Browsing by Browse by FOR 2008 "010301 Numerical Analysis"

Contour dynamics and contour-advective methods are commonly used numerical techniques for simulating inviscid fluid motions. In these methods the vorticity or potential vorticity of a flow is represented by a series of contours which are advected according to the prevailing velocity field. In some circumstances the contours may cross, eroding the accuracy of the numerical solution and violating the equations of motion. This paper describes an automated method for explicitly revealing such crossings, first considering the case of determining if two contours cross and then later the more general case of determining if and where an arbitrary number of contours cross.

Performance estimates of a parallel computer during sparse matrix factorisation aid in the identification of overheads and the tuning of software. This paper proposes a technique which allows the computer parameters of computation speed, communication speed, latency and parallel efficiency to be estimated. The technique is based upon the use of mathematical models derived from a model problem in conjunction with experimental results. By combining the mathematical models with the experimental results, sets of simultaneous equations can be derived which can be solved for the above computer parameters. The technique is explained in the context of sparse QR factorisation.

This is the first of two papers devoted to the analysis of contour crossing errors that occur in contour-advective simulations of fluid motion. Here an algorithm is presented for quantifying the error due to contour crossings. The first step is to determine the relative proximity of all possible pairs of contours. A digital representation of each contour is produced to aid in the corresponding calculation. Simple analysis of functions is then used to find any crossings between contours deemed close to each other by the digital representation method. Next, the area in error of a pair of crossing contours is calculated by identifying the polygon or polygons that approximately bound the erroneous region. Finally, some preliminary results of analysis of contour crossings that occur in contour-advective semilagrangian (CASL) simulations of single layer quasigeostrophic turbulence are presented. It is shown that the error due to contour crossings is small in the simulations considered here.

This is the second of two papers devoted to the analysis of contour crossing errors that occur in contour-advective simulations of fluid motion, where either vorticity or potential vorticity is represented by contours. We begin with a detailed discussion on some of the potential mechanisms for contour crossing. Past work has suggested that the formation of contour crossings is due to inadequate spatial resolution of contours [1]. The implementation of two schemes for preventing contour crossings within the framework of the Contour-Advective Semi-Lagrangian (CASL) algorithm is detailed here. We then present an analysis of contour crossing errors in simulations of quasigeostrophic turbulence on the f-plane and the quasigeostrophic motion of an initially circular vortex patch on the b-plane using the algorithm detailed in Part 1. We find that in general individual crossings occur at scales smaller than the inversion grid scale on which velocity is calculated, but at scales larger than that of the surgical scale that defines the smallest resolved features (vorticity) of a flow. If the resolution of a quasigeostrophic turbulence simulation on the f-plane is increased by doubling the number of grid points in each coordinate direction used in the calculation of the velocity field, then the total area in error due to contour crossings remains unchanged; a smaller number of crossings introducing larger scale area errors is replaced by a greater number of smaller local errors. Uniformly increasing the density of nodes along all contours and placement of nodes at points of close approach on contours are both effective methods for limiting contour crossings.

Groups of animals coordinate remarkable, coherent, movement patterns during periods of collective motion. Such movement patterns include the toroidal mills seen in fish shoals, highly aligned parallel motion like that of flocks of migrating birds, and the swarming of insects. Since the 1970’s a wide range of collective motion models have been studied that prescribe rules of interaction between individuals, and that are capable of generating emergent patterns that are visually similar to those seen in real animal groups. This does not necessarily mean that real animals apply exactly the same interactions as those prescribed in models. In more recent work, researchers have sought to infer the rules of interaction of real animals directly from tracking data, by using a number of techniques, including force mapping methods. In one of the simplest formulations, the force mapping methods determine the mean changes in the components of the velocity of an individual over time as a function of the relative coordinates of group mates. The force mapping methods can also be modified to estimate other closely related quantities including the mean relative direction of motion of group mates as a function of their relative coordinates. Since these methods for extracting interaction rules and related quantities from trajectory data are relatively new, the accuracy of these methods has had limited inspection. In my thesis, I examine the ability of the force mapping method to reveal prescribed rules of interaction from data generated by three individual based models for collective motion, namely the zonal model developed by Couzin et al. [20], the ODE model developed by D’Orsogna et al. [22] and an alignment only model developed by Vicsek et al. [87], as well as variants of the Couzin et al. [20] and Vicsek et al. [87] models where some interactions apply over topological scales. Topological scales refer to the distance-based neighbour rank of other members of a group, in contrast to metric scales, which are based on linear distances.

