[00187] Analysis and geometry of inextensible materials
Session Time & Room : 3C (Aug.23, 13:20-15:00) @G801
Type : Proposal of Minisymposium
Abstract : There are many objects in the world around us that can be modeled as inextensible: pipes, chains, ribbons, cloth, whips, flagella, filaments, macromolecules, soft robot links, yarn, flags, cables in the ocean, galactic motion and octopus tentacles. In a certain sense, the inextensibility interpolates between rigid bodies and incompressible fluids but in comparison to them has many genuinely new difficulties due to the presence of unknown Lagrange multipliers. We intend to bring together some of the leading experts to discuss the modern ways to handle the analytical complexity of the PDE related to inextensible materials and the beautiful underlying geometry.
[02002] Modeling and Simulation of Thin Sheet Folding
Format : Talk at Waseda University
Author(s) :
Soeren Bartels (University of Freiburg)
Abstract : The folding of a thin elastic sheet along a curved arc has various applications including the construction of bistable devices. We discuss the derivation of a plate model from three-dimensional hyperelasticity and rigidity properties of admissible deformations and minimizers. The numerical solution is based on an isoparametric discontinuous Galerkin finite element method that provides a suitable geometric approximation of the folding arc. Error estimates are presented for a linearized version of the model problem.
[02066] Periodic partitions with minimal perimeter
Format : Talk at Waseda University
Author(s) :
Matteo Novaga (University of Pisa)
Annalisa Cesaroni (University of Padova)
Abstract : I will discuss existence and regularity of fundamental domains which minimize a general perimeter functional in a homogeneous metric measure space. In the planar case I will give a detailed description of the domains which are minimal for a general anisotropic perimeter.
[02439] Inextensible elastic curves and subriemannian maniflods
Format : Talk at Waseda University
Author(s) :
Chun-Chi Lin (National Taiwan Normal UniversiNational Taiwan Normal University)
Abstract : Elastic curves are relatively simple geometric objects in differential geometry but are related to many applications in image sciences and geometric control theory.
These curves are characterized the equilibrium configurations to the energy functional,
$$\int \, (a+b\cdot|\vec\kappa|^2) \, ds, $$
where $a, b$ are positive constants, $\vec\kappa$ is the curvature vector of curves and $s$ is the arclength of curves.
The Euler-Lagrange equations are fourth-order differential equations of the curves.
On the other hand, an energy decreasing flow of inextensible elastic curves in the plane can be formulated by a second-order parabolic equation with Lagrange multipliers.
However, difficulties come out as one generalizes this approach to curves in higher dimensional Euclidean spaces or Riemannian manifolds.
In this talk, we are interested in relating inextensible elastic curves to geometric analysis of curves in subriemannian manifolds.
We will introduce a different approach for problem and demonstrate our progress and results.
[02016] Gradient flows of inextensible networks
Format : Talk at Waseda University
Author(s) :
Dmitry Vorotnikov (Universidade de Coimbra )
Abstract : We address solvability of equations of overdamped motion of inextensible networks. Problems of this kind can be expressed as PDE that involve unknown Lagrange multipliers and non-standard boundary conditions related to the moving junctions. They can also be interpreted as gradient flows on certain ''manifolds'' of probability measures. We also discuss the geometry of these manifolds as well as links of our equations to the mean curvature flow and to fluid dynamics.
