eprintid: 32028
rev_number: 23
eprint_status: archive
userid: 6853
dir: disk0/00/03/20/28
datestamp: 2022-10-06 07:51:59
lastmod: 2022-10-11 12:21:00
status_changed: 2022-10-06 07:51:59
type: doctoralThesis
metadata_visibility: show
creators_name: Hofmann, Lutz
title: Local Features and Topology in Higher-Dimensional and Time-Dependent Vector Fields
divisions: i-110300
divisions: i-708000
adv_faculty: af-11
abstract: Vector fields can be used to describe a wide variety of physical phenomena. Their structure can prominently be understood by appropriate feature extraction and visualization. While existing methods mainly treat 2D and 3D vector fields, this thesis focuses on higher-dimensional vector fields of dimension four and beyond, as well as the special case of treating time as additional dimension.
In the first part of this thesis, we introduce the dependent vectors operator, which enables to define and extract a number of local features in scalar and vector fields of arbitrary dimension. It generalizes the 3D parallel vectors operator by replacing the parallelism condition with linear dependency of a set of (possibly derived) vector fields. The resulting manifolds, whose dimension depends on the number of vector fields considered, are obtained by a generic extraction algorithm. We demonstrate their utility by introducing generalized vortex core manifolds, bifurcation manifolds, as well as ridge manifolds, which we extract from dynamical systems as well as higher-dimensional fields derived from 2D and 3D numerical datasets.
Vector field topology (VFT), a global feature describing qualitative transport structure, is treated in the second part of this thesis. First, we generalize the well-established 2D and 3D VFT to four-dimensional steady vector fields. We classify the different types of 4D critical points, propose a set of glyphs made of 4D geometry to represent each type, extract and classify 4D periodic orbits, and extract their invariant manifolds. For effective exploration of the resulting 4D topological skeleton, we introduce a 4D camera that exploits degeneracies in the 3D projections to reduce clutter and self-intersection. We exemplify the utility of our approach using analytic 4D dynamical systems that showcase properties of 4D VFT.
Finally, we make several contributions to 2D and 3D time-dependent vector field topology. In this concept, the role of invariant manifolds from steady VFT is taken on by Lagrangian coherent structures (LCS), which represent the main organizing streak manifolds over finite time intervals. Instead of the commonly employed expensive computation of the LCS by ridge extraction from the finite-time Lyapunov exponent (FTLE) field, our approach is based on the refinement of locally extracted candidate manifolds. For the 2D case, we combine and extend existing concepts for obtaining initial candidates and refinement toward hyperbolic trajectories, and show that our approach is more accurate and efficient than existing methods. In the 3D case, we present a novel method for obtaining hyperbolic path surfaces from candidate surfaces, which we extract locally from the four-dimensional space-time domain. We evaluate our approach on analytic flows, as well as data from computational fluid dynamics, using the FTLE as a ground truth superset.
date: 2022
id_scheme: DOI
id_number: 10.11588/heidok.00032028
ppn_swb: 1818592487
own_urn: urn:nbn:de:bsz:16-heidok-320284
date_accepted: 2022-09-30
advisor: HASH(0x55fc36ad7008)
language: eng
bibsort: HOFMANNLUTLOCALFEATU20211005
full_text_status: public
place_of_pub: Heidelberg
citation:   Hofmann, Lutz  (2022) Local Features and Topology in Higher-Dimensional and Time-Dependent Vector Fields.  [Dissertation]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/32028/3/Dissertation_Hofmann_2021.pdf
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/32028/2/cover.pdf