scholarly journals Geometric curve flows in low dimensional Cayley–Klein geometries

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Joseph Benson ◽  
Francis Valiquette
Keyword(s):  
Evolution Equations ◽  
Three Dimensions ◽  
Geometric Flows ◽  
Moving Frames ◽  
Arc Length ◽  
Recursion Operators ◽  
Curvature Invariants ◽  
Curve Flows ◽  
Low Dimensional ◽  
Equivariant Moving Frames

Abstract Using the method of equivariant moving frames, we derive the evolution equations for the curvature invariants of arc-length parametrized curves under arc-length preserving geometric flows in two-, three- and four-dimensional Cayley–Klein geometries. In two and three dimensions, we obtain recursion operators, which show that the curvature evolution equations obtained are completely integrable.

Nonformal recursion operators and nonlinear integrable evolution equations

1986 ◽  
Vol 12 (2) ◽  
pp. 97-100 ◽  
Author(s):  
M. Boiti ◽  
B. G. Konopelchenko
Keyword(s):  
Evolution Equations ◽  
Recursion Operators ◽  
Integrable Evolution

Generalized vortex methods for free-surface flow problems

1982 ◽  
Vol 123 ◽  
pp. 477-501 ◽  
Author(s):  
Gregory R. Baker ◽  
Daniel I. Meiron ◽  
Steven A. Orszag
Keyword(s):  
Water Waves ◽  
Evolution Equations ◽  
Bottom Topography ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Three Dimensions ◽  
Fredholm Integral Equations ◽  
Free Surfaces ◽  
Flow Problems ◽  
Dipole Vortex

The motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths. The resulting Fredholm integral equations of the second kind may be solved efficiently in both storage and work by iteration in both two and three dimensions. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interfacial waves are given.

Mesoscale Simulation of Grain Growth

2004 ◽  
Vol 467-470 ◽  
pp. 1057-1062 ◽  
Author(s):  
D. Kinderlehrer ◽  
Jee Hyun Lee ◽  
Irene Livshits ◽  
Anthony D. Rollett ◽  
Shlomo Ta'asan
Keyword(s):  
Grain Growth ◽  
Evolution Equations ◽  
Materials Science ◽  
Statistical Significance ◽  
Dissipative System ◽  
Two Dimensions ◽  
Nonlinear Evolution Equations ◽  
Three Dimensions ◽  
Mesoscale Simulation ◽  
Large Systems

Simulation is becoming an increasingly important tool, not only in materials science in a general way, but in the study of grain growth in particular. Here we exhibit a consistent variational approach to the mesoscale simulation of large systems of grain boundaries subject to Mullins Equation of curvature driven growth. Simulations must be accurate and at a scale large enough to have statistical significance. Moreover, they must be sufficiently flexible to use very general energies and mobilities. We introduce this theory and its discretization as a dissipative system in two and three dimensions. The approach has several interesting features. It consists in solving very large systems of nonlinear evolution equations with nonlinear boundary conditions at triple points or on triple lines. Critical events, the disappearance of grains and and the disappearance or exhange of edges, must be accomodated. The data structure is curves in two dimensions and surfaces in three dimensions. We discuss some consequences and challenges, including some ideas about coarse graining the simulation.

scholarly journals Symmetry and reduction in collectives: low-dimensional cyclic pursuit

2016 ◽  
Vol 472 (2194) ◽  
pp. 20160465 ◽  
Author(s):  
Kevin S. Galloway ◽  
Eric W. Justh ◽  
P. S. Krishnaprasad
Keyword(s):  
Closed Loop ◽  
Physical Space ◽  
Relative Equilibria ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Control Schemes ◽  
Loop Dynamics ◽  
Cyclic Pursuit ◽  
Steering Law ◽  
Low Dimensional

We investigate low-dimensional examples of cyclic pursuit in a collective, wherein each agent employs a constant bearing (CB) steering law relative to exactly one other agent. For the case of three agents in the plane, we characterize relative equilibria and pure shape equilibria of associated closed-loop dynamics. Re-scaling time yields a reduction of phase space to two dimensions and effective tools for stability analysis. Study of bifurcation of a family of collinear equilibria dependent on a single CB control parameter reveals the presence of a rich collection of trajectories that are periodic in shape and undergo precession in physical space. For collectives in three dimensions, with an appropriate notion of CB pursuit strategy and corresponding steering law, the two-agent case proves to be explicitly integrable. These results suggest control schemes for small teams of mobile robotic agents engaged in area coverage tasks such as search and rescue, and raise interesting possibilities for behaviour in biological contexts.

Modeling Damage Evolution in Friction Stir Welding Process

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Youliang He ◽  
Paul R. Dawson ◽  
Donald E. Boyce
Keyword(s):  
Friction Stir Welding ◽  
Void Growth ◽  
Evolution Equations ◽  
Volume Fraction ◽  
Welding Process ◽  
Three Dimensions ◽  
Element Formulation ◽  
Operational Parameters ◽  
Friction Stir ◽  
Welding Processes

The evolution of voids (damage) in friction stir welding processes was simulated using a void growth model that incorporates viscoplastic flow and strain hardening of incompressible materials during plastic deformation. The void growth rate is expressed as a function of the void volume fraction, the effective deformation rate, and the ratio of the mean stress to the strength of the material. A steady-state Eulerian finite element formulation was employed to calculate the flow and thermal fields in three dimensions, and the evolution of the strength and damage was evaluated by integrating the evolution equations along the streamlines obtained in the Eulerian configuration. The distribution of internal voids within the material was qualitatively compared with experimental results, and a good agreement was observed in terms of the spatial location of voids. The effects of pin geometry and operational parameters such as tool rotational and travel speeds on the evolution of damage were also examined.

scholarly journals The Specious Art of Single-Cell Genomics

2021 ◽  
Author(s):  
Tara Chari ◽  
Joeyta Banerjee ◽  
Lior Pachter
Keyword(s):  
Single Cell ◽  
Large Scale ◽  
Three Dimensions ◽  
Large Scale Data ◽  
Biological Discovery ◽  
Low Dimensional ◽  
Supervised Dimension Reduction ◽  
Cell Expression ◽  
Biological Patterns ◽  
Global And Local

Dimensionality reduction is standard practice for filtering noise and identifying relevant dimensions in large-scale data analyses. In biology, single-cell expression studies almost always begin with reduction to two or three dimensions to produce 'all-in-one' visuals of the data that are amenable to the human eye, and these are subsequently used for qualitative and quantitative analysis of cell relationships. However, there is little theoretical support for this practice. We examine the theoretical and practical implications of low-dimensional embedding of single-cell data, and find extensive distortions incurred on the global and local properties of biological patterns relative to the high-dimensional, ambient space. In lieu of this, we propose semi-supervised dimension reduction to higher dimension, and show that such targeted reduction guided by the metadata associated with single-cell experiments provides useful latent space representations for hypothesis-driven biological discovery.

EQUATION-FREE, EFFECTIVE COMPUTATION FOR DISCRETE SYSTEMS: A TIME STEPPER BASED APPROACH

2005 ◽  
Vol 15 (03) ◽  
pp. 975-996 ◽  
Author(s):  
J. MÖLLER ◽  
O. RUNBORG ◽  
P. G. KEVREKIDIS ◽  
K. LUST ◽  
I. G. KEVREKIDIS
Keyword(s):  
Evolution Equations ◽  
Time Integration ◽  
Discrete Systems ◽  
Computer Assisted ◽  
Discrete Problem ◽  
Arc Length ◽  
Effective Equation ◽  
Discrete Evolution Equations ◽  
Coarse Model ◽  
The Continuum

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

Singularity formation in three-dimensional motion of a vortex sheet

1995 ◽  
Vol 300 ◽  
pp. 339-366 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Integration ◽  
Finite Time ◽  
Fourier Coefficients ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Leading Order ◽  
Singularity Formation

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

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