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A recent development in Computational Fluid Dynamics (CFD) has been the meshless method calledWeakly Compressible Smoothed Particle Hydrodynamics (WCSPH), which is a Lagrangian method that tracks physical quantities of a fluid as it moves in time and space. One disadvantage of WCSPH is the small time steps required due to the use of the weakly compressible Tait equation of state, so large scale simulations using WCSPH have so far been rare and only performed on very expensive CPU-based supercomputers. As CFD simulations grow larger and more detailed, the need to use high performance computing also grows. There is therefore great interest in any computer technology that can provide the equivalent computational power of the CPU-based supercomputer for a fraction of the cost. Hence the excitement aroused in the SPH community by the Graphics Processing Unit (GPU). The GPU offers great potential for providing significant increases in computational performance due to its much smaller size and power consumption relative to the more established and traditional high performance computers comprising hundreds or thousands of CPUs. However, there are some disadvantages in programming GPUs. The memory structure of the GPU is more complex and more variable in speed, and there are other factors that can seriously affect performance, such as the thread grid dimensions which drives the occupancy of the GPU. The aim of this thesis is to describe how WCSPH can be efficiently implemented on multiple GPUs. First, some CFD methods and their success or otherwise in simulating free surfaces are discussed, and examples of previous attempts at implementing CFD algorithms on GPUs are given. The mathematical theory of WCSPH is then presented, followed by a detailed examination of the architecture of a GPU and how to program a GPU. Two different implementations of the same WCSPH algorithm are then described to simulate a well known experiment of a collapse of a column of water to highlight two possible uses of the GPU memory. The first method uses the fast shared memory of the GPU, which is recommended by the GPU manufacturer, while the second method uses the texture i memory of the GPU, which acts as a cache. It is shown that due to the theory of WCSPH, which allows particles to only interact with other particles a short distance apart, that despite the speed of the shared memory and the power of coalescing data into the shared memory, the texture memory method is currently the most efficient, but that this method of implementing WCSPH on a single GPU requires a much higher degree of complexity of programming than the shared memory method. It is also shown that the size of the thread block can have a significant effect on performance. Riemann solvers add more computational effort but can provide more accuracy. The use of Riemann solvers in WCSPH and their success or otherwise is then examined, and the results and performance of one particular WCSPH algorithm that uses an approximate Riemann solver when executed on a GPU are reported. The treatment of boundaries has been and continues to be a problem in WCSPH, and there are a number of creative proposals for boundary treatments. Some of these are described in detail before a new boundary treatment is proposed that builds upon a boundary treatment that was recently proposed, and improves its performance in execution time on a GPU by using the registers and not the slower memories of the GPU. This new boundary treatment builds a unique private grid of boundary particles for each fluid particle close to the boundary. All computation is performed in the registers, the properties of the boundary particles depend on the fluid particle only, and there is no requirement to recall data from the slower global or texture memories of the GPU. The new boundary treatment is also shown to propagate a solitary wave further, preserves the wave height more and takes less execution time to compute than the original boundary treatment this new treatment builds on. A unique and simple implementation of WCSPH on multiple GPUs is then described, and the results of a simulation of a collapse of a column of water in 3D are reported and compared against the results from a simulation of the same problem with the same WCSPH algorithm executed on a large cluster of multi core CPUs. The conclusion is that simulations on a small cluster of GPUs can achieve greater performance than from a cluster of multi core CPUs, but to achieve this the slow GPU memories, including the texture ii memory, must be avoided by using the registers as much as possible, and the architecture of the network linking the GPUs together must be exploited. The former was achieved by using the new boundary treatment proposed in this thesis and discussed above, and the latter was achieved by the use of the MPI Group functionality. The GPUs used for this thesis were already connected together in boxes of 4 by the manufacturer. The cluster used for this thesis consisted of 8 of these boxes, giving a total of 32 GPUs. These boxes of 4 GPUs were connected together through a common host, but the communication speed over the connection between the box and the host is much slower than that between the GPUs inside the box. The total communication time was minimized by grouping the GPUs inside a box together with their private unique MPI communicator, and a communication procedure was created to minimize communication over the relatively slow connection between the boxes of GPUs and the host. Finally, some conclusions are drawn and suggestions for further work are made.

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This thesis aimed at the understanding and further development of smoothed particle hydrodynamics (SPH). The first part described the implementations of non-reflecting boundary conditions for elastic- waves in SPH. The second part contains a stability analysis of the semi-discrete SPH equations and a new method for stabilising basic SPH in tension.

