We consider vortices of topological-type for a class of selfdual gauge models, with periodic boundary conditions and as the ratio of the vortex core size to the separation distance between vortex points (the scaling parameter) tends to zero. We use a gluing technique (shadowing lemma) for solutions to the corresponding semilinear elliptic equation on the plane, where the vortex points are periodically arranged. This approach is particularly convenient and natural for the study of the asymptotics as the scaling parameter tends to zero. In particular, we prove a factorization ansatz for multivortex solutions, up to an error which is exponentially small.