Han Gyu Joo

An incomplete domain decomposition preconditioning method for nonlinear nodal kinetics calculations

Methods are proposed for the efficient parallel solution of nonlinear nodal kinetics equations. Because the two-node calculation in the nonlinear nodal method is naturally parallelizable, the majority of the effort is devoted to the development of parallel methods for solving the coarse-mesh finite difference (CMFD) problem. A preconditioned Krylov subspace method (biconjugate gradient stabilized) is chosen as the iterative algorithm for the CMFD problem, and an efficient parallel preconditioning scheme is developed based on domain decomposition techniques. An incomplete lowerupper triangular factorization method is first formulated for the coefficient matrices representing each three-dimensional subdomain, and coupling between subdomains is then approximated by incorporating only the effect of the nonleakage terms of neighboring subdomains. The methods are applied to fixed-source problems created from the International Atomic Energy Agency three-dimensional benchmark problem. The effectiveness of the incomplete domain decomposition preconditioning on a multiprocessor is evidenced by the small increase in the number of iterations as the number of subdomains increases. Through the application to both CMFD-only and nodal calculations, it is demonstrated that speedups as large as 49 with 96 processors are attainable in the nonlinear nodal kinetics calculations.

High-fidelity light water reactor analysis with the numerical nuclear reactor

Abstract The Numerical Nuclear Reactor (NNR) was developed to provide a high-fidelity tool for light water reactor analysis based on first-principles models. High fidelity is accomplished by integrating full physics, highly refined solution modules for the coupled neutronic and thermal-hydraulic phenomena. Each solution module employs methods and models that are formulated faithfully to the first principles governing the physics, real geometry, and constituents. Specifically, the critical analysis elements that are incorporated in the coupled code capability are a direct whole-core neutron transport solution and an ultra-fine-mesh computational fluid dynamics/heat transfer solution, each obtained with explicit (sub-fuel-pin-cell level) heterogeneous representations of the components of the core. The considerable computational resources required for such highly refined modeling are addressed by using massively parallel computers, which together with the coupled codes constitute the NNR. To establish confidence in the NNR methodology, verification and validation of the solution modules have been performed and are continuing for both the neutronic module and the thermal-hydraulic module for single-phase and two-phase boiling conditions under prototypical pressurized water reactor and boiling water reactor conditions. This paper describes the features of the NNR and validation of each module and provides the results of several coupled code calculations.

Generation of Few-Group Diffusion Theory Constants by Monte Carlo Code McCARD

Abstract The purpose of this paper is to present the Monte Carlo (MC) method augmented by the B1 spectrum to generate few-group diffusion theory constants, to assess their qualification in terms of the core depletion analysis, and thus to validate the MC method implemented into the Seoul National University MC code, McCARD, as a few-group diffusion theory constant generator. To do so, two-step core neutronics analyses are conducted for two types of power reactors, pressurized water reactors and very high temperature gas-cooled reactors, by the McCARD/MASTER code system in which McCARD is used as a MC few-group constant generation code and MASTER as a deterministic core analysis code. The two-step calculations for the effective multiplication factors and assembly power distributions of the two types of power reactor cores by McCARD/MASTER are compared with the reference calculations from McCARD, the nuclear design report, or measurements. By showing excellent agreement between McCARD/MASTER and the reference neutronics analyses for the two types of power reactors, it is concluded that the MC method implemented in McCARD can generate few-group diffusion theory constants that are well qualified for high-accuracy two-step core neutronics calculations.

Whole Core Transport Calculation Employing Hexagonal Modular Ray Tracing and CMFD Formulation

