A vertex based discretisation scheme applied to material non-linearity within a multi-physics finite volume framework

The objective of this research is the development of novel three dimensional Finite Volume(FV) algorithms for the solution of small strain, quasi-static, Computational Solid Mechanics(CSM) problems involving non-linear material behavior, specifically materials described by an elasto-visco-plastic constitutive relationship and a von-Mises yield criterion. The motivation is to contribute the non-linear CSM capability to an integrated FV framework for the comprehensive solution of the thermo-mechanical behaviour exhibited by the shape casting of metals. A study of novel two and three dimensional FV algorithms associated with CSM is presented. The algorithms employ a variety of two and three dimensional elements and are compared with the standard Bubnov-Galerkin Finite Element Method, with regard to algorithmic procedure, linear solvers, accuracy and computational cost. A variety of benchmark solid mechanics problems involving elasto-plastic and elasto-visco-plastic material behaviour are studied. These include the plane stress analysis of a perforated tensile strip of aluminium, the plane strain analysis of a hollow metal cylinder and the three dimensional analysis of a hollow metal sphere. The control volume-unstructured mesh, vertex based, FV algorithm for CSM problems is integrated within a multi-physics FV framework PHYSICA, which includes cell-centred FV procedures for the solution of problems involving simultaneous heat transfer, solidification and fluid flow. The thermo-mechanical coupling is described in detail. A variety of thermo-mechanical benchmark problems involving thermo-elasto-plastic and thermo-elasto-visco-plastic behaviour are studied, these include the quenching and the solidification of an infinite steel plate. Finally, the completely coupled capability of the FV framework PHYSICA is validated against experimental observations obtained from the gravity die casting of a hollow aluminium cylinder

A vertex-based finite volume method applied to non-linear material problems in computational solid mechanics

A vertex-based finite volume (FV) method is presented for the computational solution of quasi-static solid mechanics problems involving material non-linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two- and three-dimensional element types. A detailed comparison between the vertex-based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted

High Order Fluctuation Splitting Schemes for Hyperbolic Conservation Laws

This thesis presents the construction, the analysis and the verification of a new form of higher than second order fluctuation splitting discretisation for the solution of steady conservation laws on unstructured meshes. This is an alternative approach to the two existing higher than second order fluctuation splitting schemes, which use submesh reconstruction (developed by Abgrall and Roe) and gradient recovery (developed by Caraemi) to obtain the loacl higher degree polynomials used to evaluate the fluctuation. The new higher than second order approach constructs the polynomial interpolant of the values of the dependent variables at an appropriate number of carefully chosen mesh nodes. As they stand, none of the higher than second order methods can guarantee the absence of spurious oscillations from the flow without the application of an additional smoothing stage. The implementation of a technique that removes unphysical oscillations (devised by Hubbard) as part of a new higher than second order approach will be outlined. The design steps and theoretical bases are discussed in depth. The new higher than second order approach is examined and analysed through application to a series of linear and nonlinear scalar problems, using a pseudo-time-stepping technique to reach steady state solution on two-dimensional structured and unstructured meshes. The results demonstrate its effectiveness in approximating the linear and nolinear scalar problems. This thesis also addresses the development and examination of a multistage high order (in space and time) fluctuation splitting scheme for two-dimensional unsteady scalar advection on triangular unstructured meshes. the method is similar in philosophy to that of multistep high order (in space and time) fluctuation splitting scheme for the approximation of time-dependent hyperbolic conservation laws. The construction and implementation of the high order multistage time-dependent method are discussed in detail and its performance is illustrated using several standard test problems. The multistage high order time-dependent method is evaluated in the context of existing fluctuation splitting approaches to modelling time-dependent problems and some suggestions for their future development are made. Results presented indicate that the multistage high orer method can produce a slightly more accurate solution than the multistep high order method

Development of a finite volume method for elastic materials and fluid-solid coupled applications

