Dans cet article, nous proposons une nouvelle bibliothèque de techniques non linéaires pour accélérer l’équation de transport en ordonnées discrètes. Deux nouveaux types de méthodes d'accélération non linéaire appelées méthode de rééquilibrage spatialement variable (SVRM) et accélération de matrice de réponse (RMA), respectivement, sont proposées et étudiées. La première méthode, SVRM, est basée sur le calcul de la variation spatiale de premier ordre de l'équation de la balance des neutrons. RMA, est une méthode DP0 qui utilise la connaissance de l'opérateur de transport pour former une relation cohérente. Deux variantes distinctes de RMA, appelées respectivement Explicit-RMA (E-RMA) et Balance (B-RMA), sont dérivées. Les propriétés de convergence des deux méthodes d'accélération sont étudiées pour deux schémas d'itération différents de l'opérateur de transport de la méthode des caractéristiques (MOC) pour une dalle 1D, en utilisant une analyse spectrale et une analyse de Fourier. Sur la base des résultats de la comparaison 1D, seuls les outils RMA et CMFD ont été implémentés dans la bibliothèque. Les performances de RMA sont comparées à celles de CMFD en utilisant les tests 3D C5G7, ZPPR et UH12. Les schémas de résolution parallèles et séquentiels sont considérés. L'analyse des résultats indique que les deux variantes de RMA ont une efficacité et une stabilité améliorées par rapport au CMFD, pour les matériaux à diffusion optique. De plus, le RMA montre une amélioration importante de la stabilité et de l'efficacité lorsque la géométrie est décomposée spatialement. Pour obtenir des performances numériques optimales, une combinaison de RMA et de CMFD est suggérée. Une enquête plus approfondie sur l'utilisation et l'amélioration de la RMA est proposée. De plus, de nombreuses idées pour étendre les fonctionnalités de la bibliothèque sont présentées. / In this paper we propose a new library of non-linear techniques for accelerating the discrete-ordinates transport equation. Two new types of nonlinear acceleration methods called Spatially Variant Rebalancing Method (SVRM) and Response Matrix Acceleration (RMA), respectively, are proposed and investigated. The first method, SVRM, is based on the computation of the zeroth and first order spatial variation of the neutron balance equation. RMA, is a DP0 method that uses knowledge of the transport operator to form a consistent relationship. Two distinct variants of RMA, called Explicit-RMA (E-RMA) and Balance (B-RMA), respectively, are derived. The convergence properties of both acceleration methods are investigated for two different iteration schemes of the method of characteristics (MOC) transport operator for a 1D slab, using spectral and Fourier analysis. Based off the results of the 1D comparison, only RMA and CMFD were implemented in the library. The performance of RMA is compared to CMFD using the C5G7, ZPPR, and UH12 3D benchmarks. Both parallel and sequential solving schemes are considered. Analysis of the results indicates that both variants of RMA have improved effectiveness and stability relative to CMFD, for optically diffusive materials. Moreover, RMA shows great improvement in stability and effectiveness when the geometry is spatially decomposed. To achieve optimal numerical performance, a combination of RMA and CMFD is suggested. Further investigation into the use and improvement of RMA is proposed. As well, many ideas for extending the features of the library are presented.

We investigate several aspects of the numerical solution of the radiative transfer equation in the context of coal combustion: the parallel efficiency of two commonly used opacity models, the sensitivity of turbulent radiation interaction (TRI) effects to the presence of coal particulate, and an improvement of the order of temporal convergence using the coarse mesh finite difference (CMFD) method. There are four opacity models commonly employed to evaluate the radiative transfer equation in combustion applications; line-by-line (LBL), multigroup, band, and global. Most of these models have been rigorously evaluated for serial computations of a spectrum of problem types [1]. Studies of these models for parallel computations [2] are limited. We assessed the performance of the Spectral-Line- Based weighted sum of gray gasses (SLW) model, a global method related to K-distribution methods [1], and the LBL model. The LBL model directly interpolates opacity information from large data tables. The LBL model outperforms the SLW model in almost all cases, as suggested by Wang et al. [3]. The SLW model, however, shows superior parallel scaling performance and a decreased sensitivity to load imbalancing, suggesting that for some problems, global methods such as the SLW model, could outperform the LBL model. Turbulent radiation interaction (TRI) effects are associated with the differences in the time scales of the fluid dynamic equations and the radiative transfer equations. Solving on the fluid dynamic time step size produces large changes in the radiation field over the time step. We have modifed the statistically homogeneous, non-premixed flame problem of Deshmukh et al. [4] to include coal-type particulate. The addition of low mass loadings of particulate minimally impacts the TRI effects. Observed differences in the TRI effects from variations in the packing fractions and Stokes numbers are difficult to analyze because of the significant effect of variations in problem initialization. The TRI effects are very sensitive to the initialization of the turbulence in the system. The TRI parameters are somewhat sensitive to the treatment of particulate temperature and the particulate optical thickness, and this effect are amplified by increased particulate loading. Monte Carlo radiative heat transfer simulations of time-dependent combustion processes generally involve an explicit evaluation of emission source because of the expense of the transport solver. Recently, Park et al. [5] have applied quasidiffusion with Monte Carlo in high energy density radiative transfer applications. We employ a Crank-Nicholson temporal integration scheme in conjunction with the coarse mesh finite difference (CMFD) method, in an effort to improve the temporal accuracy of the Monte Carlo solver. Our results show that this CMFD-CN method is an improvement over Monte Carlo with CMFD time-differenced via Backward Euler, and Implicit Monte Carlo [6] (IMC). The increase in accuracy involves very little increase in computational cost, and the figure of merit for the CMFD-CN scheme is greater than IMC. / Graduation date: 2012

