asymmetric tensor zhang - University of Chicagoweb.ci.uchicago.edu/~schultz/visweek10/zhang-asymmetric.pdf · – The pioneers in vector and tensor field analysis andThe pioneers

2D Asymmetric Tensor Field 2D Asymmetric Tensor Field 2D Asymmetric Tensor Field 2D Asymmetric Tensor Field yyAnalysis and VisualizationAnalysis and Visualization

yyAnalysis and VisualizationAnalysis and Visualization

Eugene Zhang

Oregon State University

IntroductionIntroductionIntroductionIntroduction

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

deformation gradient in solid mechanics– deformation gradient in solid mechanics


• Flow visualization has a wide range of applications in areo- and hydro-dynamics:pp y y– Climatology

Oceanography and limnology– Oceanography and limnology

– Hydraulic engineering

– Aircraft and undersea vehicle design


• Existing techniques often focus on velocity [Laramee et al. 2004, Laramee et al. 2007]

– Good for visualizing particle movement

Trajectories TopologyVector magnitude


• Basic types of non-translational motions

Rotation (+/-)Expansion

Contraction

Pure shear


• Given a vector field , the local linearization at is:

Velocity vector field:Velocity vector field: translation

Velocity gradient tensor field: y grotation, isotropic scaling (expansion

and contraction), and pure shear.


• Rotation:

• Isotropic scaling:

• Anisotropic stretching (aka pure shear):• Anisotropic stretching (aka pure shear):


• Flow motions and physical meanings: [Batchelor

1967, Lighthill 1986, Ottino 1989, Sherman 1990]

– Rotation: • VorticityVorticity

– Isotropic scaling: l h d/ t t hi i th thi d di i• volume change and/or stretching in the third dimension

– Anisotropic stretching: • rate of angular deformation, related to energy dissipation and rate

of fluid mixing


• Velocity gradient tensor has been used in vector field visualization– Singularity classification [Helman and Hesselink 1991]

Periodic orbit extraction [Ch t l 2007]– 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]


• However, tensor field structures were not investigated and applied to flow visualizationg pp– Velocity gradient is asymmetric

Past work in tensor field visualization focus on– 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]


• Symmetric tensors – two real eigenvaluestwo real eigenvalues

– two mutually perpendicular eigenvectors when not degeneratedegenerate

• Asymmetric tensors – can have complex eigenvalues

– eigenvectors not always mutually perpendicularg y y p p


• Questions:– What are features in an asymmetric tensor field?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 DecompositionTensor DecompositionTensor DecompositionTensor Decomposition

– Isotropic scaling:

– Rotation:

– Pure shear:– Pure shear:


• The set of 2x2 tensors can be parameterized by , , , and , such thaty– , and

–

• This is a four-dimensional space

• Can we focus on configuration spaces with lower-dimensions?lower dimensions?


• Eigenvalues only depend on , , and

• Eigenvectors are dependent on andEigenvectors are dependent on , , and

• Define eigenvector and eigenvalue manifolds

Eigenvector ManifoldEigenvector ManifoldEigenvector ManifoldEigenvector Manifold

• Eigenvectors of

are same as

are same asare same as

can be rewritten as


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


• Eigenvalues are constant along any latitude


• Real domains and complex domains

• Degenerate curves


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











• Bisectors never change along any longitude

• They are dual-eigenvectors (Zheng andThey are dual eigenvectors (Zheng and Pang 2005)

• Dual-eigenvectors are eigenvectors of


• Dual-eigenvectors of


• Dual-eigenvectors of


• Which side of the Equator matters

• Incorporate the Equator into asymmetricIncorporate the Equator into asymmetric tensor topology


• Degenerate (circular) points of– Number, location, index, orientationNumber, location, index, orientation

Major Eigenvectors of Symmetric Component

Major Dual-Eigenvectors


• 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


• Visualization– Black curves:Black curves:

– White curves:

Bl– Blue curves:

– Degenerate points

– The Equator

– Degenerate curvesDegenerate curves


Eigenvalue ManifoldEigenvalue ManifoldEigenvalue ManifoldEigenvalue Manifold

• Eigenvalues depend on , , and

• We are interested in relatively strengthsWe are interested in relatively strengths among the three components




Dominant component



All components


Tensor Magnitude

Combining Eigenvector and Combining Eigenvector and g gEigenvalue Manifolds

g gEigenvalue Manifolds

• CCW dominant region (red) must be in the• 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 Combining Eigenvector and g gEigenvalue Manifolds

g gEigenvalue Manifolds

ApplicationsApplicationsApplicationsApplications

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


• Sullivan flow (a tornado model)

Tensor magnitudeDominant eigenvalue + major eigenvector and major dual-Eigenvector

Vector field topology



Tensor MagnitudeDominant Eigenvalue + Major Eigenvector and Major Dual-Eigenvector

Vector Field Topology



Tensor MagnitudeDominant Eigenvalue + Major Eigenvector and Major Dual-Eigenvector

Vector Field Topology



Tensor magnitudeDominant eigenvalue + major eigenvector and major dual-eigenvector

Vector field topology


• Diesel engine


• Diesel engine

Open QuestionsOpen QuestionsOpen QuestionsOpen Questions

• What does tensor index tell us about the flow, other than zero stretching?g

• Can we generate a graph representation for asymmetric tensors, much like the vector field topology?


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


• How does the analysis carry over to 3D?– Axis of rotation is not always aligned with any ofAxis 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?p g

– What do the eigenvalue and eigenvector manifolds look like?


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

AcknowledgementAcknowledgementAcknowledgementAcknowledgement

C ll b t• 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

AcknowledgementAcknowledgementAcknowledgementAcknowledgement

• Inspirations from:– Dr. Xiaoqiang Zheng, Nvidiaq g g,

– 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

ReferencesReferencesReferencesReferences

• 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• 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)(http://www.computer.org/portal/web/csdl/doi/10.1109/TVCG.2008.68)

• 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?Questions?Questions?Questions?

Documents

asymmetric tensor zhang - University of Chicagoweb.ci.uchicago.edu/~schultz/visweek10/zhang-asymmetric.pdf · – The pioneers in vector and tensor field analysis andThe pioneers