This thesis presents the development of a parallel finite volume numerical method to analyse thermoelastic and hyperelastic materials and applied problems with mutual interaction between a fluid and a structure. The solid problem follows a cell-centred finite volume formulation for three-dimensional unstructured grids under the same framework that is frequently devoted to computational fluid dynamics. Second-order accurate schemes are used to discretise both in time and space. A direct implicit time integration promotes numerical stability when facing vibration and quasi-static scenarios. The geometrical non-linearities, encountered with the large displacements of both Saint Venant-Kirchhoff and neo-Hookean models, are tackled by means of an updated Lagrangian approach. Verification of the method is conducted with canonical cases which involve: static equilibrium, thermal stress, vibration, structural damping, large deformations, nearly incompressible materials and high memory usage. Significant savings in computation time are achieved owing to the acceleration strategies implemented within the system resolution, namely a segregated algorithm with Aitken relaxation and a block-coupled system arrangement. The similarities between the block-coupled method and the displacement-based finite element method, with regards to the matrix form of the resulting equations, allow for including Rayleigh viscous damping within a finite volume solver. The program for structures is to be coupled with the in-house fluid numerical models in order to produce a unified fluid-structure interaction platform, where an arbitrary Lagrangian-Eulerian approach is used to solve the flow in a conforming grid. As a first step, the method for incompressible Newtonian fluids is adapted to deal with structure-coupled problems. To do so, the Lagrangian-Eulerian version of the Navier-Stokes equations is presented, and automatic moving mesh techniques are developed. These techniques are designed to mitigate the mesh quality deterioration and to satisfy the space conservation law. Besides, a semi-implicit coupling algorithm, which only implicitly couples the fluid pressure term to the structure, is implemented. As a result, numerical stability for strongly coupled phenomena at a reduced computational cost is obtained. These new tools are tested on an applied case, consisting of the turbulent flow through self-actuated flexible valves. Finally, a pioneering coupled numerical model for the thermal and structural analysis of packed-bed thermocline storage tanks is developed. This thermal accumulation system for concentrated solar power plants has attracted the attention of the industry due to the economic advantage compared to the usual two-tank system. Dynamic coupling among the thermoelastic equations for the tank shell and the numerical models for all other relevant elements of the system is considered. After validating the model with experimental results, the commercial viability of the thermocline concept, regarding energetic effectiveness and structural reliability, is evaluated under real operating conditions of the power plants.Esta tesis presenta el desarrollo de un método numérico paralelo basado en volúmenes finitos para analizar materiales termoelásticos e hiperelásticos y problemas con una interacción mutua entre un fluido y una estructura. El problema del sólido sigue una formulación de volúmenes finitos centrada en las celdas para mallas no-estructuradas tridimensionales, bajo el mismo marco que se suele emplear en la dinámica de fluidos computacional. Se utilizan esquemas de segundo orden de precisión para discretizar el tiempo y el espacio. Una integración temporal directa implícita asegura estabilidad numérica al afrontar escenarios casi-estáticos o de vibración. Las no linealidades, que aparecen con los amplios desplazamientos de los modelos de Saint Venant-Kirchhoff y de neo-Hookean, son abordadas con un enfoque Lagrangiano actualizado. La verificación del método se realiza a través de casos canónicos que involucran: equilibrio estático, tensiones térmicas, vibración, amortiguación estructural, grandes deformaciones, materiales casi incompresibles y altos requerimientos de memoria. Se registra un ahorro significativo en el tiempo de cálculo gracias a las estrategias de aceleración implementadas dentro de la resolución del sistema, principalmente un algoritmo segregado con relajación Aitken y una disposición acoplada en bloques del sistema. Las similitudes entre este método acoplado en bloques y el método de los elementos finitos basados en el desplazamiento, con respecto a la forma matricial de las ecuaciones resultantes, permiten incluir la amortiguación viscosa tipo Rayleigh dentro de un solucionador de volúmenes finitos. El programa para estructuras se acoplará con los modelos numéricos internos para fluidos con el objetivo de generar una plataforma unificada de interacción fluido-estructura, donde se usa un enfoque arbitrario Lagrangiano-Euleriano sobre una malla conforme para resolver el fluido. Como primer paso, el método para flujos incompresibles Newtonianos se adapta para lidiar con problemas acoplados a una estructura. Para ello, se presenta la versión Lagrangiana-Euleriana de las ecuaciones de Navier-Stokes y se desarrollan técnicas automáticas de movimiento de malla. El diseño de estas técnicas se centra en mitigar el deterioro de la calidad de la malla y satisfacer la ley de conservación del espacio. Además, se implementa un algoritmo de acoplamiento semi-implícito, que sólo acopla implícitamente el término fluido de presión a la estructura. Como resultado, se obtiene estabilidad numérica para fenómenos fuertemente acoplados a un coste computacional reducido. Estas nuevas herramientas se prueban en un caso aplicado, que consiste el flujo turbulento a través de válvulas flexibles autoactivadas. Finalmente, se desarrolla un modelo numérico acoplado pionero para analizar estructuralmente y térmicamente los tanques termoclina de almacenamiento térmico. Este sistema de acumulación para centrales termosolares ha atraído la atención de la industria debido al ahorro económico comparado con el sistema de doble tanque habitual. Se tiene en cuenta el acoplamiento dinámico entre las ecuaciones gobernantes de la pared del tanque y las de todos los elementos relevantes del sistema. Tras validar el modelo con datos experimentales, se evalúa la viabilidad comercial de estos tanques, en cuanto a rendimiento energético y fiabilidad estructural, bajo condiciones reales de operación de las centrales.Postprint (published version

