2D second-order asymmetric tensor field analysis and visualization

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. I 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 two-dimensional manifold.

To illustrate the structures in asymmetric tensor fields, I 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 is used to design effective visualization strategies. Moreover, I extend eigenvectors continuously into the complex domains which are referred to as pseudo-eigenvectors. Evenly-spaced tensor lines following pseudo-eigenvectors 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 relates the tensor analysis to physical quantities such as rotation, angular deformation, and dilation, which provide physical interpretation of the tensor-driven vector field analysis in the context of fluid mechanics.

To demonstrate the utility of the approach, I have applied the visualization techniques and interpretation to the study of the Sullivan Vortex as well as computational fluid dynamics simulation data.