Cracking prediction of concrete structures is a significant challenge in structural mechanics. Models formulated within the framework of continuum damage mechanics are generally used to model these problems. When the material softens, however, the solution to the equilibrium problem is no longer unique. In the context of the finite element method, this leads to a pathological dependence on the finite element mesh used to discretize the computational domain. The Lip-field approach introduces a new way to avoid spurious localizations and recover mesh independence. The idea is to impose a Lipschitz regularity on the internal variables controlling material softening. The solution to the problem is sought by minimizing an incremental potential, which is a function of displacement and damage. Such potential being convex with respect to each of the variables separately, an alternating minimization approach is used to solve the problem. At present, the use of the Lip-field method for solving large-size problems (structural scale) remains quite tricky, mainly due to the prohibitive computational costs of the alternating minimization solver. The aim of the Ph.D. thesis is to propose several improvements to the theoretical and numerical formulation (e.g., parallel computing, new discretization methods) of the problem, aiming to reduce the computation time of each step of the alternating minimization process (and mainly of the constrained minimization problem for computing the damage field). The Lip-field formulation will be applied finally to the simulation of reinforced concrete structures under quasi-static and dynamic loadings.