Numerical analysis of elastica with obstacle and adhesion effects

Abstract

We consider the numerical computation of a variational problem that arises from materials science. The target functional is a type of elastic energy that is influenced by obstacles and adhesion. Owing to its strong nonlinearity and discontinuity, the Euler-Lagrange equation is very complicated, and numerical computation of its critical points is difficult. In this paper, we discretize and regularize the target energy as a functional defined on a space of polygonal curves. Moreover, we develop convergence analysis for discrete minimizers in the framework of Γ-convergence. We first show that the discrete energy functional Γ-converges to the original one. Then, we establish the compactness property for the sequence of discrete minimizers. These two results allow us to extract a convergent subsequence from the discrete minimizers. We also present some numerical examples in the last section of the paper. Existence of singular local minimizers is suggested by numerical experiments.

Asymmetric equilibria of two nested elastic rings

“…Further, beyond the elastic regime, thin sheets can develop intricate ridge networks when folded and crumpled [1]. More recently, the problem of confined elastic curves has gained interest amongst mathematicians [9,15,18,7] and control engineers [2].…”

Overhanging of membranes and filaments adhering to periodic graph substrates

“…To our knowledge, there is little mathematical literature of higher order problems with contact potentials except the aforementioned papers [17,23]. In [23], the author obtains an energy expansion as C → 0.…”

“…In [23], the author obtains an energy expansion as C → 0. The paper [17] studies a discretization of our model and proposes numerical results. As already mentioned, the papers [17,23] assume that admissible curves are graphs.…”

“…We finally give a brief remark on periodicity. This paper assumes that admissible curves and minimizers have a same period, but the paper [17] proposes a numerical example of a global minimizer of a period several times a substrate period. Hence, physically, a more natural assumption is that, if an original substrate has a period λ, then a minimizer γ has the period nλ for a positive integer n. It is not easy to determine n for a general case.…”

“…The graph setting yields strong topological and morphological constraints, and hence makes the analysis considerably simpler. In fact, there are theoretical studies [17,23,31] concerning the whole shapes of minimizers in our model, but all of them rely on the graph setting. The paper [31] particularly depends on the graph setting since its analysis crucially uses the small slope approximation.…”