Time-Dependent 2-D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures

This paper presents an approach to a time-dependent variant of the concept of vector field topology for 2-D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle-type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite-time Lyapunov exponent field. The proposed approach is evaluated on both 2-D time-dependent synthetic and vector fields from computational fluid dynamics

Do Finite-Size Lyapunov Exponents Detect Coherent Structures?

Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.Comment: 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left) that appears in the version published in Chao

Surface networks

© Copyright CASA, UCL. The desire to understand and exploit the structure of continuous surfaces is common to researchers in a range of disciplines. Few examples of the varied surfaces forming an integral part of modern subjects include terrain, population density, surface atmospheric pressure, physico-chemical surfaces, computer graphics, and metrological surfaces. The focus of the work here is a group of data structures called Surface Networks, which abstract 2-dimensional surfaces by storing only the most important (also called fundamental, critical or surface-specific) points and lines in the surfaces. Surface networks are intelligent and “natural ” data structures because they store a surface as a framework of “surface ” elements unlike the DEM or TIN data structures. This report presents an overview of the previous works and the ideas being developed by the authors of this report. The research on surface networks has fou

Local stable and unstable manifolds and their control in nonautonomous finite-time flows

It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.Comment: Under consideration for publication in the Journal of Nonlinear Science

A Local Iterative Approach for the Extraction of 2D Manifolds from Strongly Curved and Folded Thin-Layer Structures

Ridge surfaces represent important features for the analysis of 3-dimensional (3D) datasets in diverse applications and are often derived from varying underlying data including flow fields, geological fault data, and point data, but they can also be present in the original scalar images acquired using a plethora of imaging techniques. Our work is motivated by the analysis of image data acquired using micro-computed tomography (Micro-CT) of ancient, rolled and folded thin-layer structures such as papyrus, parchment, and paper as well as silver and lead sheets. From these documents we know that they are 2-dimensional (2D) in nature. Hence, we are particularly interested in reconstructing 2D manifolds that approximate the document's structure. The image data from which we want to reconstruct the 2D manifolds are often very noisy and represent folded, densely-layered structures with many artifacts, such as ruptures or layer splitting and merging. Previous ridge-surface extraction methods fail to extract the desired 2D manifold for such challenging data. We have therefore developed a novel method to extract 2D manifolds. The proposed method uses a local fast marching scheme in combination with a separation of the region covered by fast marching into two sub-regions. The 2D manifold of interest is then extracted as the surface separating the two sub-regions. The local scheme can be applied for both automatic propagation as well as interactive analysis. We demonstrate the applicability and robustness of our method on both artificial data as well as real-world data including folded silver and papyrus sheets.Comment: 16 pages, 21 figures, to be published in IEEE Transactions on Visualization and Computer Graphic

Attracting and repelling Lagrangian coherent structures from a single computation

Hyperbolic Lagrangian Coherent Structures (LCSs) are locally most repelling or most attracting material surfaces in a finite-time dynamical system. To identify both types of hyperbolic LCSs at the same time instance, the standard practice has been to compute repelling LCSs from future data and attracting LCSs from past data. This approach tacitly assumes that coherent structures in the flow are fundamentally recurrent, and hence gives inconsistent results for temporally aperiodic systems. Here we resolve this inconsistency by showing how both repelling and attracting LCSs are computable at the same time instance from a single forward or a single backward run. These LCSs are obtained as surfaces normal to the weakest and strongest eigenvectors of the Cauchy-Green strain tensor.Comment: Under consideration for publication in Chaos/AI