Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations

[EN] The evolution of the neutronic power inside of a nuclear reactor core can be approximated by means of the diffusive time-dependent simplified spherical harmonics equations (SPN). For the spatial discretization of these equations, a continuous Galerkin high order finite element method is applied to obtain a semi-discrete system of equations that is usually stiff. A semi-implicit time scheme is used for the time discretization and many linear systems are needed to be solved and previously, preconditioned. The aim of this work is to speed up the convergence of the linear systems solver with a multilevel preconditioner that uses different degrees of the polynomials used in the finite element method. Furthermore, as the matrices that appear in this type of system are very large and sparse, a matrix-free implementation of the preconditioner is developed to avoid the full assembly of the matrices. A benchmark transient tests this methodology. Numerical results show, in comparison with the block Gauss-Seidel preconditioner, an improvement in terms of number of iterations and the necessity of computational resources.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar, D.; Verdú, G. (2022). Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 68-77. https://doi.org/10.4995/YIC2021.2021.12510OCS687

Adaptive modal methods to integrate the neutron diffusion equation

Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2019). Adaptive modal methods to integrate the neutron diffusion equation. R. Company, J. C. Cortés, L. Jódar and E. López-Navarro. 26-31. http://hdl.handle.net/10251/180549S263

A block Arnoldi method for the SPN equations

[EN] The simplified spherical harmonics equations are a useful approximation to the stationary neutron transport equation. The eigenvalue problem associated with them is a challenging problem from the computational point of view. In this work, we take advantage of the block structure of the involved matrices to propose the block inverse-free preconditioned Arnoldi method as an efficient method to solve this eigenvalue problem. For the spatial discretization, a continuous Galerkin finite element method implemented with a matrix-free technique is used to keep reasonable memory demands. A multilevel initialization using linear shape functions in the finite element method is proposed to improve the method convergence. This initialization only takes a small percentage of the total computational time. The proposed eigenvalue solver is compared to the standard power iteration method, the Krylov-Schur method and the generalized Davidson method. The numerical results show that it reduces the computational time to solve the eigenvalue problem.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901. Moreover, it has been supported by the Catedra of the CSN Vicente SerradellVidal-Ferràndiz, A.; Carreño, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2020). A block Arnoldi method for the SPN equations. International Journal of Computer Mathematics. 97(1-2):341-357. https://doi.org/10.1080/00207160.2019.1602768S341357971-

Frequency-domain models in the SPN approximation for neutron noise calculations

[EN] Simulations of the neutron flux fluctuations, known as neutron noise, can be performed by means of the spherical harmonics equations (SPN) approximation with higher accuracy than with the diffusion equation. In this sense, one can solve these equations in the time-domain or in the frequency-domain. This last approach permits solving the neutron noise without performing complete time-dependent simulations for monochromatic perturbations. This work presents two formulations of the SPN equations in the frequency domain, that are obtained by using different treatments of the time derivatives of the field moments. The methodology is verified with several neutron noise problems where the numerical results are compared with the time-domain computations of FEMFFUSION code. The C5G7 noise benchmark compares both SPN formulations, showing the applicability of the diffusive SPN approximation.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2022). Frequency-domain models in the SPN approximation for neutron noise calculations. Progress in Nuclear Energy. 148:1-11. https://doi.org/10.1016/j.pnucene.2022.10423311114

Time-dependent simplified spherical harmonics formulations for a nuclear reactor system

[EN] The steady-state simplified spherical harmonics equations (SPNequations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SPN equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full systemThis work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2021). Time-dependent simplified spherical harmonics formulations for a nuclear reactor system. Nuclear Engineering and Technology. 53(12):3861-3878. https://doi.org/10.1016/j.net.2021.06.010S38613878531

Edge-wise perturbations to model vibrating fuel assemblies in the frequency-domain using FEMFFUSION: development and verification

[EN] The mechanical vibrations of fuel assemblies have shown to give high levels of neutron noise, triggering in some circumstances the necessity to operate nuclear reactors at a reduced power level. This behaviour can be modelled using the neutron noise diffusion approximation in the frequency-domain. This work presents an extension of the finite element method code FEMFFUSION, to simulate mechanical vibrations in hexagonal reactors in the frequency domain. This novel strategy in neutron noise simulation is based on introducing perturbations on the edges of the cells associated with the vibrating fuel assemblies, allowing to model the movement of these fuel assemblies accurately and efficiently, without the necessity of using locally refined meshes. Numerical results verify the edge-wise methodology in the frequency-domain against the usual cell-wise frequency-domain model and the time-domain model. The edge-wise frequency-domain methodology has also been compared to other neutronic codes, as CORESIM and PARCS.This project has received funding from the Euratom research and training program 2014-2018 under grant agreement No 754316.Vidal-Ferràndiz, A.; Carreño, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2022). Edge-wise perturbations to model vibrating fuel assemblies in the frequency-domain using FEMFFUSION: development and verification. Annals of Nuclear Energy. 175:1-12. https://doi.org/10.1016/j.anucene.2022.10924611217

