Alexandre Laurier

This work presents two novel topics regarding the Superhomogenization method: 1) the formalism for the implementation of the method with the linear Boltzmann Transport Equation, and 2) a Newton algorithm for the solution of the nonlinear problem that arises from the method. These new ideas have been implemented in a continuous finite element discretization in the MAMMOTH reactor physics application. The traditional solution strategy for this nonlinear problem uses a Picard, fixed-point iterative process whereas the new implementation relies on MOOSE’s Preconditioned Jacobian-Free Newton Krylov method to allow for a direct solution. The PJFNK-SPH can converge problems that were either intractable or very difficult to converge with the traditional iterative approach, including geometries with reflectors and vacuum boundary conditions. This is partly due to the underlying Scalable Nonlinear Equations Solvers in PETSc, which are integral to MOOSE and offer Newton damping, line search and trust region methods. The PJFNK-SPH has been implemented and tested for various discretizations of the transport equation included in the Rattlesnake transport solver. Speedups of five times for diffusion and ten to fifteen times for transport were obtained when compared to the traditional Picard approach. The three test problems cover a wide range of applications including a standard Pressurized Water Reactor lattice with control rods, a Transient Reactor Test facility control rod supercell and a prototype fast-thermal reactor. The reference solutions and initial cross sections were obtained from the Serpent 2 Monte Carlo code. The SPH-corrected cross sections yield eigenvalues that are near exact, relative to reference solutions, for reflected geometries, even with reflector regions. In geometries with vacuum boundary conditions the accuracy is problem dependent and solutions can be within a few to a few hundred pcm. The root-mean-square error in the power distribution is below 0.8% of the reference Monte Carlo. There is little benefit from SPH-corrected transport in typical scoping calculations, but for more detailed analyses it can yield superior convergence of the solution in some of the test problems. This PJFNK-SPH approach is currently being used in the modeling of the Transient Test Reactor at Idaho National Laboratory, where full reactor core SPH-corrected cross sections are employed to reduce the homogenization errors in transient or multi-physics calculations. This base implementation of the PJFNK-SPH provides an extremely robust solver and a springboard to further improve the Superhomogenization method in order to better preserve neutron currents, one of the primary deficiencies of the method.

