To model efficiently the multi-scale behaviour of large composite structures, we proposed a multiscale spectral generalised finite element method (MS-GFEM) [1]. Our approach constructs a coarse approximation space (or reduced order model) of the full-scale underlying composite problem by solving local generalized eigen problem in an A-harmonic subspace. This significant aspect leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses. Most notably our method relies on a parallel construction and is inherently adaptive, we can control the error by simply setting a threshold on the eigenvalues to decide which eigenvectors need to be included into the local spaces, hence maximising model order reduction. Geometrically nonlinear behaviour must be considered to capture large displacements, particularly when buckling occurs prior to damage onset. We recently combined the MS-GFEM method and the Newton-Raphson algorithm, including geometric nonlinearity via the co-rotational formulation [2,3]. Domain decomposition within the MS-GFEM method and the subdomain independence enables parallelisation at each Newton iteration such that the residual internal forces and an update of the total displacement can be computed at the coarse level. Using a two-scale approach [4], where local Newton steps are carried out in parallel, reduces the total number of global iterations as well as providing an efficient and scalable way to update the coarse space. However, in contrast to [4], the method provides a better subdomain interaction at the coarse level due to the partition of unity method in the overlaps. Furthermore, to improve the global efficiency of the method when applied to larger structures, it is possible to avoid the local nonlinear solves and the updates of the local bases in subdomains where the structure exhibits little or no nonlinear behaviour. In fact, the displacement of the structure will in general not be uniform, and thus many subdomains will not be affected by large displacements. A criterion on the local tangent stiffness matrices is set to ensure that no unnecessary updates are performed.