this report that there is a need for a better qualification of the noise analysis methods applied... more this report that there is a need for a better qualification of the noise analysis methods applied in BWR stability studies. A follow-up benchmark was thus proposed dedicated to the analysis of time series data and including the evaluation of both global and regional stability. With the help of Par Lansaker (Forsmark Kraftgrupp AB), Tomas Lefvert identified six interesting cases for this purpose, measured at Forsmark 1 & 2. He presented a proposal to the NEA Nuclear Science Committee, who accorded its approval. Jos Manuel Conde and Manuel Recio from the Consejo de Seguridad Nuclear, Madrid, agreed to find the necessary support to arrange the co-ordination of this benchmark through the team of Dr. Gumersindo Verd of the Polytechnic University of Valencia. The purpose of this benchmark is the comparison among the different time series analysis methods that can be applied to the study of BWR stability. While the Ringhals 1 stability benchmark included both time domain and frequency doma...
In order to analyse the steady state of a nuclear power reactor, the neutron diffusion equation i... more In order to analyse the steady state of a nuclear power reactor, the neutron diffusion equation in the approximation of several groups of energy is typically used. This problem corresponds to a differential generalized eigenvalue problem. For the spatial discretization of the neutron diffusion equation a high order finite element method is used, which makes use of Lagrange polynomials defined on Gauss-Lobato-Legendre quadrature points. These polynomials provide a natural partition of the shape functions into vertices, edges, faces and interior shape functions. The most expensive computation to solve the discretized problem consists of the calculation of the solution of linear systems of equations whose coefficient matrices are associated with each group of energy. These linear systems are large and sparse, and a preconditioned Krylov iterative method is typically used to approximate their solution. Classically, a preconditioner based on an incomplete factorization is used. Neverthel...
The paper “SPICE Model of Photomultiplier Tube Under Different Bias Conditions” is commented. We ... more The paper “SPICE Model of Photomultiplier Tube Under Different Bias Conditions” is commented. We revisit the mathematical formulation to compensate for some ambiguities in the original manuscript, and point out some inconsistencies in the results and reproducibility of the simulations, as well as in the optimized parameters originally obtained with the PSPICE simulation engine. All simulations are recalculated with the NGSPICE software using the corrected parameters and compared against the original figures. The reproducibility of our simulations is independently verified with PSPICE, as well as by numerically solving the analytical system of non-linear equations using Newton’s method within MATLAB.
The early detection of anomalies through the analysis of the neutron noise recorded by incore and... more The early detection of anomalies through the analysis of the neutron noise recorded by incore 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 to detect 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 elemen...
In the CORTEX project, several solvers are developed and applied to analyze neutron noise problem... more In the CORTEX project, several solvers are developed and applied to analyze neutron noise problems. They are based on Monte Carlo and deterministic (higher-order transport and diffusion) methods. For the study of their validity and limitations, an extensive verification and validation work has been undertaken and includes the simulation of numerical exercises and experiments. In the current paper the solvers are compared over two neutron noise benchmarks defined in a 2-D simplified UOX fuel assembly, with Monte Carlo used as a reference. In the two exercises, a global neutron noise source and a combination of stationary perturbations of the various cross sections are respectively prescribed. The higher-order neutron transport methods provide consistent results with respect to Monte Carlo. The calculations obtained from the diffusion-based solvers show discrepancies that can be significant, in particular close to the neutron noise source.
Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or g... more Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or generators in some developments and applications. This is due to their capacity of being deformed when submitted to an electric field. Some models of their actuation are available, but recently, viscoelastic models have been proposed to give an account of the dissipative behaviour of these materials. Their response to an external mechanical or electrical force field implies a relaxation process towards a new state of thermodynamic equilibrium, which can be described by a relaxation time. However, it is well known that viscoelastic and dielectric materials, as for example, polymers, exhibit a distribution of relaxation times instead of a single relaxation time. In the present approach, a continuous distribution of relaxation times is proposed via the introduction of fractional derivatives of the stress and strain, which gives a better account of the material behaviour. The application of f...
Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference
Inside a nuclear reactor core, the neutronic power distribution can be approximated by means of t... more Inside a nuclear reactor core, the neutronic power distribution can be approximated by means of the multigroup time-dependent simplified spherical harmonics equations. In particular, this work uses a formulation where the time derivatives of the even spherical harmonics moments are assumed equal to zero. This treatment yields to diffusive equations of order two that only depend on the position and time.For the spatial discretization of the equations, a continuous Galerkin high order finite element method is applied. In the time discretization, two sets of equations appear: one related to the neutron moments and the other related to the delayed neutron precursor concentrations. Moreover, these time differential equations are usually stiff. Thus, a semi-implicit time scheme must be proposed that needs to solve several linear systems in each time-step. And generally, these systems must be preconditioned.The main aim of this work is to speed up the convergence of the linear systems sol...
This work outlines an approach for localizing anomalies in nuclear reactor cores during their ste... more This work outlines an approach for localizing anomalies in nuclear reactor cores during their steady state operation, employing deep, one-dimensional, convolutional neural networks. Anomalies are characterized by the application of perturbation diagnostic techniques, based on the analysis of the so-called “neutron-noise” signals: that is, fluctuations of the neutron flux around the mean value observed in a steady-state power level. The proposed methodology is comprised of three steps: initially, certain reactor core perturbations scenarios are simulated in software, creating the respective perturbation datasets, which are specific to a given reactor geometry; then, the said datasets are used to train deep learning models that learn to identify and locate the given perturbations within the nuclear reactor core; lastly, the models are tested on actual plant measurements. The overall methodology is validated on hexagonal, pre-Konvoi, pressurized water, and VVER-1000 type nuclear reacto...
High efficient methods are required for the computation of several lambda modes associated with t... more High efficient methods are required for the computation of several lambda modes associated with the neutron diffusion equation. Multiple iterative eigenvalue solvers have been used to solve this problem. In this work, three different block methods are studied to solve this problem. The first method is a procedure based on the modified block Newton method. The second one is a procedure based on subspace iteration and accelerated with Chebyshev polynomials. Finally, a block inverse-free Krylov subspace method is analyzed with different preconditioners. Two benchmark problems are studied illustrating the convergence properties and the effectiveness of the methods proposed.
This paper presents the experience of introducing the Kahoot tool in the computer practice sessio... more This paper presents the experience of introducing the Kahoot tool in the computer practice sessions of the subject Mathematics II of Electronic and Automatic Engineering degree as a motivating element. Furthermore, an analysis of a survey passed to the students of two groups of the subject with different characteristics is made, where different questions related to the tool and its capacity to motivate the students to the revision of the material of the practices before the presential session are included. From the analysis of the obtained results, it can be concluded that, in general, the Kahoot tool is a motivating element for the class, mainly in the part of computer practices.
The neutron transport equation describes the distribution of neutrons inside a nuclear reactor co... more The neutron transport equation describes the distribution of neutrons inside a nuclear reactor core. Homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem. The generalized equivalence theory for homogenization defines discontinuity factors at the boundaries of the homogenized subdomains. In this work, the generalized equivalence theory is extended to the PN equations for one-dimensional geometries using the finite element method. Here, pin discontinuity factors are proposed instead of the usual assembly discontinuity factors and the use of the spherical harmonics approximation as an extension of the diffusion theory. An interior penalty finite element method is used to discretize and solve the problem using discontinuity factors. Numerical results show that the pr...
