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Numerical Methods for Nuclear Nonlinear Coupled System

Numerical Methods for Nuclear Nonlinear Coupled System. J. Gan, Y. Xu T. J. Downar School of Nuclear Engineering Purdue University October, 2002. Outline. Numerical methods for nonlinear system: Marching forward method Nested loop method Newton – Krylov method (intro at end)

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Numerical Methods for Nuclear Nonlinear Coupled System

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

  4. step n+1 step n TRAC-M PARCS A. Original Marching Forward 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

  5. N N N Y Y Y step n step n+1 TRAC-M Converged? Converged? Converged? PARCS B. Gauss Seidel style Nested iteration Iterative Methods for Coupled System – Nested Loop • Alternative nested loop method: n: time step index; i: nested loop index Convergence criteria constrained, more accurate !

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

  7. Nested Iteration Gauss Seidel style Iteration Parallel style Iteration Iterate step 2 and 3 until converged It can be also viewed as : Fix point iteration for The nonlinear system Iterate step 2 until converged

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

  9. 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 New Module in TRAC-M nestedloop.F90 • 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

  10. Test Case: OECD MSLB EX2 TMI-I nuclear power plant Transient Simulation: EOC, 177 FA, 2772 MWth, 24 axial layers PARCS and TRAC-M Thermal Hydraulic Channel Mapping TRAC-M Core Nodalization

  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 Excluding the SCRAM time phase • 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

  15. Cost Comparison (cont.) Total Cost Summary for Various Numerical Solvers time step size: 0.5 sec, time steps: 50 Excluding the SCRAM time phase • 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

  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 rpowrn rpowro 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 =Rdropthe superscript i Methodology: find ΔmKm(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-1Jvj For I=1,2,…j Do h i,j :=(wj,vi) wj:=wj - h i,jvi Enddo hj+1,j=||wj||2. If hj+1,j=0, set m :=j, andbreak v j+1=wj/ h j+1,j Enddo Compute ym the minizer of || βe1-Hmy||2, and Δm= Δ0+Vmym

  25. Matrix Free Inexact Newton Method With the following approximation of Jivj, vjKm(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)

  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(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!

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

  28. 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)

  29. GMRES_MFP • MATRIX - FREE PRECONDITIONING GMRES • Iteration j: • estimate • wj: = P-1 J vj ≈ [P-1 F(xi+εvj) - r0] / ε GMRES Iteration Newton_perturb (xi + εvj ) TRAC-M PARCS • Back to GMRES: • P-1 F(xi+εvj) = (xi + εvj )-f(xi + εvj ) Once Nested Coupling Loop No GMRES Convergence? Yes finish Implementation in TRACM/PARCS

  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 Excluding the SCRAM time phase • 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

  33. Cost Comparison (cont.) Total Cost Summary for Various Numerical Solvers time step size: 0.5 sec, time steps: 50 Excluding the SCRAM time phase • 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

  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

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