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]
• Velocity gradient tensor can provide
complementary information to velocity vector
field (non-translational)
Introduction
Tensor field
• Velocity gradient tensor can provide
complementary information to velocity vector
field
Introduction
Vector fieldTensor field
• Velocity gradient tensor can provide
complementary information to velocity vector
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 ++=∇
Tensor Field Decomposition
– 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
Tensor Field Decomposition
−+
−+
=
=
∂∂
∂∂
∂∂
∂∂
θθθθ
γγγcossin
sincos
01
10
10
01srddc
ba
yG
xG
yF
xF
Asymmetric tensor visualization
Tensor Field Decomposition
– 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
Tensor Field Decomposition
– Directional information is contained in
−+
−+
=
=
∂∂
∂∂
∂∂
∂∂
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
Tensor Field Decomposition
– Directional information is contained in
−+
−+
=
=
∂∂
∂∂
∂∂
∂∂
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
• Eigenvector Manifold
• Eigenvalues constant along each latitude
• Angular components of eigenvectors linearly depend on
Deformation Pattern and Orientation
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin πθ 20 <≤
22
πϕπ ≤≤−
θ
Eigenvalues
• Eigenvector Manifold
Deformation Pattern and Orientation
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
Real domains
• Eigenvector Manifold
Deformation Pattern and Orientation
Real domains
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Real domains
Major eigenvector
Minor eigenvector
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Complex domains
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Complex domains
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Complex domains
Major dual-eigenvector
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Degenerate curves
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Degenerate curves
Simple shear
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Circular points
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Circular points
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Pure shear
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Pure shear
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
• Eigenvector Manifold
Deformation Pattern and Orientation
Pure shear
Major eigenvector
Minor eigenvector
−+
−θθ
θθϕϕ
cossin
sincoscos
01
10sin
Deformation Pattern and Orientation
• Eigenvector Manifold
Deformation Pattern and Orientation
• 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]