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Material coherence from trajectories via Burau eigenanalysis of braids
Melissa Yeung1, David Cohen-Steiner2, Mathieu Desbrun1
1Computing + Mathematical Sciences, Caltech, Pasadena, California 91125, USA.
Chaos (Woodbury, N.Y.)
|April 3, 2020
Summary
This study introduces a new computational method using topological braids to analyze material coherence from passive tracer trajectories. This approach efficiently identifies coherent regions in dynamical systems, even with large datasets.
Area of Science:
- Fluid dynamics
- Computational physics
- Materials science
Background:
- Understanding material coherence is crucial for analyzing dynamical systems.
- Previous methods for analyzing tracer trajectories are computationally intensive.
Purpose of the Study:
- To develop a scalable numerical tool for studying material coherence.
- To identify coherent regions within dynamical systems using Lagrangian trajectories.
Main Methods:
- Utilizing 2D Lagrangian trajectories of passive tracers.
- Applying the Burau representation of topological braids.
- Solving an eigenvalue problem to find levelsets corresponding to Nielsen-Thurston decomposition.
Main Results:
- Eigenvectors of the Burau representation reveal components of the dynamical system's decomposition.
- The method successfully detects and identifies clusters of space-time trajectories.
- Identified clusters correspond to coherent regions within the dynamical system.
Conclusions:
- The braid-based approach offers a scalable and efficient method for analyzing material coherence.
- This tool enables the study of large datasets of trajectories.
- Provides a novel way to understand the structure of dynamical systems.
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