Numerical modelling of local scour with computational methods

Evaluating bed morphological evolution (specifically the scoured bed level) accurately using computational modelling is critical for analyses of the stability of many marine and coastal structures, such as piers, groynes, breakwaters, submarine pipelines and even telecommunication cables. This thesis considers the coupled hydrodynamic and morphodynamic modelling of the local scour around hydraulic structures, such as near a vertical pile or near a horizontal pipe. The focus in this study is on applying a fluid-structure interaction (FSI) approach to simulate the morphodynamical behaviour of the bed deformation, replacing the structural (i.e. solid mechanics) equation by the sediment continuity equation or Exner equation. Specifically, this works presents a novel method of mesh movement with anisotropic mesh adaptivity based on optimization for simulating local scour near structures with discontinuous Garlerkin (DG) discretisation methods for solving the flow field. Amongst the other goals of this work is the validation of the proposed procedure with previously performed laboratory as well as two- and three-dimensional numerical experiments. Additionally, performance is considered using an implementation of the methodology within Fluidity (http://fluidityproject.github.io/), an open-source, multi-physics, computational fluid dynamics (CFD) code, capable of handling arbitrary multi-scale unstructured tetrahedral meshes and including algorithms to perform dynamic anisotropic mesh adaptivity and mesh movement. The flexibility over mesh structure and resolution that these optimisation capabilities provide makes it potentially highly suitable for accounting the extreme bed morphological evolution close to a fixed solid structure under the action of hydrodynamics. Galerkin-based finite element methods have been used for the hydrodynamics (including discontinuous Galerkin discretisations) and morphological calculations, and automatic mesh deformation has been utilised to account for bed evolution changes while preserving the validity and quality of the mesh. Finally, the work extends the scope in regards of computational methods and considers scour modelling with pure Lagrangian and meshless methods such as smoothed particle hydrodynamics (SPH), which have also become of interest in the analysis and modelling of coastal sediment transport, particularly in scour-related processes. The SPH modelling is considered in a two-phase, flow-sediment fully Lagrangian scour simulation where the discrete-particle interaction forces between phases are resolved at the interface and continuous changes in the bed profile are obtained naturally.Open Acces

