time-dependent visualization of lagrangian coherent structures by grid advection
DESCRIPTION
Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection. February 2009 – TopoInVis Filip Sadlo @ VISUS – Universit ät Stuttgart, Germany Alessandro Rigazzi, Ronald Peikert @ CGL – ETH Zurich, Switzerland. Lagrangian Coherent Structures (LCS). - PowerPoint PPT PresentationTRANSCRIPT
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Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection
February 2009 – TopoInVis
Filip Sadlo @ VISUS – Universität Stuttgart, GermanyAlessandro Rigazzi, Ronald Peikert @ CGL – ETH Zurich, Switzerland
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Vector Field Topology Lagrangian Coherent Structures
Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 2
Crit. pts. & streamlines Instantaneous view Fast
Ridges in Lyapunov exponent Transient view Slow (trajectory per point & time) Grid Advection
Lagrangian Coherent Structures (LCS)
Shadden et al. 2005
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Vector Field Topology Lagrangian Coherent Structures
Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 3
Crit. pts. & streamlines Instantaneous view Fast
Ridges in Lyapunov exponent Transient view Slow (trajectory per point & time) Grid Advection
Lagrangian Coherent Structures (LCS)
Shadden et al. 2005
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Confluences Glaciers
LCS = Interfaces LCS = Moraines
LCS in Nature
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from: www.scienceclarified.com/Ga-He/Glacier.htmlfrom: www.publicaffairs.water.ca.gov/swp/swptoday.cfm
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Finite-Time Lyapunov Exponent (FTLE)
FTLE: “growth of perturbation after advection time T”
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01, , ln /FTLE t TT
x
0t T
0t
x0t
0t T
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FTLE Computation
By pre-sampled flow map
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0 0
:tt t t x x x
0tx tx
Shadden et al. 2005
t0=FTLE
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FTLE Computation
By pre-sampled flow map
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0
0 2 t Tt x
0 0
:tt t t x x x
0tx tx
Shadden et al. 2005
t0=FTLE
max2TA A A
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FTLE Computation
By pre-sampled flow map
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0
0max 0 2
1, , ln t TtFTLE t T
Tx x
0 0
:tt t t x x x
max2TA A A
0tx tx
Shadden et al. 2005
t0=FTLE
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FTLE Computation
By pre-sampled flow map
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0
0max 0 2
1, , ln t TtFTLE t T
Tx x
0 0
:tt t t x x x
max2TA A A
0tx tx
t0=FTLE
Shadden et al. 2005
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Related Work
(1) “Distinguished Material Surfaces and Coherent Structures in Three-dimensional Fluid Flows”, G. Haller, 2001
(2) “Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications”, C. Garth et al., 2007
(3) “Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction”, F. Sadlo et al., 2007
(4) “Ridges in Image and Data Analysis”, D. Eberly, 1996
(5) “Automatic Detection of Open and Closed Separation and Attachment Lines”, D. Kenwright, 1998
(6) “Fast and Robust Extraction of Separation Line Features”, X. Tricoche et al., 2005
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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
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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
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Flow Separation: Forward Advection of Sampling Grid
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Initial Grid Grid Adaptation
Grid Advection Grid Adaptation
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Grid Growing
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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
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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)
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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%
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Intake of Power Plant
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Intake of Power Plant
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Adapted sampling grid 8 advection steps since
last resample
Initial sampling grid Negative-time path lines
Resulting ridge
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Grid Advection
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Grid Advection
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Intake of Power Plant: Performance
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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)
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End
Thanks for your attentionQuestions?
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Grid Advection
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Moraines and LCS
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“Appearing as dark lines on the surface, moraines indicate how many smaller glaciers feed into the system”
-> LCS, dynamical systems
from: www.fs.fed.us/r10/tongass/forest_facts/resources/geology/icefields.htm
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Flow Separation / Flow Attachment
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Flow Attachment: Backward Advection of Sampling Grid
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Initial Grid Grid Adaptation
Grid Advection Grid Adaptation
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Intake of Power Plant: LCS Error
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(a) ridge by advected grid
(b) ridge by uniform grid
(c) distance between (a) and (b) color-coded on (a)
(a)
(b)
(c)