Model-free data-driven computational mechanics (DDCM) is a new paradigm for simulations in solid mechanics. As in traditional approaches, the boundary-value problem is formulated with physics-based PDEs, such as the balance of momentum and compatibility equations, which define the admissibility conditions. However, DDCM does not use phenomenological constitutive laws to close the problem. Instead, it uses directly data on material response, originating from either experiments or lower-scale simulations, in order to avoid constitutive modeling's biases. The problem is posed in a phase space where the admissibility conditions define a manifold and the material behavior is represented by a set of material response points. DDCM solvers search element-wise for the admissible state that best matches the material data. This requires scanning the material dataset, a process which, despite being easily parallelizable, remains the computational bottleneck of the method. Many materials, such as metals, display both a well-defined linear-elastic behavior at small strains and a clear threshold after which non-linear phenomena kick in, making the modeling of the ensuing mechanical response much more challenging. Building on this fact, the present method of mesh data-refinement (``d-refinement'') aims to run more efficient simulations by, starting from a full linear-elastic finite-element model, iteratively enriching those elements that surpass the non-linearity threshold with a dataset that represents the non-linear mechanical response. As such, those computationally onerous phase-space searches are conducted only where and when necessary. This scheme is particularly well-suited for problems that present localization of strains or stress concentrations; in these cases, the computation speed is greatly improved and can match or surpass traditional solvers that do rely on a constitutive model.