A whole core transport module for hexagonal cores is developed and implemented in the DeCART code. The module consists of a whole core ray tracing kernel for solving the two-dimensional (2-D) method of characteristics (MOC) equation and a coarse-mesh finite difference (CMFD) kernel to accelerate the MOC transport iteration. Whole core ray tracing is realized by incorporating a hexagonal-assembly-based modular ray tracing scheme. The complete path linking constraint forthe modular ray is achieved by adjusting the ray angle and the ray spacing in the range of [0, 30°], and the complete reflection constraint at the problem boundary is satisfied by defining the corresponding reflection angles at the reflection surfaces. The hexagonal CMFD kernel employs unstructured nodes that can treat the irregular-shaped gap cells as well as the regular hexagonal cells. Some features such as cell ray approximation and modified cycle ray scheme are employed to reduce the memory requirements for the segment information and the boundary angular fluxes, respectively. The solution accuracy and execution performance of the hexagonal module are examined for the C5G7 hexagonal variation problems that are established by modifying the original C5G7MOX 3-D extension benchmark. The CMFD kernel shows a significant speedup of 60 in the 2-D core problem. The cell ray approximation does not violate the original solution accuracy when using the default ray spacing of the DeCART code. The modified cycle ray scheme shows its superiority over the simple core ray sweeping scheme in terms of the memory requirement and the original cycle ray scheme in terms of the computing time. Compared with the Monte Carlo solutions, the DeCART solution agrees to within 40 pcm for the eigenvalue and 2% for the pin power distribution.

Axial SPN and Radial MOC Coupled Whole Core Transport Calculation

The Simplified PN (SPN) method is applied to the axial solution of the two-dimensional (2-D) method of characteristics (MOC) solution based whole core transport calculation. A sub-plane scheme and the nodal expansion method (NEM) are employed for the solution of the one-dimensional (1-D) SPN equations involving a radial transverse leakage. The SPN solver replaces the axial diffusion solver of the DeCART direct whole core transport code to provide more accurate, transport theory based axial solutions. In the sub-plane scheme, the radial equivalent homogenization parameters generated by the local MOC for a thick plane are assigned to the multiple finer planes in the subsequent global three-dimensional (3-D) coarse mesh finite difference (CMFD) calculation in which the NEM is employed for the axial solution. The sub-plane scheme induces a much less nodal error while having little impact on the axial leakage representation of the radial MOC calculation. The performance of the sub-plane scheme and SPN nodal transport solver is examined by solving a set of demonstrative problems and the C5G7MOX 3-D extension benchmark problems. It is shown in the demonstrative problems that the nodal error reaching upto 1,400 pcm in a rodded case is reduced to 10pcm by introducing 10 sub-planes per MOC plane and the transport error is reduced from about 150pcm to 10pcm by using SP3. Also it is observed, in the C5G7MOX rodded configuration B problem, that the eigenvalues and pin power errors of 180 pcm and 2.2% of the 10 sub-planes diffusion case are reduced to 40 pcm and 1.4%, respectively, for SP3 with only about a 15% increase in the computing time. It is shown that the SP5 case gives very similar results to the SP3 case.

Consistent comparison of the codes RELAP5/PARCS and TRAC-M/PARCS for the OECD MSLB coupled code benchmark

Abstract A generalized interface module was developed for coupling any thermal-hydraulic code to any spatial kinetic code. In the design used here the thermal-hydraulic and spatial kinetic codes function as independent processes and communicate using the Parallel Virtual Machine software. This approach helps maximize flexibility while minimizing modifications to the respective codes. Using this interface, the U.S. Nuclear Regulatory Commission (NRC) three-dimensional neutron kinetic code, Purdue Advanced Reactor Core Simulator (PARCS), has been coupled to the NRC system analysis codes RELAP5 and Modernized Transient Reactor Analysis Code (TRAC-M). Consistent comparison of code results for the Organization for Economic Cooperation and Development/Nuclear Energy Agency main steam line break benchmark problem using RELAP5/PARCS and TRAC-M/PARCS was made to assess code performance.

APPLICATION OF BACKWARD DIFFERENTIATION FORMULA TO SPATIAL REACTOR KINETICS CALCULATION WITH ADAPTIVE TIME STEP CONTROL

The backward differentiation formula (BDF) method is applied to a three-dimensional reactor kinetics calculation for efficient yet accurate transient analysis with adaptive time step control. The coarse mesh finite difference (CMFD) formulation is used for an efficient implementation of the BDF method that does not require excessive memory to store old information from previous time steps. An iterative scheme to update the nodal coupling coefficients through higher order local nodal solutions is established in order to make it possible to store only node average fluxes of the previous five time points. An adaptive time step control method is derived using two order solutions, the fifth and the fourth order BDF solutions, which provide an estimate of the solution error at the current time point. The performance of the BDF- and CMFD-based spatial kinetics calculation and the adaptive time step control scheme is examined with the NEACRP control rod ejection and rod withdrawal benchmark problems. The accuracy is first assessed by comparing the BDF-based results with those of the Crank-Nicholson method with an exponential transform. The effectiveness of the adaptive time step control is then assessed in terms of the possible computing time reduction in producing sufficiently accurate solutions that meet the desired solution fidelity.

