Modified Block Newton method for the lambda modes problem

[EN] To study the behaviour of nuclear power reactors it is necessary to solve the time dependent neutron diffusion equation using either a rectangular mesh for PWR and BWR reactors or a hexagonal mesh for VVER reactors. This problem can be solved by means of a modal method, which uses a set of dominant modes to expand the neutron flux. For the transient calculations using the modal method with a moderate number of modes, these modes must be updated each time step to maintain the accuracy of the solution. The updating modes process is also interesting to study perturbed configurations of a reactor. A Modified Block Newton method is studied to update the modes. The performance of the Newton method has been tested for a steady state perturbation analysis of two 2D hexagonal reactors, a perturbed configuration of the IAEA PWR 3D reactor and two configurations associated with a boron dilution transient in a BWR reactor.This work has been partially supported by the Spanish Ministerio de Educación y Ciencia under projects ENE2008-02669 and MTM2007-64477-AR07, the Generalitat Valenciana under project ACOMP/2009/058, and the Universidad Politécnica de Valencia under project PAID-05-09-4285.González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ. (2013). Modified Block Newton method for the lambda modes problem. Nuclear Engineering and Design. 259:230-239. https://doi.org/10.1016/j.nucengdes.2011.06.045S23023925

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

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

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

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

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

A finite element method for neutron noise analysis in hexagonal reactors

[EN] The early detection of anomalies through the analysis of the neutron noise recorded by in-core and ex-core instrumentation gives the possibility to take proper actions before such problems lead to safety concerns or impact plant availability. The study of the neutron fluctuations permits detecting and differentiate anomalies depending on their type and possibly to characterize and localize such anomalies. This method is non-intrusive and does not require any external perturbation of the system. To effectively use the neutron noise for reactor diagnostics it is essential to accurately model the effects of the anomalies on the neutron field. This paper deals with the development and validation of a neutron noise simulator for reactors with different geometries. The neutron noise is obtained by solving the frequency-domain two-group neutron diffusion equation in the first order approximation. In order to solve this partial differential equation a code based on a high order finite element method is developed. The novelty of this simulator resides on the possibility of dealing with rectangular meshes in any kind of geometry, thus allowing for complex domains and any location of the perturbation. The finite element method also permits automatic refinements in the cell size (h-adaptability) and in its polynomial degree (p-adaptability) that lead to a fast convergence. In order to show the possibilities of the neutron noise simulator developed a perturbation in a hexagonal two-dimensional reactor is investigated in this paper.This project has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 754316. Also, this work has been partially supported by Spanish Ministerio de Economía y Competitividad under project BES-2015-072901 and financed with the help of a Primeros Proyectos de Investigacin (PAID-06-18), Vicerrectorado de Investigación, Innovación y Transferencia of the Universitat Politecnica de València (UPV).Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Carreño, A.; Verdú Martín, GJ.; Demazière, C. (2021). A finite element method for neutron noise analysis in hexagonal reactors. EPJ Web of Conferences (Online). 247:1-8. https://doi.org/10.1051/epjconf/202124721007S1824