Efficient computational methods have long been a major focus and challenge in the field of particle transport, and the unstructured grid discrete ordinates (SN) method is one of the effective numerical approaches for neutron transport simulations. However, the massive computational and storage demands of three-dimensional (3-D) unstructured grid SN calculations constitute a bottleneck constraining the large-scale multi-physics coupling simulations. The serial nature of SN sweeps and the complex data dependencies within 3-D unstructured grids result in limited algorithmic parallelism, complex task scheduling, and high communication latency. These factors pose significant challenges to achieving efficient, scalable parallel computing for large-scale application problems. To address these issues, we develop an SN angle-space dynamic scheduling algorithm and a load-balanced energy group local communication algorithm. Furthermore, we advance a multi-level parallel computations integrating space, angle, and energy groups. Numerical test results demonstrate the high scalability of our multi-level parallel algorithm, and the parallel efficiency at 65536 cores relative to 2048 cores reaches 85%. This supports efficient simulations of neutron transport problems on 3-D unstructured grids, scalable to 100000 cores and over 100 billion degrees of freedom.

A Taylor basis discontinuous Galerkin finite element scheme is presented for the discrete ordinates (SN) transport equation in the two-dimensional r-z coordinate system. The basis functions based on the Taylor series expansion are hierarchical and independent of mesh shape, thus providing a unified framework for regular, deformed, and hybrid grids of triangles and quadrilaterals, which may be beneficial for some coupled simulation applications. The spatial distribution of the solution is represented by linear, bilinear, quadratic, and cubic polynomial expansion functions, respectively. Finite element integration is calculated in the physical element by Gaussian quadrature. By extending the basis function evaluation module, arbitrary higher-order DG schemes can be implemented. We develop a unified DG-SN code for the fixed-source, k-eigenvalue, alpha-eigenvalue, and time-dependent transport equations. Based on mesh priority and lagged angular flux, a sweep sorting algorithm is applied for decoupling dependency cycles between severely deformed grids. Numerical verification is performed for several transport problems, and the results indicate that our method achieves the theoretical convergence rate and strong robustness on non-orthogonal grids in r-z geometry. For the tested multi-medium transport problems, higher-order schemes exhibit certain computational advantages over lower-order schemes on coarse grids.

In previous work, we developed the unified gas dynamics scheme (UGKS) and its corresponding time-implicit variant (IUGKS) for neutron transport. The multi-scale characteristics of these schemes are particularly well-suited for transport equations, which show good performances in large-scale conditions. However, only the isotropic scattering model equation was considered in these studies, which does not meet the application requirements in complex engineering cases. In this paper, we extend our studies of isotropic neutron IUGKS to encompass the transport equation with arbitrary anisotropic scattering. We first clarify the construction principle of IUGKS-PN scheme according to the general form of anisotropic scattering models. However, the construction of the scheme is complex and the versatility is not strong. To address these challenges, a more concise scheme IUGKS-AM is proposed based on the isotropic IUGKS framework, in which the higher-order moments of the neutron distribution function are estimated. We theoretically analyze the asymptotic property of IUGKS-AM, and verify the numerical performances in anisotropic model by numerical experiments. Finally, several complex anisotropic scattering models are computed by using the numerical scheme, demonstrating its general applicability. This study significantly enhances the practical utility of multi-scale neutron transport UGKS in engineering applications.

