Asymmetric Tensor Analysis for Flow Visualization

Abstract

Abstract-The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectory-based vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field that can represent either a 2D compressible flow or the projection of a 3D compressible or incompressible flow onto a 2D manifold. To illustrate the structures in asymmetric tensor fields, we introduce the notions of eigenvalue manifold and eigenvector manifold. These concepts afford a number of theoretical results that clarify the connections between symmetric and antisymmetric components in tensor fields. In addition, these manifolds naturally lead to partitions of tensor fields, which we use to design effective visualization strategies. Moreover, we extend eigenvectors continuously into the complex domains which we refer to as pseudoeigenvectors. We make use of evenly spaced tensor lines following pseudoeigenvectors to illustrate the local linearization of tensors everywhere inside complex domains simultaneously. Both eigenvalue manifold and eigenvector manifold are supported by a tensor reparameterization with physical meaning. This allows us to relate our tensor analysis to physical quantities such as rotation, angular deformation, and dilation, which provide a physical interpretation of our tensor-driven vector field analysis in the context of fluid mechanics. To demonstrate the utility of our approach, we have applied our visualization techniques and interpretation to the study of the Sullivan Vortex as well as computational fluid dynamics simulation data.

Airflow measurements at a wavy air–water interface using PIV and LIF

“…In two dimensions,  contains two independent components and can, therefore, be represented by a vector . The "strength of anisotropic stretching", or "shearing strength", defined by Zhang et al (2009) as the magnitude of can then be expressed as:…”

2D Asymmetric Tensor Field Topology

“…There has been relatively little work in asymmetric (second-order) tensor fields [24,23]. Dodd [5] develops a method to represent the geometry of a symmetric tensor field in terms of its geodesics.…”

“…They also introduce the concept of dual-eigenvectors which allow directional information contained in the tensor field to be visualized even when real-valued eigenvectors do not exist. Zhang et al [23] introduce the notions of eigenvalue manifold and eigenvector manifold, which are supported by tensor re-parameterizations with physical meaning and lead to effective visualization techniques. To the best of our knowledge, the topology of asymmetric tensor fields has not been defined nor systematically studied.…”

“…We define the topology of 2D asymmetric tensor fields, whose analysis can benefit a wide range of applications in solid and fluid mechanics [23,24]. Topology-based analysis has achieved much success in the processing and visualization of scalar fields [1,6], vector fields [2,8,14,13], and symmetric second-order tensor fields [3, Zhongzang Lin Oregon State University, Corvallis, Oregon, e-mail: lin@eecs.oregonstate.edu…”

Image-Space Tensor Field Visualization Using a LIC-like Method

“…As these methods are based on vector fields, their application to a symmetric tensor field's major eigenvector is possible, with the limitation that the eigenvector field does not provide an orientation. For asymmetric tensor-fields, other approaches exist [30]. Besides the non-trivial application of image-space LIC to tensor data, image-space LIC has certain other drawbacks.…”