Constructing planar domain parameterization with HB-splines via quasi-conformal mapping

Abstract

Constructing a high-quality parameterization of a computational domain is a fundamental research problem in isogeometric analysis, which has been extensively investigated so far. However, most of the current approaches employ non-uniform rational B-splines (NURBS) as the geometric representation of the physical domain. NURBS introduce redundant degrees of freedom due to their tensor-product structure. In this paper, we propose a new parameterization method for planar domains by adopting hierarchical B-splines (HB-splines) as the geometric representation that possess local refinement abilities. Starting from an initial parameterization such as a harmonic map, our method repeats the following two steps until a bijective parameterization with low distortion is achieved. First, a non-linear optimization model is proposed to compute a quasi-conformal map represented by HB-splines, and an efficient algorithm is provided to deal with this model by alternatively solving two quadratic optimization problems. Second, the parameterization is refined locally through HB-splines based on the bijectivity and conformal distortion of the parameterization. Several examples are demonstrated to verity the effectiveness and advantages of the proposed approach.