The global tally problem has a wide range of applications in major research fields such as Monte Carlo simulations of pin-by-pin reactor models and time-dependent particle transport problems in multi-physics coupling calculations. Due to the uneven power distribution of the simulated system, the statistical errors of all tallies are unevenly distributed, resulting in some low global efficiency. For this kind of problem with global characteristics, it is necessary to develop global variance reduction techniques to obtain the accurate distribution of target tallies in the entire space. A large number of global variance reduction algorithms have been studied based on the consideration of flattening global tally error distribution, so as to improve global efficiency. This work focuses on the combination of two efficient global variance reduction algorithms, namely, the uniform tally density algorithm and the weight window algorithm, which belong to source bias and transport process bias, respectively. In tally, a method is proposed to adjust the weight window parameters by using the bias factor of the uniform tally density algorithm. Then, the weight window method will be used to reduce the weight fluctuation caused by the uniform tally density method. In this way, an organic combination of these two algorithms can be realized. A series of comparative tests are carried out based on the Hoogenboom-Martin pressurized water reactor benchmark, and it is verified that the hybrid global variance reduction algorithm proposed in this work is better than the single weight window algorithm or the uniform tally density algorithm. In terms of reducing the maximum error, the global efficiency of the hybrid algorithm is 2.6 times and 3 times that of the weight window algorithm and the uniform tally density algorithm, respectively. In addition, through the comparative analysis of computational asymmetry degree and computational efficiency, it is verified that the uniform tally density algorithm has better performance than the classical uniform fission site algorithm, and the performance advantages of the uniform tally density algorithm are quantitatively evaluated based on some new indicators. The results show that the hybrid global variance reduction algorithm proposed in this work can solve the global tally problem efficiently, thereby further promoting research in related fields.

In our previous work, the Unified Gas Kinetic Scheme (UGKS) and an efficient time-implicit scheme (known as IUGKS) were constructed for neutron transport. Both UGKS and IUGKS have been proved to be numerical schemes with asymptotic preserve (AP) property in neutron transport simulations. However, these schemes are built based on isotropic scattering models. To extend these methods to complex engineering applications, in this paper, the IUGKS-PN scheme for anisotropic scattering models is developed firstly, and it is demonstrated that the scheme keeps good AP property in the anisotropic scattering problems. Furthermore, to relieve the complication in IUGKS-PN for high-order PN expansion, the IUGKS-AM (IUGKS with anisotropic-scattering modification) is further developed with the anisotropic scattering correction. In the IUGKS-AM, there is no need to solve the higher-order moment equation which is used in IUGKS-PN. The AP property of IUGKS-AM is also derived, while the numerical performances of IUGKS-AM is compared with IUGKS-P1 in detail. In the numerical tests, the accuracy and efficiency of IUGKS-AM are verified with typical one- and two-dimensional benchmarks, and the numerical performance of the scheme in complex problems is further verified with a three-dimensional case. The results show that the scheme can be used for scattering model with complicated functional form. This study improves the application capability of IUGKS in engineering particle transport simulations.

Neutron transport calculations are extremely challenging due to the high computational cost of large and complex problems. A multilevel octree grid algorithm (MLTG) of discrete ordinates method was developed to improve the modeling accuracy and simulation efficiency on 3-D Cartesian grids. The Balakovo-3 VVER-1000 neutron dosimetry benchmark is calculated to verify and validate this numerical technique. A simplified S2 synthetic acceleration is used in the MLTG calculation method to improve the convergence of the source iterations. For the triangularly arranged fuel pins, we adopt a source projection algorithm to generate pin-by-pin source distributions of hexagonal assemblies. MLTG provides accurate geometric modeling and flexible fixed source description at a lower cost than traditional Cartesian grids. The total number of meshes is reduced to 1.9 million from the initial 9.5 million for the Balakovo-3 model. The numerical comparisons show that the MLTG results are in satisfactory agreement with the conventional SN method and experimental data, within the root-mean-square errors of about 4% and 10%, respectively. Compared to uniform fine meshing, approximately 70% of the computational cost can be saved using the MLTG algorithm for the Balakovo-3 computational model.

