39 research outputs found
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-Ferrà ndiz, A.; Ginestar Peiro, D.; Verdú MartÃn, 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
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-Ferrà ndiz, A.; Carreño, A.; Ginestar Peiro, D.; Verdú MartÃn, 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-
NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc
[EN] SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.This work was partially funded by the Spanish Agencia Estatal de Investigacion AEI http://ciencia.gob.es under grants TIN2016-75985-P AEI and PID2019-107379RB-I00 AEI (including European Commission FEDER funds).Campos, C.; Roman, JE. (2021). NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc. ACM Transactions on Mathematical Software. 47(3):1-29. https://doi.org/10.1145/3447544S12947
Integration methods for the time dependent neutron diffusion equation and other approximations of the neutron transport equation
[ES] Uno de los objetivos más importantes en el análisis de la seguridad en el campo de la ingenierÃa nuclear es el cálculo, rápido y preciso, de la evolución de la potencia dentro del núcleo del reactor. La distribución de los neutrones se puede describir a través de la ecuación de transporte de Boltzmann. La solución de esta ecuación no puede obtenerse de manera sencilla para reactores realistas, y es por ello que se tienen que considerar aproximaciones numéricas.
En primer lugar, esta tesis se centra en obtener la solución para varios problemas estáticos asociados con la ecuación de difusión neutrónica: los modos lambda, los modos gamma y los modos alpha. Para la discretización espacial se ha utilizado un método de elementos finitos de alto orden. Diversas caracterÃsticas de cada problema espectral se analizan y se comparan en diferentes reactores.
Después, se investigan varios métodos de cálculo para problemas de autovalores y estrategias para calcular los problemas algebraicos obtenidos a partir de la discretización espacial. La mayorÃa de los trabajos destinados a la resolución de la ecuación de difusión neutrónica están diseñados para la aproximación de dos grupos de energÃa, sin considerar dispersión de neutrones del grupo térmico al grupo rápido. La principal ventaja de la metodologÃa que se propone es que no depende de la geometrÃa del reactor, del tipo de problema de autovalores ni del número de grupos de energÃa del problema.
Tras esto, se obtiene la solución de las ecuaciones estacionarias de armónicos esféricos. La implementación de estas ecuaciones tiene dos principales diferencias respecto a la ecuación de difusión neutrónica. Primero, la discretización espacial se realiza a nivel de pin. Por tanto, se estudian diferentes tipos de mallas. Segundo, el número de grupos de energÃa es, generalmente, mayor que dos. De este modo, se desarrollan estrategias a bloques para optimizar el cálculo de los problemas algebraicos asociados.
Finalmente, se implementa un método modal actualizado para integrar la ecuación de difusión neutrónica dependiente del tiempo. Se presentan y comparan los métodos modales basados en desarrollos en función de los diferentes modos espaciales para varios tipos de transitorios. Además, también se desarrolla un control de paso de tiempo adaptativo, que evita la actualización de los modos de una manera fija y adapta el paso de tiempo en función de varias estimaciones del error.[CA] Un dels objectius més importants per a l'anà lisi de la seguretat en el camp de l'enginyeria nuclear és el cà lcul, rà pid i precÃs, de l'evolució de la potència dins del nucli d'un reactor. La distribució dels neutrons pot modelar-se mitjançant l'equació del transport de Boltzmann. La solució d'aquesta equació per a un reactor realÃstic no pot obtenir's de manera senzilla. És per això que han de considerar-se aproximacions numèriques.
En primer lloc, la tesi se centra en l'obtenció de la solució per a diversos problemes està tics associats amb l'equació de difusió neutrònica: els modes lambda, els modes gamma i els modes alpha. Per a la discretització espacial s'ha utilitzat un mètode d'elements finits d'alt ordre. Algunes de les caracterÃstiques dels problemes espectrals s'analitzaran i es compararan per a diferents reactors.
Tanmateix, diversos solucionadors de problemes d'autovalors i estratègies es desenvolupen per a calcular els problemes obtinguts de la discretització espacial. La majoria dels treballs per a resoldre l'equació de difusió neutrònica estan dissenyats per a l'aproximació de dos grups d'energia i sense considerar dispersió de neutrons del grup tèrmic al grup rà pid. El principal avantatge de la metodologia exposada és que no depèn de la geometria del reactor, del tipus de problema d'autovalors ni del nombre de grups d'energia del problema.
Seguidament, s'obté la solució de les equacions estacionà ries d'harmònics esfèrics. La implementació d'aquestes equacions té dues principals diferències respecte a l'equació de difusió. Primer, la discretització espacial es realitza a nivell de pin a partir de l'estudi de diferents malles. Segon, el nombre de grups d'energia és, generalment, major que dos. D'aquesta forma, es desenvolupen estratègies a blocs per a optimitzar el cà lcul dels problemes algebraics associats.
Finalment, s'implementa un mètode modal amb actualitzacions dels modes per a integrar l'equació de difusió neutrònica dependent del temps. Es presenten i es comparen els mètodes modals basats en l'expansió dels diferents modes espacials per a diversos tipus de transitoris. A més a més, un control de pas de temps adaptatiu es desenvolupa, evitant l'actualització dels modes d'una manera fixa i adaptant el pas de temps en funció de và ries estimacions de l'error.[EN] One of the most important targets in nuclear safety analyses is the fast and accurate computation of the power evolution inside of the reactor core. The distribution of neutrons can be described by the neutron transport Boltzmann equation. The solution of this equation for realistic nuclear reactors is not straightforward, and therefore, numerical approximations must be considered.
