAC-M Once Nested Coupling Loop

30. Consistency of Numerical Results

31. Analysis of Time Step Size (cont.) Marching Forward Method vs. Newton – Krylov Method

32. Cost Comparison Total 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 Solver Diff. Power (%) CPU-Time (sec)Iterations TRAC-MPARCSNESTEDNEWTONGMRES Marching Forward0.15369.31.0298.61.0250-- Nested Loop Method-253.73.7582.72.0971-- Newton Method0.0511602.68.71686.76.62351693774 Excluding the SCRAM time phase

33. 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 Solver Diff. Power (%) CPU-Time (sec)Iterations TRAC-MPARCSNESTEDNEWTONGMRES Marching Forward2.9715.90.290.50.350-- Nested Loop Method 1.2661.10.9165.960.6223-- Newton Method1.35334.94.81564.65.21251192480

34. 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

35. 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