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

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

On a general implementation of hh- and pp-adaptive curl-conforming finite elements

Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, specially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR. We cover into detail how to implement arbitrary order (i.e., $p$-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., $h$-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation