Efficiency Enhancements of Coupled RELAP5/PANBOX Calculations Using Adaptive Methods

Herbert Finnemann 
Siemens/KWU – Germany

and

Robert Scheichl
University of Bath – UK

Introduction
RELAP5/PANBOX - in short R/P/C - is a nuclear plant safety analysis code system comprised of the core simulation package PANBOX coupled directly to the best-estimate plant simulation code RELAP51. 
Three main options exist by which the coupling between the core and the plant can be simulated. These options are:
•	Neutronic Feedback from COBRA Thermalhydraulics
Using this option, the core thermalhydraulic boundary conditions of core pressure, inlet temperature, and inlet mass flow rate are passed from RELAP5 to COBRA, the core thermalhydraulic module of PANBOX. With these boundary conditions and the PANBOX power profile, COBRA recalculates the core thermalhydraulic conditions using a one-channel per fuel assembly grid. This option has the advantage that thermal margins may be calculated within COBRA (s. below).
•	Neutronic Feedback from RELAP5 Thermalhydraulics
With this option, no COBRA calculation is made at all. The core thermalhydraulic data is passed from RELAP5 to PANBOX, and this data is mapped and interpolated onto the 3-D core nodalization. The newly calculated PANBOX power distribution is collapsed onto the RELAP5 nodalization, and this information is passed back to RELAP5. 
•	Neutronic Feedback from RELAP5 Thermalhydraulics, with parallel COBRA calculation
This option is identical to that described in the preceding section, except that a COBRA core thermalhydraulic solution is also calculated in parallel to the RELAP5/PANBOX calculation. This option is useful for calculating thermal margins with COBRA based on the identification of hot pin positions and the resulting automatic renodalization of the COBRA channel geometry. 
The program is thus capable to perform global neutronic/thermalhydraulic calculations and to evaluate important local safety-related parameters. Further work is especially directed towards improvements of computational efficiency.

Adaptive Neutron Kinetics
In the recent past, the efficiency of the code system was significantly improved by the systematic use of adaptive algorithms2,3.
•	Dimensionally Adaptive Neutron Kinetics 
In addition to the standard 3-D neutron kinetics model in PANBOX, 1-D and point neutron kinetic models are available. The coefficients for these lower dimensional models are generated from the 3-D data base, using a variant of the quasi-static approximation. The user of R/P/C has the option to switch between 3-D, 1-D and point kinetic models during various periods of a transient calculation. These periods may be selected through a combination of manual input and an automatic adaptive algorithm, with significant savings in computing time. Results indicating code accuracy and efficiency for the various options are given in Ref. 3.
•	Parallel Adaptive Nodal Kinetics 
Modern developments try to enhance the efficiency of the nodal solution process in 3-D by using multi-level techniques4,5. 
The following short discussion will be mainly concerned with the acceleration of higher-order nodal methods. It is based on the formally rigorous nodal balance equations.

	(1 b)

	(1 c)

The incoming currents j± appearing on the RHS of (1 a) are given as outgoing currents of adjacent boxes. The coupling coefficients  (1 b) can be calculated by diffusion or higher-order transport theory approximation. In addition, the accuracy of the final result is determined by the order of the numerical integration used. The whole process can be divided in several subprocesses with different integration order, in such a way that the higher-order process - both with respect to physical model and numerical integration - determines the accuracy of the converged solution. Presently the overall iterative solution process employs implicit time integration methods and a quadratic approximation (NEM-M02) of the current equations (2), to describe the global coupling.

	(2)

Similar to other nonlinear nodal methods the coefficients of the fluxes in (2) are adapted from higher-order current equations to retain the accuracy of the more sophisticated models. These calculations are naturally parallelizable6.

A prototype parallel version of R/P/C has already been implemented earlier7 for use with multiprocessor systems and workstations clusters. Recently this work was extended using multigrid (MG) and Krylov subspace methods8 (preconditioned BiCGStab) for the iterative solution of the NEM-M0 equations. The speedup for the BiCGStab method compares favorably with the earlier approach (s. Table 1). Preliminary results using the MG method8 indicate that a comparison of the two methods may turn out in favor of MG. This is an interesting point for further investigations.

Table 1: Parallel Speedup / Efficiency for the Iterative Solution of the NEM-M0 Equations on an SGI Power Challenge MP

Number of Processors	NEACRP Rod Ejection Benchmark	
	Old version7	Prec. BiCGStab8	
1	1.0	/		100 %	1.0	/	100 %	
2	1.8			90 %	2.1		105 %	
4	2.7			68 %	4.1		102 %	
6	3.5			58 %	6.0		100 %	
8	3.7			46 %	7.7		 96 %	
10		-		8.9		 89 %	
12		-		10.5		 88 %	

Summary
The features of R/P/C make it a valuable general tool for the analysis of postulated accident scenarios. The application range of RELAP5 is extended by the added ability to simulate 3-D neutron kinetics phenomena and to calculate thermal margins. The lower dimensional models may be used in an adaptive algorithm with the 3-D kinetics model to save computation time, or they may be used as independent models to replace the point kinetics model within RELAP5. Additional savings in CPU time can be achieved by using multi-level adaptive techniques and the parallelization of compute-intensive parts of the code system. 

References
1.	C. J. JACKSON, H. FINNEMANN
”Verification of the Coupled RELAP5/PANBOX” System with the NEACRP LWR Core Transient Benchmark.” International Conference on Mathematics and Computations, Reactor Physics, and Environmental Analyses, Portland, Oregon, (April 30-May 4, 1995).
2.	C. J. JACKSON, D. G. CACUCI, H. FINNEMANN,
”Dimensionally Adaptive Neutron Kinetics for Multidimensional Reactor Safety 
Transients-I: New Features of RELAP5/PANBOX”
Nucl. Sci. Eng., 131, 143 (1999)

3.	C. J. JACKSON, D. G. CACUCI, H. FINNEMANN,
”Dimensionally Adaptive Neutron Kinetics for Multidimensional Reactor Safety Transients-II: Dimensionally Adaptive Switching Algorithms”
Nucl. Sci. Eng., 131, 164 (1999)

4.	A. BRANDT,
”Multi-Level Adaptive Solutions to Boundary-Value Problems,” Mathematics of Computation, 31, No. 138, pp. 333-390 (1977);

5.	H. FINNEMANN, R. BÖER, R.MÜLLER, Y. I. KIM, 
”Multi-Level Techniques for the Acceleration of Nodal Reactor Calculations,” Proc. Int. Top. Mtg. Advances in Mathematics, Computations, and Reactor Physics, Vol. 3, Pittsburgh (1991).

6.	H. G. JOO and T. J. DOWNAR,
”An Incomplete Domain Decomposition Preconditioning Method for Nonlinear Nodal Kinetics Calculations,” Nucl. Sci. Eng., 123, 403 (1996)

7.	C. HAMMER, M. PAFFRATH, R. BÖER, H. FINNEMANN, C. J. JACKSON, 
”Performance of the Coupled Thermalhydraulics/Neutron Kinetics Code R/P/C on Workstation Clusters and Multiprocessor System,” Proceedings of the International Conference on Physics of Reactors, PHYSOR96, Sept. 16-20, 1996, Mito Japan.

8.	R. SCHEICHL, 
”Parallel Solvers for the Transient Multigroup Neutron Diffusion Equations,” Mathematics Preprint 98/30 (1999), University of Bath, UK.
http://www.maths.bath.ac.uk/MATHEMATICS/preprints.html