Technical Report
 

Asymmetric tensor analysis for flow visualization

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https://ir.library.oregonstate.edu/concern/technical_reports/5q47rq32d

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  • The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight into the vector field 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 two-dimensional manifold. The structure in the eigenvalue field is illustrated using the eigenvalue manifold, which enables novel visualizations that depict the relative strengths among the physical components in the vector field, such as isotropic scaling, rotation, and anisotropic stretching. Our eigenvector analysis is based on the concept of the eigenvector manifold, which affords additional insight on 2D asymmetric tensors fields beyond previous analyses. Our results include a simple and intuitive geometric realization of the dual-eigenvectors, a novel symmetric discriminant that measures the signed distance of a tensor from being symmetric, the classification of degenerate (circular) points, and the extension of the Poincaré-Hopf index theorem to continuous asymmetric tensor fields defined on closed two-dimensional manifolds. We also extend eigenvectors continuously into the complex domains which we refer to as pseudo-eigenvectors. We make use of evenly spaced tensor lines following pseudo-eigenvectors to illustrate the local linearization of tensors everywhere inside complex domains simultaneously. Both eigenvalue manifold and eigenvector manifold are supported by a tensor reparamterization that has physical meaning. This allows us to relate our tensor analysis to physical quantities such as vorticity, deformation, expansion, contraction, which provide 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.
  • Keywords: Asymmetric tensors, Tensor field visualization, Flow analysis, Surfaces
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