On the FiniteTime Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents TopoInVis 2011 Filip Sadlo Markus Üffinger Thomas Ertl Daniel Weiskopf VISUS University of Stuttgart ID: 774338
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On the Finite-Time Scope for Computing Lagrangian Coherent Structures fromLyapunov Exponents TopoInVis 2011 Filip Sadlo , Markus Üffinger , Thomas Ertl, Daniel Weiskopf VISUS - University of Stuttgart
Different Finite-Time ScopesFinite-Time Scope for LCS from Lyapunov Exponents 2 Aletsch Glacier Image region: 5 km Flow speed: 100 m/y Time scope: 109 s But: “same river”! Rhone in Lake Geneva Image region: 1 kmFlow speed: 10 km/h Time scope: 102 s Lagrangian coherent structures
LCS by Ridges in FTLE Lagrangian coherent structures (LCS) can be obtained as Ridges 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)Finite-Time Scope for LCS from Lyapunov Exponents3Shadden et al. 2005 T T>0 repelling LCST<0 attracting LCS
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 dependentFinite-Time Scope for LCS from Lyapunov Exponents4 T = 0.5 s T = 3 s CFD example
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 propertyNo transport barriers for too small T Lower bound can be motivated by advection propertyFinite-Time Scope for LCS from Lyapunov Exponents5 T = 2 s T = 8 s Double gyre example
Testing Advection Property: State of the ArtShadden 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 propertyReason: tangential flow discrepancy not tested:Problem: tangential speed of ridge not available(Ridges are purely geometric, not by identifiable particles that advect)Finite-Time Scope for LCS from Lyapunov Exponents6 u u ? FTLE ridge
Testing Advection PropertyOur approach (only for 2D fields) If both ridges in forward and reverse FTLE satisfy advection property, then also their intersections I ntersections represent identifiable points that have to advect Approach 1:Velocity of intersection ui = (i1 - i0) / tRequire limt0 ui = u( (i0 + i1)/2, t + t / 2 )Finite-Time Scope for LCS from Lyapunov Exponents7 forw. FTLE ridge rev. FTLE ridge t t + t path line t i 0 i 1 Find corresponding intersection: Advect i 0 (by path line) and get nearest intersection ( i 1 ) 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
Testing Advection PropertyOur approach (only for 2D fields) If both ridges in forward and reverse FTLE satisfy advection property, then also their intersections I ntersections represent identifiable points that have to advect Approach 2:Use comparably large t (several cells) and measure Analyze for all intersectionsWe used average Finite-Time Scope for LCS from Lyapunov Exponents8 forw. FTLE ridge rev. FTLE ridge t t + t path line t i 0 i 1 Find corresponding intersection: Advect i 0 (by path line) and get nearest intersection ( i 1 ) Allow prescription of threshold on discrepancy
Overall MethodA fully automatic selection of T is not feasible P arameterization of FTLE visualization depends on goal, typically by trial-and-error User selects sampling grid, filtering thresholds, T min and Tmax , etc.Our technique takes over these parameters and providesPlotLocal and global minimaSmallest T that satisfies prescribed discrepancy…Finite-Time Scope for LCS from Lyapunov Exponents9
Example: Buoyant Flow with Obstacles Finite-Time Scope for LCS from Lyapunov Exponents 10 T = 0.2 s T = 0.4 s T = 1.0 s d iscrepancy in FTLE sampling cell size 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
ConclusionWe presented a technique for analyzing the advection quality w.r.t. to T selecting T w.r.t. to a prescribed discrepancyWe confirmed findings of Shadden et al. 2005 Advection property increases with increasing T and ridge sharpnessHowever, ridge extraction accuracy seems to be a major limiting factor Needs future work on accuracy of height ridgesWe only test intersectionsCould be combined with Shadden et al. 2005Comparison of accuracy of both approachesExtend to 3D fieldsFinite-Time Scope for LCS from Lyapunov Exponents11
Thank you for your attention! Finite-Time Scope for LCS from Lyapunov Exponents 12