A simple scheme for the approximation of self-avoiding inextensible curves

Abstract

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 and the tangent-point functional.Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization.Finally 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.Date: April 9, 2018. 2010 Mathematics Subject Classification. 65N12 (57M25 65N15 65N30).

Finite element simulation of nonlinear bending models for thin elastic rods and plates

“…Finally, we do not change the stability of the iteration compared to a fully implicit time stepping scheme since the potential does not have obvious convexity properties. Using the differentiability properties of the functional TP and assuming that the flow metric is the H 2 scalar product, it has been shown in [17] that the conditional energy decay property…”

“…For the efficient iterative solution we adopt ideas from [45,41]. We discuss the treatment of bilayer plates following [14,13], illustrate a method that enforces injectivity of deformations in the case of rods following [19,17], and propose methods for the numerical solution of bending deformations with shearing effects following ideas from [12]. The problems considered in this article have similarities with problems related to the length-preserving elastic flow of curves and the surface area and volume preserving Willmore-Helfrich flow of closed surfaces but require different numerical methods.…”

Discrete Gradient Flows for General Curvature Energies

“…For theoretical results we refer to Polden (1996); Dziuk et al (2002); Grunau (2007, 2009) ;Schätzle (2010); Dall'Acqua et al (2017), while numerical approximations have been considered in Dziuk et al (2002); Deckelnick and Dziuk (2009); Barrett et al (2010Barrett et al ( , 2012. A scheme for the elastic flow of inextensible curves has been proposed and analysed in Bartels (2013); Bartels et al (2018). Here we stress that in Deckelnick and Dziuk (2009) a first stability result for a discretisation of elastic flow was presented, together with an error analysis.…”

Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms

“…This is further aggravated by the presence of a higher order term like (1.2) which rules out the (direct) use of piecewise affine finite elements. An alternative, more accessible penalization was recently proposed and studied in [6], but only for beams (effectively 1d). Here, after precisely outlining the model we work with in Section 2, we show that instead of (1.3), alternatives in the form of double integrals can be used (Section 3), for example…”

“…Therefore, this penalty term does not interfere with the existence of minimizers, even without a term like (1.2) regularizing the energy, since for fixed e 2 , it just acts as a compact perturbation. By contrast, the tangent point functional used in [15] in a similar role (for d = 1) is not so easy to extend to higher dimensions, depends on derivatives of y and also penalizes local curvature. We show that terms like (1.4) also lead to a limiting constraint equivalent to the standard Ciarlet-Necˇas condition (1.1) (Theorem 3.3) while the energy minima converge as before, at least for the regularized problem with fixed s .…”