605 research outputs found
Uniform convergence of discrete curvatures from nets of curvature lines
We study discrete curvatures computed from nets of curvature lines on a given
smooth surface, and prove their uniform convergence to smooth principal
curvatures. We provide explicit error bounds, with constants depending only on
properties of the smooth limit surface and the shape regularity of the discrete
net.Comment: 21 pages, 8 figure
Shape selection in non-Euclidean plates
We investigate isometric immersions of disks with constant negative curvature
into , and the minimizers for the bending energy, i.e. the
norm of the principal curvatures over the class of isometric
immersions. We show the existence of smooth immersions of arbitrarily large
geodesic balls in into . In elucidating the
connection between these immersions and the non-existence/singularity results
of Hilbert and Amsler, we obtain a lower bound for the norm of the
principal curvatures for such smooth isometric immersions. We also construct
piecewise smooth isometric immersions that have a periodic profile, are
globally , and have a lower bending energy than their smooth
counterparts. The number of periods in these configurations is set by the
condition that the principal curvatures of the surface remain finite and grows
approximately exponentially with the radius of the disc. We discuss the
implications of our results on recent experiments on the mechanics of
non-Euclidean plates
Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets
The edges of torn plastic sheets and growing leaves often display
hierarchical buckling patterns. We show that this complex morphology (i)
emerges even in zero strain configurations, and (ii) is driven by a competition
between the two principal curvatures, rather than between bending and
stretching. We identify the key role of branch-point (or "monkey-saddle")
singularities in generating complex wrinkling patterns in isometric immersions,
and show how they arise naturally from minimizing the elastic energy.Comment: 6 pages, 6 figures. This article supersedes arXiv:1504.0073
Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization
In this article, we study an analog of the Bj\"orling problem for isothermic
surfaces (that are more general than minimal surfaces): given a real analytic
curve in , and two analytic non-vanishing orthogonal
vector fields and along , find an isothermic surface that is
tangent to and that has and as principal directions of
curvature. We prove that solutions to that problem can be obtained by
constructing a family of discrete isothermic surfaces (in the sense of Bobenko
and Pinkall) from data that is sampled along , and passing to the limit
of vanishing mesh size. The proof relies on a rephrasing of the
Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its
discretization which is induced from the geometry of discrete isothermic
surfaces. The discrete-to-continuous limit is carried out for the Christoffel
and the Darboux transformations as well.Comment: 29 pages, some figure
Distributed branch points and the shape of elastic surfaces with constant negative curvature
We develop a theory for distributed branch points and investigate their role
in determining the shape and influencing the mechanics of thin hyperbolic
objects. We show that branch points are the natural topological defects in
hyperbolic sheets, they carry a topological index which gives them a degree of
robustness, and they can influence the overall morphology of a hyperbolic
surface without concentrating energy. We develop a discrete differential
geometric (DDG) approach to study the deformations of hyperbolic objects with
distributed branch points. We present evidence that the maximum curvature of
surfaces with geodesic radius containing branch points grow
sub-exponentially, in contrast to the exponential growth
for surfaces without branch points. We argue that, to optimize
norms of the curvature, i.e. the bending energy, distributed branch points are
energetically preferred in sufficiently large pseudospherical surfaces.
Further, they are distributed so that they lead to fractal-like recursive
buckling patterns.Comment: 59 pages, 20 figures. Major revisions including new proofs with
weakened hypotheses, expanded discussion and additional references. Some
images are not at their original resolution to keep them at a reasonable
size. Comments are very welcome and much appreciate
Optimization of gridshell bar orientation using a simplified genetic approach
Gridshells are defined as structures that have the shape and rigidity of a
double curvature shell but consist of a grid instead of a continuous surface.
This study concerns those obtained by elastic deformation of an initially flat
two-way grid. This paper presents a novel approach to generate gridshells on an
imposed shape under imposed boundary conditions. A numerical tool based on a
geometrical method, the compass method, is developed. It is coupled with
genetic algorithms to optimize the orientation of gridshell bars in order to
minimize the stresses and therefore to avoid bar breakage during the
construction phase. Examples of application are shown
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