Neutron transport calculations are extremely challenging due to the high computational cost of large and complex problems. A multilevel octree grid algorithm (MLTG) of discrete ordinates method was devel-oped to improve the modeling accuracy and simulation efficiency on 3-D Cartesian grids. The Balakovo-3 VVER-100 0 neutron dosimetry benchmark is calculated to verify and validate this numerical technique. A simplified S2 synthetic acceleration is used in the MLTG calculation method to improve the convergence of the source iterations. For the triangularly arranged fuel pins, we adopt a source projection algorithm to generate pin-by-pin source distributions of hexagonal assemblies. MLTG provides accurate geometric modeling and flexible fixed source description at a lower cost than traditional Cartesian grids. The total number of meshes is reduced to 1.9 million from the initial 9.5 million for the Balakovo-3 model. The numerical comparisons show that the MLTG results are in satisfactory agreement with the conventional SN method and experimental data, within the root-mean-square errors of about 4% and 10%, respectively. Compared to uniform fine meshing, approximately 70% of the computational cost can be saved using the MLTG algorithm for the Balakovo-3 computational model.(c) 2022 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

When multiphysics coupling calculations contain time-dependent Monte Carlo particle transport simulations, these simulations often account for the largest part of the calculation time, which is insufferable in certain important cases. This study proposes an adaptive strategy for automatically adjusting the sample size to fulfil more reasonable simulations. This is realized based on an extension of the Shannon entropy concept and is essentially different from the popular methods in time-independent Monte Carlo particle transport simulations, such as controlling the sample size according to the relative error of a target tally or by experience. The results of the two models show that this strategy can yield almost similar results while significantly reducing the calculation time. Considering the efficiency, the sample size should not be increased blindly if the efficiency cannot be enhanced further. The strategy proposed herein satisfies this requirement.

Maintaining a reasonable balance between computational accuracy and overhead is important for neutron transport simulations of engineering problems. This paper presents a goal-oriented mesh adaptive algorithm applied to the multigroup discrete ordinates equation for fixed source and criticality problems. The dual-weighted residual (DWR) approach estimates numerical solution errors and drives local mesh refinement for specific targets, such as detector response, integral flux, and multiplication factor. We employ a reconstruction method to evaluate the spatial residuals of the fluxes obtained by the weighted difference scheme. To improve the performance of adaptive algorithms, new estimation models are proposed for adjoint fluxes needed by the DWR theory, including a regional goal model for fixed source problems and an inconsistent fission source model for k-eigenvalue problems. Additionally, we analyze the impact of the truncation of flux reconstruction and isotropic approximation of adjoint fluxes on grid error indicators and adaptive calculations. Numerical results demonstrate that for the quantities of interest, our adaptive approach saves more than 70% of computational effort and run time when obtaining a level of high accuracy comparable to that of uniform fine grids.

Multi-physics coupling calculation has applications in many important research fields. If particle transport process is included in this calculation, Monte Carlo method is often used to simulate this process and usually a large amount of calculation time is needed. So, efficient Monte Carlo algorithm for time-dependent particle transport problem is important for an efficiently coupling calculation, which inevitably relies on large-scale parallel calculation. Based on the characteristic of time-dependent particle transport problem, two methods are proposed in this paper to achieve high- efficiency calculation. One is a tally-reducing algorithm which is used in the coupling of transport simulation and burnup calculation. By reducing the quantity of data which should be reduced necessarily, this method can reduce the calculation time largely. It can be seen that a new coupling mode for these two processes in MPI environment has a larger value when model scale is larger than the sample size. The other method is an adaptive method of setting the sample size of Monte Carlo simulation. The law of large number assures that the Monte Carlo method will obtain an exact solution when the sample scale tends to infinity. But generally, no one knows which sample scale is big enough for obtaining a solution with target precision in advance. So, the common strategy is to set a huge-enough sample scale by experience and conduct the posterior check for all results. Apparently, this way cannot be efficient because the calculation will go on after the precision of solution has reached an object value. Another popular method is to set the sample size to rely on the relative error of some single calculation. The sample size is enlarged without a break until the relative error is less than some presetting value. This method is not suitable either, because Monte Carlo particle transport simulation will gives feedbacks to other process which is composed of many tallies. It is inappropriate to adjust the sample size according to the relative error of any calculation. Relying on the generalization of the Shannon entropy concept and an on-the-fly diagnosis rule for a entropy value sequence, the adaptive method proposed in this paper can reduce the original huge sample scale to a reasonable level. By numerically testing some non-trivial examples, both algorithms can reduce the calculation time largely, with the results kept almost unchanged, so the efficiency is high in these cases.

