We present a novel method to compute bijective domain parameterizations with low distortion for isogeometric analysis. Our key insight is that instead of solving the difficult mapping optimization problem, we fit the spline function to a piecewise linear map between computational and parametric domains while ensuring bijection. The basis of our key insight is that the bijection of the piecewise linear mapping is theoretically guaranteed for 2D Tutte (1963) and 3D Campen et al. (2016). Due to the continuity difference between the spline and piecewise linear maps, we develop a two-stage optimization strategy for robust map fitting. The first stage enforces the Jacobian similarity and the second optimizes the boundary difference. Besides, these two stages always guarantee bijection by explicit checks in combination with line search. We demonstrate the efficacy of our method on various complex models in both 2D and 3D. Compared to state-of-the-art methods, our method achieves higher robustness for computing bijective and low distortion domain parameterizations.

我们提出了一种新颖的方法来计算具有低畸变的双射域参数化，以进行等几何分析。我们的主要见解是，除了解决困难的映射优化问题之外，我们还将样条函数拟合到计算域和参数域之间的分段线性映射，同时确保了双射。我们关键洞察力的基础是，对于2T Tutte（1963）和3D Campen等人，理论上保证了分段线性映射的双射。（2016）。由于样条曲线图和分段线性图之间的连续性差异，我们开发了一种用于鲁棒图拟合的两阶段优化策略。第一阶段执行雅可比相似性，第二阶段优化边界差。此外，这两个阶段始终通过结合行搜索的显式检查来保证双射。我们证明了我们的方法在2D和3D各种复杂模型上的功效。与最新技术相比，我们的方法在计算双射和低失真域参数化方面具有更高的鲁棒性。