Shape optimization of self-avoiding curves

Abstract

This paper presents a softened notion of proximity (or self-avoidance) for curves. We then derive a sensitivity result, based on shape differential calculus, for the proximity. This is combined with a gradient-based optimization approach to compute three-dimensional, parameterized curves that minimize the sum of an elastic (bending) energy and a proximity energy that maintains self-avoidance by a penalization technique. Minimizers are computed by a sequential-quadratic-programming (SQP) method where the bending energy and proximity energy are approximated by a finite element method. We then apply this method to two problems. First, we simulate adsorbed polymer strands that are constrained to be bound to a surface and be (locally) inextensible. This is a basic model of semi-flexible polymers adsorbed onto a surface (a current topic in material science). Several examples of minimizing curve shapes on a variety of surfaces are shown. An advantage of the method is that it can be much faster than using molecular dynamics for simulating polymer strands on surfaces. Second, we apply our proximity penalization to the computation of ideal knots. We present a heuristic scheme, utilizing the SQP method above, for minimizing rope-length and apply it in the case of the trefoil knot. Applications of this method could be for generating good initial guesses to a more accurate (but expensive) knot-tightening algorithm.

Gradient-based optimization of 3D MHD equilibria

“…Condition can be avoided by introducing a constraint on the global radius of curvature (Gonzalez & Maddocks 1999; Walker 2016) of each of the curves, , which will now be defined. The radius of the unique circumcircle containing any three points , , and (figure 1), can be computed from where is the area of a triangle with vertices , , and .…”

“…For this reason, the concept of the global radius of curvature has been employed for the shape optimization of finite-thickness knots (Gonzalez & Maddocks 1999; Carlen et al. 2005; Walker 2016).…”

“…where w R is a weight which sets the gradient length scale of the objective, R min is the minimum allowable major radius and A P is the area of S P . Condition (b) can be avoided by introducing a constraint on the global radius of curvature (Gonzalez & Maddocks 1999;Walker 2016) of each of the curves, x φ 0 (θ ), which will now be defined. The radius of the unique circumcircle containing any three points x 1 , x 2 , and x 3 (figure 1), can be computed from…”

“…is satisfied, this implies that the curve can be 'thickened' to a tube without self-contact surrounding C such that at any given point, the cross-section of the tube is a circle in the plane perpendicular to the local tangent vector of radius R (Gonzalez & Maddocks 1999). For this reason, the concept of the global radius of curvature has been employed for the shape optimization of finite-thickness knots (Gonzalez & Maddocks 1999;Carlen et al 2005;Walker 2016).…”

Sobolev Gradients for the Möbius Energy

“…Both theoretical and numerical results have been obtained on linear combinations of the bending energy and the Möbius energy [51,61,95]. More generally, in order to find minimizers of an elastic energy within an isotopy class, each knot energy can be employed in two ways: either as regularizer as it was done, e.g., in [4][5][6]29,32,40,94,97], or by using it to encode a hard bound into the domain, which was done with the knot thickness in [39,76,96]. Fig.…”