The neutron transport equation is usually solved against a stationary background medium. When the background material is moving, the transport equation will need to be modified. Solving the transport equation with moving material is quite complicated, especially for the curved coordinate system because of the double angular redistributions. In this study, the discretization method of the simplified transport equation considering the moving-material effect is implemented in three-dimensional cylindrical geometry. Directly solving this modified transport equation with the standard solution technique is problematic since the advection term introduced by moving material may render the transport solver numerically unstable. The speed ratio lambda is defined for stability analysis. A forced-stable method is proposed in this study to achieve good numerical stability for any material speeds and time-step sizes. The accuracy of this new method is verified using manufactured solutions. Steady numerical results demonstrate that the effects introduced by background motion cannot be neglected as the material speed starts to approach one-tenth of the neutron speed. Moreover, transient analysis indicates that the moving background has a considerable impact on the criticality of a system.

It is difficult to control discretization errors and reduce computational cost at the same time for multi-scale neutron transport problems, and the adaptive mesh refinement technique is one of the powerful and effective methods for high-resolution transport calculation. We propose a multilevel mesh adaptivity algorithm and develop a discrete ordinates calculation framework on 3-D Cartesian meshes. Data management and transport sweep are optimized based on the multilevel pyramid data structure containing pointer-type and array-type variables. A new spatial-moment-ratio error indicator is derived to measure the leading term of distribution functions and to drive the local mesh refinement. The numerical properties of spatial moment factors are better than the normalized gradient of angular flux. Compared with uniform refinement, our adaptivity algorithm reaches the same accuracy level with a computational cost saving of 60%-80% for heterogeneous problems. (C) 2022 Elsevier Ltd. All rights reserved.

If particle transport is included in some multi-physics calculation, Monte Carlo method is often used to simulate it and occupies the largest amount of calculation time. So, efficient dynamic Monte Carlo simulation for time-dependent particle transport problem is important, which is inevitably relied on large-scale parallel calculation. Two methods are proposed in this paper. The one is a tally-reduce algorithm which is used in the coupling of transport simulation and burn-up calculation. By reduces the amount of data which should be reduced necessary, this method can decrease the tally-reduce time largely. It can be seen as a new coupling mode for these two processes in MPI environment and will has larger value in cases when model scale is larger relatively compared to sample size. The other method is an adaptive method for setting the sample size of Monte Carlo simulation. Relying on the generalization of the Shannon entropy concept and an on-the-fly diagnosis rule for a entropy value sequence, the adaptive method proposed in this paper can decrease the original huge sample scale to a reasonable level. By numerical test for some non-trivial examples, both algorithms can decrease the calculation time largely while make the results almost unchanged.

在极端工程问题的物理过程中,背景介质通常具有与中子相当的运动速率,因此必须考虑介质运动对中子输运的影响.严格考虑流场对粒子输运的影响是异常复杂的工作,通过引入适当近似所导出的具有守恒形式的中子-流场耦合输运方程可较好地捕捉到移动背景介质下中子输运行为的主要特征.基于SN方法对耦合输运方程的角度变量进行离散,并首次给出了三维柱坐标系下耦合输运方程双重角度再分布项的离散方案.基于所研究的方法,先后在稳态问题和瞬态问题中分析了背景介质运动对中子输运的影响.数值结果表明:当介质运动速率达到中子飞行速率的1/10时,背景介质运动对中子通量分布的影响不可忽略;背景介质运动对系统的临界性能有重要影响.

