  We present a novel method for isogeometric analysis (IGA) to directly work on
geometries constructed by Boolean operations including difference (i.e.,
trimming), union and intersection. Particularly, this work focuses on the union
operation, which involves multiple independent, generally non-conforming and
trimmed spline patches. Given a series of patches, we overlay one on top of
another in a certain order. While the invisible part of each patch is trimmed
away, the visible parts of all the patches constitute the entire computational
domain. We employ the Nitsche's method to weakly couple independent patches
through visible interfaces. Moreover, we propose a minimal stabilization method
to address the instability issue that arises on the interfaces shared by small
trimmed elements. We show in theory that our proposed method recovers stability
and guarantees well-posedness of the problem as well as optimal error
estimates. In the end, we numerically verify the theory by solving the
Poisson's equation on various geometries that are obtained by the union
operation.
