can be also done using Greenwood and Tripp’s model [@greenwood1967elastic]. The deterministic multi-asperity model, compared to statistical multi-asperity models [@greenwood1966prcl; @bush1975w; @carbone2008jmps], takes into account long-range interactions; and it enables to handle accurately very light loads, which are hardly reachable for conventional direct numerical simulations using finite element [@pei2005finite; @yastrebov2011cras] or modifications of boundary element methods [@stanley1997; @polonsky1999numerical; @campana2006practical; @bemporad2015optimization].

The paper is organized as follows. In Section \[sec:1\] we define the problem, recall the roughness generation method and explain the method to extract asperities’ data. In Section \[sec:2\], the numerical model to solve contact problems is presented. The results are presented in Section \[sec:3\], namely (1) the analysis of the true contact area evolution, the two relevant regimes and transition between them; (2) the study of contact pressure and contact area as well as their spatial and statistical distributions. We conclude by a short discussion of main results in Section \[sec:4\].

\[sec:1\]Problem set-up and roughness
=====================================

Set-up
------

Using the deterministic multi-asperity model with long-range interaction we will consider a problem of contact between locally axisymmetric parabolic solids \[see Fig. \[fig:setup\](a)\] of curvature radii $R_1$ and $R_2$, the solids are assumed to be elastic, homogeneous and isotropic with elastic 