This thesis presents a two-dimensional water model investigation and development of a multiscale method for the modelling of large systems, such as virus in water or peptide immersed in the solvent. We have implemented a two-dimensional ‘Mercedes Benz’ (MB) or BN2D water model using Molecular Dynamics. We have studied its dynamical and structural properties dependence on the model’s parameters. For the first time we derived formulas to calculate thermodynamic properties of the MB model in the microcanonical (NVE) ensemble. We also derived equations of motion in the isothermal–isobaric (NPT) ensemble. We have analysed the rotational degree of freedom of the model in both ensembles. We have developed and implemented a self-consistent multiscale method, which is able to communicate micro- and macro- scales. This multiscale method assumes, that matter consists of the two phases. One phase is related to micro- and the other to macroscale. We simulate the macro scale using Landau Lifshitz-Fluctuating Hydrodynamics, while we describe the microscale using Molecular Dynamics. We have demonstrated that the communication between the disparate scales is possible without introduction of fictitious interface or approximations which reduce the accuracy of the information exchange between the scales. We have investigated control parameters, which were introduced to control the contribution of each phases to the matter behaviour. We have shown, that microscales inherit dynamical properties of the macroscales and vice versa, depending on the concentration of each phase. We have shown, that Radial Distribution Function is not altered and velocity autocorrelation functions are gradually transformed, from Molecular Dynamics to Fluctuating Hydrodynamics description, when phase balance is changed. In this work we test our multiscale method for the liquid argon, BN2D and SPC/E water models. For the SPC/E water model we investigate microscale fluctuations which are computed using advanced mapping technique of the small scales to the large scales, which was developed by Voulgarakisand et. al.

The behaviour of flow that is inherently non-linear in nature is dependent on the values of certain parameters. At critical parameter values this can result in a change in stability, known as a bifurcation point. Identification of bifurcation points is necessary to understand these non-linearities and tracking the path of the bifurcation point as it varies with two parameters allows the behavioural response of the system under a range of different conditions to be determined. This thesis ultimately presents a new and innovative, computationally inexpensive method to directly locate and track bifurcation points. It achieves this through the systematic developments made to existing continuation and bifurcation tracking methods. Firstly, an existing numerical continuation method has been extended to allow continuation in shape parameters for compressible turbulent flows at high Reynolds numbers around an aerofoil for the first time. This continuation method used the time-independent form of the system but continuation was also performed on the time-discrete system using the Recursive Projection Method (RPM). This is the first time that RPM has been applied to this type of flow in order to perform continuation. The original time-independent continuation method used a coupled form of the mean-flow and turbulence model equations, however it is commonplace to use a decoupled formulation. The effect of using this decoupled formulation within a continuation method has not before been investigated, and so was implemented to discover whether the assumptions employed to enable a decoupled method are valid throughout the entire parameter range of interest. Results from the application of the newly developed method to flows about aero foils are presented. Results are presented for both the Spalart-Allmaras and Menter SST turbulence model for continuation in shape parameters for the first time. Whilst results are presented using the Spalart-Allmaras turbulence model and a number of different aerofoils that are known to exhibit the flow behaviour of interest for the decoupled continuation, RPM and bifurcation tracking methods. The results show that continuation methods can be used to identify the dependence of equilibrium solutions on geometrical parameters and that these equilibrium solutions provide a good approximation to the time-averaged unsteady values whilst the solution is stable. Furthermore, a decoupled method can be used during the stable region to enable a computationally more efficient continuation method. The Recursive Projection Method allows continuation to be performed with a large time-discrete RANS system. This extends the applicability of the underlying time integration scheme and improves the convergence rate. The results of the bifurcation tracking methods show that they represent a viable option as a means of understanding the dependency of the bifurcation point as two parameters are varied. The novel bifurcation tracking methods developed in this work offer a low-cost way of achieving this.

This thesis is an investigation of the tangled vortex lines that arise in the interference of complex waves in three dimensions; they are nodal lines of the intensity where both the real and imaginary components of the wavefield cancel out, and are singularities of the complex phase about which it sweeps out a quantised total change. We investigate the behaviour of this tangle as expressed in random degenerate eigenfunctions of the 3-torus, 3-sphere and quantum harmonic oscillator as models for wave chaos, in which many randomly weighted interfering waves produce a statistically characteristic vortex ensemble. The geometrical and topological nature of these vortex tangles is examined via large scale numerical simulations of random wave fields; local geometry is recovered with sufficient precision to confirm the connection to analytical random wave models, but we also recover the (high order) torsion that appears analytically inaccessible, and quantify the different length scales along which vortex lines decorrelate. From our simulations we recover statistics also on much larger scales, confirming a fractality of individual vortices consistent with random walks but also comparing and contrasting the scaling of the full vortex ensemble with other models of filamentary tangle. The nature of the tangling itself is also investigated, geometrically where possible but in particular topologically by testing directly whether vortex curves are knotted or linked with one another. We confirm that knots and links exist, but find their statistics greatly influenced by the nature of the random wave ensemble; vortices in the 3-torus are knotted far less than might be expected from their scales of geometrical de correlation, but in the 3-sphere and harmonic oscillator exhibit more common and more complex topology. We discuss how this result relates to the construction of each system, and finish with brief discussion of some selected topological observations.

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