We prove that whenever T is a power-bounded o-minimal theory, t-stratifications exist for definable maps and sets in T-convex fields. To this effect, a thorough analysis of definability in T-convex fields is carried out. One of the conditions required for the result above is the Jacobian property, whose proof in this work is a long and technical argument based on an earlier proof of this property for valued fields with analytic structure. An example is given to illustrate that t-stratifications do not exist in general when T is not power-bounded. We also show that if T is power-bounded, the theory of all T-convex fields is b-minimal with centres. We also address several applications of tstratifications. For this we exclusively work with a power-bounded T. The first application establishes that a t-stratification of a definable set X in a T-convex field induces t stratifications on the tangent cones of X. This is a contribution to local geometry and singularity theory. Regarding R as a model of T, the remaining applications are derived by considering the stratifications induced on R by t-stratifications in non-standard models. We prove that each such induced stratification is a C1-Whitney stratification; this in turn leads to a new proof of the existence of Whitney stratifications for definable sets in R. We also deal with interactions between tangent cones of definable sets in R and stratifications.

In this thesis we study ergodic theoretical properties of high-dimensional systems coupled on graphs. The local dynamics at each node is hyperbolic and coupled with other nodes according to the edges of the graph. We focus our attention on the case of graphs with heterogeneous degrees meaning that most of the nodes make a small number of interactions, while a few hub nodes have very high degree. For such high-dimensional systems there is a regime of the interaction strength for which the coupling is small for poorly connected systems, and large for the hub nodes. In particular, global hyperbolicity might be lost. We show that, under certain hypotheses, the dynamics of hub nodes can be very well approximated by a low-dimensional system for exponentially long time in the size of the network and that the system exhibit hyperbolic behaviour in this time window. Even if this describes only a long transient, we argue that this is the behaviour that one expects to observe in experiments. Such a description allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections). The HCM we study provide a new paradigm to explain why and how the dynamics of a network dynamical system can change across layers.

Recent studies of transition to turbulence in linearly stable shear flows have been dominated by finding, and characterising, so-called exact coherent structures which exist as either equilibrium or travelling wave solutions of the governing Navier-Stokes equations. These structures have been shown to act as either edge states between laminar and turbulent flow or become associated with attractors for turbulent flows, making the quest to understand these structures equally as mathematically interesting as it is industrially relevant. In this thesis, vortex-wave interaction theory, an asymptotic approach based on the assumption of large Reynolds number, is used as a means of finding such structures in plane Couette flow. A new, iterative, numerical method will be developed to solve the governing interaction equations that will allow us to circumvent many of the difficulties in calculating such structures (in particular at high Reynolds numbers) and perform analyses into the state-space of solutions, as well as explore their stability properties. A number of previously found exact coherent structures will be shown to be vortex-wave interaction states, including the so-called upper branch sinuous mode and the class of solutions called mirror-symmetric modes. A number of receptivity type problems will also be considered. Namely, the use of periodic blowing/suction on the channel walls and the introduction of variations in the wall shape. The effect of these two techniques on the skin friction drag and the wave amplitude will be discussed in the context of whether they are beneficial in terms of laminar flow control. Further, it will be shown that the use of periodic blowing/suction leads to a new class of synchronous mirror-symmetric modes and variations in the wall shape lead to a new class of inhomogeneous vortex-wave interaction states.

The acoustic performance of dissipative silencers, including the effects of both a mean flow in the airway and an induced internal steady flow in the absorbent, are analysed. Finite element models, based upon the modified Galerkin method, have been derived for the determination of the noise attenuation of silencers, both by forced response and eigenvalue analysis. The corresponding computer programs, incorporating subroutines from the NAG Finite Element library, have been developed.

Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this manuscript we present the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. An analysis of the boundary value problem with non standard condition is provided as well as the numerical evidence showing the advantages of the new method. Furthermore, we present and analyse a stabilisation method for the presented discretisation that allows the use of the same polynomial degrees for velocity and pressure discrete spaces. The original definition of the domain decomposition preconditioners is one-level, this is, the preconditioner is built only using the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why we have also introduced, and tested numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider two finite element discretisations, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations for the nearly incompressible elasticity problems and Stokes equations.

