advanced topic: asymmetric tensor analysis and...

Advanced Topic: Asymmetric

Tensor Analysis and Visualization

Some materials provided by Prof. Eugene Zhang

• Asymmetric tensors are important but have

received relatively little attention

– Velocity gradient tensor

• fluid mechanics, weather prediction, …

– Deformation gradient tensor

• solid mechanics, earthquake modeling, …

Introduction

• Velocity vector field visualization can directly

reflect the movement of particles (translation)

Introduction

Vector field

• Velocity gradient tensor can provide

complementary information to velocity vector

field (non-translational)

Introduction

Rotation (+/-)

Expansion = Positive Isotropic Scaling

Anisotropic stretching or pure shear

Contraction = Negative Isotropic Scaling

Vector field

• Given a vector field , the local

linearization at is:

Introduction

=

),(

),(),(

yxG

yxFyxV

),( 00 yx

−−

∂∂

∂∂

∂∂

∂∂

+=0

0

0000

0000

00),(),(),(

),(),(),(

00 yy

xx

yxyGyxx

G

yxyFyxx

FyxVLV yx

translation

Velocity gradient: rotation, isotropic scaling,

anisotropic stretching

• Rotation

• Isotropic scaling

• Anisotropic stretching

Introduction

−=

∂∂

∂∂

∂∂

∂∂

01

10r

yG

xG

yF

xF

γ

=

∂∂

∂∂

∂∂

∂∂

10

01d

yG

xG

yF

xF

γ

−=

∂∂

∂∂

∂∂

∂∂

θθθθ

γcossin

sincoss

yG

xG

yF

xF

Introduction

• Flow motions and physical meanings: [Batchelor

1967, Fischer et al. 1979, Ottino 1989, Sherman 1990]

– Rotation: • Vorticity

– Isotropic scaling:

• volume change and/or stretching in the third dimension

– 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 (source, sink, saddle, etc) [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]



field (non-translational)

Introduction

Tensor field



field

Introduction

Vector fieldTensor field



field

Introduction

vector fieldtensor field vector/tensor field

• Much work exists for symmetric tensors

– Analysis: Delmarcello and Hesselink [1993],

Tricoche et al. [2001], Zheng and Pang [2004,

2005], …

– Visualization: Zheng and Pang [2003], Feng et al.

[2004], [Zhang et al. 2007], …

Introduction

• Much work exists for symmetric tensor fields:

– Analysis: Delmarcello and Hesselink [1993],

Tricoche et al. [2001], Zheng and Pang [2004,

2005a], …

– Visualization: Zheng and Pang [2003], Feng et al.

[2004], [Zhang et al. 2007], …

• Asymmetric tensor fields:

– Analysis and visualization: Zheng and Pang

[2005b]

Introduction

• Why are asymmetric tensor fields more

challenging than symmetric tensor fields?

Introduction

• Why are asymmetric tensor fields more

challenging than symmetric tensor fields?

– Real-valued eigenvalues

– Orthonormal eigenvectors

Introduction

• Zheng and Pang’s pioneering work in 2D

asymmetric tensor fields

Introduction

Image courtesy: Xiaoqiang Zheng and Alex Pang

Complex domains

(major dual-eigenvectors)

Real domains

(major eigenvectors)

Degenerate curves

Circular points

• Interesting yet unanswered questions:

Introduction

Image courtesy: Xiaoqiang Zheng and Alex Pang

Physical meanings?

Better visualization?

What is asymmetric tensor field topology?

• Sum of three fields

– Id = identity

– R = anti-symmetric

– S = traceless and symmetric

• Behaviors of tensor fields are determined by the interaction of these three fields

Tensor Field Decomposition

SRIdV srd γγγ ++=∇


– Isotropic scaling:

– Rotation:

– Anisotropic stretching:

−+

−+

=

=

∂∂

∂∂

∂∂

∂∂

θθθθ

γγγcossin

sincos

01

10

10

01srddc

ba

yG

xG

yF

xF

2

dad

+=γ

2

bcr

−=γ

2

)()( 22 cbdas

++−=γ

−+= −

da

cb1tanθ


−+

−+

=

=

∂∂

∂∂

∂∂

∂∂

θθθθ

γγγcossin

sincos

01

10

10

01srddc

ba

yG

xG

yF

xF

Asymmetric tensor visualization


– Directional information is contained in

−+

−+

=

=

∂∂

∂∂

∂∂

∂∂

θθθθ

γγγcossin

sincos

01

10

10

01srddc

ba

yG

xG

yF

xF

−+

−θθ

θθγγ

cossin

sincos

01

10sr

The traceless part of the tensor



−+

−+

=

=

∂∂

∂∂

∂∂

∂∂

θθθθ

γγγcossin

sincos

01

10

10

01srddc

ba

yG

xG

yF

xF

}cossin

sincoscos

01

10{sin

cossin

sincos

01

10

22

−+

−+=

−+

−

θθθθ

ϕϕγγ

θθθθ

γγ

sr

srThe traceless part of the tensor



−+

−+

=

=

∂∂

∂∂

∂∂

∂∂

θθθθ

γγγcossin

sincos

01

10

10

01srddc

ba

yG

xG

yF

xF

}cossin

sincoscos

01

10{sin

cossin

sincos

01

10

22

−+

−+=

−+

−

θθθθ

ϕϕγγ

θθθθ

γγ

sr

sr

)(tan 1

r

sγ

γϕ −=

The traceless part of the tensor

• Eigenvector Manifold

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin

πθ 20 <≤22

πϕπ ≤≤−

Deformation Pattern and Orientation


• Eigenvalues constant along each latitude

• Angular components of eigenvectors linearly depend on


−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin πθ 20 <≤

22

πϕπ ≤≤−

θ

Eigenvalues



−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin

Real domains



Real domains

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Real domains

Major eigenvector

Minor eigenvector

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Complex domains

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Complex domains

Major dual-eigenvector

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Degenerate curves

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Degenerate curves

Simple shear

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Circular points

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Pure shear

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin



Pure shear

Major eigenvector

Minor eigenvector

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin

• Why exhibit symmetric tensor field patterns?

Circular Points

wedge

trisector

• Major and minor dual-eigenvectors of

• are given by major and minor eigenvectors of

Circular Points

−+

−θθ

θθϕϕ

cossin

sincoscos

01

10sin

Major dual-eigenvector

+−+

++

)2

cos()2

sin(

)2

sin()2

cos(cos

|sin|

sinπθπθ

πθπθϕ

ϕϕ

Circular Points

Major Eigenvectors of Symmetric Component

Major Dual-Eigenvectors

+−+

++

)2

cos()2

sin(

)2

sin()2

cos(

πθπθ

πθπθ

− θθθθ

cossin

sincos

Combined Eigenvalues and Eigenvectors

Stretching dominant

Rotation dominant

Hybrid Visualization

Stretching

Rotation

Applications

A Simulated Earthquake Deformation Data

46

Applications

Application to Simulated Earthquake Deformation Data

47

Applications

Application to Simulated Earthquake Deformation Data

Hyperstreamlines Hybrid visualization

48

Applications

Application to a cooling jacket simulation

49

Application to a cooling jacket simulation

Applications

Hyperstreamlines Hybrid visualization

50

Applications

Applications to diesel engine flow

Hyperstreamlines only Hybrid without tensor magnitude

Hybrid with tensor magnitude

Side view

51

Tensor In Complex domains

• The symmetric tensor is as follows

The final size of the glyph is scaled by α

52

Elliptical pattern

[Zheng and Pang Vis05]

advanced topic: asymmetric tensor analysis and...

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