Complexity bounds on supermesh construction for quasi-uniform meshes

Projecting fields between different meshes commonly arises in computational physics. This operation requires a supermesh construction and its computational cost is proportional to the number of cells of the supermesh $n$. Given any two quasi-uniform meshes of $n_A$ and $n_B$ cells respectively, we show under standard assumptions that n is proportional to $n_A + n_B$. This result substantially improves on the best currently available upper bound on $n$ and is fundamental for the analysis of algorithms that use supermeshes

Galerkin projection of discrete fields via supermesh construction

Interpolation of discrete FIelds arises frequently in computational physics. This thesis focuses on the novel implementation and analysis of Galerkin projection, an interpolation technique with three principal advantages over its competitors: it is optimally accurate in the L2 norm, it is conservative, and it is well-defined in the case of spaces of discontinuous functions. While these desirable properties have been known for some time, the implementation of Galerkin projection is challenging; this thesis reports the first successful general implementation. A thorough review of the history, development and current frontiers of adaptive remeshing is given. Adaptive remeshing is the primary motivation for the development of Galerkin projection, as its use necessitates the interpolation of discrete fields. The Galerkin projection is discussed and the geometric concept necessary for its implementation, the supermesh, is introduced. The efficient local construction of the supermesh of two meshes by the intersection of the elements of the input meshes is then described. Next, the element-element association problem of identifying which elements from the input meshes intersect is analysed. With efficient algorithms for its construction in hand, applications of supermeshing other than Galerkin projections are discussed, focusing on the computation of diagnostics of simulations which employ adaptive remeshing. Examples demonstrating the effectiveness and efficiency of the presented algorithms are given throughout. The thesis closes with some conclusions and possibilities for future work

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

Modelling multiple-material flows on adaptive unstructured meshes.

The ability to distinguish between regions with different material properties is essential when numerically modelling many physical systems. Using a dual control volume mesh that avoids the problem of corner coupling, the HyperC face value scheme is extended to multiple dimensions and applied as a device for material advection on unstructured simplex meshes. The new scheme performs well at maintaining sharp interfaces between materials and is shown to produce small advection errors, comparable to those of standard material advection methods on structured meshes. To further minimise numerical diffusion of material interfaces a total variation bounded flux limiter, UltraC, is defined using a normalised variable diagram. Combining the material tracking scheme with dynamically adapting meshes, the use of a minimally dissipative bounded projection algorithm for interpolating fields from the old mesh to the new, optimised mesh is demonstrated that conserves the mass of each material. More generally, material conservation during the advection process is ensured through the coupling of the material tracking scheme to the momentum and mass equations. A new element pair for the discretisation of velocity and pressure is proposed that maintains the stability of the system while conserving the mass of materials. When modelling multiple materials the use of independent advection algorithms for each material can lead to the problem of non-physical material overlap. A novel coupled flux limiter is developed to overcome this problem. This automatically generalises to arbitrary numbers of materials. Using the fully coupled (and rigorously verified) multi-material model, several geophysically relevant simulations are presented examining the generation of waves by landslides. The model is validated by demonstrating close agreement between model predictions and experimental results of wave generation, propagation and run-up. The simulations also showcase the powerful capabilities of an unstructured, adaptive multi-material model in real world scenarios

Consistent Point Data Assimilation in Firedrake and Icepack

We present methods and tools that significantly improve the ability to estimate quantities and fields which are difficult to directly measure, such as the fluidity of ice, using point data sources, such as satellite altimetry. These work with both sparse and dense point data with estimated quantities and fields becoming more accurate as the number of measurements are increased. Such quantities and fields are often used as inputs to mathematical models that are used to make predictions so improving their accuracy is of vital importance. We demonstrate how our methods and tools can increase the accuracy of results, ensure posterior consistency, and aid discourse between modellers and experimenters. To do this, we bring point data into the finite element method ecosystem as discontinuous fields on meshes of disconnected vertices. Point evaluation can then be formulated as a finite element interpolation operation (dual-evaluation). Our new abstractions are well-suited to automation. We demonstrate this by implementing them in Firedrake, which generates highly optimised code for solving PDEs with the finite element method. Our solution integrates with dolfin-adjoint/pyadjoint which allows PDE-constrained optimisation problems, such as data assimilation, to be solved through forward and adjoint mode automatic differentiation. We demonstrate our new functionality through examples in the fields of groundwater hydrology and glaciology

Methods for tracking material properties within an unstructured, adaptive mesh computational modelling framework, with application to simulating the development of seismic anisotropy at spreading centres and transform faults

