Computing planar and volumetric B-spline parameterizations for IGA by robust mapping fitting

Abstract

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.