The neutron transport equation is usually solved against stationary background materials. When the background material is moving, the transport equation will need to be modified. Two historical approaches, "effective" cross-sections and "co-moving" frame, have been used in the past decades. The neutron transport community prefers to use the second method. However, the modified transport equation in the "co-moving" frame is extremely complicated. In this study, we discuss these modifications for non-relativistic neutrons. Transport equations considering background motions have been derived under the three-dimensional (3-D) Cartesian, cylindrical, and spherical coordinates, which can be conveniently reduced to lower-dimensional systems. Moreover, methods of transforming the coupled equation between different phase spaces have been studied. (C) 2021 Elsevier Ltd. All rights reserved.

A parallel DSA algorithm based on the red-black sweep PCG method is proposed to accelerate the neutron transport calculation in the 3-D cylindrical geometry. This parallel acceleration technique shares the same spatial domain decomposition strategy with the KBA method. The red-black sweep iteration is formed by moving the r- and theta-axial components of the diffusion operator into the right hand side of the diffusion equation. Using the initial values refreshed by several red-black sweep iterations simplifies the matrix-vector multiplication in the CG procedure. Through the homogeneous problems, how the times of the red-black sweep effects the DSA algorithm is analyzed. A wide range of problems including FBR benchmark, PWR Pin-by-pin calculation and PWR shielding calculation have been performed to test the effectiveness and efficiency of the new method. The solution time is reduced by 3 similar to 10 times in different problems. This new acceleration algorithm has well parallel scalability. (C) 2019 Elsevier Ltd. All rights reserved.

Simulation of the ex-core detector response function (DRF) is of great importance for reactor power monitoring, control and protection systems. However, the 3D DRF calculation requires vast majorities of computational resources due to its characteristics such as deep penetration, strong anisotropic scattering, large computational geometry, etc. In this paper, the step characteristics (SC) spatial differencing scheme and GMRES algorithm have been implemented in the 3D S-N code to ensure the simulation efficient. The adjoint first-collision (FC) source method is developed to mitigate the ray effect in the DRF simulation. The Kobayashi benchmarks are calculated to verify the correctness of the FC module. Numerical results based on the Kori-1 reactor ex-core detector indicate that the SN-FC method has the comparable accuracy with multi-group Monte-Carlo method but achieves much higher efficiency. (C) 2019 Elsevier Ltd. All rights reserved.

In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D multi-group neutron transport equation with delayed neutron is developed. The explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order CrankNicholson scheme, become the subsets of the current IUGKS. In neutron transport, the microscopic angular flux and the macroscopic scalar flux are fully coupled in an implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step. Mathematically, the current scheme has the asymptotic preserving (AP) property in recovering automatically the diffusion solution in the continuum regime. Since the explicit scanning along neutron traveling direction within the computational domain is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other schemes, such as the explicit UGKS, the commonly used finite difference, and finite volume methods. This study shows that the IUGKS can be used faithfully to study neutron transport in practical engineering applications. AMS subject classifications: 82D75, 82C70, 82C40

堆外探测器响应函数表征了堆芯活性区各位置处的裂变中子对堆外探测器响应的贡献,通过共轭SN输运计算可快速得到堆外探测器的响应函数.然而,堆外探测器远离堆芯且相对于堆芯体积很小,SN方法的计算结果会受到射线效应的影响.为解决堆外探测器响应函数计算中的射线效应问题,研究了共轭首次碰撞源射线效应消除方法.此外,为克服共轭首次碰撞源方法在三维堆芯计算中面临的计算量大、内存需求高等问题,研究了共轭首次碰撞源的并行化计算方法和动态内存管理方法.基于韩国Kori-1压水堆的计算结果表明:共轭首次碰撞源SN方法和多群蒙特卡罗方法具有相当的计算精度,但计算效率高1个量级.