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

Abstract

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.