Volumetric spline parameterization for isogeometric analysis

Given the spline representation of the boundary of a three dimensional domain, constructing a volumetric spline parameterization of the domain (i.e., a map from a unit cube to the domain) with the given boundary is a fundamental problem in isogeometric analysis, and the quality of parameterization has a great influence on the accuracy and efficiency in subsequent analysis. In this paper, we propose a collocation-based approach for constructing volumetric parameterization with good quality. Firstly, a harmonic map is computed between a unit cube and the computational domain. Then the task of finding a good parameterization is modeled as a max–min optimization problem with a collocation-based formulation of the bijectivity condition of a map, and an algorithm based on the coarse-to-fine and divide-and-conquer strategy is proposed to solve the optimization problem efficiently. Finally, to further improve the quality of the parameterization, the MIPS (Most Isometric Parameterizations) method is adopted to reduce the conformal distortion of the solved map. We provide several examples to demonstrate the feasibility of our approach and to compare our approach with several state-of-the-art methods. The experimental results show that our algorithm can produce parameterizations with higher quality than previous approaches especially for complex domains.