Two-dimensional metric tensor visualization using pseudo-meshes

Tchon, Ko-Foa; Dompierre, Julien; Vallet, Marie-Gabrielle; Guibault, François; Camarero, R.

doi:10.1007/s00366-006-0012-3

Cited by 11 publications

(8 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By introducing an appropriate rescaling, we alter this ODE in a fashion so that its numerical solution yields a globally smooth set of strainlines. To achieve this, we follow the scaling suggested in Tchon et al 27 for general tensor lines by letting r 0 ðsÞ ¼ f rðsÞ; r 0 ðs À DÞ ð Þ ;…”

Section: Computing Smooth Strainlinesmentioning

confidence: 99%

Computing Lagrangian coherent structures from their variational theory

Farazmand

Haller²

2012

Chaos

185

161

View full text Add to dashboard Cite

Using the recently developed variational theory of hyperbolic Lagrangian coherent structures (LCSs), we introduce a computational approach that renders attracting and repelling LCSs as smooth, parametrized curves in two-dimensional flows. The curves are obtained as trajectories of an autonomous ordinary differential equation for the tensor lines of the Cauchy-Green strain tensor. This approach eliminates false positives and negatives in LCS detection by separating true exponential stretching from shear in a frame-independent fashion. Having an explicitly parametrized form for hyperbolic LCSs also allows for their further in-depth analysis and accurate advection as material lines. We illustrate these results on a kinematic model flow and on a direct numerical simulation of two-dimensional turbulence.

show abstract

Section: Computing Smooth Strainlinesmentioning

confidence: 99%

Computing Lagrangian coherent structures from their variational theory

Farazmand

Haller²

2012

Chaos

185

161

View full text Add to dashboard Cite

show abstract

“…Their computation, however, requires a number of nonstandard steps that complicate its implementation. These steps include (i) an accurate computation of eigenvalues and eigenvectors of tensor fields [16]; (ii) trajectory integration for direction fields as opposed to vector fields [17]; (iii) detection of singularities (regions of repeated eigenvalues) of tensor fields and identification of their topological type [8]; and (iv) selection of Poincaré sections (PSs) for locating closed direction-field trajectories (see [18] or appendix A).…”

Section: Introductionmentioning

confidence: 99%

Efficient computation of null geodesics with applications to coherent vortex detection

Serra¹,

Haller²

2017

Proc. R. Soc. A.

View full text Add to dashboard Cite

Recent results suggest that boundaries of coherent fluid vortices (elliptic coherent structures) can be identified as closed null-geodesics of appropriate Lorentzian metrics defined on the flow domain.Here we derive an automated method for computing such null-geodesics based on the geometry of the underlying geodesic flow. Our approach simplifies and improves existing procedures for computing variationally defined Eulerian and Lagrangian vortex boundaries. As an illustration, we compute objective vortex boundaries from satellite-inferred ocean velocity data. A MATLAB implementation of our method is available as supplementary material. *

show abstract

“…These two topologically distinct singularities serve as the organizing skeleton around which the rest of the SORF smoothly vary. These topologies have been observed numerically [15,16] but apparently not classified before.…”

Section: Behavior Near Singularitiesmentioning

confidence: 74%

“…• In dealing with points at which the eigenvector field is not defined, an approach is to mollify the field by multiplying with a sufficiently smooth field which is zero at such points (e.g., the square of the difference in the two eigenvalues [15]).…”

Section: Computational Issues Of Finding Foliationsmentioning

confidence: 99%

Globally optimal stretching foliations of dynamical systems reveal the organizing skeleton of intensive instabilities

Balasuriya¹,

Bollt²

2021

Preprint

View full text Add to dashboard Cite

Understanding instabilities in dynamical systems drives to the heart of modern chaos theory, whether forecasting or attempting to control future outcomes. Instabilities in the sense of locally maximal stretching in maps is well understood, and is connected to the concepts of Lyapunov exponents/vectors, Oseledec spaces and the Cauchy-Green tensor. In this paper, we extend the concept to global optimization of stretching, as this forms a skeleton organizing the general instabilities. The 'map' is general but incorporates the inevitability of finite-time as in any realistic application: it can be defined via a finite sequence of discrete maps, or a finite-time flow associated with a continuous dynamical system. Limiting attention to two-dimensions, we formulate the global optimization problem as one over a restricted class of foliations, and establish the foliations which both maximize and minimize global stretching. A classification of nondegenerate singularities of the foliations is obtained. Numerical issues in computing optimal foliations are examined, in particular insights into special curves along which foliations appear to veer and/or do not cross, and foliation behavior near singularities. Illustrations and validations of the results to the Hénon map, the double-gyre flow and the standard map are provided.

show abstract

Two-dimensional metric tensor visualization using pseudo-meshes

Cited by 11 publications

References 13 publications

Computing Lagrangian coherent structures from their variational theory

Computing Lagrangian coherent structures from their variational theory

Efficient computation of null geodesics with applications to coherent vortex detection

Globally optimal stretching foliations of dynamical systems reveal the organizing skeleton of intensive instabilities

Contact Info

Product

Resources

About