This work presents an iterative framework for the robust and efficient solution of frictionless contact problems [1]. The algorithm follows a simple two-step fixed-point structure: (1) compute the displacement by solving a linear system with a fixed stiffness matrix, and (2) update the multiplier-like (dual) variable. Hence, only the right-hand side is updated at each iteration. The choice of the dual variable update depends on how the contact condition is enforced: the standard Lagrange multiplier formulation of the non-penetration constraint recovers the classical Uzawa method, whereas a penalized enforcement of the contact condition leads to a penalty-based operator-splitting approach. In this work, we focus on Uzawa-type methods, which have the advantage of avoiding the direct solution of ill-conditioned saddle-point systems in the Lagrange multiplier formulation. A longstanding limitation, however, is their slow convergence and the relatively narrow range of admissible algorithmic parameter. Several studies have proposed acceleration techniques [2], but these improvements remain effective only within the classical convergence bounds of the standard Uzawa method. A key contribution of this work is the integration of a Crossed Secant acceleration strategy [3], which significantly improves convergence and renders the formulation effectively parameter-unconstrained, ensuring robustness across a wide range of algorithmic parameters and enhancing its practical applicability. Benchmarking against the classical Uzawa algorithm shows computational time reductions of up to 15-50 times, depending on the application, and outperforms the acceleration achieved by existing techniques in the literature. The method’s performance and versatility are demonstrated on a range of academic and industrial 3D linear elasticity test cases. These results pave the way for reliable applications in more complex scenarios, including nonlinear material behavior, frictional contact, and large-scale simulations, with potential extensions to high-performance computing environnements. REFERENCES [1] V. Yastrebov, Numerical Methods in Contact Mechanics, ISTE Ltd and John Wiley & Sons, 2013. [2] Y. Kanno, An Accelerated Uzawa Method for Application to Frictionless Contact Problem, Optim. Letters, 2020. [3] I. Ramière & T. Helfer, Iterative residual-based vector methods to accelerate fixed point iterations, Comp. & Math. with Appl., 2015