Robust Extraction and Analysis Towards Complete 3D Tensor Field Topology

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the topological analysis of 3D symmetric tensor fields focus on the local behaviors of tensor fields at degenerate points, which usually form curves. In this paper, we make a number of observations about tensor field topology that are more global in nature. For instance, a degenerate curve can be a knot, and two degenerate curves may be linked. We explore the conditions under which this might occur. In addition, we introduce the notion of eigenvalue manifold which provides a more global description of tensor field topology. As part of our investigation, we include additional tensors into tensor field topology, such as the boundary between the linear and planar types of tensors, as well as traceless tensors. Robust extraction of degenerate curves is a challenging task, despite recent progress. We convert the problem of finding degenerate curves into solving a system of algebraic equations. This approach allows us to borrow techniques from the computer-aided design (CAD) community as well as the algebraic geometry community. The end result is a two-step pipeline that first locates mesh cells in which degenerate points can occur. Existing methods are then used to extract the degenerate curves inside these cells. This approach provides the guarantee that no cells containing a degenerate curve will be missed due to numerical issues with existing degenerate curve extraction methods. We also find the surface-type of tensor field topology using this approach. Finally, we apply our analysis to a simulated seismic wave field propagation from an earthquake as well as a selection of fluid flow data sets. We describe the reaction and resulting insights from domain experts in the fields of fluid mechanics and seismology.