Asymmetric Tensors [pdf]

2D Asymmetric Tensor Field
Analysis and Visualization
Eugene Zhang
Oregon State University

Introduction
• Asymmetric tensors can model the gradient
of a vector field
– velocity gradient in fluid dynamics
– deformation gradient in solid mechanics

Introduction
• Flow visualization has a wide range of
applications in areo- and hydro-dynamics:
y
– Climatology
– Oceanography and limnology
– Hydraulic engineering
– Aircraft and undersea vehicle design

Introduction
• Existing techniques often focus on velocity
[Laramee et al. 2004, Laramee et al. 2007]
– Good for visualizing particle movement
Trajectories
Vector magnitude
Topology

Introduction
• Basic types of non-translational motions
Rotation (+/-)
Expansion
Pure shear
Contraction

Introduction
• Given a vector field , the local
linearization at is:
Velocity vector field:
translation
Velocity gradient tensor field:
yg
rotation, isotropic scaling (expansion
and contraction), and pure shear.

Introduction
• Rotation:
• Isotropic scaling:
• Anisotropic stretching (aka pure shear):

Introduction
• Flow motions and physical meanings: [Batchelor
1967, Lighthill 1986, Ottino 1989, Sherman 1990]
– Rotation:
• Vorticity
– Isotropic scaling:
• volume change and/or stretching t in the third dimensioni
– Anisotropic stretching:
• rate of angular deformation, related to energy dissipation and rate
of fluid mixing

Introduction
• Velocity gradient tensor has been used in
vector field visualization
– Singularity classification [Helman and Hesselink 1991]
– Periodic orbit extraction [Chen et al. 2007]
– Attachment and separation detection [Kenwright 1998]
– Vortex core identification [Sujudi and Haimes 1995, Jeong and
Hussain 1995, Peikert and Roth 1999, Sadarjoen and Post 2000]

Introduction
• However, tensor field structures were not
investigated and applied to flow visualization
– Velocity gradient is asymmetric
– Past work in tensor field visualization focus on
symmetric tensors [Delmarcelle and Hesselink 1994, Hesselink et
al. 1997, Tricoche et al. 2001, Tricoche et al. 2003, Hotz et al. 2004,
Zheng and Pang 2004, Zheng et al. 2005, Zhang et al. 2007]

Introduction
• Symmetric tensors
– two real eigenvalues
– two mutually perpendicular eigenvectors when not
degenerate
• Asymmetric tensors
– can have complex eigenvalues
– eigenvectors not always mutually perpendicularp

Introduction
• Questions:
– What are features in an asymmetric tensor field?
– How to visualize these features?
Wh t l b t th fl f th
– What can we learn about the flow from these
features?

Tensor Decomposition
– Isotropic scaling:
– Rotation:
– Pure shear:

Tensor Decomposition
• The set of 2x2 tensors can be parameterized
by , , , and , such that
– , and
–
• This is a four-dimensional space
• Can we focus on configuration spaces with
lower-dimensions?

Tensor Decomposition
• Eigenvalues only depend on , , and
• Eigenvectors are dependent on , , and
• Define eigenvector and eigenvalue manifolds

Eigenvector Manifold
• Eigenvectors of
are same as
are same as
can be rewritten as

Eigenvector Manifold
Image credit: http://math.etsu.edu/MultiCalc/Chap3/Chap3-4/sphere1.gif

Eigenvector Manifold
• Eigenvalues are constant along any latitude

Eigenvector Manifold
• Real domains and complex domains
• Degenerate curves

Eigenvector Manifold
• We can focus on any longitude and understand
how eigenvectors change

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold

Eigenvector Manifold
• Bisectors never change along any longitude
• They are dual-eigenvectors (Zheng and
Pang 2005)
• Dual-eigenvectors are eigenvectors of

Eigenvector Manifold
• Dual-eigenvectors of

Eigenvector Manifold
• Dual-eigenvectors of

Eigenvector Manifold
• Which side of the Equator matters
• Incorporate the Equator into asymmetric
Incorporate the Equator into asymmetric
tensor topology

Eigenvector Manifold
• Degenerate (circular) points of
– Number, location, index, orientation
Major Eigenvectors of
Symmetric Component
Major Dual-Eigenvectors

