Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines

We address the problem of constructing a high-quality parameterization of a given planar physical domain, defined by means of a finite set of boundary curves. We look for a geometry map represented in terms of Powell-Sabin B-splines. Powell-Sabin splines are C1 quadratic splines defined on a triangulation, and thus the parameter domain can be any polygon. The geometry map is generated by the following three-step procedure. First, the shape of the parameter domain and a corresponding triangulation are determined, in such a way that its number of corners matches the number of corners of the physical domain. Second, the boundary control points related to the Powell-Sabin B-spline representation are chosen so that they parameterize the boundary curve of the physical domain. Third, the remaining inner control points are obtained by solving a nimble optimization problem based on the Winslow functional. The proposed domain parameterization procedure is illustrated numerically in the context of isogeometric Galerkin discretizations based on Powell-Sabin splines. It turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution.