Analysis, Simulation, and Modeling of Elastic Curves 2019 - Schedule

Venue:

Seminarraum 226, 2nd floor, Hermann-Herder-Str. 10, 79104 Freiburg (map).

Timetable:

 Monday Tuesday Wednesday 09:30 - 10:20 Audoly Hornung Blatt 10:20 - 11:10 Paroni Reiter Mennucci 11:10 - 11:40 Break Break Break 11:40 - 12:30 v.d. Heijden Deckelnick Neukamm 12:30 - 14:00 Lunch Lunch Lunch 14:00 - 14:50 Maddocks Palmurella 14:50 - 15:20 Break Break 15.20 - 15.50 Wacker Menzel (until 16:10) 15.50 - 16.20 Wolff-Vorbeck 16.20 - 16.50 Doljesch Kuwert (16:10 - 16:40)

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).