Conformal parameterization for multiply connected domains: combining finite elements and complex analysis

Conformal parameterization plays an important role in isogeometric analysis. Genus zero surfaces with multiple boundary components (multiply connected domains) can be conformally mapped onto planar domains with circular holes (circle domains). This work introduces a novel method to compute such conformal mappings combining finite element and complex analysis methods. First, the surface is mapped to planar annulus with concentric circular slits using holomorphic differentials, which is carried out using a finite element method based on Hodge decomposition; second the slit domain is conformally mapped to a circle domain by a Laurent series method. Compared with existing algorithms, the proposed method is more efficient and robust. Numerical experiments demonstrate the efficiency and efficacy of the method.