A Three-Level Global-Local Coupling Strategy with Non-Matching Interfaces for Crack Propagation Analysis in Beam-Solid Structures

Global-local non-intrusive coupling is a well-established strategy for the analysis of structures with localised non-linear behaviour, but its practical applicability is limited by the requirement that the local patch coincides with a node of the global mesh. This work proposes a three-level non-intrusive global-local coupling strategy in which an intermediate 1D beam model is inserted between the global and the local domains, restoring full freedom in the placement of the local patch without modifying the global model. The formulation transfers displacements and reactions between two non-coincident interfaces through explicit projection operators, and the additional level introduces only three beam elements and four nodes per coupling region. The methodology is validated on a cantilever beam (linear, notched and crack-propagation variants) and on a three-storey framed structure with X-FEM crack propagation, against a monolithic 3D reference and against the standard matching-interface global-local method. For the framed structure, the non-matching scheme reduces the total degrees of freedom by 97.46% with respect to the monolithic solution and cuts the computation time by 68.19%, compared to 53.86% for the matching-interface version. Displacement errors remain below 10% and iteration counts drop by up to one third with respect to the matching scheme. The proposed formulation is therefore accurate, computationally efficient, and geometrically flexible in scenarios where the global model cannot be modified.