Spectrum analysis of , , and constructions for extraordinary points

G-splines are smooth spline surface representations that support control nets with arbitrary unstructured quadrilateral layout. Supporting any distribution of extraordinary points (EPs) is necessary to satisfactorily meet the demands of real-world engineering applications. G-splines impose \(G^1\) constraints across the edges emanating from the EPs, which leads to discretizations with global \(C^1\) continuity in physical space when used in isogeometric analysis (IGA). In this work, we perform spectrum analyses of G-splines for the first time. Our results suggest that G-splines do not have outliers near the boundary when uniform elements and control nets are used. When EPs are considered, G-splines result in significantly higher spectral accuracy than the D-patch framework. In addition, we develop G-spline discretizations that use bi-quartic elements around EPs instead of bi-quintic elements around EPs as it was the case in our preceding work. All the other elements are bi-cubic. Our evaluations of surface quality, convergence studies of linear elliptic boundary-value problems, and spectral analyses suggest that using bi-quartic elements around EPs is preferable for IGA since they result in similar performance as using bi-quintic elements around EPs while being more computationally efficient.