American Institute of Mathematical Sciences

Advanced
January  2015, 2(1): 83-93. doi: 10.3934/jcd.2015.2.83

Attraction-based computation of hyperbolic Lagrangian coherent structures

Daniel Karrasch 1, , Mohammad Farazmand 2, and  George Haller 1,

1. 

ETH Zürich, Institute of Mechanical Systems, Leonhardstrasse 21, 8092 Zürich, Switzerland, Switzerland
2. 

ETH Zürich, Institute of Mechanical Systems, Rämistrasse 101, 8092 Zürich, Switzerland

Received  May 2014 Revised  October 2014 Published  August 2015

Recent advances enable the simultaneous computation of both attracting and repelling families of Lagrangian Coherent Structures (LCS) at the same initial or final time of interest. Obtaining LCS positions at intermediate times, however, has been problematic, because either the repelling or the attracting family is unstable with respect to numerical advection in a given time direction. Here we develop a new approach to compute arbitrary positions of hyperbolic LCS in a numerically robust fashion. Our approach only involves the advection of attracting material surfaces, thereby providing accurate LCS tracking at low computational cost. We illustrate the advantages of this approach on a simple model and on a turbulent velocity data set.
Keywords: mixing, Lagrangian Coherent Structures, Transport, tracking, stability., nonautonomous dynamical systems.
Mathematics Subject Classification: Primary: 37C60; Secondary: 37N1.
Citation: Daniel Karrasch, Mohammad Farazmand, George Haller. Attraction-based computation of hyperbolic Lagrangian coherent structures. Journal of Computational Dynamics, 2015, 2 (1) : 83-93. doi: 10.3934/jcd.2015.2.83
References:
[1]

M. Farazmand, D. Blazevski and G. Haller, Shearless transport barriers in unsteady two-dimensional flows and maps,, Physica D, 278-279 (2014), 278.  doi: 10.1016/j.physd.2014.03.008.  Google Scholar

[2]

M. Farazmand and G. Haller, Computing Lagrangian coherent structures from their variational theory,, Chaos, 22 (2012).  doi: 10.1063/1.3690153.  Google Scholar

[3]

M. Farazmand and G. Haller, Attracting and repelling Lagrangian coherent structures from a single computation,, Chaos, 23 (2013).  doi: 10.1063/1.4800210.  Google Scholar

[4]

M. Farazmand and G. Haller, How coherent are the vortices of two-dimensional turbulence?,, submitted preprint, ().   Google Scholar

[5]

G. Haller, Lagrangian Coherent Structures,, Annual Review of Fluid Mechanics, 47 (2015), 137.  doi: 10.1146/annurev-fluid-010313-141322.  Google Scholar

[6]

G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows,, Physica D, 241 (2012), 1680.  doi: 10.1016/j.physd.2012.06.012.  Google Scholar

[7]

G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent,, Chaos, 21 (2011).  doi: 10.1063/1.3579597.  Google Scholar

[8]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence,, Physica D, 147 (2000), 352.  doi: 10.1016/S0167-2789(00)00142-1.  Google Scholar

[9]

D. Karrasch, Attracting Lagrangian coherent structures on Riemannian manifolds,, Chaos, 25 (2015).   Google Scholar

[10]

A. M. Mancho, D. Small, S. Wiggins and K. Ide, Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields,, Physica D, 182 (2003), 188.  doi: 10.1016/S0167-2789(03)00152-0.  Google Scholar

[11]

K. Onu, F. Huhn and G. Haller, LCS Tool: A computational platform for Lagrangian coherent structures,, Journal of Computational Science, 7 (2015), 26.  doi: 10.1016/j.jocs.2014.12.002.  Google Scholar

[12]

R. Peikert and F. Sadlo, Height Ridge Computation and Filtering for Visualization,, in Visualization Symposium, (2008), 119.  doi: 10.1109/PACIFICVIS.2008.4475467.  Google Scholar

[13]

B. Schindler, R. Peikert, R. Fuchs and H. Theisel, Ridge Concepts for the Visualization of Lagrangian Coherent Structures,, in Topological Methods in Data Analysis and Visualization II (eds. R. Peikert, (2012), 221.  doi: 10.1007/978-3-642-23175-9_15.  Google Scholar

[14]

K.-F. Tchon, J. Dompierre, M.-G. Vallet, F. Guibault and R. Camarero, Two-dimensional metric tensor visualization using pseudo-meshes,, Engineering with Computers, 22 (2006), 121.  doi: 10.1007/s00366-006-0012-3.  Google Scholar

show all references

References:
[1]

M. Farazmand, D. Blazevski and G. Haller, Shearless transport barriers in unsteady two-dimensional flows and maps,, Physica D, 278-279 (2014), 278.  doi: 10.1016/j.physd.2014.03.008.  Google Scholar

[2]

M. Farazmand and G. Haller, Computing Lagrangian coherent structures from their variational theory,, Chaos, 22 (2012).  doi: 10.1063/1.3690153.  Google Scholar

[3]

M. Farazmand and G. Haller, Attracting and repelling Lagrangian coherent structures from a single computation,, Chaos, 23 (2013).  doi: 10.1063/1.4800210.  Google Scholar

[4]

M. Farazmand and G. Haller, How coherent are the vortices of two-dimensional turbulence?,, submitted preprint, ().   Google Scholar

[5]

G. Haller, Lagrangian Coherent Structures,, Annual Review of Fluid Mechanics, 47 (2015), 137.  doi: 10.1146/annurev-fluid-010313-141322.  Google Scholar

[6]

G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows,, Physica D, 241 (2012), 1680.  doi: 10.1016/j.physd.2012.06.012.  Google Scholar

[7]

G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent,, Chaos, 21 (2011).  doi: 10.1063/1.3579597.  Google Scholar

[8]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence,, Physica D, 147 (2000), 352.  doi: 10.1016/S0167-2789(00)00142-1.  Google Scholar

[9]

D. Karrasch, Attracting Lagrangian coherent structures on Riemannian manifolds,, Chaos, 25 (2015).   Google Scholar

[10]

A. M. Mancho, D. Small, S. Wiggins and K. Ide, Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields,, Physica D, 182 (2003), 188.  doi: 10.1016/S0167-2789(03)00152-0.  Google Scholar

[11]

K. Onu, F. Huhn and G. Haller, LCS Tool: A computational platform for Lagrangian coherent structures,, Journal of Computational Science, 7 (2015), 26.  doi: 10.1016/j.jocs.2014.12.002.  Google Scholar

[12]

R. Peikert and F. Sadlo, Height Ridge Computation and Filtering for Visualization,, in Visualization Symposium, (2008), 119.  doi: 10.1109/PACIFICVIS.2008.4475467.  Google Scholar

[13]

B. Schindler, R. Peikert, R. Fuchs and H. Theisel, Ridge Concepts for the Visualization of Lagrangian Coherent Structures,, in Topological Methods in Data Analysis and Visualization II (eds. R. Peikert, (2012), 221.  doi: 10.1007/978-3-642-23175-9_15.  Google Scholar

[14]

K.-F. Tchon, J. Dompierre, M.-G. Vallet, F. Guibault and R. Camarero, Two-dimensional metric tensor visualization using pseudo-meshes,, Engineering with Computers, 22 (2006), 121.  doi: 10.1007/s00366-006-0012-3.  Google Scholar

[1]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[2]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[3]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[4]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[5]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[6]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[7]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[8]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[9]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
[10]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[11]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[12]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[13]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[14]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[15]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[16]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[17]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[18]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161
[19]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[20]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

 Impact Factor: 

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences