FTLE and LCSPranav Mantini

Contents• Introduction• Visualization• Lagrangian Coherent Structures• Finite-Time Lyapunov Exponent Fields• Example• Future Plan

Time-Varying Vector Fields

• Vector Field defines a vector(v(x)) at every point x on the grid

• In time variant vector field the vector defined at the points on the grid change with time(v(x,t)).

• Creating complex patterns and requires sophisticated techniques for analysis and visualization

Applications• Thorough analysis of

flows plays an important role in many different processes,o Airplane o Car designo Environmental researcho And medicine

• Deepwater Horizon Oil Spill

Mathematical Framework

• A time-varying vector field is a map

• Satisfies the conditions

o ) =

Visualization• Traditionally visualized using Vector Field

Topology.• Gives a simplified representation of a vector field

a dynamical system, with respect to the regions of different behavior.

• VFT deals with the detection, classification and global analysis of critical points

• VFT are significantly helpful for visualizing the time independent vector fields.

Visualization• In time varying vector fields, pathline diverge

from stream lines and the critical points move.• Forces to visualize only at a single point of time.• Coherent structures provide a more meaningful

representation

Lagrangian Coherent Structures

• LCS has gained attention in visualizing time dependent vector fields

• A set of LCS can represents regions that exhibits similar behavior

• example, a recirculation region can be delimited from the overall flow and can represent an isolated LCS

• LCS boundaries can be obtained by computing height ridges of the finite-time Lyapunov exponent fields

Real World Correspondence

Confluences Glaciers

LCS = Interfaces LCS = Moraines

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

Finite-Time Lyapunov Exponent

Fields• Scalar Value • Quantifies the amount of

stretching between two particles flowing for a given time

• High FTLE values correspond to particles that diverge faster than other particles in the flow field

High FTLE Values

Finite-Time Lyapunov Exponent

Fields• Advect each sample point at time with the flow

for time , resulting in a flow map • maps a sample point x to its advection position

Finite-Time Lyapunov Exponent

Fields• Give an arbitrary point

• Aim of the FTLE is to estimate the maximal growth of during the time period

•

+L2 Norm on both sides

=

𝑥 (𝑡 0 )

Finite-Time Lyapunov Exponent

Fields• Maximum value can be computed from• maximum stretching would then be the square

root of the largest eigenvalue of • FTLE is calculated as

Lagrangian Coherent Structures

• Ridge lines in these fields correspond to LCS• Height ridges are locations where a scalar field

has a local extremum in at least one direction• Ridge criterion can be formulated using the

gradient and the Hessian of the scalar field• Eigenvectors belonging to the largest eigenvalues

of the Hessian point along the ridge, and the smallest point orthogonal to the ridge.

Example• Velocity Field• Time-dependent double gyre• Domain  [0, 2] x [0, 1]

FTLE and LCS

• FTLE

• FTLE

LCS

Crowd Flow Segmentation & Stability

Analysis• (CVPR), 2007

Future Plan• It is obvious that the

LCS are influenced by the 3D geometry.

• It might be interesting to see how the change in geometry influences the LCS

Build Vector Field, Find LCS

Change Geometry

Estimate LCS

Future Plan• Week – 1&2: Estimate LCS for an example Vector

Field• Week – 3: Build a Vector Field from real world

scenario• Week – 4: Estimate LCS• Week – 5….: Try to estimate LCS based on

geometry and other information, in the absence of a vector field

References• “Visualizing Lagrangian Coherent Structures and

Comparison to Vector Field Topology” Filip Sadlo and Ronald Peikert Computer Graphics Laboratory, Computer Science Department

• Efficient Computation and Visualization of Coherent Structures in Fluid Flow Application. Christoph Garth, Florian Gerhardt, Xavier Tricoche, Hans Hagens

• http://mmae.iit.edu/shadden/LCS-tutorial/

Any Questions