Core Lines in 3D Second-Order Tensor Fields

Abstract

Vortices are important features in vector fields that show a swirling behavior around a common core. The concept of a vortex core line describes the center of this swirling behavior. In this work, we examine the extension of this concept to 3D second-order tensor fields. Here, a behavior similar to vortices in vector fields can be observed for trajectories of the eigenvectors. Vortex core lines in vector fields were defined by Sujudi and Haimes to be the locations where stream lines are parallel to an eigenvector of the Jacobian. We show that a similar criterion applied to the eigenvector trajectories of a tensor field yields structurally stable lines that we call tensor core lines. We provide a formal definition of these structures and examine their mathematical properties. We also present a numerical algorithm for extracting tensor core lines in piecewise linear tensor fields. We find all intersections of tensor core lines with the faces of a dataset using a simple and robust root finding algorithm. Applying this algorithm to tensor fields obtained from structural mechanics simulations shows that it is able to effectively detect and visualize regions of rotational or hyperbolic behavior of eigenvector trajectories.