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In this paper, a discontinuous Galerkin finite element method (DGFEM) based on B-spline wavelet on the interval (BSWI) is developed and applied to solve the first-order neutron transport equation with discrete ordinate (SN) angular discretization. Then, using the scaling function of BSWI function as the basis function instead of the usual Lagrange basis function, the neutron transport equation based on BSWI discontinuous finite element is derived. Owing to the high precision characteristic of BSWI function, the method proposed in this paper can obtain fast convergence and statistical numerical accuracy under the condition that few spatial elements are used, and comparing with the results of the traditional discontinuous finite element method. Finally, some numerical examples are utilized to verify the accuracy and efficiency of the proposed method, and the results show that the method has high efficiency, reliable and stability, and the calculation accuracy is higher than that of the conventional discontinuous finite element method.

The Nuclear Thermal-hydraulic Laboratory (NutheL) at Xi’an Jiaotong University developed a Temporal and Spatial Neutronics Analysis Module (TaSNAM) based on the User Defined Function (UDF) and User Define Scalar (UDS) in Fluent. The neutron diffusion equations were solved based on the Finite volume method (FVM). In this paper, the TaSNAM is extended the original function to solve 3D problems and obtain neutron parameters based on the neutron parameters library, and improved to TaSNAM 2.0. The 3D-IAEA benchmark problem was selected and simulated to validate. The calculated results show good agreement with the reference values and the difference from the reference value of keff is 40pcm. A three-dimensional steady-state simulation of the whole core in AP1000 was completed employing the TaSNAM 2.0. The calculated results of important parameters agree well with the design values and the detailed three dimensional distributions of neutron flux and power were achieved.

This paper focuses on the application and demonstration of the GTF-based acceleration introduced in the first part of the article to three-dimensional problems. Two novelties are here introduced over the state-of-the-art. First, an operative definition of generalized transfer functions for three-dimensional distributions is introduced. Second, the GTF-based acceleration is used to speed up the Picard iteration scheme applied to 3D problems based on Pressurized Water Reactor (PWR) designs. Consistently with the mono-dimensional results presented in Part I of this article, the GTF-based acceleration allows to achieve convergence within a lower number of iterations, with a speed-up factor up to 5.5, compared to the original Picard iteration.

Fully Ceramic Micro-encapsulated (FCM) fuel is an important candidate for the accident tolerant fuel (ATF). Compared with traditional fuel, the double heterogeneity of FCM fuel makes the effective multi-group cross section calculation more challenging. In this paper, an improved disadvantage factor method is proposed to deal with the self-shielding effect of FCM fuel in the resonance energy range and non-resonance energy range, to achieve the equivalent homogenization of the FCM fuel. A new equivalent particle–matrix model was constructed by using the particle Dancoff factor to overcome the problem that the traditional volume-weight equivalent model could not consider the macro heterogeneity between fuel rods. Based on the new one-dimensional equivalent sphere model, the ultrafine group slowing down equation is solved to obtain the ultrafine group disadvantage factor in the resonance energy region. In the non-resonance energy region, the multi-group disadvantage factors of the fast group and thermal group are obtained by using the eigenvalue calculation in the new equivalent particle–matrix model. The proposed method has been implemented in the high fidelity neutronics program NECP-X and tested with a set of cases. The results show its good agreement with the Monte Carlo reference for both the reactivity and self-shielding cross sections.

The accurate calculation of kinetic parameters, including the effective delayed neutron fraction and the neutron mean generation time, is very important in the study of time dependent neutron population behavior. To calculate these parameters, it is necessary to calculate the flux and adjoint flux in the phase space. In this paper, the kinetic parameters of the new core configuration of the heavy water zero power reactor (HWZPR) are calculated using two deterministic and probabilistic methods. The quantitative and qualitative discussion is presented to illustrate the advantages and disadvantages of each method. The maximum difference between the two accounting methods was less than 3%. The effect of different energy structures on the criticality and kinetic parameters has been investigated in the deterministic approach. Additionally, the calculated kinetic parameters of the HWZPR validated experimentally with a reactivity inject response. Good consistency has been observed between different methods of computation and experimental values.

In this paper, we derive a mathematical formulation of the higher order adjoint NEM-M2B2 equations by preconditioning the nodal interface neutron currents equations of the forward equations system, and by using the Lagrangian Multipliers analysis method. In the NEM-M2B2 system of equations, the quadratic transverse leakage approximation is used to model the leakage of neutrons between each node in the system. The solution of the adjoint equation can be used to perform adjoint-based predictive sensitivity/perturbation analysis. As an example, we use the mathematical adjoint solution as sensitivity weighting for predicting the response of the IAEA-3D benchmark’s eigenvalue to a perturbation in the independent parameters of the system (i.e., cross-sections). We also derive perturbation equations associated with the particular NEM-M2B2 model we are using. These perturbation-equations are used in predicting the model eigenvalue change without resorting to recalculating the forward NEM-M2B2 system of equations again (labeled as exact calculations). They also enabled construction of a reactivity sensitivity map showing the importance of each calculation node of the benchmark depending on its spatial and spectral coordinates. Perturbations were imposed on both the absorption cross-sections (fast and thermal) and the scattering cross-section of the IAEA-3D benchmark problem. Several verification steps were taken to ensure that the developed mathematical adjoint solver is adequate for adjoint analysis (e.g., commutativity checks, and comparison against exact calculations).