Stabilization techniques for the nonlinear analytic nodal method

The nonlinear analytic nodal method, which is formulated by combining the nonlinear itera- tion technique and the analytic nodal method (ANM), requires analytic solutions of the two-node prob- lems. When the method is applied to problems that contain near-critical nodes in which there is essentially no net leakage, the two-node ANM solution for such nodes results in highly ill-conditioned matrices and potential numerical instabilities, especially in single precision arithmetic. Two stabilization techniques are introduced to resolve the instability problem by employing alternate basis functions for near-critical nodes. The first uses the exact ANM solution for a critical node, and the second employs the nodal expan- sion method. Both techniques are shown to perform well; however, the solution accuracy can be mildly sensitive to the criterion used to invoke the stabilized coupling kernel. I. INTRODUCTION Nonlinear nodal methods 1 have been widely used to solve static and transient reactor physics problems. A well- known advantage of the nonlinear method is the reduc- tion in computer storage that results from not having to save the expansion coefficients. However, an equally im- portant advantage is a reduction in execution time be- cause the solution variables in the global coarse-mesh finite difference ~CMFD! portion of the algorithm are only the node-averaged fluxes for which efficient solution schemes have been developed. Furthermore, the two- node portion of the algorithm lends itself naturally to par- allel solution, and when solved in conjunction with a parallel CMFD method, significant execution time re- ductions can be achieved with the nonlinear nodal method on multiprocessors. 2 One of principal issues in the implementation of the nonlinear nodal method is the interface current tech- nique used in the two-node coupling relation. As dis- cussed in the review by Lawrence, 3 two principal classes of transverse-integrated methods have been developed over the years—the polynomial and the analytic meth- ods. In the polynomial nodal expansion method 4 ~NEM!,

Solution of the BEAVRS benchmark using the nTRACER direct whole core calculation code

The BEAVRS (Benchmark for Evaluation and Validation of Reactor Simulation) benchmark is solved by the nTRACER direct whole core calculation code to assess its accuracy and to examine the solution dependence on modeling parameters. A sophisticated nTRACER core model representing the BEAVRS core is prepared after a series of sensitivity study to ensure solution accuracy. The resulting solutions for several hot-zero-power (HZP) states are compared first with the corresponding Monte Carlo solutions, which consist of the McCARD solutions for the assembly problems and the OpenMC solutions for the core problems, and then with the measured data which include the control rod worths (CRWs) and incore detector signals as well as the critical boron concentrations (CBC). The core depletion calculation is performed for the initial and second cycles with a set of approximated power histories and the calculated CBCs are compared with the measured data. The comparison results show that the criticality, control rod bank worths at HZP and the boron let-down curves of two cycles agree well with the measurements within 180 pcm and 25 ppm, respectively.

Multiobjective Loading Pattern Optimization by Simulated Annealing Employing Discontinuous Penalty Function and Screening Technique

Abstract The problem of multiobjective fuel loading pattern (LP) optimization employing high-fidelity three-dimensional (3-D) models is resolved by introducing the concepts of discontinuous penalty function, dominance, and two-dimensional (2-D)–based screening into the simulated annealing (SA) algorithm. Each constraint and objective imposed on a reload LP design is transformed into a discontinuous penalty function that involves a jump to a quadratic variation at the point of the limiting value of the corresponding core characteristics parameter. It is shown that with this discontinuous form the sensitivity of the penalty coefficients is quite weak compared to the stochastic effect of SA. The feasible LPs found during SA update the set of candidate LPs through a dominance check that is done by examining multiple objectives altogether. The 2-D–based screening technique uses a precalculated database of the 2-D solution errors and is shown to be very effective in saving the SA computation time by avoiding 3-D evaluations for the unfavorable LPs that are frequently encountered in SA. Realistic applications of the proposed method to a pressurized water reactor reload LP optimization with the dual objectives of maximizing the cycle length and minimizing the radial peaking factor demonstrate that the method works quite well in practice.