The stationary neutron transport equation describes the neutron population and thus, the generate... more The stationary neutron transport equation describes the neutron population and thus, the generated heat, inside a nuclear reactor core. Obtaining the solution of this equation requires to solve a generalized eigenvalue problem efficiently. The majority of the eigenvalue solvers use the factorization of the system matrices to construct preconditioners, such as the ILU decomposition or the ICC decomposition, to speed up the convergence of the methods. The storage of the involved matrices and incomplete factorization demands high quantities of computational memory although a the sparse format is used. This makes the computational memory the limiting factor for this kind of calculations in some personal computers. In this work, we propose a matrix-free preconditioned eigenvalue solver that does not need to have the matrices allocated in memory explicitly. This method is based on the block inverse-free preconditioned Arnoldi method (BIFPAM) with the innovation that uses a preconditioner ...
Boiling channels and systems may oscillate owing to the behaviour of the liquid-steam mixture use... more Boiling channels and systems may oscillate owing to the behaviour of the liquid-steam mixture used for removing the thermal power. A thermal-hydraulic system may be unstable under particular operating conditions. Two kinds of power oscillation have been observed in BWR cores. One is an in-phase (core-wide) and the other is an out-of-phase (regional) oscillation. Since the above feature can make detection more difficult, the latter oscillation is potentially more severe. The problem is well known since the design of the first BWR system. However, to improve the safety systems of these reactors, it is necessary to be able to detect in a reliable way these oscillations from the neutronic signals. The purpose of this work is to characterize the unstable behaviour of a BWR. Within this study, it has been performed a number of perturbation analysis. The coupled codes RELAP5Mod3.3/PARCS have used for the simulation of the transients. Validation has been performed against Peach Bottom-2 Low...
A discrete ordinates method has been developed to approximate the neutron transport equation for ... more A discrete ordinates method has been developed to approximate the neutron transport equation for the computation of the lambda modes of a given configuration of a nuclear reactor core. This method is based on discrete ordinates method for the angular discretization, resulting in a very large and sparse algebraic generalized eigenvalue problem. The computation of the dominant eigenvalue of this problem and its corresponding eigenfunction has been done with a matrix-free implementation using both, the power iteration method and the Krylov-Schur method. The performance of these methods has been compared solving different benchmark problems with different dominant ratios.
this report that there is a need for a better qualification of the noise analysis methods applied... more this report that there is a need for a better qualification of the noise analysis methods applied in BWR stability studies. A follow-up benchmark was thus proposed dedicated to the analysis of time series data and including the evaluation of both global and regional stability. With the help of Par Lansaker (Forsmark Kraftgrupp AB), Tomas Lefvert identified six interesting cases for this purpose, measured at Forsmark 1 & 2. He presented a proposal to the NEA Nuclear Science Committee, who accorded its approval. Jos Manuel Conde and Manuel Recio from the Consejo de Seguridad Nuclear, Madrid, agreed to find the necessary support to arrange the co-ordination of this benchmark through the team of Dr. Gumersindo Verd of the Polytechnic University of Valencia. The purpose of this benchmark is the comparison among the different time series analysis methods that can be applied to the study of BWR stability. While the Ringhals 1 stability benchmark included both time domain and frequency doma...
In order to analyse the steady state of a nuclear power reactor, the neutron diffusion equation i... more In order to analyse the steady state of a nuclear power reactor, the neutron diffusion equation in the approximation of several groups of energy is typically used. This problem corresponds to a differential generalized eigenvalue problem. For the spatial discretization of the neutron diffusion equation a high order finite element method is used, which makes use of Lagrange polynomials defined on Gauss-Lobato-Legendre quadrature points. These polynomials provide a natural partition of the shape functions into vertices, edges, faces and interior shape functions. The most expensive computation to solve the discretized problem consists of the calculation of the solution of linear systems of equations whose coefficient matrices are associated with each group of energy. These linear systems are large and sparse, and a preconditioned Krylov iterative method is typically used to approximate their solution. Classically, a preconditioner based on an incomplete factorization is used. Neverthel...
