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

[EN] The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P and financed with the help of a Primeros Proyectos de Investigacion (PAID-06-18) from Vicerrectorado de Investigacion, Innovacion y Transferencia of the Universitat Politecnica de Valencia.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2021). Adaptive time-step control for modal methods to integrate the neutron diffusion equation. Nuclear Engineering and Technology. 53(2):399-413. https://doi.org/10.1016/j.net.2020.07.004S39941353

Schwarz type preconditioners for the neutron diffusion equation

[EN] Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem. In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account. The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the pre-conditioner is less than the improvement in the convergence rate. (C) 2016 Elsevier B.V. All rights reserved.The work has been partially supported by the spanish Ministerio de Economía y Competitividad under projects ENE 2014-59442-P and MTM2014-58159-P, the Generalitat Valenciana under the project PROMETEO II/2014/008 and the Universitat Politècnica de València under the project FPI-2013. The work has also been supported partially by the Swedish Research Council (VR-Vetenskapsrådet) within a framework grant called DREAM4SAFER, research contract C0467701.Vidal-Ferràndiz, A.; González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ.; Demazière, C. (2017). Schwarz type preconditioners for the neutron diffusion equation. Journal of Computational and Applied Mathematics. 309:563-574. https://doi.org/10.1016/j.cam.2016.02.056S56357430

Moving meshes to solve the time-dependent neutron diffusion equation in hexagonal geometry

To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. Here the spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. This means that the accuracy of the solution can be improved refining the spatial mesh (h-refinement) and also increasing the degree of the polynomial expansions used in the finite element method (p-refinement). Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying one-dimensional and three-dimensional benchmark problems. (C) 2015 Elsevier B.V. All rights reserved.This work has been partially supported by the Spanish Ministerio de Ciencia e Innovacion under project ENE2011-22823, the Generalitat Valenciana under projects II/2014/08 and ACOMP/2013/237, and the Universitat Politecnica de Valencia under project UPPTE/2012/118.Vidal-Ferràndiz, A.; Fayez Moustafa Moawad, R.; Ginestar Peiro, D.; Verdú Martín, GJ. (2016). Moving meshes to solve the time-dependent neutron diffusion equation in hexagonal geometry. Journal of Computational and Applied Mathematics. 291:197-208. https://doi.org/10.1016/j.cam.2015.03.040S19720829

3D calculation of the lambda eigenvalues and eigenmodes of the two-group neutron diffusion equation by coarse-mesh nodal methods

[EN] This paper shows the development and roots of the lambda eigenvector and eigenmodes calculation by coarse-mesh finite difference nodal methods. In addition, this paper shows an inter-comparison of the eigenvalues and power profiles obtained by different 3D nodal methods with two neutron energy groups. The methods compared are: the nodal collocation method (Verdú et al., 1998, 1993; Hebert, 1987) with different orders of the Legendre expansions, the modified coarse-mesh nodal method explained in this paper, and the method implemented in the PARCS code by Wysocki et al. (2014, 2015). In this paper we have developed a program NODAL-LAMBDA that uses a two-group modified coarse-mesh finite difference method with albedo boundary conditions. Some of the approximations performed originally by Borressen (1971) have been discarded and more exact expressions have been used. We compare for instance the eigenvalues and power profiles obtained with Borressen original approach of 1.5 group and with 2 groups. Also, some improvements in the albedo boundary conditions suggested by (Turney 1975; Chung et al., 1981) have been incorporated to the code as an option. The goal is to obtain the eigenvalues and the sub-criticalities (ki¿k0) of the harmonic modes that can be excited during an instability event in a fast way and with an acceptable precision.The authors of this paper are indebted to IBERDROLA generacion nuclear for provide support and funding to perform this work.Muñoz-Cobo, J.; Miró Herrero, R.; Wysocki, A.; Soler Domingo, A. (2019). 3D calculation of the lambda eigenvalues and eigenmodes of the two-group neutron diffusion equation by coarse-mesh nodal methods. Progress in Nuclear Energy. 110:393-409. https://doi.org/10.1016/j.pnucene.2018.10.008S39340911

Solution of the Lambda modes problem of a nuclear power reactor using an h-p finite element method

Lambda modes of a nuclear power reactor have interest in reactor physics since they have been used to develop modal methods and to study BWR reactor instabilities. An h–p-Adaptation finite element method has been implemented to compute the dominant modes the fundamental mode and the next subcritical modes of a nuclear reactor. The performance of this method has been studied in three benchmark problems, a homogeneous 2D reactor, the 2D BIBLIS reactor and the 3D IAEA reactor.This work has been partially supported by the Spanish Ministerio de Ciencia e Innovacion under project ENE2011-22823, the Generalitat Valenciana under projects PROMETEO/2010/039 and ACOMP/2013/237, and the Universitat Politecnica de Valencia under project UPPTE/2012/118.Vidal Ferràndiz, A.; Fayez Moustafa Moawad, R.; Ginestar Peiro, D.; Verdú Martín, GJ. (2014). Solution of the Lambda modes problem of a nuclear power reactor using an h-p finite element method. Annals of Nuclear Energy. 72:338-349. https://doi.org/10.1016/j.anucene.2014.05.026S3383497

Final results from the PERUSE study of first-line pertuzumab plus trastuzumab plus a taxane for HER2-positive locally recurrent or metastatic breast cancer, with a multivariable approach to guide prognostication

Background: The phase III CLinical Evaluation Of Pertuzumab And TRAstuzumab (CLEOPATRA) trial established the combination of pertuzumab, trastuzumab and docetaxel as standard first-line therapy for human epidermal growth factor receptor 2 (HER2)-positive locally recurrent/metastatic breast cancer (LR/mBC). The multicentre single-arm PERtUzumab global SafEty (PERUSE) study assessed the safety and efficacy of pertuzumab and trastuzumab combined with investigator-selected taxane in this setting. Patients and methods: Eligible patients with inoperable HER2-positive LR/mBC and no prior systemic therapy for LR/mBC (except endocrine therapy) received docetaxel, paclitaxel or nab-paclitaxel with trastuzumab and pertuzumab until disease progression or unacceptable toxicity. The primary endpoint was safety. Secondary endpoints included progression-free survival (PFS) and overall survival (OS). Prespecified subgroup analyses included subgroups according to taxane, hormone receptor (HR) status and prior trastuzumab. Exploratory univariable analyses identified potential prognostic factors; those that remained significant in multivariable analysis were used to analyse PFS and OS in subgroups with all, some or none of these factors. Results: Of 1436 treated patients, 588 (41%) initially received paclitaxel and 918 (64%) had HR-positive disease. The most common grade 653 adverse events were neutropenia (10%, mainly with docetaxel) and diarrhoea (8%). At the final analysis (median follow-up: 5.7 years), median PFS was 20.7 [95% confidence interval (CI) 18.9-23.1] months overall and was similar irrespective of HR status or taxane. Median OS was 65.3 (95% CI 60.9-70.9) months overall. OS was similar regardless of taxane backbone but was more favourable in patients with HR-positive than HR-negative LR/mBC. In exploratory analyses, trastuzumab-pretreated patients with visceral disease had the shortest median PFS (13.1 months) and OS (46.3 months). Conclusions: Mature results from PERUSE show a safety and efficacy profile consistent with results from CLEOPATRA and median OS exceeding 5 years. Results suggest that paclitaxel is a valid alternative to docetaxel as backbone chemotherapy. Exploratory analyses suggest risk factors that could guide future trial design