Department of Biomedical Engineering and Mechanics
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A collaboration between School of Biomedical Engineering and Sciences and the Department of Engineering Science and Mechanics to form the Department of Biomedical Engineering and Mechanics.
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Browsing Department of Biomedical Engineering and Mechanics by Subject "2-dimensional maps"
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- The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifoldsLekien, F.; Ross, Shane D. (American Institute of Physics, 2010-03-01)We generalize the concepts of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Moumlbius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh.
- Topological chaos, braiding and bifurcation of almost-cyclic setsGrover, Piyush; Ross, Shane D.; Stremler, Mark A.; Kumar, Pankaj (American Institute of Physics, 2012-12-01)In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or " ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.