RECENT DEVELOPMENTS IN COMPUTATIONAL REACTOR ANALYSIS

Transcription

1 RECENT DEVELOPMENTS IN COMPUTATIONAL REACTOR ANALYSIS Dean Wang April 30, Nuclear Reactor Physics

2 Outline 2 Introduction and Background Coupled T-H/Neutronics Safety Analysis Numerical schemes for T-H transport equations and neutron kinetics diffusion equations Recent Developments: Implementation and Assessments of High-Resolution Numerical Methods in TRACE High-resolution schemes Flux Limiters Implementation and assessments Computational Challenges in Reactor Analysis

3 History of TRACE/PARCS Development 3 TRACE is the latest best-estimate reactor system thermal-hydraulics analysis code developed by NRC for analyzing transient and steady-state neutronicthermal-hydraulic behavior in light water reactors. PARCS is a three-dimensional reactor core neutron kinetics code which solves the steady-state and time-dependent, multi-group neutron diffusion equations. I Joined ISL 7/7/03 V5.0-stable branch Branch code tree for assessment 6/7/06 V5.0 Release 7/31/07 V5.0p1-stable branch Peer Review 8/07 8/08 V5.0p1 Release 10/17/08 V /4/10 PARCS 3.0 V5.0p2 Release 6/2/10 I Joined ORNL 12/5/10 V5.0p3 Release 5/11/12 V5.0p3ho 11/2012 V5.0p4 with ho Release 8/2014 I Joined UML 9/14 Main Development trunk

4 Coupled T-H/Neutronics 4 Radial Mapping Time-Step: n TRACE PARCS Axial Mapping Time-Step: n+1 TRACE PARCS Thermal-hydraulics (TRACE) Two-fluid, two-phase 6 field equations Space: Finite volume, 1st order upwind differencing Time: SETS or Semi-Implicit. Semi- Implicit is subject to the Courant limit, while SETS is more diffusive Neutronics (PARCS) Two-group nodal diffusion kinetics equations Space: CMFD with local two-node correction Time: the theta method with exponential transformation Coupling between T-H/Neutronics (TRACE/ PARCS) Space: radial and axial Time: explicit

5 Cross Sections for Neutronics 5 D. Wang, et al., Cross Section Generation Guidelines for TRACE-PARCS, NUREG/CR-7164, June 2013.

6 Reactor Modeling and Simulation Physical System Physical Laws SS and Transients LOCA AOO/ATWS Stability Mathematical Model Variables Field Equations Constitutive Equations Initial and Boundary Conditions Numerical Solution Numerical Methods Implicit or explicit First-order or highorder Computational Algorithms = 1 + 2

7 Two-Phase Flow Transport Equations 7 Mass: Momentum: Energy:

8 1 st -Order Upwind Finite-Volume Method 8 j-1 j j+1 j-3/2 j-1/2 j +1/2 u Upwind: Local Truncation Error:

9 Standard 2 nd -Order Schemes 9 They don t work well for two-phase flow!

10 Numerical Diffusion 10 Most reactor system analysis codes use the 1 st -order upwind scheme such as TRACE, RELAP, COBRA, etc. While very robust, 1 st -order upwinding leads to excessive numerical diffusion (damping). 2 nd -order methods can effectively reduce numerical diffusion, but often produce spurious oscillations and slow down convergence It has to be carefully treated for reactor analysis, particularly for BWR stability analysis and boron tracking

11 Fix for Standard 2 nd -Order Schemes 11 Flux Limiters:

12 Why These Flux Limiters Work? TVD Region and Monotonicity Limiter Functions MUSCL1.5 OSPRE Van Albada ENO 2 nd -order TVD Region 1.5 Φ(r) φ(r)=1 TVD Region r Theorem: Any TVD numerical method is monotonicity preserving

13 Desired Properties of Flux Limiters 13 Φ(1) = 1. This is a necessary requirement for 2 nd - order accuracy on smooth solu=ons Φ(r)/r = Φ(1/r). This symmetric property ensures that a flux-limiter has the same actions on forward and backward gradients Φ(r) is located in the 2 nd - order TVD region Φ(r) MUSCL1.5 OSPRE Van Albada ENO Limiter Functions r 2 nd -order TVD Region 13

14 14 Linear Transport Equation

15 Why Are These Limiters Important? 15 Provide a solution to the important and long-lasting numerical issue in the 1d system analysis codes Effectively reduce numerical damping without incurring spurious oscillations. They have a promising application in BWR stability analysis. We used to develop a non-uniform nodalization to mitigate numerical diffusion. Higher numerical accuracy can be achieved with these flux limiters on a uniform coarse nodalization. We initially introduced 2 nd -order limiter schemes into TRACE (ENO and others) Relatively simple algorithm Easy implementation within TRACE Very promising results

16 Implementation of Flux-Limiters in TRACE 16 Central Difference with flux limiters for the convection terms in the mass and energy conservation equations. The momentum equation in TRACE is in nonconservative form, in which 2 nd -order central difference is used for the convection term. Time integration scheme is semi-implicit

17 Assessment of Flux Limiters Convergence Performance 17 CPU time taken to achieve the steady-state convergence. The time-step size on the top of each column is the maximum time-step size allowed to achieve the SS convergence. 600 VA (and MUSCL) has very comparable performance with the upwind method in terms of CPU time. CPU time (s) Van Albada Upwind Time-Step Size (s)

18 18 Applications of Flux Limiter Schemes Ringhals Cycle 14 Coupled TRACE/PARCS BWR plant model 8 stability test points of Ringhals Cycle 14 Upwind largely underpredicts decay ratio of core-wide and out-of-phase oscillations 2 nd -order flux limiter VA has improved the overall accuracy by 7% TRACE DR TRACE DR TRACE V5.0p3 - Upwind Core-Wide DR Out-Of-Phase DR Uncertainty ±14% Benchmark DR TRACE V5.0p3 -VA Core-Wide DR Out-Of-Phase DR Uncertainty ±7% Benchmark DR P09

19 Oskarshamn Feedwater Transient Event 19 Oskarshamn-2 experienced a stability event on February 25, A loss of feedwater preheaters and control system logic failure resulted in high feedwater flow and low temperature Then, automatic power and flow control moved the reactor into a low flow - high power regime Very challenging coupled T-H/Neutronics problem The 2 nd -order flux limiter shows improvement in the prediction of nonlinear power oscillations with DR > 1.

20 Summary and Recommendations 20 A number of flux limiters were selected and implemented in TRACE for assessments. In this study, assessments were mainly focused on evaluating their performance for BWR stability analysis. VA, OSPRE, and MUSCL seem to work best in terms of numerical stability, accuracy, and convergence. Good numerical accuracy on a coarse nodalization can be achieved with these 2 nd -order flux limiter schemes. It is worth noting that 2 nd -order accuracy in both time and space can be achieved when using the Lax-Wendroff scheme with these flux limiters. These high-resolution methods are now officially released with TRACE V5.0p4.

21 2 nd -order Accurate both in Space and 21 Time where D. Wang, et al., "Solving Two-Phase Flow Transport Equations Using the Lax-Wendroff Scheme," Trans. AM. Nucl. Soc., 2015

22 Challenges in Computational Reactor Safety Analysis 22 Coupling between T-H and Neutronics Speedup of calculations Beyond CMFD Beyond 1D T-H

23 Implicit Coupling between T-H and Neutronics? 23 Fully implicit coupling Tightly coupled T-H, fuel heat conduction, and Neutronics equations Nonlinear system: need local linearization Jacobian matrix, Challenges Large matrix Complex coupling terms Storage Larger time-step for speedup? higher order time integration is needed J. K. Watson, PhD thesis, 2010 Is it really worth going implicit coupling?

24 Speedup of Reactor Safety Analysis CPU Time (s) Speedup # of CHANs in TRACE Parallelize XSEC and BiCGSTAB Modules Using OpenMP Peach Bottom TT2 (2s transient) BiCGSTAB XSEC Numbers of Cores TRACE PARCS - CMFD PARCS - Nodal PARCS - T/H PARCS - XS PARCS - Pin Power Coupled T-H/Neutronics computation is time-consuming BWR stability analysis AOO/ATWS Station blackout Speedup Relax time-step using high-order implicit time integration? n Not very promising SSP n Physical time scales, e.g. BWR flow/ power oscillations n Advanced iterative solvers for large linear systems Locally adaptive time stepping Parallelization and GPU computing

25 GPU Computing 25 NVIDIA Quadro K4200 CUDA Cores: 1344 GPU Memory: 4GB Memory Bandwidth: 173GB/s

26 Beyond CMFD? 26 Now the CMFD is widely used for solving core neutron diffusion equations or as a preconditioner for neutron transport calculations. However, the CMFD may encounter numerical stability problems and fails to converge. Solution: high-order Galerkin spectral element methods may provide viable approaches for solving neutron kinetics problems with better accuracy and computational efficiency.

27 Beyond 1D T-H and Low-Order CFD 27 SMRs, SFR and FHRs largely rely on natural circulation to remove decay heat Thermal stratification and mixing requires 3D modeling and simulation capability However, large scales and local complexities of the reactor system pose a paramount challenge for low-order CFD analysis, In particular, long-term transient problems. Solution: highly efficient CFD solvers

28 28 DOE CASL Advanced M&S

29 29 DOE NEAMS

30 What We Are Working on Now and 30 Plan to Do 2 nd -order numerical methods in BOTH time and space for TRACE V5.0p4. Implementation of high-resolution schemes in COBRA-TF (CASL). Locally adaptive time stepping schemes for twophase flow simulation. Development of new acceleration schemes for neutron transport calculations.

31 References 31 D. Wang, Reduce Numerical Diffusion in TRACE Using the High-Resolution Numerical Method ENO, Trans. AM. Nucl. Soc.107, 2012 D. Wang, et al., Implementation and Assessment of High-Resolution Numerical Methods in TRACE, Nuclear Engineering and Design 263 (2013) D. Wang, et al., "Solving Two-Phase Flow Transport Equations Using the Lax-Wendroff Scheme," Trans. AM. Nucl. Soc., 2015 N.P. Waterson, H. Deconinck, Design Principles for Bounded Higher-Order Convection Schemes a Unified Approach, Journal of Computational Physics, 224 (2007), P.K. Sweby, High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SIAM J. Num. Anal., 21 (1984), pp C.W. SHU and S. OSHER, Efficient Implementation of Essentially Non-oscillatory Shock Capturing Scheme, II, J. Comput. Phys., 83: 32-78, R.J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH Zurich, Birkhauser, R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, M.O. Deville, et al., High-Order Methods for Incompressible Fluid Flow, Cambridge University Press, C. Hirsch, Numerical Computation of Internal and External Flows, 2 nd Edition, Wiley, J.S. Hesthaven, et al., Spectral Methods for Time-Dependent Problems, Cambridge J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer 2008.

32 32 Thank You!