A High Order Finite Element Coupled Multi-Physics Approach To MRI Scanner Design

Magnetic Resonance Imaging (MRI) scanners are becoming increasingly popular with many clinical experts for use in both medical research and clinical imaging of patients, due to their ability to perform high-resolution non-intrusive imaging examinations. Recently, however, there has been an increasing demand for higher resolution scanners that are capable of performing quicker scans with increased patient comfort. With this demand for more advanced MRI systems, there also follows a number of challenges facing designers. Understanding the physical phenomena behind MRI is crucial in the development of scanners that are capable of producing accurate images of the patient with maximum comfort and minimal noise signatures.MRI scanners utilise strong static magnetic fields coupled with rapidly time varying gradient magnetic fields to generate images of the patient. In the presence of these time varying fields, the conducting components of MRI scanners generate eddy currents, which give rise to Lorentz forces and cause the conductors to vibrate.These vibrations cause acoustic waves to form that propagate through the air and result in audible noise which can cause significant discomfort for the patient. They also generate Lorentz currents which feedback into the electromagnetic field and this process results in a fully coupled non-linear acousto-magneto-mechanical system.The determination of the coupling mechanisms involved in such a system is a nontrivial task and so, in order to understand the behaviour of MRI systems during operation, advanced computational tools and techniques are required. Moreover, there exists certain small scale physical phenomena that arise in the coupled system which require high resolutions to obtain accurate results.In this thesis, a new computational framework for the treatment of acoustomagneto-mechanical coupling that arises in low-frequency electro-magneto-mechanical systems, such as MRI scanners, is proposed. The transient Newton-Raphson strategy involves the solution of a monolithic system, obtained from the linearisation of the coupled system of equations and two approaches are considered: (i) the linearised approach and (ii) the non-linear approach.In (i), physically motivated by the excitation from static and time varying current sources of MRI scanners, the fields may be split into a dominant static component and a much smaller dynamic component. The resulting linearised system is obtained by performing the linearisation of the fields about this dominant static component.This approach permits solutions in the frequency domain, for understanding the response of MRI systems under various excitations, and provides a computationally efficient way to solve this challenging problem, as it allows the tangent stiffness matrix to be inverted independently of time or frequency.In (ii), there is no approximation from a physical standpoint and the linearization is performed about the current solution. This approach requires that solutions are obtained in the time domain and thus the focus is then put on transient solutions to the coupled system of equations to address the following two important questions: 1) How good is the agreement between the computationally efficient linearised approach compared with the intensive non-linear approach?; and 2) Over what range of MRI operating conditions can the linearised approach be expected to provide acceptable results for MRI scanner design?Motivated by the need to solve industrial problems rapidly, solutions will be restricted to problems consisting of axisymmetric geometries and current sources. This treatment also discusses, in detail, the computational requirements for the solution of these coupled problems on unbounded domains and the accurate discretisation of the fields using hp-finite elements. A set of academic and industrially relevant examples are studied to benchmark and illustrate both approaches, in a hp- finite element context, as well as performing rigorous comparisons between the approaches

Advanced numerical methods for mantle convection models

Numerical modelling of Earth's mantle is a complex, and computationally demanding task due to, amongst others, the broad spectrum of temporal and spatial scales playing a role in mantle flow, large uncertainties in the physical properties of mantle material, with large and localised transitions in viscosity and density. This thesis introduces and analyses a number of numerical techniques that may bring a significant contribution in meeting some of these challenges. Firstly, we introduce a novel time integration scheme for free surface movement in mantle convection models that is more accurate and stable for large time steps. Secondly, we extend the capabilities of anisotropic mesh optimisation, which allows efficient focussing of mesh resolution, to handle cylindrical and spherical shell domains and demonstrate that a significant reduction in the required number of degrees of freedom is possible while maintaing accuracy. Finally, to verify correctness, and evaluate and compare properties of various numerical schemes, we derive an extensive suite of analytical solutions to the Stokes equations governing mantle flow in cylindrical and spherical shell domains, with physically relevant boundary conditions. As a numerical benchmark they also serve to facilitate comparisons of different geodynamical models, and the further development of numerical techniques to improve these.Open Acces