The paper “SPICE Model of Photomultiplier Tube Under Different Bias Conditions” is commented. We ... more The paper “SPICE Model of Photomultiplier Tube Under Different Bias Conditions” is commented. We revisit the mathematical formulation to compensate for some ambiguities in the original manuscript, and point out some inconsistencies in the results and reproducibility of the simulations, as well as in the optimized parameters originally obtained with the PSPICE simulation engine. All simulations are recalculated with the NGSPICE software using the corrected parameters and compared against the original figures. The reproducibility of our simulations is independently verified with PSPICE, as well as by numerically solving the analytical system of non-linear equations using Newton’s method within MATLAB.
The early detection of anomalies through the analysis of the neutron noise recorded by incore and... more The early detection of anomalies through the analysis of the neutron noise recorded by incore 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 to detect 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 elemen...
In the CORTEX project, several solvers are developed and applied to analyze neutron noise problem... more In the CORTEX project, several solvers are developed and applied to analyze neutron noise problems. They are based on Monte Carlo and deterministic (higher-order transport and diffusion) methods. For the study of their validity and limitations, an extensive verification and validation work has been undertaken and includes the simulation of numerical exercises and experiments. In the current paper the solvers are compared over two neutron noise benchmarks defined in a 2-D simplified UOX fuel assembly, with Monte Carlo used as a reference. In the two exercises, a global neutron noise source and a combination of stationary perturbations of the various cross sections are respectively prescribed. The higher-order neutron transport methods provide consistent results with respect to Monte Carlo. The calculations obtained from the diffusion-based solvers show discrepancies that can be significant, in particular close to the neutron noise source.
Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or g... more Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or generators in some developments and applications. This is due to their capacity of being deformed when submitted to an electric field. Some models of their actuation are available, but recently, viscoelastic models have been proposed to give an account of the dissipative behaviour of these materials. Their response to an external mechanical or electrical force field implies a relaxation process towards a new state of thermodynamic equilibrium, which can be described by a relaxation time. However, it is well known that viscoelastic and dielectric materials, as for example, polymers, exhibit a distribution of relaxation times instead of a single relaxation time. In the present approach, a continuous distribution of relaxation times is proposed via the introduction of fractional derivatives of the stress and strain, which gives a better account of the material behaviour. The application of f...
Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference
Inside a nuclear reactor core, the neutronic power distribution can be approximated by means of t... more Inside a nuclear reactor core, the neutronic power distribution can be approximated by means of the multigroup time-dependent simplified spherical harmonics equations. In particular, this work uses a formulation where the time derivatives of the even spherical harmonics moments are assumed equal to zero. This treatment yields to diffusive equations of order two that only depend on the position and time.For the spatial discretization of the equations, a continuous Galerkin high order finite element method is applied. In the time discretization, two sets of equations appear: one related to the neutron moments and the other related to the delayed neutron precursor concentrations. Moreover, these time differential equations are usually stiff. Thus, a semi-implicit time scheme must be proposed that needs to solve several linear systems in each time-step. And generally, these systems must be preconditioned.The main aim of this work is to speed up the convergence of the linear systems sol...
This work outlines an approach for localizing anomalies in nuclear reactor cores during their ste... more This work outlines an approach for localizing anomalies in nuclear reactor cores during their steady state operation, employing deep, one-dimensional, convolutional neural networks. Anomalies are characterized by the application of perturbation diagnostic techniques, based on the analysis of the so-called “neutron-noise” signals: that is, fluctuations of the neutron flux around the mean value observed in a steady-state power level. The proposed methodology is comprised of three steps: initially, certain reactor core perturbations scenarios are simulated in software, creating the respective perturbation datasets, which are specific to a given reactor geometry; then, the said datasets are used to train deep learning models that learn to identify and locate the given perturbations within the nuclear reactor core; lastly, the models are tested on actual plant measurements. The overall methodology is validated on hexagonal, pre-Konvoi, pressurized water, and VVER-1000 type nuclear reacto...
