Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.Open Acces

On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinite-dimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for plane-parallel geometries with isotropic scattering, but the approach and proofs generalize to multi-dimensional problems and more general scattering operators verbatim

Least-Squares FEM: Literature Review

During the last years the interest in least squares finite element methods (LSFEM) has grown continuously. Least squares finite element methods offer some advantages over the widely used Galerkin variational principle. One reason is the ability to cope with first order differential operators without special treatment as required by the Galerkin FEM. The other reason comes from the numerical point of view, where the LSFEM leads to symmetric positive definite matrices which can be solved very efficiently under some conditions. This report gives an overview about the recent literature which appeared in the field of least squares finite element methods and summarises the essential results and facts about the LSFEM.Während der letzten Jahre hat das Interesse an Least Squares Finite Element Methoden (LSFEM) stetig zugenommen. Least Squares Finite Element Methoden bieten einige Vorteile gegenüber dem etablierten Galerkin Variationsansatz. So können Differentialoperatoren erster Ordnung ohne besondere numerische Techniken, wie z.B. Stabilisierung, direkt behandelt werden. Ein anderer Grund für den Einsatz der LSFEM liegt in den entstehenden algebraischen Gleichungssystemen, die immer symmetrisch positiv definit sind und unter bestimmten Vorraussetzungen eine effiziente Lösung ermöglichen.Dieser Bericht gibt einen Überblick über die aktuelle Literatur zur LSFEM und faßt die entscheidenden Ergebnisse zusammen

Scalable angular adaptivity for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which for the first time shows evidence of $\mathcal{O}(n)$ scaling in both runtime and memory usage, where $n$ is the number of adapted angles. This adaptivity uses Haar wavelets, which perform structured $h$-adaptivity built on top of a hierarchical P$_0$ FEM discretisation of a 2D angular domain, allowing different anisotropic angular resolution to be applied across space/energy. Fixed angular refinement, along with regular and goal-based error metrics are shown in three example problems taken from neutronics/radiative transfer applications. We use a spatial discretisation designed to use less memory than competing alternatives in general applications and gives us the flexibility to use a matrix-free multgrid method as our iterative method. This relies on scalable matrix-vector products using Fast Wavelet Transforms and allows the use of traditional sweep algorithms if desired

Nonlinear Diffusion Acceleration in Voids for the Least-Squares Transport Equation

In this dissertation we present advances to the nonlinear diffusion acceleration for void regions using second order forms of the transport equation. We consider the weighted least-squares and the self-adjoint angular flux transport equations. We show that these two equations are closely related through the definition of the weight function and how the nonlinear diffusion acceleration can be extended to hold in void regions. Using a Fourier analysis we show the convergence properties of our method for homogeneous and heterogeneous problems. We use several problems to study the numerical behavior and the influence of different discretization schemes. Second order forms of the transport equation allow the use of continuous finite elements (CFEM). CFEM discretization is computationally cheaper and easier to implement on unstructured meshes, allowing more detailed geometries. The selfadjoint transport operators result normally in symmetric positive-definite matrices which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage. In this dissertation we study the weighted least-squares and compare to the self-adjoint angular flux transport equations with void treatment, both well defined in voids. The nonlinear diffusion acceleration (NDA) is an effective scheme to increase convergence for highly diffusive problems, but can also ensure conservation for nonconservative transport schemes. However, for second order transport equations, the scheme was not yet defined in voids. In this dissertation we derived modifications to the NDA to handle problems containing void regions. A Fourier analysis showed that the newly developed modifications accelerates unconditionally for scattering ratios smaller than one. Extensive testing on various parameters was performed to ensure that the modifications are stable and efficient. Numerical tests with Reed’s problem showed that the NDA scheme results in a non-constant flux shape in the void regions. Further investigations revealed that this coarse mesh problem is caused by the interface coupling between void and material regions. The separation of the low-order equation at the interface ameliorates these problems. We give a proof-of-concept for a high-order CFEM/low-order DFEM scheme as well as for an artificial diffusion scheme to restore causality and obtain an improved scalar flux solution in the void. The NDA void modifications were then tested on a modified C5G7 problem, a challenging reactor physics benchmark. The results were compared to first order transport and the self-adjoint angular flux equation with void treatment. The results indicated that the weighted least-squares equations give adequate results while maintaining a symmetric positive-definite matrix

Even-Parity S_(N) Adjoint Method Including SP_(N) Model Error and Iterative Efficiency

In this Dissertation, we analyze an adjoint-based approach for assessing the model error of SP_(N) equations (low fidelity model) by comparing it against S_(N) equations (high fidelity model). Three model error estimation methods, namely, direct , residual, and adjoint methods are proposed. In order to compare the SP_(N) solution against S_(N), we also proposed angular intensity reconstruction schemes for reconstructing S_(N) angular intensity from SP_(N) solutions. The methodology is then applied to a vehicle atmosphere re-entry problem and the convergence behavior of the SP_(N) and Even-parity S_(N) are compared with that of the Least-squares S_(N) method. The results show that all the three model error estimation methods are equivalent up to a readily computable compensation and the Least-squares S_(N) method is far superior than the Even-parity S_(N) and SP_(N) methods when applied to such a near-void problem. Various forms of SP_(N) equations, together with their appropriate iterative solution schemes and acceleration techniques are evaluated in terms of iterative efficiency. The Fourier analyses and numerical test results indicate the Canonical form solved with DSA or AnMG preconditioned source iteration offering the best iterative performance

Improved Fully-Implicit Spherical Harmonics Methods for First and Second Order Forms of the Transport Equation Using Galerkin Finite Element

In this dissertation, we focus on solving the linear Boltzmann equation -- or transport equation -- using spherical harmonics (PN) expansions with fully-implicit time-integration schemes and Galerkin Finite Element spatial discretizations within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The presentation is composed of two main ensembles. On one hand, we study the first-order form of the transport equation in the context of Thermal Radiation Transport (TRT). This nonlinear application physically necessitates to maintain a positive material temperature while the PN approximation tends to create oscillations and negativity in the solution. To mitigate these flaws, we provide a fully-implicit implementation of the Filtered PN (FPN) method and investigate local filtering strategies. After analyzing its effect on the conditioning of the system and showing that it improves the convergence properties of the iterative solver, we numerically investigate the error estimates derived in the linear setting and observe that they hold in the non-linear case. Then, we illustrate the benefits of the method on a standard test problem and compare it with implicit Monte Carlo (IMC) simulations. On the other hand, we focus on second-order forms of the transport equation for neutronics applications. We mostly consider the Self-Adjoint Angular Flux (SAAF) and Least-Squares (LS) formulations, the former being globally conservative but void incompatible and the latter having -- in all generality -- the opposite properties. We study the relationship between these two methods based on the weakly-imposed LS boundary conditions. Equivalences between various parity-based PN methods are also established, in particular showing that second-order filters are not an appropriate fix to retrieve void compatibility. The importance of global conservation is highlighted on a heterogeneous multigroup k-eigenvalue test problem. Based on these considerations, we propose a new method that is both globally conservative and compatible with voids. The main idea is to solve the LS form in the void regions and the SAAF form elsewhere. For the LS form to be conservative in void, a non-symmetric fix is required, yielding the Conservative LS (CLS) formulation. From there, a hybrid SAAF-- CLS method can be derived, having the desired properties. We also show how to extend it to near-void regions and time-dependent problems. While such a second-order form already existed for discrete-ordinates (SN) discretizations (Wang et al. 2014), we believe that this method is the first of its kind, being well-suited to both SN and PN discretizations