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.