On coupling resolved and unresolved physical processes in finite element discretisations of geophysical fluids

At the heart of modern numerical weather forecasting and climate modelling lie simulations of two geophysical fluids: the atmosphere and the ocean. These endeavours rely on numerically solving the equations that describe these fluids. A key challenge is that the fluids contain motions spanning a range of scales. As the small-scale processes (unresolved by the numerical model) affect the resolved motions, they need to be described in the model, which is known as parametrisation. One major class of methods for numerically solving such partial differential equations is the finite element method. This thesis focuses on the coupling of such parametrised processes to the resolved flow within finite element discretisations. Four sets of research are presented, falling under two main categories. The first is the development of a compatible finite element discretisation for use in numerical weather prediction models, so as to avoid the bottleneck in computational scalability associated with the convergence at the poles of latitude-longitude grids. We present a transport scheme for use with the lowest-order function spaces in such a compatible finite element method, which is motivated by the coupling of the resolved and unresolved processes within the model. This then facilitates the use of the lower-order spaces within Gusto, a toolkit for studying such compatible finite element discretisations. Then, we present a compatible finite element discretisation of the moist compressible Euler equations, parametrising the unresolved moist processes. This is a major step in the development of Gusto, extending it to describe its first unresolved processes. The second category with which this thesis is concerned is the stochastic variational framework presented by Holm [Variational principles for stochastic fluid dynamics, P. Roy. Soc. A-Math. Phy. 471 (2176), (2015)]. In this framework, the effect of the unresolved processes and their uncertainty is expressed through a stochastic component to the advecting velocity. This framework ensures the circulation theorem is preserved by the stochastic equations. We consider the application of this formulation to two simple geophysical fluid models. First, we discuss the statistical properties of an enstrophy-preserving finite element discretisation of the stochastic quasi-geostrophic equation. We find that the choice of discretisation and the properties that it preserves affects the statistics of the solution. The final research presented is a finite element discretisation of the stochastic Camassa-Holm equation, which is used to numerically investigate the formation of ‘peakons’ within this set-up, finding that they do still always form despite the noise’s presence.Open Acces

Ultra-fast screening of stress-sensitive (naturally fractured) reservoirs using flow diagnostics

Quantifying the impact of poro-mechanics on reservoir performance is critical to the sustainable management of subsurface reservoirs containing either hydrocarbons, groundwater, geothermal heat, or being targeted for geological storage of fluids (e.g., CO2 or H2). On the other hand, accounting for poro-mechanical effects in full-field reservoir simulation studies and uncertainty quantification workflows in complex reservoir models is challenging, mainly because exploring and capturing the full range of geological and mechanical uncertainties requires a large number of numerical simulations and is hence computationally intensive. Specifically, the integration of poro-mechanical effects in full-field reservoir simulation studies is still limited, mainly because of the high computational cost. Consequently, poro-mechanical effects are often ignored in reservoir engineering workflows, which may result in inadequate reservoir performance forecasts. This thesis hence develops an alternative approach that couples hydrodynamics using existing flow diagnostics simulations for single- and dual-porosity models with poro mechanics to screen the impact of coupled poro-mechanical processes on reservoir performance. Due to the steady-state nature of the calculations and the effective proposed coupling strategy, these calculations remain computationally efficient while providing first-order approximations of the interplay between poro-mechanics and hydrodynamics, as we demonstrate through a series of case studies. This thesis also introduces a new uncertainty quantification workflow using the proposed poro-mechanical informed flow diagnostics and proxy models. These computationally efficient calculations allow us to quickly screen poro-mechanics and assess a broader range of geological, petrophysical, and mechanical uncertainties to rank, compare, and cluster a large ensemble of models to select representative candidates for more detailed full-physics coupled reservoir simulations.James Watt Scholarshi