Adaptive time-step control for the modal method to integrate the multigroup neutron diffusion equation

[EN] The distribution of the power inside a reactor core can be described by the time dependent multigroup neutron diffusion equation. One of the approaches to integrate this time-dependent equation is the modal method, that assumes that the solution can be described by the sum of amplitude function multiplied by shape functions of modes. These shape functions can be computed by solving a _-modes problems. The modal method has a great interest when the distribution of the power cannot be well approximated by only one shape function, mainly, when local perturbations are applied during the transient. Usually, the shape functions of the modal methods are updated for the time-dependent equations with a constant time-step size to obtain accurate results. In this work, we propose a modal methodology with an adaptive control time-step to update the eigenfunctions associated with the modes. This algorithm improves efficiency because of time is not spent solving the systems to a level of accuracy beyond relevance and reduces the step size if they detect a numerical instability. Step size controllers require an error estimation. Different error estimations are considered and analyzed in a benchmark problem with a out of phase local perturbation.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2021). Adaptive time-step control for the modal method to integrate the multigroup neutron diffusion equation. EPJ Web of Conferences (Online). 247:1-8. https://doi.org/10.1051/epjconf/202124707010S1824

Block hybrid multilevel method to compute the dominant lambda-modes of the neutron diffusion equation

[EN] The dominant lambda-modes associated with a nuclear reactor configuration describe the neutron steady-state distribution and its criticality. Furthermore, they are useful to develop modal methods to study reactor instabilities. Different eigenvalues solvers have been successfully used to obtain such modes, most of them are implemented reducing the original generalized eigenvalue problem to an ordinary one. Thus, it is necessary to solve many linear systems making these methods not very efficient, especially for large problems. In this work, the original generalized eigenvalue problem is considered and two block iterative methods to solve it are studied: the block inverse-free preconditioned Arnoldi method and the modified block Newton method. All of these iterative solvers are initialized using a block multilevel technique. A hybrid multilevel method is also proposed based on the combination of the methods proposed. Two benchmark problems are studied illustrating the convergence and the competitiveness of the methods proposed. A comparison with the Krylov-Schur method and the Generalized Davidson is also included.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2018). Block hybrid multilevel method to compute the dominant lambda-modes of the neutron diffusion equation. Annals of Nuclear Energy. 121:513-524. https://doi.org/10.1016/j.anucene.2018.08.010S51352412

Spatial modes for the neutron diffusion equation and their computation

[EN] Different spatial modes can be defined for the neutron diffusion equation such as the k; a and c-modes. These modes have been successfully used for the analysis of nuclear reactor characteristics. In this work, these modes are studied using a high order finite element method to discretize the equations and also different methods to solve the resulting algebraic eigenproblems, are compared. Particularly, Krylov subspace methods and block-Newton methods have been studied. The performance of these methods has been tested in several 3D benchmark problems: a homogeneous reactor and several configurations of NEACRP reactor.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2014-59442-P, MTM2014-58159-P and BES-2015-072901.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2017). Spatial modes for the neutron diffusion equation and their computation. Annals of Nuclear Energy. 110:1010-1022. https://doi.org/10.1016/j.anucene.2017.08.018S1010102211

Neutronic Simulation of Fuel Assembly Vibrations in a Nuclear Reactor

This is an Accepted Manuscript of an article published by Taylor & Francis in Nuclear Science and Engineering on 2020, available online: http://www.tandfonline.com/10.1080/00295639.2020.1756617[EN] The mechanical vibrations of core internals such as fuel assemblies (FAs) cause oscillations in the neutron flux that require in some circumstances nuclear power plants to operate at a reduced power level. This work simulates and analyzes the changes of the neutron flux throughout a nuclear core due to the oscillation of a single FA without considering thermal-hydraulic feedback. The amplitude of the FA vibration is bounded to a few millimeters, and this implies the use of fine meshes and accurate numerical solvers due to the different scales of the problem. The results of the simulations show a main oscillation of the neutron flux with the same frequency as the FA vibration along with other harmonics at multiples of the vibration frequency much smaller in amplitude. Also, this work compares time domain analysis and frequency domain analysis of the mechanical vibrations. Numerical results show a close match between these two approaches for the fundamental frequency.This project has received funding from the Euratom research and training programme 2014-2018 under grant agreement number 754316. Also, this work has been partially supported by Spanish Ministerio de Economia y Competitividad under project BES-2015-072901 and financed with the help of Primeros Proyectos de Investigacion (PAID-06-18), Vicerrectorado de Investigacion, Innovacion y Transferencia of the Universitat Politecnica de Valencia (UPV).Vidal-Ferràndiz, A.; Carreño, A.; Ginestar Peiro, D.; Demazière, C.; Verdú Martín, GJ. (2020). Neutronic Simulation of Fuel Assembly Vibrations in a Nuclear Reactor. Nuclear Science and Engineering. 194(11):1067-1078. https://doi.org/10.1080/00295639.2020.1756617S106710781941