The ability to accurately and efficiently track material properties in a Lagrangian sense during geodynamical flows, as well as evaluate how they evolve through both space and time, is of vital importance to our understanding of the structure, dynamics and evolution of Earth's mantle and lithosphere. An approach for achieving this, widely advocated by the geodynamics community, is the so-called particle-in cell technique. With this scheme, material properties are tracked by a large number of particles that are advected with the flow field. These properties can represent a wide range of parameters (e.g. material composition or strain) and the information they carry can be accessed during a simulation to feed back into the flow equations (e.g. composition controlling material density), as well as generate diagnostic fields for analysis (e.g. the generation of lattice-preferred orientation). In Chapter 2, we develop a particle-in-cell scheme within an adaptive, unstructured, anisotropic mesh computational modelling framework called Fluidity. Regions of geodynamic interest often vary throughout a simulation, and the combination of the particle-in-cell scheme with a state of the art adaptive mesh algorithm enables both the mesh resolution and particle density to capture areas of interest with high resolution (or density), while reducing resolution (or density) in areas of little geodynamic interest. This ensures that a high level of accuracy is maintained throughout the computational domain, while also greatly improving numerical efficiency. The implementation of this scheme saw several inherent challenges which had to be overcome, including the treatment of particles during mesh adaptivity, the transfer of particle between processors, and the interpolation of values between particles and nodes of the mesh. In Chapter 3, validation of the implemented particle-in-cell scheme is undertaken for a series of well-known thermo-chemical convection benchmark tests. For each benchmark, particles track material composition as they are advected throughout the computational domain, with material composition feeding back onto the density field and influencing the buoyancy of materials. Results from the particle scheme are compared with results from a field-based control volume, flux-limited conservative difference scheme known as HyperC. Both schemes perform favourably across the series of benchmark tests, with HyperC displaying superior mass conservation and the particle scheme exhibiting superior shape preservation, resulting in a smoother material interface and the visualization of finer scale features. The particle-in-cell scheme is favoured, as it is more flexible in its application to tracking generalized material properties, enabling it to be applied to a wide range of geodynamical problems, such as the tracking of material texture. In Chapter 4, we develop the software package PyDRex, which is capable of converting tracked deformation parameters into predictions of material lattice-preferred orientation and seismic anisotropy. The generation of lattice-preferred orientation and subsequent observations of seismic anisotropy within Earth's mantle yields some of the most direct constrains available on both past and present-day deformation. By simulating these processes with geodynamics models, it is possible to generate synthetic seismic anisotropy predictions, which can be compared with observations in an attempt to constrain the prevalent flow regime in the upper mantle. In Chapter 5, we utilize the PyDRex software package with Fluidity and develop three sets of oceanic plate boundary models, being 2-D and 3-D mid-ocean ridge models, and 3-D mid-ocean ridge models with a transform fault offset. We compare anisotropy predictions from these models with seismic observations of anisotropy, allowing inferences to be made on the prevalent flow regime and distribution of material anisotropy surrounding oceanic plate boundaries

Modelling the transient drainage of liquid in foams

Froth flotation is the largest tonnage separation process worldwide and is used for paper deinking, water purification and, particularly, mineral separation. One of the key aspects of the performance of flotation cells is the behaviour of the liquid within the froth, as it is crucial to the purity of the product and a major influence on the overall recovery. Nonlinearities in models for liquid motion in the froth make them complex to solve and existing numerical solutions have been in two dimensions at most. In order to predict the performance of industrial flotation cell designs, a three-dimensional solution for these equations is desirable. Moreover, the understanding of the process would be enhanced if a transient model were used to predict the dynamics of foam drainage. In this work, the equations for the liquid drainage have been rearranged in order to make them analogous to a compressible version of the Navier-Stokes equations, coupled to an equation of state. A model for predicting the movement of the flowing foam has also been developed, which is able to solve for the foam velocity in two and three dimensions. This has allowed the transient behaviour of liquid in flotation foams to be modelled using Fluidity, a general purpose finite element method code that allows simulations to be carried out on unstructured adaptive meshes. This is an important feature for improving the computational cost of modelling these systems, as there are boundary layers present in the process, whose size is independent of the scale of the flotation system being modelled. These models have allowed, for the first time, to carry out numerical investigations of drainage for arbitrary flotation tank geometries in up to three dimensions, and have been verified against analytical solutions and compared to laboratory scale experiments with satisfactory agreement