Topological analysis and visualization of asymmetric tensor fields

摘要

Visualizing asymmetric tensors is an important task in understanding fluid dynamics. In this paper, we describe topological analysis and visualization techniques for asymmetric tensor fields on surfaces based on analyzing the impact of the symmetric and antisymmetric components of the tensor field on its eigenvalues and eigenvectors. At the core of our analysis is a reparameterization of the space of 2 x 2 tensors, which allows us to understand the topology of tensor fields by studying the manifolds of eigenvalues and eigenvectors. We present a partition of the eigenvalue manifold using a Voronoi diagram, which allows to segment a tensor field based on its relatively strengths in isotropic scaling, rotation, and anisotropic stretching. Our analysis of eigenvectors is based on the observation that the dual-eigenvectors of a tensor depend solely on the symmetric constituent of the tensor. The anti-symmetric component acts on the eigenvectors by rotating them either clockwise or counterclockwise towards the nearest dual-eigenvector. The orientation and the amount of the rotation are derived from the ratio between the symmetric and anti-symmetric components. We observe that symmetric tensors form the boundary between regions of clockwise flows and regions of counterclockwise flows. Crossing such a boundary results in discontinuities in the major and minor dual eigenvectors. Thus we define symmetric tensors as part of the tensor field topology in addition to degenerate tensors. These observations inspire us to illustrate the topology of the symmetric component and anti-symmetric component simultaneously. We demonstrate the utility of our techniques on an important application from computational fluid dynamics, namely, engine simulation.