Detection And Visualization Of Unsteady Structures In Complex Time-Dependent Flow Fields

Oceanic and atmospheric flows are typical time-dependent vector fields, which belong to nonlinear dynamical system. In most cases, a phenomenon in such type of flow fields is difficult to be described and analyzed in an analytic form. Therefore, developing scientific visualization methods for observing and studying unsteady structures in these complex flow fields has a significant and practical meaning.Bifurcations is a key problem in the study of nonlinear dynamical system. Investigation of bifurcations dynamical helps reveal unsteady processes in the system, and thus plays an important role in research on the dynamical mechanism for causing instability in vector field. In addition, some kinds of bifurcations, such as Hopf bifurcations and periodical bifurcations, can lead to chaos. Therefore, we make bifurcations a research emphasis, and explore methods for detecting and visualizing unsteady structures in complex flow fields via extraction of bifurcations.In terms of data visualization, detecting and visualizing bifurcations mean topology tracking in time-dependent flows, which is a study focus in scientific visualization. To implement topology tracking methods, vector filed topology analysis which can extract and classify topological features is needed. Because limit cycle is an important topological feature, and more importantly, its occurrence is often accompanied by creation of bifurcations, detecting it is a indispensable condition of detecting bifurcation, studying chaos and unsteady process in complex time-dependent flow fields.Vector field topology simplification is another research topic brought into focus in recent years. Mostly, for those complex flow fields with numerous feature structures, such as turbulence flows, only a few important feature structures should be investigated. Thus, before we start topology tracking to detect unsteady structures, it is necessary to employ topological simplification which can eliminate the topological structures simplification related to minor features but preserve the main topological skeleton.This paper address the two basic problems mentioned above, limit cycle extraction and vector field topology simplification. First, in order to overcome the drawbacks of the algorithm presented by Wischgoll and Scheuermann in 2001 for detecting and visualizing the limit cycle in planar vector fields, an innovative algorithm based on clustering critical points is provided and obtains much better results. Second, this paper presents an implicit topological simplification method based on grid-combining and closely related to features. In this algorithm, a concept of Grid Granularity is introduced as well. Via combining those grids within the less important regions defining the minor features of a flow field, the algorithm performs a topological simplification with high data compression and feature sensitivity.Finally, a convincing result provided by a method, Exactly Locating and Visualizing Hopf Bifurcations in Time-dependent Planar Vector Fields, which is developed by the author's research group for a pilot study of unsteady processes in complex time-dependent flow fields, testifies the validity of the approach to detecting limit cycles in this paper.