Construction of analysis-suitable G1 planar multi-patch parameterizations

The construction of smooth surfaces of complex shapes is at the heart of computer-aided design (CAD). Many different approaches generating C1-smooth surfaces are available and well-studied. Isogeometric analysis (IGA) has sparked new interest in these methods, since it allows to incorporate CAD based parameterizations into numerical simulations. In IGA one can utilize shape functions of global C1 continuity (or of higher continuity) over multi-patch geometries. Such functions can then be used to discretize high order partial differential equations, such as the biharmonic equation. However, the requirements posed by the IGA simulation are often different from the requirements in CAD. The construction of C1-smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations to ensure that the resulting C1 isogeometric spaces possess optimal approximation properties. For this purpose, we select so-called analysis-suitable G1 (AS-G1) parameterizations, proposed in Collin et al. (2016). In this work, we show through examples that it is possible to construct AS-G1 multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-G1 multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that C1 isogeometric spaces over AS-G1 multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.