875 research outputs found
Global Topology of 3D Symmetric Tensor Fields
There have been recent advances in the analysis and visualization of 3D
symmetric tensor fields, with a focus on the robust extraction of tensor field
topology. However, topological features such as degenerate curves and neutral
surfaces do not live in isolation. Instead, they intriguingly interact with
each other. In this paper, we introduce the notion of {\em topological graph}
for 3D symmetric tensor fields to facilitate global topological analysis of
such fields. The nodes of the graph include degenerate curves and regions
bounded by neutral surfaces in the domain. The edges in the graph denote the
adjacency information between the regions and degenerate curves. In addition,
we observe that a degenerate curve can be a loop and even a knot and that two
degenerate curves (whether in the same region or not) can form a link. We
provide a definition and theoretical analysis of individual degenerate curves
in order to help understand why knots and links may occur. Moreover, we
differentiate between wedges and trisectors, thus making the analysis more
detailed about degenerate curves. We incorporate this information into the
topological graph. Such a graph can not only reveal the global structure in a
3D symmetric tensor field but also allow two symmetric tensor fields to be
compared. We demonstrate our approach by applying it to solid mechanics and
material science data sets.Comment: IEEE VIS 202
A Visual Approach to Analysis of Stress Tensor Fields
We present a visual approach for the exploration of stress tensor fields. In contrast to common tensor visualization methods that only provide a single view to the tensor field, we pursue the idea of providing various perspectives onto the data in attribute and object space. Especially in the context of stress tensors, advanced tensor visualization methods have a young tradition. Thus, we propose a combination of visualization techniques domain experts are used to with statistical views of tensor attributes. The application of this concept to tensor fields was achieved by extending the notion of shape space. It provides an intuitive way of finding tensor invariants that represent relevant physical properties. Using brushing techniques, the user can select features in attribute space, which are mapped to displayable entities in a three-dimensional hybrid visualization in object space. Volume rendering serves as context, while glyphs encode the whole tensor information in focus regions. Tensorlines can be included to emphasize directionally coherent features in the tensor field. We show that the benefit of such a multi-perspective approach is manifold. Foremost, it provides easy access to the complexity of tensor data. Moreover, including
well-known analysis tools, such as Mohr diagrams, users can familiarize themselves gradually with novel visualization methods. Finally, by employing a focus-driven hybrid rendering, we significantly reduce clutter, which was a major problem of other three-dimensional tensor visualization methods
The Asymmetric Merger of Black Holes
We study event horizons of non-axisymmetric black holes and show how features
found in axisymmetric studies of colliding black holes and of toroidal black
holes are non-generic and how new features emerge. Most of the details of black
hole formation and black hole merger are known only in the axisymmetric case,
in which numerical evolution has successfully produced dynamical space-times.
The work that is presented here uses a new approach to construct the geometry
of the event horizon, not by locating it in a given spacetime, but by direct
construction. In the axisymmetric case, our method produces the familiar
pair-of-pants structure found in previous numerical simulations of black hole
mergers, as well as event horizons that go through a toroidal epoch as
discovered in the collapse of rotating matter. The main purpose of this paper
is to show how new - substantially different - features emerge in the
non-axisymmetric case. In particular, we show how black holes generically go
through a toroidal phase before they become spherical, and how this fits
together with the merger of black holes.Comment: 28 pages, 10 figures, uses REVTEX. Improved quality figures and
additional color images are provided at http://www.phyast.pitt.edu/~shusa/EH
Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold
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 degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can lead to the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis
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