Parameter-unbounded convergence of Crossed-Secant accelerated Uzawa and penalty-splitted algorithms for frictionless contact

Abstract

We propose a unified iterative framework for the solution of frictionless mechanical contact problems, which relies exclusively on the solution of standard stiffness systems. The framework is built upon a two-step fixed-point algorithm: first, the displacement is computed for given contact forces; second, the contact forces are updated based on the displacement solution. The choice of the dual update scheme depends on the numerical contact formulation under consideration. Specifically, the Uzawa iterative scheme is obtained for the Lagrange multiplier formulation, while a penalty-based operator-splitting strategy is proposed for the penalty contact formulation. The main interest of such displacement-force splitting strategy is to involve only standard rigidity matrices in the solving step: no saddle-point or penalized ill-conditionned coefficient matrices have to be handled, so no specialized preconditioning is required. Moreover only the right-hand side of the system is updated throughout the iterations, which enables matrix factorization reuse or efficient iterative solvers initialization. The main limitation of such splitting iterative strategies lies in the inherently slow convergence of the underlying fixed-point iterations. Moreover, convergence is guaranteed only within a narrow range of numerical parameter values. This work addresses both issues by applying the Crossed-Secant fixed-point acceleration strategy, which substantially improves the convergence rate and renders the iterative schemes effectively parameter-unconstrained. To the best of our knowledge, this contribution provides the first computational demonstration of efficient, parameter-unbounded convergence for such contact formulations. The substantial practical benefits of the proposed approach are illustrated through representative three-dimensional academic and industrial frictionless contact problems.