D-PANA: a convergent block-relaxation solution method for the discretized dual formulation of the Signorini–Coulomb contact problem†

Signorini's law of unilateral contact and Coulomb's friction law constitute a simple and useful framework for the analysis of unilateral frictional contact problems of a linearly elastic body with a rigid support. For quasi-static, monotone-loadings, the discrete dual formulation of this problem leads to a quasi-variational inequality, whose unknowns, after condensation, are the normal and tangential contact forces at nodes of the initial contact area. A new block-relaxation solution technique is proposed here. At the typical iteration step, shown to be a contraction for small friction coefficients, two quadratic programming problems are solved one after the other: the former is a friction problem with given normal forces, the latter is a unilateral contact problem with prescribed tangential forces. The contraction principle is used to establish the well-posedness of the discrete formulation, to prove the convergence of the algorithm, and to obtain an estimate of the convergence rate.