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

TL;DR

This paper compares various optimization algorithms for solving the frictionless normal contact problem between rough surfaces using boundary element method, introducing a fast NNLS-based approach and a multi-resolution warm start for improved efficiency.

Contribution

It introduces a fast NNLS algorithm for contact problems and a multi-resolution warm start method, enhancing computational efficiency and accuracy in contact simulations.

Findings

NNLS method is two orders faster than Greedy CG.

The multi-resolution warm start improves convergence.

The approach is effective for self-similar fractal surfaces.

Abstract

This paper revisits the fundamental equations for the solution of the frictionless unilateral normal contact problem between a rough rigid surface and a linear elastic half-plane using the boundary element method (BEM). After recasting the resulting Linear Complementarity Problem (LCP) as a convex quadratic program (QP) with nonnegative constraints, different optimization algorithms are compared for its solution: (i) a Greedy method, based on different 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 ($iv$) the Non-Negative Least Squares (NNLS) algorithm, possibly warm-started by accelerated gradient projection steps or taking advantage of a loading history. The latter method is two orders of magnitude faster than the Greedy CG method and one order of magnitude faster than the constrained CG algorithm. Finally, we propose another type of warm start based on a refined criterion for the identification of the initial trial contact domain that can be used in conjunction with all the previous optimization algorithms. This method, called Cascade Multi-Resolution (CMR), takes advantage of physical considerations regarding the scaling of the contact predictions by changing the surface resolution. The method is very efficient and accurate when applied to real or numerically generated rough surfaces, provided that their power spectral density function is of power-law type, as in case of self-similar fractal surfaces.