High efficient methods are required for the computation of several lambda modes associated with t... more High efficient methods are required for the computation of several lambda modes associated with the neutron diffusion equation. Multiple iterative eigenvalue solvers have been used to solve this problem. In this work, three different block methods are studied to solve this problem. The first method is a procedure based on the modified block Newton method. The second one is a procedure based on subspace iteration and accelerated with Chebyshev polynomials. Finally, a block inverse-free Krylov subspace method is analyzed with different preconditioners. Two benchmark problems are studied illustrating the convergence properties and the effectiveness of the methods proposed.
This paper presents the experience of introducing the Kahoot tool in the computer practice sessio... more This paper presents the experience of introducing the Kahoot tool in the computer practice sessions of the subject Mathematics II of Electronic and Automatic Engineering degree as a motivating element. Furthermore, an analysis of a survey passed to the students of two groups of the subject with different characteristics is made, where different questions related to the tool and its capacity to motivate the students to the revision of the material of the practices before the presential session are included. From the analysis of the obtained results, it can be concluded that, in general, the Kahoot tool is a motivating element for the class, mainly in the part of computer practices.
The neutron transport equation describes the distribution of neutrons inside a nuclear reactor co... more The neutron transport equation describes the distribution of neutrons inside a nuclear reactor core. Homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem. The generalized equivalence theory for homogenization defines discontinuity factors at the boundaries of the homogenized subdomains. In this work, the generalized equivalence theory is extended to the PN equations for one-dimensional geometries using the finite element method. Here, pin discontinuity factors are proposed instead of the usual assembly discontinuity factors and the use of the spherical harmonics approximation as an extension of the diffusion theory. An interior penalty finite element method is used to discretize and solve the problem using discontinuity factors. Numerical results show that the pr...
The stationary neutron transport equation describes the neutron population and thus, the generate... more The stationary neutron transport equation describes the neutron population and thus, the generated heat, inside a nuclear reactor core. Obtaining the solution of this equation requires to solve a generalized eigenvalue problem efficiently. The majority of the eigenvalue solvers use the factorization of the system matrices to construct preconditioners, such as the ILU decomposition or the ICC decomposition, to speed up the convergence of the methods. The storage of the involved matrices and incomplete factorization demands high quantities of computational memory although a the sparse format is used. This makes the computational memory the limiting factor for this kind of calculations in some personal computers. In this work, we propose a matrix-free preconditioned eigenvalue solver that does not need to have the matrices allocated in memory explicitly. This method is based on the block inverse-free preconditioned Arnoldi method (BIFPAM) with the innovation that uses a preconditioner ...
Boiling channels and systems may oscillate owing to the behaviour of the liquid-steam mixture use... more Boiling channels and systems may oscillate owing to the behaviour of the liquid-steam mixture used for removing the thermal power. A thermal-hydraulic system may be unstable under particular operating conditions. Two kinds of power oscillation have been observed in BWR cores. One is an in-phase (core-wide) and the other is an out-of-phase (regional) oscillation. Since the above feature can make detection more difficult, the latter oscillation is potentially more severe. The problem is well known since the design of the first BWR system. However, to improve the safety systems of these reactors, it is necessary to be able to detect in a reliable way these oscillations from the neutronic signals. The purpose of this work is to characterize the unstable behaviour of a BWR. Within this study, it has been performed a number of perturbation analysis. The coupled codes RELAP5Mod3.3/PARCS have used for the simulation of the transients. Validation has been performed against Peach Bottom-2 Low...
A discrete ordinates method has been developed to approximate the neutron transport equation for ... more A discrete ordinates method has been developed to approximate the neutron transport equation for the computation of the lambda modes of a given configuration of a nuclear reactor core. This method is based on discrete ordinates method for the angular discretization, resulting in a very large and sparse algebraic generalized eigenvalue problem. The computation of the dominant eigenvalue of this problem and its corresponding eigenfunction has been done with a matrix-free implementation using both, the power iteration method and the Krylov-Schur method. The performance of these methods has been compared solving different benchmark problems with different dominant ratios.
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