such as compactness property investigated by M. Avellaneda and Lin in [@Avellaneda1987], and recently a thorough study for Neumann boundary condition by Shen et al. [@Kenig2013; @Shen2016]. As a model problem for the multiscale phenomenon, it also attracts great attention among scientific computation community. Due to the high oscillation emerging in the solution, classical computational method, such as finite element methods (FEM) can not reveal the fine scale information. Because of strong practical background, several modern multiscale computation methods have been developed since 1990s. We can classify those as three groups in methodology: modify FEM piecewise polynomial basis to enhance the expression ability, such as MsFEM [@Hou1999; @Efendiev2000; @Chen2002] and LOD [@Malqvist2014; @Henning2014]; utilize scale separation property to decompose original solution into coarsen and fine parts, VMM [@Hughes1995; @Brezzi1997; @Larson2007]; improve the accuracy of homogenized solution by involving smallscale information, HMM [@Weinan2005; @Abdulle2012]. In those methods Dirichlet problem is chosen when conducting numerical experiments and error analysis, while Robin problem or more general contact problems are scarcely investigated.

The notion of hemivariational inequalities was first introduced by Panagiotopoulos in the early 1980s [@Panagiotopoulos1983]. Since then, hemivariational inequalities receive broad applications in nonsmooth mechanics, contact mechanics, physics, and economics [@Naniewicz1995; @Panagiotopoulos1993; 