Visualization Research Center University of Stuttgart
On the Finite-Time Scope for Computing Lagrangian Coherent Structures fromLyapunov Exponents
TopoInVis 2011Filip Sadlo, Markus Üffinger, Thomas Ertl, Daniel WeiskopfVISUS - University of Stuttgart
Finite-Time Scope for LCS from Lyapunov Exponents 2
Different Finite-Time Scopes
Aletsch GlacierImage region: 5 kmFlow speed: 100 m/y Time scope: 109 s
But: “same river”!
Rhone in Lake GenevaImage region: 1 kmFlow speed: 10 km/h Time scope: 102 s
Lagrangian coherent structures
Finite-Time Scope for LCS from Lyapunov Exponents 3
LCS by Ridges in FTLE
• Lagrangian coherent structures (LCS)can be obtained asRidges in finite-time Lyapunov exponent (FTLE) field
FTLE = 1/|T| ln ( / )
Lyapunov exponent (LE)LE = lim T 1/|T| ln ( / )
LCS behave like material lines (advect with flow)
Shadden et al. 2005
T
T>0 repelling LCST<0 attracting LCS
Finite-Time Scope for LCS from Lyapunov Exponents 4
Finite-Time Scope: Upper Bound
• “Time scope T can’t be too large”• For T : FTLE = LE Well interpretable
• But LCS tend to grow as T grows
Sampling problems & visual clutter
Upper bound is application dependent
T = 0.5 s T = 3 s
CFD
exam
ple
Finite-Time Scope for LCS from Lyapunov Exponents 5
Finite-Time Scope: Lower Bound
• “Time scope T must not be too small” (for topological relevance)• For T 0: FTLE major eigenvalue of (u + (u)T)/2 Ridges of “instantaneous FTLE” cannot satisfy advection property
• No transport barriers for too small T
Lower bound can be motivated by advection property
T = 2 s T = 8 s
Doub
le g
yre
exam
ple
Finite-Time Scope for LCS from Lyapunov Exponents 6
Testing Advection Property: State of the Art
• Shadden et al. 2005• Measure cross-flow of instantaneous velocity through FTLE ridges Theorem 4.4:
• Larger time scopes T better advection property • Sharper ridges better advection property
• But: zero cross-flow is necessary but not sufficient for advection property• Reason: tangential flow discrepancy not tested:
• Problem: tangential speed of ridge not available(Ridges are purely geometric, not by identifiable particles that advect)
u
u
?FTLE ridge
Finite-Time Scope for LCS from Lyapunov Exponents 7
Testing Advection Property
• Our approach (only for 2D fields)• If both ridges in forward and reverse FTLE satisfy advection property,
then also their intersections Intersections represent identifiable points that have to advect
• Approach 1:• Velocity of intersection ui = (i1 - i0) / t
• Require limt0 ui = u( (i0 + i1)/2, t + t / 2 )
forw. FTLE ridge
rev. FTLE ridget t + t
path lineti0 i1
Find corresponding intersection:• Advect i0 (by path line) and get
nearest intersection (i1)• Allow prescription of threshold on
discrepancy
Problem:• Accuracy of ridge extraction in
order of FTLE sampling cell size Ridge extraction error dominates
for small t
Finite-Time Scope for LCS from Lyapunov Exponents 8
Testing Advection Property
• Our approach (only for 2D fields)• If both ridges in forward and reverse FTLE satisfy advection property,
then also their intersections Intersections represent identifiable points that have to advect
• Approach 2:• Use comparably large t (several cells) and measure • Analyze for all intersections• We used average
forw. FTLE ridge
rev. FTLE ridget t + t
path lineti0 i1
Find corresponding intersection:• Advect i0 (by path line) and get
nearest intersection (i1)• Allow prescription of threshold on
discrepancy
Finite-Time Scope for LCS from Lyapunov Exponents 9
Overall Method
• A fully automatic selection of T is not feasible• Parameterization of FTLE visualization depends on goal, typically by trial-and-error
User selects sampling grid, filtering thresholds, Tmin and Tmax, etc.
Our technique takes over these parameters and provides• Plot• Local and global minima• Smallest T that satisfies prescribed discrepancy• …
Finite-Time Scope for LCS from Lyapunov Exponents 10
Example: Buoyant Flow with Obstacles
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
discre
pancy
in FT
LE cel
l size
FTLE advection time T
average discrepancyminimum discrepancy
intersections count
T = 0.2 s T = 0.4 s T = 1.0 s
disc
repa
ncy
in F
TLE
sam
plin
g ce
ll siz
e
• Accuracy of ridge extraction in order of FTLE sampling cell size
• Discrepancy can even grow with increasing T because ridges get sharper, introducing aliasing
• LCS by means of FTLE ridges is highly sampling dependent,in space and time
FTLE vs. advected repelling ridges (black) after t’ = 0.05 s
Finite-Time Scope for LCS from Lyapunov Exponents 11
Conclusion
• We presented a technique for• analyzing the advection quality w.r.t. to T• selecting T w.r.t. to a prescribed discrepancy
• We confirmed findings of Shadden et al. 2005• Advection property increases with increasing T and ridge sharpness
• However, ridge extraction accuracy seems to be a major limiting factor• Needs future work on accuracy of height ridges
• We only test intersections Could be combined with Shadden et al. 2005
• Comparison of accuracy of both approaches
• Extend to 3D fields
Finite-Time Scope for LCS from Lyapunov Exponents 12
Thank you for your attention!