A Descent Scheme for Thick Elastic Curves with Self-contact and Container Constraints

We present a numerical method to simulate thick elastic curves that accounts for self-contact and container (obstacle) constraints under large deformations. The base model includes bending and torsion effects, as well as inextensibility. A minimizing movements, descent scheme is proposed for computing energy minimizers, under the non-convex inextensibility, self-contact, and container constraints (if the container is non-convex). At each pseudo time-step of the scheme, the constraints are linearized, which yields a convex minimization problem (at every time-step) with affine equality and inequality constraints. First order conditions are established for the descent scheme at each time-step, under reasonable assumptions on the admissible set. Furthermore, under a mild time-step restriction, we prove energy decrease for the descent scheme, and show that all constraints are satisfied to second order in the time-step, regardless of the total number of time-steps taken. Moreover, we give a modification of the scheme that regularizes the inequality constraints, and establish convergence of the regularized solution. We then discretize the regularized problem with a finite element method using Hermite and Lagrange elements. Several numerical experiments are shown to illustrate the method, including an example that exhibits massive amounts of self-contact for a tightly packed curve inside a sphere.