In each cell $\{i,j\}$, $i,j \in [0,N-1]$ all contacting asperities $\mathcal I_{ij} \subset \mathcal I$ contribute to the coarse-grained contact area fraction $A_{ij}$, and to the coarse-grained contact pressure $p_{ij}$, where $$A'_{ij} = \frac{1}{\Delta x^2}\sum\limits_{k\in\mathcal I_{ij}} \pi a_k^2,\quad p_{ij} = \frac{1}{\Delta x^2} \sum \limits_{k\in\mathcal I_{ij}} F_{k},
  \label{eq:coarse}$$ where $a_k$ is computed using  and $\mathcal I_{ij}$ is a set of asperities with in-plane coordinates located in the given cell.

\[sec:3\]Results
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Two regimes of the contact-area growth
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From the physical evidence, under most conditions and for most materials, the frictional force (static friction) increases practically linearly with the normal load independently on the nominal contact area, which is reflected by the Amontons-Coulomb’s friction law. Accordingly to the adhesive theory of contact [@bowden_tabor], the frictional force is proportional to the true contact area. Therefore, the linear growth of the contact area could be considered as a necessary condition of the model validity. The model of Greenwood and Tripp [@greenwood1967elastic], which represents the fully analytical treatment of the elastic indentation problem of a rough surface including elastic interaction between contacting asperities in the statistical sense , predicts the linearity between the force and the contact area, as well as the recent full 