ALEX 2018 Workshop: Abstracts

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

Sören Bartels${}^{(1)}$ and Philipp Reiter${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$

(1) University of Freiburg, Department of Applied Mathematics (Germany)

(2) University of Georgia, Department of Mathematics (USA)

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 the bending energy $\mathrm{B}$ and the tangent-point functional $\mathrm{TP}$, i.e.,

$$E(u)=\kappa \mathrm{B}(u)+\varrho \mathrm{TP}(u)=\frac{\kappa }{2}{\int }_I{|u^{\mathrm{\prime \prime }}(x)|}^{2}dx+\varrho {\iint }_{I\times I}\frac{\mathrm{d}x\mathrm{d}y}{r{(u(y),u(x))}^q}$$

with the tangent-point radius $r(u(y),u(x))$ which is the radius of the circle that is tangent to the curve $u$ at the point $u(y)$ and that intersects with $u$ in $u(x)$.

We define evolutions via the gradient flow for $E$ within a class of arclength parametrized curves, i.e., given an initial curve $u^0\in H^2(I;{ℝ}^3)$ we look for a family $u:[0,T]\to H^2(I;{ℝ}^3)$ such that, with an appropriate inner product ${(\cdot ,\cdot )}_X$ on $H^2(I;{ℝ}^3)$,

$${({\partial }_{t}u,v)}_X=-\delta E(u)[v],u(0)=u^0,$$

subject to the linearized arclength constraints

$${[{\partial }_{t}u]}^{\prime }\cdot u^{\prime }=0,v^{\prime }\cdot u^{\prime }=0.$$

Our numerical approximation scheme for the evolution problem is specified via a semi-implicit discretization, i.e., for a step-size $\tau >0$ and the associated backward difference quotient operator $d_t$, we compute iterates ${(u^k)}_{k=0,1,\mathrm{\dots }}\subset H^2(I;{ℝ}^3)$ via the recursion

$${(d_t{u}^k,v)}_X+\kappa ({[u^k]}^{\mathrm{\prime \prime }},v^{\mathrm{\prime \prime }})=-\varrho \delta \mathrm{TP}(u^{k-1})[v]$$

with the constraints

$${[d_t{u}^k]}^{\prime }\cdot {[u^{k-1}]}^{\prime }=0,v\cdot {[u^{k-1}]}^{\prime }=0.$$

The scheme leads to sparse systems of linear equations in the time steps for cubic $C^1$ splines and a nodal treatment of the constraints. The explicit treatment of the nonlocal 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. The results are published in the article [2] and provide in combination with the spatial discretization estimates of [1] a quite complete numerical analysis.

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