Elastische Knoten
Zusammenfassung der Projektergebnisse
This project has been devoted to the study of knotted loops made of springy wire. Our model is based on the assumption that their thickness is infinitesimal and that their behavior can essentially be characterized by the geometry of the centerline, or more specifically of a framing that tracks the twisting of the material about the centerline. Equilibrium shapes correspond to minimizers of the bending energy or the elastic energy, respectively. Prescribing arc-length parametrization reflects the inextensibility of the wire. The technical challenge however consists in incorporating impermeability. To this purpose one can regularize the energy by a self-repulsive functional. This is by its nature not only nonlinear and nonconvex, but in first place highly nonlocal. Limits of minimizers as the regularization parameter approaches zero, so-called elastic knots, should not depend on the particular choice of a self-repulsive functional. There are several candidates for this purpose: perhaps the most natural choice is the ropelength functional, i.e., the quotient of length over thickness. It is particularly convenient for qualitative arguments as it models a uniform tube about the centerline. However, it is nonsmooth which renders the definition of a gradient flow quite intrinsic. For numerical simulations we employ the family of tangent-point potentials as they are smooth, require only a two-dimensional integration domain, and yield in fact an approximation of ropelength. In a joint work with Henryk Gerlach and Heiko von der Mosel we have been able to rigorously characterize global minimizers in certain special cases without any initial assumption on the geometry. Substantial progress has been made with respect to discretization and the design of numerical schemes in collaboration with Sören Bartels. Here we aim at finding (at least local) minimizers as stationary configurations of suitable gradient flows. A numerical H 2 -flow based on a finite element discretization has remarkable properties, leading to evolutions which move considerably faster than a corresponding L 2 -flow. In particular we have been able to derive a stability result which guarantees both the decay of energy along the flow and maintenance of arclength parametrization. However, gaining insight in the shapes of limit configurations requires a careful choice of certain variables, e.g., reducing the regularization parameter also requires refining the mesh. Numerical experiments suggest that the geometry of the respective energy landscape is quite complex. Moreover, even in cases where we already know the limit configuration as the regularization parameter tends to zero, its shape may not be well approximated for numerically feasible choices of that regularization parameter. In fact, more theoretical insight is required in order to check the validity of simulations. As in many similar cases, it is difficult to numerically detect global minimizers.
Projektbezogene Publikationen (Auswahl)
- On non-convex anisotropic surface energy regularized via the Willmore functional: the two-dimensional graph setting. ESAIM: COCV, 23(3):1047–1071, 2017
Paola Pozzi and Philipp Reiter
(Siehe online unter https://doi.org/10.1051/cocv/2016024) - The elastic trefoil is the doubly covered circle. Arch. Rat. Mech. Anal., 225(1):89–139, 2017
Henryk Gerlach, Philipp Reiter, and Heiko von der Mosel
(Siehe online unter https://doi.org/10.1007/s00205-017-1100-9) - A simple scheme for the approximation of self-avoiding inextensible curves. IMA J. Numer. Anal., 38(2):543–565, 2018
Sören Bartels, Philipp Reiter, and Johannes Riege
(Siehe online unter https://doi.org/10.1093/imanum/drx021) - Open and closed random walks with fixed edgelengths in Rd. Journal of Physics A: Mathematical and Theoretical, 51(43):434002, 2018
Jason Cantarella, Kyle Chapman, Philipp Reiter, and Clayton Shonkwiler
(Siehe online unter https://doi.org/10.1088/1751-8121/aade0a) - Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves
Sören Bartels and Philipp Reiter