First, the thesis is focused on the attainment of the solution for several steady-state problems associated with neutron diffusion problem: the -modes, the -modes and the -modes problems. A high order finite element method is used for the spatial discretization. Several characteristics of each type of spectral problem are compared and analyzed on different reactors.
Thereafter, several eigenvalue solvers and strategies are investigated to compute efficiently the algebraic eigenvalue problems obtained from the discretization. Most works devoted to solve the neutron diffusion equation are made for the approximation of two energy groups and without considering up-scattering. The main property of the proposed methodologies is that they depend on neither the reactor geometry, the type of eigenvalue problem nor the number of energy groups.
After that, the solution of the steady-state simplified spherical harmonics equations is obtained. The implementation of these equations has two main differences with respect to the neutron diffusion. First, the spatial discretization is made at level of pin. Thus, different meshes are studied. Second, the number of energy groups is commonly bigger than two. Therefore, block strategies are developed to optimize the computation of the algebraic eigenvalue problems associated.
Finally, an updated modal method is implemented to integrate the time-dependent neutron diffusion equation. Modal methods based on the expansion of the different spatial modes are presented and compared in several types of transients. Moreover, an adaptive time-step control is developed that avoids setting the time-step with a fixed value and it is adapted according to several error estimations.Carreño Sánchez, AM. (2020). Integration methods for the time dependent neutron diffusion equation and other approximations of the neutron transport equation [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/144771TESI
The Biglobal Instability of the Bidirectional Vortex
State of the art research in hydrodynamic stability analysis has moved from classic one-dimensional methods such as the local nonparallel approach and the parabolized stability equations to two-dimensional, biglobal, methods. The paradigm shift toward two dimensional techniques with the ability to accommodate fully three-dimensional base flows is a necessary step toward modeling complex, multidimensional flowfields in modern propulsive applications. Here, we employ a two-dimensional spatial waveform with sinusoidal temporal dependence to reduce the three-dimensional linearized Navier-Stokes equations to their biglobal form. Addressing hydrodynamic stability in this way circumvents the restrictive parallel-flow assumption and admits boundary conditions in the streamwise direction. Furthermore, the following work employs a full momentum formulation, rather than the reduced streamfunction form, accounting for a nonzero tangential mean flow velocity. This approach adds significant complexity in both formulation and implementation but renders a more general methodology applicable to a broader spectrum of mean flows. Specifically, we consider the stability of three models for bidirectional vortex flow. While a complete parametric study ensues, the stabilizing effect of the swirl velocity is evident as the injection parameter, kappa, is closely examined
Adaptive heterogeneous parallelism for semi-empirical lattice dynamics in computational materials science.
With the variability in performance of the multitude of parallel environments available today, the conceptual overhead created by the need to anticipate runtime information to make design-time decisions has become overwhelming. Performance-critical applications and libraries carry implicit assumptions based on incidental metrics that are not portable to emerging computational platforms or even alternative contemporary architectures. Furthermore, the significance of runtime concerns such as makespan, energy efficiency and fault tolerance depends on the situational context. This thesis presents a case study in the application of both Mattsons prescriptive pattern-oriented approach and the more principled structured parallelism formalism to the computational simulation of inelastic neutron scattering spectra on hybrid CPU/GPU platforms. The original ad hoc implementation as well as new patternbased and structured implementations are evaluated for relative performance and scalability. Two new structural abstractions are introduced to facilitate adaptation by lazy optimisation and runtime feedback. A deferred-choice abstraction represents a unified space of alternative structural program variants, allowing static adaptation through model-specific exhaustive calibration with regards to the extrafunctional concerns of runtime, average instantaneous power and total energy usage. Instrumented queues serve as mechanism for structural composition and provide a representation of extrafunctional state that allows realisation of a market-based decentralised coordination heuristic for competitive resource allocation and the Lyapunov drift algorithm for cooperative scheduling
High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis
The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis
A multigrid accelerated eigensolver for the Hermitian Wilson-Dirac operator in lattice QCD
Eigenvalues of the Hermitian Wilson-Dirac operator are of special interest in
several lattice QCD simulations, e.g., for noise reduction when evaluating
all-to-all propagators. In this paper we present a Davidson-type eigensolver
that utilizes the structural properties of the Hermitian Wilson-Dirac operator
to compute eigenpairs of this operator corresponding to small eigenvalues.
The main idea is to exploit a synergy between the (outer) eigensolver and its
(inner) iterative scheme which solves shifted linear systems. This is achieved
by adapting the multigrid DD-AMG algorithm to a solver for shifted
systems involving the Hermitian Wilson-Dirac operator. We demonstrate that
updating the coarse grid operator using eigenvector information obtained in the
course of the generalized Davidson method is crucial to achieve good
performance when calculating many eigenpairs, as our study of the local
coherence shows. We compare our method with the commonly used software-packages
PARPACK and PRIMME in numerical tests, where we are able to achieve significant
improvements, with speed-ups of up to one order of magnitude and a near-linear
scaling with respect to the number of eigenvalues. For illustration we compare
the distribution of the small eigenvalues of on a lattice
with what is predicted by the Banks-Casher relation in the infinite volume
limit