357 research outputs found
Subdivision Shell Elements with Anisotropic Growth
A thin shell finite element approach based on Loop's subdivision surfaces is
proposed, capable of dealing with large deformations and anisotropic growth. To
this end, the Kirchhoff-Love theory of thin shells is derived and extended to
allow for arbitrary in-plane growth. The simplicity and computational
efficiency of the subdivision thin shell elements is outstanding, which is
demonstrated on a few standard loading benchmarks. With this powerful tool at
hand, we demonstrate the broad range of possible applications by numerical
solution of several growth scenarios, ranging from the uniform growth of a
sphere, to boundary instabilities induced by large anisotropic growth. Finally,
it is shown that the problem of a slowly and uniformly growing sheet confined
in a fixed hollow sphere is equivalent to the inverse process where a sheet of
fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless,
quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl
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Discrete Differential Geometry
Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session
Integrable Background Geometries
This work has its origins in an attempt to describe systematically the
integrable geometries and gauge theories in dimensions one to four related to
twistor theory. In each such dimension, there is a nondegenerate integrable
geometric structure, governed by a nonlinear integrable differential equation,
and each solution of this equation determines a background geometry on which,
for any Lie group , an integrable gauge theory is defined. In four
dimensions, the geometry is selfdual conformal geometry and the gauge theory is
selfdual Yang-Mills theory, while the lower-dimensional structures are
nondegenerate (i.e., non-null) reductions of this. Any solution of the gauge
theory on a -dimensional geometry, such that the gauge group acts
transitively on an -manifold, determines a -dimensional
geometry () fibering over the -dimensional geometry with
as a structure group. In the case of an -dimensional group acting
on itself by the regular representation, all -dimensional geometries
with symmetry group are locally obtained in this way. This framework
unifies and extends known results about dimensional reductions of selfdual
conformal geometry and the selfdual Yang-Mills equation, and provides a rich
supply of constructive methods. In one dimension, generalized Nahm equations
provide a uniform description of four pole isomonodromic deformation problems,
and may be related to the Toda and dKP equations via a
hodograph transformation. In two dimensions, the Hitchin
equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while
the Hitchin equation leads to a Euclidean analogue of
Plebanski's heavenly equations.Comment: for Progress in Twistor Theory, SIGM
Arnold-type Invariants of Curves on Surfaces
Recently V. Arnold introduced Strangeness and invariants of generic
immersions of an oriented circle to . Here these invariants are
generalized to the case of generic immersions of an oriented circle to an
arbitrary surface . We explicitly describe all the invariants satisfying
axioms, which naturally generalize the axioms used by V. Arnold.Comment: 25 pages, 11 figure
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