Accurate and Efficient Integral Surfaces in Time-Dependent Vector Fields
Christoph Garth, Hari Krishnan, Xavier Tricoche, T. Bobach, and Ken Joy
Integral surfaces, as a natural generalization of integral lines, are a basic yet powerful tool for insightful vector field visualization. Representing a continuum of integral lines, they constitute a surface and allow for the application of surface shading techniques. In comparison to streamlines or other line-based techniques, they support depth perception and greatly facilitate the visual understanding of complex three-dimensional structures.
Integral surfaces appear quite naturally in many problems that involve vector fields; the most prominent example is flow visualization that centers on the visual analysis of datasets generated by simulation (Computational Fluid Dynamics) or measurement. In the context of flows, the groundbreaking drawing work of Dallmann has shown that many types of flow patterns may be well understood in terms of so-called flow sheets. These sheets represent precisely chosen integral surfaces that emanate from specific lines on the surface of objects embedded in a flow, and often take on the role of flow separators that divide regions of differing behavior. In the broader context of dynamical systems, integral surfaces appear as separatrices of three-dimensional vector fields, where they represent the two-dimensional stable or unstable manifolds associated with critical points. However, even if not coupled to such specific interpretations, integral surfaces have been shown to possess great illustrative power. In the case of flows, they can accurately depict folding, shearing and twisting and impart an intuitive understanding of the flow geometry.
In recent years, several approaches have been presented for the computation and graphical representation of integral surfaces in CFD datasets. However, most this work has focused on the computation of stream surfaces, i.e. integral surfaces in steady vector fields, as opposed to path surfaces, which denote the time-dependent case. Much of this prior work has concentrated on the generation of a viable graphical representation using advancing front methods, based on adaptive refinement heuristics that are aimed at producing good surface meshes.
However, the accuracy of the resulting surface, which is a crucial property for some application scenarios, has not been examined in detail. This is in part attributed to the generally complex form of the resulting algorithms that make such discussion difficult.
In this work, we present a novel algorithm for the computation of integral surfaces in a general context. Our method is based on the adaptively-refined advancing front paradigm, and is applicable to both stationary and time-varying vector fields. Treatment of the latter is achieved in a streaming fashion, thus allowing our method to work even on extremely large datasets with thousands of time steps.