Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection

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

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

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

Confluences Glaciers

LCS = Interfaces LCS = Moraines

LCS in Nature

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 4

from: www.scienceclarified.com/Ga-He/Glacier.htmlfrom: www.publicaffairs.water.ca.gov/swp/swptoday.cfm

Finite-Time Lyapunov Exponent (FTLE)

FTLE: “growth of perturbation after advection time T”

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 5

01, , ln /FTLE t TT

x

0t T

0t

x0t

0t T

FTLE Computation

By pre-sampled flow map

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 6

0 0

:tt t t x x x

0tx tx

Shadden et al. 2005

t0=FTLE

FTLE Computation

By pre-sampled flow map

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 7

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

FTLE Computation

By pre-sampled flow map

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 8

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

FTLE Computation

By pre-sampled flow map

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 9

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

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

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

Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection 18

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

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

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

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)