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This paper revisits the fundamental equations for the solution of thefrictionless unilateral normal contact problem between a rough rigid surfaceand a linear elastic half-plane using the boundary element method (BEM). Afterrecasting the resulting Linear Complementarity Problem (LCP) as a convexquadratic program (QP) with nonnegative constraints, different optimizationalgorithms are compared for its solution: (i) a Greedy method, based ondifferent solvers for the unconstrained linear system (Conjugate Gradient CG,Gauss-Seidel, Cholesky factorization), (ii) a constrained CG algorithm, (iii)the Alternating Direction Method of Multipliers (ADMM), and ($iv$) theNon-Negative Least Squares (NNLS) algorithm, possibly warm-started byaccelerated gradient projection steps or taking advantage of a loading history.The latter method is two orders of magnitude faster than the Greedy CG methodand one order of magnitude faster than the constrained CG algorithm. Finally,we propose another type of warm start based on a refined criterion for theidentification of the initial trial contact domain that can be used inconjunction with all the previous optimization algorithms. This method, calledCascade Multi-Resolution (CMR), takes advantage of physical considerationsregarding the scaling of the contact predictions by changing the surfaceresolution. The method is very efficient and accurate when applied to real ornumerically generated rough surfaces, provided that their power spectraldensity function is of power-law type, as in case of self-similar fractalsurfaces.

Optimization algorithms for the solution of the frictionless normal contact between rough surfaces