Le but de ce projet de maitrise etait d’implementer la procedure SPH a l’interieur du code de physique de reacteur du Idaho National Laboratory (INL), qui fonctionne sur le code d’elements finis MOOSE. Precedant ce projet, le INL ne possedait pas de procedure de correction de sections efficaces, autre que l’utilisation de la methode SPH par le code DRAGON. La creation de ce projet vient du manque de flexibilite du code DRAGON dont le INL avait besoin. L’objectif premier de ce projet fut d’implementer la methode SPH pour l’equation de diffusion neutronique avec la normalisation de flux traditionnelle, Selengut et “True Selengut” et d’en tester les capacites. Le deuxieme visait la derivation des equations de transport SPH ainsi que leur implementation pour produire les premiers resultats sur des problemes complexes. En se basant sur des articles theorisant la correction en transport, nous avons implemente la correction SPH pour les equations de transport en calcul SN et PN. La correction SPH fut testee sur des assemblages de reacteur a eau ressurisee ou les resultats obtenus avec la correction de transport sont simliaires mais non superieurs a ceux obtenus avec l’equation de diffusion. Par contre, nous pensons que l’implementation de la correction des equations de transport permettera d’obtenir des meilleurs resultats dans les problemes ou la resolution en methode SN ou PN sont necessaires. Une consequence additionnelle de cette recherche fut l’implementation d’une nouvelle methode de resolution du probleme SPH non lineaire. Jusqu’au moment present, la procedure SPH fut resolue au travers de la methode de Picard, soit une methode iterative de pointfixe, tandis que la nouvelle implementation utilise la methode “Preconditioned Jacobian-Free Newtron Krylov” (PJFNK) qui etait deja presente au sein de MOOSE pour directement resoudre le probleme non lineaire. Cette nouvelle methode de resolution presente une reduction de temps de calcul d’un facteur approchant 50 et qui genere des facteurs SPH equivalents a ceux obtenus avec la methode iterative avec un critere de convergence tres strict, soit � < 10−8. La methode SPH resolue avec PJFNK permet aussi de resoudre des problemes qui contiennent des conditions frontieres de vide ou des materiaux reflecteurs, des cas ou la methode iterative traditionnelle ne peut converger. Dans les cas ou la methode PJFNK ne permet pas de converger, nous avons elabore une methode hybride qui combine la methode iterative et PJFNK. Pour ce faire, la methode iterative est utilisee pour forcer la condition initiale de la methode PJFNK a se situer a l’interieur du rayon de convergence des methodes de Newton.----------Abstract The goal of this thesis was to implement the SPH homogenization procedure within the MOOSE finite element framework at INL. Before this project, INL relied on DRAGON to do their SPH homogenization which was not flexible enough for their needs. As such, the SPH procedure was implemented for the neutron diffusion equation with the traditional, Selengut and true Selengut normalizations. Another aspect of this research was to derive the SPH corrected neutron transport equations and implement them in the same framework. Following in the footsteps of other articles, this feature was implemented and tested successfully with both the PN and SN transport calculation schemes. Although the results obtained for the power distribution in PWR assemblies show no advantages over the use of the SPH diffusion equation, we believe the inclusion of this transport correction will allow for better results in cases where either PN or SN are required. An additional aspect of this research was the mplementation of a novel way of solving the non-linear SPH problem. Traditionally, this was done through a Picard, fixed-point iterative process whereas the new implementation relies on MOOSE’s Preconditioned Jacobian-Free Newton Krylov (PJFNK) method to allow for a direct solution to the non-linear problem. This novel implementation showed a decrease in calculation time by a factor reaching 50 and generated SPH factors that correspond to those obtained through a fixed-point iterative process with a very tight convergence criteria: � < 10−8. The use of the PJFNK SPH procedure also allows to reach convergence in problems containing important reflector regions and void boundary conditions, something that the traditional SPH method has never been able to achieve. At times when the PJFNK method cannot reach convergence to the SPH problem, a hybrid method is used where by the traditional SPH iteration forces the initial condition to be within the radius of convergence of the Newton method. This new method was tested on a simplified model of INL’s TREAT reactor, a problem that includes very important graphite reflector regions as well as vacuum boundary conditions with great success. To demonstrate the power of PJFNK SPH on a more common case, the correction was applied to a simplified PWR reactor core from the BEAVRS benchmark that included 15 assemblies and the water reflector to obtain very good results. This opens up the possibility to apply the SPH correction to full reactor cores in order to reduce homogenization errors for use in transient or multi-physics calculations.

In this thesis, the measurements of the production cross section of a Z boson produced in association with high-transverse-momentum jets and decaying into a charged- lepton pair are presented. The data used for these measurements are analyzed from proton-proton collisions at the Large Hadron Collider. The dataset corresponds to an integrated luminosity of 139 fb−1 collected at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector between 2015 and 2018. The cross-section measurements are performed separately for the electron and muon decay channels of the Z boson and then combined in order to increase the precision of the final Z(→ l+l−)+jets measurements. Events with at least one jet with pT ≥ 500 GeV/c are selected and populate a high-pT region. In this region, the production of an on-shell Z boson radiated by a quark is enhanced and results in events with a small angle between the Z boson and the associated quark jet such that the two objects are measured to be collinear. A fraction of the events in the high-pT region are also described by Z+1 jet events where the Z boson and the jet recoil against each-other such that the two objects are measured to be back-to-back. The large dataset has made it possible for the first time to separately study these two event topologies where a Z boson and a jet are either collinear or back-to-back. This is achieved by measuring the angular correlation between the Z boson and the closest jet. The production of a Z boson in association with jets, Z+jets, provides a near inexhaustible amount of information about the physics mechanism for the production of vector bosons at the Large Hadron Collider. This analysis allows to measure the cross section of Z+jets as a function of characteristic observables. The resulting differential cross sections are compared with state-of-the-art theoretical predictions. The differential cross sections themselves provide a powerful way to test the Standard Model and in particular quantum chromo-dynamics. The data are found to agree with the latest next-to-next-to-leading-order and next-to-leading-order theoretical predictions for Z+jets production.