Analysis, Simulation, and Modeling of Elastic Curves 2019 - Schedule
Home
| Schedule and Abstracts
| Participants and Registration
| Venue
Venue:
Seminarraum 226, 2nd floor, Hermann-Herder-Str. 10, 79104 Freiburg (map).
Timetable:
Abstracts:
Basile Audoly
Systematic derivation of one‐dimensional strain‐gradient models for non‐linear elastic structures
Work in collaboration with Claire Lestringant (ETH, Switzerland).
We present a general method for deriving one‐dimensional models for non‐linear structures, that
capture the strain energy associated not only with the macroscopic strain as in traditional structural
models, but also with the strain gradient. The method is based on a two‐scale expansion, and is
asymptotically exact. It is applied to various types of non‐linear structures featuring localization, such
as a soft elastic beam that necks in the presence of surface tension, or a tape spring that snaps. In
contrast to traditional models, the one‐dimensional models obtained in this way can account for
significant and non‐uniform deformations of the cross‐sections, and can therefore capture
localization phenomena accurately.
Roberto Paroni
The energy of a Möbius band
In 1929 Sadowsky gave a constructive proof for the existence of a developable Möbius band
and posed the problem of determining the equilibrium configuration of a Möbius strip
formed from an unstretchable material. He tackled this latter problem variationally and he
deduced the bending energy for a strip whose width is much smaller than the length. This
energy, now known as Sadowsky's energy, depends on the curvature and torsion of the
centerline of the band and it is singular at the points where the curvature vanishes.
In this talk, we re‐examine the derivation of the Sadowsky’s energy by means of the theory
of Gamma‐convergence. We obtain an energy that is never singular and agrees with the
classical Sadowsky functional only for “large” curvature of the centerline of the strip.
The talk is based on ongoing joint work with L. Freddi, P. Hornung, and M.G. Mora.
Gert van der Heijden
Quasi‐rods
We use the term quasi‐rods for a class of variational problems on curves that includes the standard
linearly elastic rod but also more complicated elastic structures with kinematic constraints that can
be reduced, by elimination of the constraints, to effective elastic rods with exotic constitutive
relations.
Elisabeth Wacker
Total Curvature of Curves in the $C^1$‐Closure of Knot Classes
The Fáry-Milnor theorem gives a lower bound on the total curvature $TC$ of a simple closed curve $\tilde \gamma$:
$$
TC(\tilde \gamma) \geq 2 \pi \mu( [\tilde{\gamma}])
$$
(see [Milnor 1950]). Here, $\mu( [\tilde \gamma])$ denotes the bridge number of the knot class $ [\tilde \gamma]$ that is the infimum of the number of minima which can be achieved in any one dimensional projection of any curve in $[\tilde \gamma]$.
Our goal is to transfer this bound to elements of the $C^1$- closure of the knot class in the following sense:
Let $(\gamma_{k})_{k \in \mathbb{N}}$
be a sequence of simple closed curves in the same knot class, i.e. $[\gamma_k]=[\gamma_{k+1}]$ for all $k \in \mathbb{N}$, such that
$\gamma_k \longrightarrow \gamma$ in $C^1$.
Notice that the limit $\gamma$ might contain self-intersections. Then
$$
TC(\gamma) \geq 2 \pi \mu( [\gamma_k])
$$
holds for $\gamma \in W^{2,2}$ with only finitely many, isolated self-intersections.
[Milnor 1950] Milnor, J. W.: On the total curvature of knots. In: The
Annals of Mathematics, 2nd Ser., Vol. 52, No. 2. (1950), S. 248-257.
doi.org/10.2307/1969467.
Pascal Dolejsch
Elastic Curves Confined to Spheres
Elastic rods that are inextensible and torsion-free can be
modelled as embedded 1D curves. The relaxation of arbitrary initial
shapes towards a minimally bent configuration, the elastica, can be
modelled by a gradient-flow method. We therefore minimize the
integrated curvature while preventing the curve from extending or
contracting by additional constraints. This method, also including a
mechanism to avoid self-intersections of the curve, has been previously
studied. Now, we examine penalty algorithms to confine the curves to
spherical domains. This allows to describe possible elastica shapes and
minimal energies.
Peter Hornung
Narrow Elastic Ribbons
In the 1930s Sadowsky showed the existence of a developable Moebius strip and proposed a
natural energy functional governing the behaviour of narrow elastic strips. Wunderlich later
formally justified the energy functional proposed by Sadowsky. A rigorous derivation in
terms of Gamma‐convergence was recently been given by Kirby and Fried.
