Numerical analysis of elastica with obstacle and adhesion effects

TL;DR

This paper develops a numerical method for a complex elastic energy problem influenced by obstacles and adhesion, proving convergence of discrete solutions and providing numerical examples.

Contribution

It introduces a discretization and regularization approach for a nonlinear variational problem with obstacle and adhesion effects, along with convergence analysis via $ ext{Gamma}$-convergence.

Findings

Discrete energy $ ext{Gamma}$-converges to the original energy

Convergence of discrete minimizers is established

Numerical examples suggest existence of singular local minimizers

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 $\Gamma$-convergence. We first show that the discrete energy functional $\Gamma$-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.