2D second-order asymmetric tensor field analysis and visualization

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

Asymmetric tensor visualization with glyph and hyperstreamline placement on 2D manifolds

Asymmetric tensor fields present new challenges for visualization techniques such as hyperstreamline placement and glyph packing. This is because the physical behaviors of the tensors are fundamentally different inside real domains where eigenvalues are real and complex domains where eigenvalues are complex. We present a hybrid visualization approach in which hyperstreamlines are used to illustrate the tensors in the real domains while glyphs are employed for complex domains. This enables an effective visualization of the flow patterns everywhere and also provides a more intuitive illustration of elliptical flow patterns in the complex domains. The choice of the types of representation for different types of domains is motivated by the physical interpretation of asymmetric tensors in the context of fluid mechanics, i.e., when the tensor field is the velocity gradient tensor. In addition, we encode the tensor magnitude to the size of the glyphs and density of hyperstreamlines. We demonstrate the effectiveness of our visualization techniques with real-world engine simulation data

Topological analysis and visualization of asymmetric tensor fields

Visualizing asymmetric tensors is an important task in understanding fluid dynamics. In this paper, we describe topological analysis and visualization techniques for asymmetric tensor fields on surfaces based on analyzing the impact of the symmetric and antisymmetric components of the tensor field on its eigenvalues and eigenvectors. At the core of our analysis is a reparameterization of the space of 2 x 2 tensors, which allows us to understand the topology of tensor fields by studying the manifolds of eigenvalues and eigenvectors. We present a partition of the eigenvalue manifold using a Voronoi diagram, which allows to segment a tensor field based on its relatively strengths in isotropic scaling, rotation, and anisotropic stretching. Our analysis of eigenvectors is based on the observation that the dual-eigenvectors of a tensor depend solely on the symmetric constituent of the tensor. The anti-symmetric component acts on the eigenvectors by rotating them either clockwise or counterclockwise towards the nearest dual-eigenvector. The orientation and the amount of the rotation are derived from the ratio between the symmetric and anti-symmetric components. We observe that symmetric tensors form the boundary between regions of clockwise flows and regions of counterclockwise flows. Crossing such a boundary results in discontinuities in the major and minor dual eigenvectors. Thus we define symmetric tensors as part of the tensor field topology in addition to degenerate tensors. These observations inspire us to illustrate the topology of the symmetric component and anti-symmetric component simultaneously. We demonstrate the utility of our techniques on an important application from computational fluid dynamics, namely, engine simulation.Keywords: Flow analysis, Tensor field visualization, Asymmetric tensorsKeywords: Flow analysis, Tensor field visualization, Asymmetric tensor

Asymmetric tensor analysis for flow visualization

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, Surface

Topological analysis of asymmetric tensor fields for flow visualization

Most existing flow visualization techniques focus on the analysis and visualization of the vector field that describes the flow. In this paper, we employ a rather different approach by performing tensor field analysis and visualization on the gradient of the vector field, which can provide additional and complementary information to the direct analysis of the vector field. Our techniques focus on the topological analysis of the eigenvector and eigenvalues of 2 x 2 tensors. At the core of our analysis is a reparameterization of the tensor space, which allows us to understand the topology of tensor fields by studying the manifolds of eigenvalues and eigenvectors. We present a partition of the eigenvalue manifold using a Voronoi diagram, which allows the segmentation of a tensor field based on relative strengths with respect to isotropic scaling, rotation, and anisotropic stretching. Our analysis of eigenvectors is based on two observations. First, the dual-eigenvectors of a tensor depend solely on the symmetric constituent of the tensor. The anti-symmetric component acts on the eigenvectors by rotating them either clockwise or counterclockwise towards the nearest dual-eigenvector. The orientation and the amount of the rotation are derived from the ratio between the symmetric and anti-symmetric components. Second, The boundary between regions of clockwise rotation and counterclockwise rotation is located where the tensor field is purely symmetric. Crossing such a boundary results in discontinuities in the dual eigenvectors. Thus we define symmetric tensors as part of tensor field topology in addition to degenerate tensors. These observations inspire our visualization techniques in which the topology of the symmetric component and anti-symmetric component are shown simultaneously. We also provide physical interpretations of our analysis, and demonstrate the utility of our visualization techniques on two applications from computational fluid dynamics, namely, engine simulation and cooling jacket design.Keywords: Surfaces, Asymmetric tensors, Tensor field visualization, Flow analysisKeywords: Surfaces, Asymmetric tensors, Tensor field visualization, Flow analysi