The main advantage of using Nodal Integral Methods (NIM) in solving partial differential equations (PDEs) is that they lead to an accurate solution over relatively coarse meshes. Due to the use of the transverse integration procedure in the derivation, most of the NIMs developed were restricted to PDEs with isotropic diffusion terms only. Recently, the NIM has been extended to arbitrary geometries using iso-parametric mapping approach, which resulted in the transformation of isotropic diffusion to anisotropic diffusion. This required the development of a method to approximate the cross-derivative terms that is consistent with the order of accuracy of traditional NIMs. In this paper, the 3D, time-dependent, anisotropic convection-diffusion equation is solved numerically using the NIM. A new method to approximate the cross-derivative terms based on the actual discrete unknowns of the NIM – namely, the line-averaged or surface-averaged variables – is developed. Also, the previously developed approximation for the cross-derivative terms in 2D (Kumar et al., 2013) that is based on the corner point values is extended to 3D. Six numerical test cases are solved to test the accuracy and efficiency of both approaches. The accuracy of NIM for the anisotropic diffusion equation is maintained even for coarse meshes for both cross-derivative approximations. However, the surface-averaged-based approximation is more accurate and efficient than the point-value-based approximation in all cases. The NIM using the surface-averaged-based approximation is second-order accurate in space and time. On the other hand, the NIM using the point-value-based approximation is between first and second-order accurate in space, and second-order accurate in time.

To study the influence of the reactivity feedback coefficient uncertainty on reactor core power control system, taking a lead-cooled fast reactor as the object, the core power control system with a BP neural network PID controller is designed. Drawing the input samples with Latin hypercube sampling method, the uncertainty quantitative analysis consists of the Chebyshev inequality and relative standard deviation calculation. Based on the method in this paper, the dynamic simulation of reactor core is performed. The results show that the uncertainties of the system output control performance parameters are less than 1% when introducing 50pcm and 100 pcm step reactivity. The uncertainty of the system output parameters changes and returns to zero with the action of the controller.

In this paper, a new adaptive genetic algorithm (AGA) is proposed for the loading pattern optimization (LPO) of the pressurized water reactor (PWR) core. The tournament selection, the two-point crossover and the mutation based on randomly swapping positions between two fuel assemblies (FAs) are applied. New calculation formulae are introduced to adjust effectively the crossover and mutation probabilities according to the fitness value of the individual. The proposed algorithm is implemented in the LPO for the first core of the 1000 MWe PWR. The objective function is to minimize the maximum radial power peaking factor (RPPF) at the equilibrium of Xe under the constraint condition for the cycle length. The maximum RPPF of the obtained LP is decreased than that of the loading pattern (LP) by the designer. The results show that the proposed AGA is effective to improve the convergence rate of genetic algorithms (GAs) in the LPO.

The quasi-static (QS) method had been devised for efficient analyses of reactor transients, and two different approaches are widely implemented, which are namely Improved Quasi-Static Method (IQM) and Predictor-Corrector Quasi-Static Method (PCQM). Conventionally, both methods regard the shape to be static while deducing the variation of the amplitude. In this paper, the formulation of standard IQM and PCQM methods based on a multi-group neutron diffusion equation is presented along with a description for implementing such methods to feedback-encompassed nonlinear transient calculation. The idea of relaxation in the quasi-static treatment of the shape through polynomial interpolation is proposed, and comparison among the ramifications of having different conjectures, e.g., static, linear, and quadratic relaxation, has been made based on a thermal-hydraulics-coupled one-dimensional two-group reactor representing the pressurized water reactor. It is found that performance of both IQM and PCQM could be clearly improved with polynomial relaxations even for strongly non-linear reactor problems.

In this paper we report validation studies of the Monte Carlo code OpenMC by the ASTRA critical facility benchmark. A high-fidelity reactor model is built using the regular lattice coated fuel particle and the body centered cube pebble spheres representations. Two nuclear data libraries, ENDFB-7.0 and ENDFB-7.1 are considered to investigate the validity of nuclear cross section libraries. In addition, three homogenization models namely the core-hom model, the ref-hom model and the CR-hom model are built to provide references for analyzing the spatial approximations employed in deterministic codes. The core-hom model equivalents the octahedral core by a cylindrical zone, the ref-hom model homogenizes the side reflector into a cylindrical region, and the CR-hom model approximates the tube-cluster-shaped control rods (CR) into a concentric annulus. Results demonstrate that ENDFB-7.1 library outperforms ENDFB-7.0 by predicting criticality more accurately. The difference between the two libraries is 1224 pcm for the criticality calculation at the experimental pebble bed height (268.9 cm). With ENDFB-7.1 the high-fidelity model well-predicts the experimental data by 620 pcm error in keff and less than 6.5% difference in CR worth. The homogenization of the reflector brings about a pronounced error of 1143 pcm. In contrast, the core-hom model causes a comparably minor discrepancy of 241 pcm to the keff, while its influence on the CR worth appears most significant for CRs located close to the core zone. The CR-hom model yields a −2.1% maximum accuracy loss in CR worth from the high-fidelity model. After all, both the high-fidelity model and the core-hom model succeed in capturing the CR interference effect and in predicting the trend of CR differential worth. This work is not only meaningful for demonstrating the validity of OpenMC, but also useful to investigate the numerical accuracy of codes used in emerging work related to PBHTGR.