2,687 research outputs found
Covariant un-reduction for curve matching
The process of un-reduction, a sort of reversal of reduction by the Lie group
symmetries of a variational problem, is explored in the setting of field
theories. This process is applied to the problem of curve matching in the
plane, when the curves depend on more than one independent variable. This
situation occurs in a variety of instances such as matching of surfaces or
comparison of evolution between species. A discussion of the appropriate
Lagrangian involved in the variational principle is given, as well as some
initial numerical investigations.Comment: Conference paper for MFCA201
The variational particle-mesh method for matching curves
Diffeomorphic matching (only one of several names for this technique) is a
technique for non-rigid registration of curves and surfaces in which the curve
or surface is embedded in the flow of a time-series of vector fields. One seeks
the flow between two topologically-equivalent curves or surfaces which
minimises some metric defined on the vector fields, \emph{i.e.} the flow
closest to the identity in some sense.
In this paper, we describe a new particle-mesh discretisation for the
evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms
are very natural for this problem because Lagrangian particles (particles
moving with the flow) can represent the movement of the shape whereas the
vector field is Eulerian and hence best represented on a static mesh. We
explain the derivation of the method, and prove conservation properties: the
discrete method has a set of conserved momenta corresponding to the
particle-relabelling symmetry which converge to conserved quantities in the
continuous problem. We also introduce a new discretisation for the geometric
current matching condition of (Vaillant and Glaunes, 2005). We illustrate the
method and the derived properties with numerical examples.Comment: I uploaded the wrong paper before! Here is the correct on
Geometric theory of flexible and expandable tubes conveying fluid: equations, solutions and shock waves
We present a theory for the three-dimensional evolution of tubes with
expandable walls conveying fluid. Our theory can accommodate arbitrary
deformations of the tube, arbitrary elasticity of the walls, and both
compressible and incompressible flows inside the tube. We also present the
theory of propagation of shock waves in such tubes and derive the conservation
laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the
tubes, and compute several examples of particular solutions. The theory is
derived from a variational treatment of Cosserat rod theory extended to
incorporate expandable walls and moving flow inside the tube. The results
presented here are useful for biological flows and industrial applications
involving high speed motion of gas in flexible tubes
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