<p>The broad implementation of three-dimensional full-core modeling, with pin-resolved detail, for computational simulation and analysis of nuclear reactors highlights the importance of accuracy and efficiency in simulation codes for accurate and precise analysis. The primary objective of this dissertation is to develop a high-fidelity code capable of solving time-dependent neutron transport problems with 3D whole-core pin-resolved detail in nuclear reactor cores. Additionally, the dissertation explores the optimization of the code's parallelism to enhance its computational efficiency. To reduce the computational intensity associated with the direct 3D calculation of the neutron transport equation, a high-fidelity neutron transport code called PANDAS-MOC is developed using the 2D/1D approach. The 2D radial solution is obtained using the 2D Method of Characteristics (MOC), the axial 1D solution is determined through the Nodal Expansion Method (NEM), and then two solutions are coupled using transverse leakages to find the 3D solution. The convergence of the iterative scheme is accelerated using the multi-level coarse finite different mesh (ML-CMFD) technique. The code's validation and verification are carried out using the C5G7-TD benchmark exercises.</p> <p><br></p> <p>The significant and innovative aspect of this work involves parallelizing and optimizing the PANDAS-MOC code. Three parallel models are developed and evaluated based on the distributed memory and shared memory architecture: MPI parallel model (PMPI), Segment OpenMP threading hybrid model (SGP), and Whole-code OpenMP threading hybrid model (WCP). When computing the steady state of the C5G7 3D core with the same resources, the obtained speedup relationship between the three models is PMPI \(>\) WCP \(>\) SGP, whereas the WCP model only consumed 60\% of the memory of the PMPI model. Furthermore, the hybrid reduction in the ML-CMFD solver and the parallelism design of the MOC sweep are significant issues that decreased the speedup of WCP. Therefore, this study also addresses further optimizations of these two modules.</p> <p><br></p> <p>Concerning the MOC parallelism, two improvements are discussed: No-atomic schedule and Additional Axial Decomposition (AAD) parallelism. The No-atomic schedule evenly distributed the workload among threads and removes the \textit{omp atomic} clause from the code by predefining the MOC calculation sequence for each launched OpenMP thread while ensuring a thread-safe parallel environment. It can significantly reduce the calculation time and improve parallel efficiency. Furthermore, AAD divides the axial layers and OpenMP threads into multiple groups and restricts each thread to work on the layers designated to the same group. </p> <p>Meanwhile, Flag-Save-Update reduction is designed to increase the computational efficiency of the hybrid MPI/OpenMP reduction operations in the ML-CMFD module. It is accomplished by using the global arrays and status flags and establishing a tree configuration of all threads, and it includes no implicit and explicit barriers. In the case of the C5G7 3D core, the parallel efficiency of the MOC solver is about 0.872 when using 32 threads (=\#MPI \(\times\)\#OpenMP), and the Flag-Save-Update reduction yielded better speedup than the traditional hybrid MPI/OpenMP reduction, and its superiority is more obvious as more OpenMP threads are utilized. As a result, the WCP model outperforms the PMPI model for the overall steady-state calculation.</p> <p><br></p> <p>This research also investigates parallelizable preconditioners to accelerate the convergence of the generalized minimal residual method (GMRES) in the CMFD solver. Preconditioners such as Incomplete LU factorization (ILU), Symmetric Successive Over-relaxation (SOR), and Reduced Symmetric Successive Over-Relaxation (RSOR), are implemented in PANDAS-MOC. Except for RSOR, others are unsuitable for hybrid MPI/OpenMP parallel machines due to their inherent sequential nature and dependency on computation order. Their counterparts using the Red-Black ordering algorithm, namely RB-SOR, RB-RSOR, and RB-ILU, are formatted and examined on benchmark reactors such as TWIGL-2D, C5G7-2D, C5G7-3D, and their corresponding subplane models (TWIGL-2D(5S), C5G7-2D(5S), C5G7-3D(5S)), with relaxed convergence criteria (\(10^{-3}\)). Results show that all preconditioners significantly reduce the required number of iterations to converge the GMRES solutions, and RB-SOR is the best one for most reactors. In the case of C5G7-3D(5S), preconditioners exhibit similar sublinear speedup but demonstrate varying runtimes across all tests for both MG-GMRES and 1G-GMRES. However, the speedup results in 1G-GMRES are more than twice as high as those in MG-GMRES. RB-RSOR has an optimal efficiency of 0.6967 at (4,8), while RB-SOR and RB-ILU have optimal efficiencies of 0.6855 and 0.7275 at (32,1), respectively.</p>