Measuring Visual Complexity of Cluster-Based Visualizations

Handling visual complexity is a challenging problem in visualization owing to the subjectiveness of its definition and the difficulty in devising generalizable quantitative metrics. In this paper we address this challenge by measuring the visual complexity of two common forms of cluster-based visualizations: scatter plots and parallel coordinatess. We conceptualize visual complexity as a form of visual uncertainty, which is a measure of the degree of difficulty for humans to interpret a visual representation correctly. We propose an algorithm for estimating visual complexity for the aforementioned visualizations using Allen's interval algebra. We first establish a set of primitive 2-cluster cases in scatter plots and another set for parallel coordinatess based on symmetric isomorphism. We confirm that both are the minimal sets and verify the correctness of their members computationally. We score the uncertainty of each primitive case based on its topological properties, including the existence of overlapping regions, splitting regions and meeting points or edges. We compare a few optional scoring schemes against a set of subjective scores by humans, and identify the one that is the most consistent with the subjective scores. Finally, we extend the 2-cluster measure to k-cluster measure as a general purpose estimator of visual complexity for these two forms of cluster-based visualization

Calculation of Scalar Isosurface Area and Applications

The problem of calculating iso-surface statistics in turbulent flows is interesting for a number of reasons, some of them being combustion modeling, entrainment through turbulent/non-turbulent interfaces, calculating mass flux through iso-scalar surfaces and mapping of scalar fields. A fundamental effect of fuid turbulence is to wrinkle scalar iso-surfaces. A review of the literature shows that iso-surface calculations have primarily been done with geometric methods, which have challenges when used to calculate surfaces that have high complexity, such as in turbulent flows. In this thesis, we propose an alternative integral method and test it against analytical solutions. We present a parallelized algorithm and code to enable in-simulation calculation of isosurface area. We then use this code to calculate area statistics for data obtained from Direct Numerical Simulations and make predictions about the variation of the iso-scalar surface area with Taylor Peclet numbers between 9.8 and 4429 and Taylor Reynolds numbers between 98 and 633

Alignment and Structure in MHD Dynamos

Magnetic fields are ubiquitous within astrophysical settings. There is strong evidence to suggest that some of these magnetic fields, for example the Sun’s, are maintained through a dynamo process whereby energy is exchanged between a flow and a magnetic field. Magnetohydrodynamics (MHD) is the branch of mathematics where this interaction is studied. The initial amplification of a weak seed field is often modelled using the kinematic dynamo approximation where the flow is not influenced by the magnetic field. This approximation to the early behaviour of a nonlinear dynamo problem, where the magnetic field grows exponentially during a kinematic phase and then saturates into a nonlinear regime, has the benefit of being far less computationally intensive.In this thesis, I examine three different topics within MHD dynamos. First, I examine how measuring alignment of the flow and magnetic field during a kinematic dynamo can reveal changes to the magnetic field structure. This I show to be useful both within individual simulations and when comparing magnetic fields within parameter studies. Secondly, I examine nonlinear dynamos where the flow and magnetic field are strongly aligned and have almost identical energies. I reproduce, and give an explanation for, a previously unexplained behaviour. Furthermore, I show that aligned flow and magnetic fields can exist for increasingly complex forcings and as such the aligned state is remark-ably robust. Finally, I consider a number of different nonlinear dynamos for a family of forcings with different magnetic field structures during their kinematic phase. Using Minkowski Functions to quantify the structures, I show that, where the magnetic field becomes sufficiently strong, the magnetic fields become (or remain) ribbon-like in the nonlinear regime. As such, the influence that stagnation points in the flow have on the magnetic field structure is less than in the kinematic dynamo equivalent