Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

1. Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection February 2009 – TopoInVis Filip Sadlo @ VISUS – Universität Stuttgart, Germany Alessandro Rigazzi, Ronald Peikert @ CGL – ETH Zurich, Switzerland

2. Lagrangian Coherent Structures (LCS) Lagrangian Coherent Structures Ridges in Lyapunov exponent Transient view Slow (trajectory per point & time) Grid Advection Vector Field Topology • Crit. pts. & streamlines • Instantaneous view • Fast Shadden et al. 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

3. Lagrangian Coherent Structures (LCS) Lagrangian Coherent Structures Ridges in Lyapunov exponent Transient view Slow (trajectory per point & time) Grid Advection Vector Field Topology • Crit. pts. & streamlines • Instantaneous view • Fast Shadden et al. 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

4. LCS in Nature Glaciers LCS = Moraines Confluences LCS = Interfaces from: www.publicaffairs.water.ca.gov/swp/swptoday.cfm from: www.scienceclarified.com/Ga-He/Glacier.html Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

5. Finite-Time Lyapunov Exponent (FTLE) FTLE: “growth of perturbation after advection time T” Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

6. FTLE Computation • By pre-sampled flow map f t0= FTLE Shadden et al. 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

7. FTLE Computation • By pre-sampled flow map f t0= FTLE Shadden et al. 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

8. FTLE Computation • By pre-sampled flow map f t0= FTLE Shadden et al. 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

9. FTLE Computation • By pre-sampled flow map f t0= FTLE Shadden et al. 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

10. Related Work • “Distinguished Material Surfaces and Coherent Structures in Three-dimensional Fluid Flows”, G. Haller, 2001 • “Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications”, C. Garth et al., 2007 • “Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction”, F. Sadlo et al., 2007 • “Ridges in Image and Data Analysis”, D. Eberly, 1996 • “Automatic Detection of Open and Closed Separation and Attachment Lines”, D. Kenwright, 1998 • “Fast and Robust Extraction of Separation Line Features”,X. Tricoche et al., 2005 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

11. Motivation • Goal: Fast computation of FTLE animations (variation of t0) • Observation 1 • Interpretation of LCS can often be restricted to a region of interest • E.g. often related to boundary effects such as attachment/separation • Observation 2 • G. Haller 2001: LCS are material surfaces • FTLE ridges advect with the vector field • Strategy 1 • Restrict FTLE computation to regions of interest • Strategy 2 • Exploit temporal coherence by advection of the sampling grid Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

12. Initial Grid • Separation/Attachment • Initial grid at lines of separation/attachment (Kenwright, Tricoche) • Or along complete boundaries • Any other region which is assumed to contain part of LCS • E.g. blades of a turbine Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

13. Flow Separation: Forward Advection of Sampling Grid Initial Grid Grid Adaptation Grid Advection Grid Adaptation Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

14. Grid Growing Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

15. Resampling • Grid advection leads to distorted cells • Affected gradient estimation (FTLE, ridge extraction) • Distorted ridges and artifacts (also due to MC-ridge extraction) • Resampling • New sampling grid around existing ridges inside region of interest • Need to re-compute trajectories • Expensive • Problem • Resampling leads to temporal incoherence due to vanishing distortion • Resampling triggered by error threshold Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

16. Error Measure • LCS error: difficult to measure (difference of ridge surfaces) • Similar to appoach by Garth et al.: • Base on error of FTLE, not its ridges (LCS) Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

17. Resampling - Algorithm • Resample each r advection steps • Error estimation by linearization • r can get estimated too large • Either take back advection steps if RMS exceeded tolerance • Or prescribe a reduced tolerance e.g. by 15% Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

18. Intake of Power Plant Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

19. Intake of Power Plant • Initial sampling grid • Negative-time path lines • Adapted sampling grid • 8 advection steps since last resample • Resulting ridge Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

20. Grid Advection Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

21. Grid Advection Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

22. Intake of Power Plant: Performance Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

23. Conclusion • Efficient method for computing time series of time-dependent quantities based on trajectories • Temporal incoherence of resulting ridges • Can be limited by prescription of maximum error • If large error is allowed, method can be used for preview mode • Future work • Better ridge extraction (no linear interpolation for MC edge intersections) Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

24. End Thanks for your attention Questions? Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

25. Grid Advection Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

26. “Appearing as dark lines on the surface, moraines indicate how many smaller glaciers feed into the system” -> LCS, dynamical systems Moraines and LCS from: www.fs.fed.us/r10/tongass/forest_facts/resources/geology/icefields.htm Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

27. Flow Separation / Flow Attachment Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

28. Flow Attachment: Backward Advection of Sampling Grid Initial Grid Grid Adaptation Grid Advection Grid Adaptation Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

29. Intake of Power Plant: LCS Error (a) (b) (a) ridge by advected grid (b) ridge by uniform grid (c) distance between (a) and (b) color-coded on (a) (c) Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection