Overlapping Multipatch Isogeometric Method With Minimal Stabilization

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 nonconforming and trimmed, spline patches. Given a series of patches, we overlay them one on top of one 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 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. To conclude, we numerically verify the theory by solving the Poisson's equation on various geometries that are obtained by the union operation.