My analysis suggests that force maps constructed in a standard form (where mean changes in components of individuals’ velocity are mapped as a function of the relative positions of neighbours) capture the qualitative, and sometimes quantitative, features of interaction rules including repulsion and attraction effects, and the presence of blind zones. However, the features of standard force maps may also be affected by emergent group level patterns of movement, and the sizes of the regions over which repulsion and attraction effects are apparent can be distorted as group size varies, dependent on how individuals respond when interacting with multiple neighbours.

I also examined the effectiveness of force maps tailored to examine orientation/alignment interactions, and interactions that apply over topological scales, via appropriate choices of dependent variables including: differences in directions of motion between a focal individual and its neighbour (angular differences), relative positions of, or distances to, neighbours, and the distance-based rank of neighbours. Force maps that illustrated mean changes in direction as a function of metric and/or topological distances to, and angular differences with, neighbours can provide relatively clear and consistent information about the presence of a direct underlying tendency for individuals to align with neighbours. However, these force maps also tend to overestimate the domain over which orientation/alignment interactions apply, as compared to explicit interactions between pairs. Force maps of this type that take into account the relative position of neighbours, rather than just their distance from an individual, can also reveal orientation behaviour, but often less clearly. In addition, such more detailed force maps seem to be strongly affected by emergent patterns of motion.

In related analysis, I examined the relationship between regions of high alignment in local alignment plots (plots that illustrate statistics of the relative directions of neighbours as a function of their relative coordinates), and the prescribed regions over which orientation interaction rules applied. In the case that simulated individuals only applied orientationbased rules for adjusting their velocity, regions of high alignment were correlated in size with prescribed orientation zones. However, this relationship did not hold when data from a more complex model was analysed, and a region of high alignment could appear in a local alignment plot even when there was no underlying behavioural rule for orientation. Ultimately, this suggests that local alignment plots are reflective of emergent behaviour, but not necessarily of the underlying interactions driving such behaviour.

Following on from my investigation of force maps, I then applied some of these maps to examine potential orientation interactions in small shoals of fish of three different species (Xray tetras, eastern mosquitofish and crimson spotted rainbowfish) from existing experimental data sets. The analysis suggested that all three species have an explicit underlying behavioural rule for aligning their movements with neighbours.

The models used throughout my thesis exhibit multiple forms of parameter and initial condition dependent emergent patterns of collective motion. In the final portion of my thesis, I examine the emergent states of one of these models - the aforementioned modification of the Couzin et al. [20] model in three spatial dimensions. The model was modified such that interactions relating to aligning directions of motion (orientation interactions) and moving towards more distant group mates (attraction interactions) operated over topological (distance based neighbour rank) scales, rather than metric (linear distance) scales. Collision avoidance (repulsion) interactions operated over a metric scale, close to each individual, as in the original model. I examined the emergent group level patterns of movement generated by the model as the numbers of neighbours that contributed to orientation and attraction based adjustments to velocity were varied. Like the metric form of the model, simulated groups could fragment, move in a swarm-like manner, move together in parallel, and mill. However, milling was extremely rare, with emergent states classified as milling not necessarily exhibiting the same clearly ordered rotational structure as typical examples from the metric form of the model. The model also generated other cohesive group movements that were not easily classified in terms of swarming, milling, or aligned motion, and in some cases these groups exhibited directed motion without strong alignment of individuals. Groups that did not fragment tended to stay relatively compact in terms of both mean and nearest neighbour distances. Even if a group did fragment, individuals remained relatively close to near neighbours, avoiding complete isolation.

This broad study attempts to provide greater insight into the accuracy of force maps and some of the factors that may contribute to their inaccuracy, and highlight some of the richness behind fundamental methods for analysing and modelling collective motion.