Visualization Research Center University of Stuttgart On the Finite-Time Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents TopoInVis 2011 Filip Sadlo, Markus Üffinger, Thomas Ertl, Daniel Weiskopf VISUS - University of Stuttgart
Different Finite-Time Scopes Finite-Time Scope for LCS from Lyapunov Exponents2 Aletsch Glacier Image region: 5 km Flow speed: 100 m/y Time scope: 10 9 s But: “same river”! Rhone in Lake Geneva Image region: 1 km Flow speed: 10 km/h Time scope: 10 2 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 Exponents3 Shadden et al T T>0 repelling LCS T<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 dependent Finite-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 property No transport barriers for too small T Lower bound can be motivated by advection property Finite-Time Scope for LCS from Lyapunov Exponents5 T = 2 s T = 8 s Double gyre example
Testing Advection Property: State of the Art Shadden et al 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) Finite-Time Scope for LCS from Lyapunov Exponents6 u u ? FTLE ridge
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 u i = (i 1 - i 0 ) / t Require lim t 0 u i = u( (i 0 + i 1 )/2, t + t / 2 ) Finite-Time Scope for LCS from Lyapunov Exponents7 forw. FTLE ridge rev. FTLE ridge t t + t path line tt i0i0 i1i1 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 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 Finite-Time Scope for LCS from Lyapunov Exponents8 forw. FTLE ridge rev. FTLE ridge t t + t path line tt i0i0 i1i1 Find corresponding intersection: Advect i 0 (by path line) and get nearest intersection (i 1 ) Allow prescription of threshold on discrepancy
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, T min and T max, 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 Exponents9
Example: Buoyant Flow with Obstacles Finite-Time Scope for LCS from Lyapunov Exponents10 T = 0.2 sT = 0.4 sT = 1.0 s discrepancy 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
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 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 Comparison of accuracy of both approaches Extend to 3D fields Finite-Time Scope for LCS from Lyapunov Exponents11
Thank you for your attention! Finite-Time Scope for LCS from Lyapunov Exponents12