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.

Introduction

Computing bijective domain parameterizations from parametric domains (generally unit cubes or squares) to computational domains is a fundamental task in the isogeometric analysis (IGA) Hughes et al. (2005). As shown in Cohen et al. (2010); Pilgerstorfer and Jüttler (2014), the subsequent accuracy and computational efficiency for solving partial differential equations are significantly affected by the parameterization quality. Thus, in addition to being bijective, domain parameterizations are also required to contain low distortion.

This paper focuses on computing bijective B-spline parameterizations with given bijective boundary correspondences between computational and parametric domains. This problem is common. For example, in partition-based methods for high-genus computational domains Chen et al. (2019); Xu et al., 2015, Xu et al., 2018; Xiao et al. (2018), the final step is to generate a bijective parameterization for each genus-zero block. In general, it first computes the boundary correspondence, and as a consequence, constructing the interior mapping is our problem. Besides, for the single-patch methods, boundary correspondences are manually specified or automatically optimized Liu et al. (2020); Zheng et al. (2019), then optimizing the interior control points is just our problem.

This problem is challenging due to the nonlinear, non-convex bijection constraint and the high complexity of the computational domains. The existing methods are usually initialized with a mapping that satisfies the given boundary constraint and violates the bijective constraint, and then attempt to get rid of the flips Xu et al. (2013a); Wang and Qian (2014); Martin et al. (2009); Nguyen and Jüttler (2010); Su et al. (2019). Here, a flip indicates that the determinant of the Jacobian matrix of the parameterization is negative somewhere. However, these methods are not guaranteed to produce a valid bijective parameterization.

To ensure bijection, we propose to start from a bijective parameterization that violates the given boundary correspondence and then optimize the boundary control points to approach the specified positions while always ensuring no flips. If the optimized boundary reaches the target, the bijective boundary map and the flip-free interior map forms a bijective result Moré and Rheinboldt (1973). A parameterization, which is bijective and violates the boundary correspondence, is easy to be obtained, such as the identity map. The critical challenge is to optimize the boundary difference to zero. One naive method is to optimize $L_2$-norm error for the boundary control points (Fig. 1, left). However, it rarely works due to the conflict of paths of the boundary control points, as observed by Fu et al. (2015); Fu and Liu (2016).

To this end, we propose a novel method to compute bijective B-spline parameterizations. Central to our method is a novel map fitting process that attempts to optimize the boundary difference to zero. We observe that there is a theoretical guarantee on bijection of piecewise linear maps from the computational domains to the parametric domains Tutte (1963); Campen et al. (2016). Therefore, instead of optimizing $L_2$-norm error for the boundary control points, we fit the B-spline function to a bijective piecewise linear map between computational and parametric domains to reduce the boundary difference. However, these two functions have different continuity. To this end, we propose a two-stage fitting process. In the first stage, the Jacobian difference is minimized, and, as a consequence, the two boundaries are similar. Then the second stage enforces the similar boundaries to be the same by minimizing the in-between $L_2$-norm error. Since the two boundaries are similar after the first stage, there is almost no path conflict between boundary control points in our second stage (Fig. 1, right).

Our method can compute bijective and low isometric distortion parameterizations for various complex volumetric domains (Fig. 2). We demonstrate the feasibility and efficacy of our method on a data set containing 222 complex models. Compared to state-of-the-art methods, our method achieves much lower distortion domain parameterizations.