G1-smooth planar parameterization of complex domains for isogeometric analysis

The construction of high-quality parameterizations for complex domains remains a significant challenge in isogeometric analysis. To address this issue, we propose a G1-smooth parameterization method for planar domains with arbitrary topology. Firstly, we generate a coarse decomposition of the given complex shape without internal singularities utilizing the extracted skeleton of the domain. We then perform a finer domain partition to obtain more accurate boundary representations. Both rectangular and triangular Bézier patches are employed to represent the geometry. Secondly, by imposing G1 continuity constraints on the adjacent patches, we construct an initial parameterization that is globally G1 continuous. Finally, we enhance the parameterization quality of each patch separately using the quasi-conformal mapping technique. Various examples are presented to justify the capability of the proposed method and to demonstrate its superiority over existing multi-patch parameterizations.