In this talk we present in some detail a different derivation. Our analysis makes no a priori
assumptions beyond a natural energy bound.
The functional we obtain agrees with the classical Sadowsky functional, but only when the
curvature of the centreline of the strip is large enough.
This is joint work with L. Freddi, M.G. Mora and R. Paroni.
Philipp Reiter
A bending‐torsion model for elastic rods
A physical wire can be modeled by a framed curve. We assume that its behavior is driven by
a linear combination of bending energy and twist energy. The latter tracks the rotation of
the frame about the centerline of the curve.
In order to obtain a more realistic setting, we have to preclude self‐intersections of the curve
which can be achieved by adding a self‐avoiding term, namely the tangent‐point potential.
We discuss the discretization of this model and present some numerical simulations.
This is joint work with Sören Bartels.
Klaus Deckelnick
Boundary value problems for the one‐dimensional Helfrich functional
We consider the one‐dimensional Helfrich functional in the class of graphs
subject to either Dirichlet or Navier boundary conditions. The talk presents results
concerning existence, uniqueness and qualitative properties of minimisers.
This is joint
work with Anna Dall'Acqua (Ulm).
Francesco Palmurella
A Resolution of the Poisson Problem for Elastic Plates
The Poisson problem consists in finding a surface immersed
in the Euclidean space minimising Germain's elastic energy (known as
Willmore energy in geometry) with assigned boundary, boundary Gauss
map and area; it constitutes a non‐linear model for the equilibrium
state of thin, clamped elastic plates.
We present a solution, and discuss its partial boundary regularity, to
a variationally equivalent version of this problem when the boundary
curve is simple and closed, as in the most classical version of the
Plateau problem.
This is a joint work with F. Da Lio & T. Rivière.
Julia Menzel
Long Time Existence of Solutions to an Elastic Flow of Networks
In this talk we consider the $L^2$ gradient flow of the elastic energy of networks in
$\mathbb{R}^2$ which leads to a fourth order evolution law with non‐trivial nonlinear boundary
conditions. Hereby we study configurations consisting of a finite union of curves
that meet in triple junctions and may or may not have endpoints fixed in the plane.
Starting from a suitable initial network we prove that the ow exists globally in
time or at least one of the following happens: as the time approaches the maximal
time of existence, the length of at least one curve tends to zero or at one of the
triple junctions of the network all the angles between the concurring curves tend
to zero or to $\pi$.
This is joint work with Harald Garcke and Alessandra Pluda.
Ernst Kuwert
Asymptotic estimates for the Willmore flow with small Energy
For the Willmore flow in the almost umbilical case, we prove asymptotic estimates for several
geometric quantities. Some of these relate to a rigidity result of DeLellis‐Müller and an estimate for
the isoperimetric deficit due to Röger‐Schätzle (joint work with Julian Scheuer).
Simon Blatt
The gradient flow of $p$‐elastic energies
I want to speak about ongoing work on the negative $L^2$ gradient flow of p‐elastic energies for
curves. After a quick overview over known results for this equation we will discuss two methods to
get longtime existence for these evolution equations, a regularization method and de Giorgi’s
method of minimizing movements.
This is joint work with Christoher Hopper und Nicole Vorderobermeier.
Andrea Mennucci
Some examples of Riemannian metrics of curves
There has been wide interest in recent years regarding Riemannian metrics of curves. Many models
have been proposed in the literature and studied , both theoretically and in applications. I will
present examples of first and second order Riemannian metrics of curves, and discuss the merits and
downfalls of these.
Handout
Stefan Neukamm
Effective bending–torsion theory for rods with micro‐heterogeneous prestrain
We investigate rods made of nonlinearly elastic, composite–materials that feature a micro‐heterogeneous prestrain that oscillates on a scale that
is small compared to the length of the rod. As a main result we derive a homogenized bending–torsion theory for rods as $\Gamma$‐limit from 3D nonlinear elasticity by simultaneous
homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvature‐torsion tensor that captures the macroscopic effect of the micro‐heterogeneous prestrain. We device a formula that allows to compute the spontaneous curvature‐ torsion tensor by
means of a weighted average of the given prestrain, with weights depending on the geometry of the composite encoded by correctors. We observe a size‐effect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of the ratio between microstructure‐scale and thickness.
This joint work with R. Bauer (TU Dresden) and M. Schäffner (U Leipzig).
