---
abstract: 'This paper is devoted to studying a type of contact problems modeled by hemivariational inequalities with small periodic coefficients appearing in PDEs, and the PDEs we considered are linear, second order and uniformly elliptic. Under the assumptions, it is proved that the original problem can be homogenized, and the solution weakly converges. We derive an $O(\epsilon^{1/2})$ estimation which is pivotal in building the computational framework. We also show that Robin problems— a special case of contact problems, it leads to an $O(\epsilon)$ estimation in $L^2$ norm. Our computational framework is based on finite element methods, and the numerical analysis is given, together with experiments to convince the estimation.'
author:
- 'Changqing Ye[^1], Junzhi Cui[^2]'
bibliography:
- 'references.bib'
date: '[^3]'
title: 'A First-order Two-scale Analysis for Contact Problems with Small Periodic Configurations'
---

Introduction
============

In composite material design and performance optimization, the controlling PDEs within the models frequently involve small periodic coefficients (e.g., [@Yang2017; @Dong2018; @Liu2013; @Liu2015]). For those problems, the periodic homogenization theory is the basis, and many PDE experts contributed considerable works to build this theory. For examples: qualitative results such as asymptotic expansion [@Bensoussan1978; @Oleuinik1992], H-convergence [@Tartar2010], $\Gamma$-convergence [@DalMaso1993], and two-scale convergence [@Allaire1992]; regularity results 