Suspension mechanics—the flow of a fluid with small fragments of solid material suspended in it—is an area of wide applicability in both industry and nature. Examples include the transport of silt in rivers, the manufacture of toothpaste, and inkjet printing where pigments remain solid within the ink. One widespread method to simulate these flows is Stokesian Dynamics, a truncated multipole expansion of the Stokes equations. It is computationally efficient while making a reasonable approximation to the hydrodynamic interactions between particles; however, all particles are identical spheres and the background matrix must be Newtonian. This project extends Stokesian Dynamics to include differently-sized spheres. This allows us to study a variety of previously inaccessible suspension problems. In many suspensions, e.g. toothpaste, the suspending fluid itself is non-Newtonian and exhibits viscoelastic properties. We have extended Stokesian Dynamics to incorporate a simple model of viscoelasticity by using the small spheres as 'beads' in bead--spring dumbbells. Different spring laws are then tested in shear, and their rheological behaviour is compared to continuum constitutive models. Next, we replicate experiments in which a large sphere is dropped through a suspension of neutrally buoyant smaller spheres undergoing oscillatory shear flow. We qualitatively replicate the principal experimental observation—that at the moment of shear reversal, the suspension microstructure hinders the falling; while at the instant of fastest shear, it enhances the falling. We propose a physical mechanism explaining the observations. Finally, we extend Stokesian Dynamics to properly implement interparticle frictional contact. Contact forces are a critical component of shear thickening in suspensions such as cornflour, yet are usually implemented in an ad hoc way, resulting in inaccurate predictions or high computational cost. The new method allows us to investigate contact models quickly and efficiently, and suggests an important factor in models of strongly shear-thickening fluids.

An overset meshing approach is an effective method of simulating fluid flow involving multiple moving bodies. It consists of minor meshes representing solid objects, which overlap a Cartesian background grid, allowing bodies to move arbitrarily whilst retaining communication between grids. However, a hole beneath each overlapping mesh must be cut from the background grid, leaving a small overlap. Current hole-cutting methods tend to be complex with some requiring extensive user knowledge and input. Since the hole must be re-cut regularly for moving body problems, it can become very time-consuming. An original approach for performing a hole-cut has been implemented by employing the Cartesian cut-cell method. This method would ordinarily be used to cut the boundary of a solid object from a single Cartesian grid, as an alternative to the overset grids approach. Thus, the treatment of cut-cells has been modified for its new purpose of hole-cutting. The cut-cell method is already a well-established technique for cutting a Cartesian grid, and is fully automated. It has not been used for hole-cutting previously within the literature and offers a very different hole-cut to existing techniques; It cuts though cells rather than around them, simplifying the cutting process and providing a smooth cut. This approach has been applied to an incompressible Navier-Stokes solver for viscous single fluid flow. Unstructured, triangular minor meshes are used due to their ability to represent complex geometries accurately. An explicit time integration method is used on these minor meshes, but an implicit integration method is implemented on the Cartesian background mesh. This new hybrid of integration methods was found to significantly reduce the CPU time in comparison to using a fully explicit method. The solver has been validated using benchmark tests, including a lid driven cavity and flow past a stationary/oscillating cylinder. The results obtained were found to be in good quantitative agreement with published numerical results. The solver was developed for 2-phase flow problems. However, during the initial validation test, convergence issues were encountered, which meant a sufficient solution could not be obtained.

In this thesis we consider a class of conservation based moving mesh methods applied to hyperbolic conservation laws. We mainly concentrate on the one dimensional case with the examples of the linear advection equation, inviscid Burgers’ equation and the Buckley-Leverett equation. The moving mesh methods are generated using the conservation of mass as a method for determining the mesh velocity at the computational nodes. We use the notion of the reference space as a mathematical tool to analyse the moving mesh methods allowing us to show the accuracy, stability conditions and convergence. In addition we use the reference space as a technique for constructing new moving mesh methods which share the accuracy and stability properties of the fixed mesh scheme they are derived from. At the end of the thesis we use the knowledge gained from the scalar conservation laws to construct moving mesh methods for the isothermal equations.

No description available.

This thesis uses an example generation task to explore undergraduate students' understanding of basic sequence properties in Real Analysis. First, a review of the literature looks at three areas of research: the transition to studying mathematics at the tertiary level, examples and the process of example generation, and the learning of Real Analysis. It notes a lack of research on how students interact with simpler de nitions in Analysis, and suggests that an example generation task is an ideal research tool for this purpose. Then, two pilot studies are reported. The rst gave 101 students an example generation task during a lecture. In this task, students were asked to generate examples of sequences that satis ed certain combinations of properties. In the second pilot study a similar task was given to six students in an interview setting with a `think-aloud' protocol. These pilot studies found that many students gave sequences that did not satisfy the requested properties, whilst other students gave examples that were not sequences. The thesis then reports on a main study in which the example generation task was completed by 15 students during an interview, and 147 students during classes. The interview data is analysed phenomenographically, with results presented along four dimensions of variation, where each dimension describes di erent ways of experiencing an aspect of sequence example generation: Using De nitions, Representation of Sequences, Sequence Construction Strategies, and Justi cations. The larger-scale class data is then analysed by Rasch Analysis to objectively rank the questions in order of their di culty, and to show that the interview-based responses re ect those in the wider cohort. By asking students to generate their own examples of sequences, this thesis has furthered what is known about student understanding in two areas. The rst area is how students understand content related to sequences in Analysis. The thesis considers students' understanding of how sequences can be represented, how sequence property de nitions can be combined and how de nitions a ect sequences in di erent ways. The second area is how students interact with example generation tasks, the approaches that are e ective when students are trying to generate examples, and the ways students justify or check their answers.