Numerical analysis of elastica with obstacle and adhesion effects

ABSTRACT

We consider numerical analysis of a variational problem arising from materials science. The target functional is a type of Euler’s elastica 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 discuss convergence analysis for discrete minimizers in the framework of $\mathrm{\Gamma }$-convergence. We first show that the discrete energy functional $\mathrm{\Gamma }$-converges to the original one under the constraint that $W^{1,\infty }$-norm is bounded. Then, we establish the compactness property for the sequence of discrete minimizers under the same constraint. These two results allow us to extract a convergent subsequence from the discrete minimizers. We also present some numerical examples in the last part of the paper. Existence of singular local minimizers is suggested by numerical experiments.

KEYWORDS:

• Euler’s elastica energy
• elastica
• obstacle problem
• finite difference method
• $\mathrm{\Gamma }$-convergence

AMS SUBJECT CLASSIFICATIONS:

• 74G15
• 74G65

Acknowledgements

The author would like to thank Dr. Tatsuya Miura for bringing this topic to the author’s attention and for his valuable discussions. In particular, he suggested the author that there may be singular examples as presented in Section 5.2. Also, the author would like to thank the anonymous reviewers for valuable comments and suggestions to improve the quality of the paper.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the Program for Leading Graduate Schools, MEXT, Japan, and by JSPS KAKENHI [grant number 15J07471], Japan.