Eigenvector Manifold
• Poincaré-Hopf theorem (asymmetric
tensors):
– Given a continuous asymmetric tensor field
defined on a closed surface S such that
has only isolated degenerate points ,
then

Eigenvector Manifold
• Visualization
– Black curves:
– White curves:
– Blue curves:
– Degenerate points
– The Equator
– Degenerate curves

Eigenvector Manifold

Eigenvalue Manifold
• Eigenvalues depend on , , and
• We are interested in relatively strengths
We are interested in relatively strengths
among the three components

Eigenvalue Manifold

Eigenvalue Manifold

Eigenvalue Manifold
Dominant component

Eigenvalue Manifold

Eigenvalue Manifold
All components

Eigenvalue Manifold
Tensor Magnitude

Combining Eigenvector and
Eigenvalue Manifolds
• CCW-dominant region (red) must be in the
northern hemisphere (red)
• CW-dominant region (green) must be in the
southern hemisphere (red)

Combining Eigenvector and
Eigenvalue Manifolds

Applications
• Sullivan flow (a tornado model) [Sullivan:1959]

Applications
• Sullivan flow (a tornado model)
Vector field topology
Dominant eigenvalue +
major eigenvector and
major dual-Eigenvector
Tensor magnitude

Applications
• Sullivan flow (a tornado model)
Vector Field Topology
Dominant Eigenvalue +
Major Eigenvector and
Major Dual-Eigenvector
Tensor Magnitude

Applications
• Sullivan flow (a tornado model)
Vector Field Topology
Dominant Eigenvalue +
Major Eigenvector and
Major Dual-Eigenvector
Tensor Magnitude

Applications
• Sullivan flow (a tornado model)
Vector field topology
Dominant eigenvalue +
major eigenvector and
major dual-eigenvector
Tensor magnitude

Applications
• Diesel engine

Applications
• Diesel engine

Open Questions
• What does tensor index tell us about the
flow, other than zero stretching?
• Can we generate a graph representation for
asymmetric tensors, much like the vector
field topology?

Open Questions
• How do we integrate information from vector
and tensor field analysis?

Open Questions
• How does the analysis carry over to 3D?
– Axis of rotation is not always aligned with any of
the eigenvector directions, which means all of
them could be moving when going from real
domains into complex domains
– How to deal with 3D anisotropic stretching?
– What do the eigenvalue and eigenvector manifolds
look like?

Open Questions
• What is the topology of higher-order tensor
fields, symmetric or not, 2D or 3D, static or
time-varying? And why do we care?

Acknowledgement
• Collaborators
– Dr. Harry Yeh, Professor in fluid mechanics, Oregon
State University
– Darrel Palke, Intel
– Zhongzang Lin, Ph.D. student at Oregon State
University
– Dr. Guoning Chen, postdoctoral researcher at
University of Utah
– Dr. Robert S. Laramee, Lecturer (Assistant Professor)
at Swansea University, UK

Acknowledgement
• Inspirations from:
– Dr. Xiaoqiang Zheng, Nvidia
– Dr. Alex Pang, Professor in computer science, UC
Santa Cruz
– The pioneers in vector and tensor field analysis and
– The pioneers in vector and tensor field analysis and
visualization

References
• Xiaoqiang Zheng and Alex Pang, “2D Asymmetric Tensor Analysis”,
IEEE Vis 2005 (http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1532770)
• Eugene Zhang, Harry Yeh, Zhongzang Lin, and Robert S. Laramee,
“Asymmetric Tensor Analysis for Flow Visualization”, IEEE Trans. on
Visualization and Computer Graphics
(http://www.computer.org/portal/web/csdl/doi/10.1109/TVCG.2008.68)
org/portal/web/csdl/doi/10 1109/TVCG • Darrel Palke, Guoning Chen, Zhongzang Lin, Harry Yeh, Robert S.
Laramee, and Eugene Zhang, “Asymmetric Tensor Visualization with
Glyph and Hyperstreamline Placement on 2D Manifolds”, Tech
Report, Oregon State University
(http://ir.library.oregonstate.edu/jspui/handle/1957/13549)

Questions?