Abstract: Aiming at the problem of constructing a high-quality parameterization of complex 3D computational domains with given boundary conditions in isogeometric analysis, a volumetric spline parameterization method via high-dimensional quasi-conformal mapping was proposed. The core of this method in the paper was to establish a nonlinear optimization model that described the bijectivity, angular distortion, and volume distortion of the mapping. First, using high-dimensional quasi-conformal mapping theory, a new formula for measuring angular distortion in high-dimensional spaces was derived. Then, the exponential variable and volume constant were introduced into the optimization model, and the geometrical meaning of the Jacobi matrix was exploited to achieve the incorporation of volume distortions while preserving mapping bijectivity. Finally, Alternating Direction Method of Multipliers (ADMM) combined with L-BFGS ( Limited memory Broyden Fletcher Goldfarb Shanno) decomposed the original problem into tractable subproblems and solved them alternatively. The proposed method in this paper guarantees global bijectivity on the experimental model; the orthogonality is improved by about 3.1% compared to the Pan method; and the volume uniformity is improved by about 32.7% compared to the Liu method. Experiments demonstrate that the proposed method in this paper achieves high-quality parameterization results, ensuring the bijectivity of the mapping while reducing angular and volume distortion.

Key words: isogeometric analysis, volumetric parameterization, high-dimensional quasi-conformal mapping, bijective, angular distortion, volume distortion.

摘要： 针对等几何分析中给定边界条件后构造复杂三维计算域的高质量参数化这一问题，提出了一种基于高维拟共形映射理论的三维区域参数化方法。该方法的核心是建立一个非线性优化模型，同时描述映射的双射性、角度扭曲和体积扭曲；首先，利用高维拟共形映射理论推导出新的衡量三维空间角度扭曲公式；其次，在优化模型中引入指数变量和体积常数，利用雅各比矩阵的几何意义，实现在保持映射双射性的同时并加入体积扭曲。最后，将交替方向乘子法(ADMM)的框架与L-BFGS( Limited memory Broyden Fletcher Goldfarb Shanno)结合将原问题分解为易于处理的子问题并进行交替求解。本文方法在实验模型上保证全局双射性；相较于ADMM-LRP(ADMM algorithm for Low-Rank Parameterization)，正交性提高了约3.1%；相较于TTS(Tet-To-Spline optimization strategy)，体积一致性提高了约32.7%。结果表明，本文方法能获得高质量的参数化结果，确保双射性，并减小角度扭曲和体积扭曲。

关键词: 等几何分析, 体参数化, 高维拟共形映射, 双射性, 角度扭曲, 体积扭曲