Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection February 2009 – TopoInVis Filip VISUS – Universität Stuttgart,

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

Vector Field Topology Lagrangian Coherent Structures Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection2  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 Advection3  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 = InterfacesLCS = Moraines LCS in Nature Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection4 from:

Finite-Time Lyapunov Exponent (FTLE) FTLE: “growth of perturbation after advection time T” Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection5

FTLE Computation  By pre-sampled flow map  Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection6 Shadden et al t0=t0= FTLE

FTLE Computation  By pre-sampled flow map  Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection7 Shadden et al t0=t0= FTLE

FTLE Computation  By pre-sampled flow map  Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection8 Shadden et al t0=t0= FTLE

FTLE Computation  By pre-sampled flow map  Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection9 t0=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 Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection10

Motivation  Goal: Fast computation of FTLE animations (variation of t 0 )  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 Advection11

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 Advection12

Flow Separation: Forward Advection of Sampling Grid Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection13 Initial GridGrid Adaptation Grid AdvectionGrid Adaptation

Grid Growing Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection14

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 Advection15

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 Advection16

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 Advection17

Intake of Power Plant Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection18

Intake of Power Plant Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection19  Adapted sampling grid  8 advection steps since last resample  Initial sampling grid  Negative-time path lines  Resulting ridge

Grid Advection Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection20

Grid Advection Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection21

Intake of Power Plant: Performance Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection22

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 Advection23

End Thanks for your attention Questions? Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection24

Grid Advection Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection25

Moraines and LCS Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection26 “Appearing as dark lines on the surface, moraines indicate how many smaller glaciers feed into the system” -> LCS, dynamical systems from:

Flow Separation / Flow Attachment Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection27

Flow Attachment: Backward Advection of Sampling Grid Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection28 Initial GridGrid Adaptation Grid AdvectionGrid Adaptation

Intake of Power Plant: LCS Error Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection29 (a) ridge by advected grid (b) ridge by uniform grid (c) distance between (a) and (b) color-coded on (a) (a) (b) (c)