We present here a novel numerical method for the kinematic limit analysis of masonry structures. A given masonry structure is first discretized with a regular mesh of elements idealized as rigid and infinitely resistant blocks. At the macro-scale, the kinematics is described by means of a discontinuous velocity field. The velocity field remains continuous within each element but presents discontinuities along the interfaces [1], which represent potential cracks. According to the hypotheses of the upper bound theorem, velocity discontinuities must respect the associated flow rule. With this aim, the homogenization model based on the Reissner-Mindlin plate theory presented in [2] is included in this method. At the micro-scale (i.e. the scale of the masonry texture), a representative volume element is identified and its kinematics is described in terms of strains, curvatures and rotations rates, which globally constitute the microscopic variables. The associated flow rule is thus written in terms of microscopic variables, which are in turn linked to the velocity discontinuities at the macro-scale. In this way, the kinematic limit analysis problem can be expressed as a conic programming problem. Its solution consists of an upper bound of the load bearing capacity which takes into account the actual texture of the material but remains strongly mesh dependent. To overcome this issue, we propose here a local mesh refinement technique inspired by [3]. Once a first velocity field has been found, the mesh is refined in proximity to the observed discontinuities, whereas adjacent blocks for which the velocity remains continuous are merged, allowing to improve the representation of the actual collapse shape and reduce the load factor iteration after iteration. This approach seems providing advantages in comparison with meta-heuristic mesh adaptation schemes already used in the limit analysis of masonry structures. Numerical examples are shown to demonstrate the potential of the method. [1] SW Sloan, PW Kleeman, Upper bound limit analysis using discontinuous velocity fields, Comput Methods Appl Mech Eng, 127:293–314, 1995 [2] A Checchi, K Sab, A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels, Int J Solids Struct, 44(18-19):6055–6079, 2007 [3] K Li, T Yu, TQ Bui, Efficient kinematic upper-bound limit analysis for hole/inclusion problems by adaptive XIGA with locally refined NURBS, Eng Anal Bound Elem, 133:138–152, 2021