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Approximating gradient flow evolutions of self-avoiding inextensible curves and elastic knots

Tuesday, June 25, 2019 - 9:00am - 10:00am
Keller 3-180
Soeren Bartels (Albert-Ludwigs-Universität Freiburg)
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of a bending energy and a so-called tangent-point functional. We define evolutions via the gradient flow for the total energy within a class of arclength parametrized curves, i.e., given an initial curve we look for a family of inextensible curves that solves the nonlinear evolution equation.

Our numerical approximation scheme for the evolution problem is specified via a semi-implicit discretization of the equation with an explicit treatment of the tangent-point functional and a linearization of the arclength condition. The scheme leads to sparse systems of linear equations in the time steps for cubic splines and a nodal treatment of the constraints. The explicit treatment of the nonlocal and nonlinear tangent-point functional avoids working with fully populated matrices and furthermore allows for a straightforward parallelization of its computation.

Based on estimates for the second derivative of the tangent-point functional and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization.

We present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in certain knot classes, so-called elastic knots.

This is joint work with Philipp Reiter (University of Halle).