AI 中文总结
本研究提出基于Bézier提取的有理T样条曲面双射性分析框架，结合伯恩斯坦系数性质推导充分与必要条件，辅以分层细分策略，可高效准确验证双射性，兼容等几何分析工作流。
AI 中文摘要
确保基于样条的参数化的双射性是几何建模与等几何分析的基础，因为无效映射可能会导致自相交、雅可比矩阵奇异以及数值不稳定性。T样条虽能通过局部细化提供更高的灵活性，但这种灵活性也使双射性验证变得更为困难。本研究提出一种基于Bézier提取的有理T样条曲面双射性分析的严格且高效框架，核心思路是将T样条表示重新表述为一系列逐元素有理Bézier面片，映射的Gram行列式在这些面片上可表示为伯恩斯坦多项式，这使得能利用伯恩斯坦基函数的凸包与正性性质对局部正则性进行基于系数的分析。基于该公式，从伯恩斯坦系数的非负性推导出双射性的充分条件，同时基于角点系数的符号一致性得出必要条件；对于这些条件无法得出结论的情况，引入分层细分策略，逐步定位模糊区域并通过细化解决。所提方法提供了一种可验证且自适应的双射性验证流程，避免了密集数值采样，且计算效率高。对复杂T样条几何的数值实验表明，该方法能准确检测有效构型与近退化构型，同时可有效扩展至包含数千个有理Bézier面片的大型模型，该框架与标准等几何分析工作流完全兼容。
英文摘要
Ensuring the bijectivity of spline-based parameterizations is fundamental in geometric modeling and isogeometric analysis, as invalid mappings may lead to self-intersections, singular Jacobians, and numerical instability. While T-splines offer enhanced flexibility through local refinement, this flexibility also makes bijectivity verification significantly more challenging. In this work, we propose a rigorous and efficient framework for bijectivity analysis of rational T-spline surfaces based on Bézier extraction. The key idea is to reformulate the T-spline representation into a collection of element-wise rational Bézier patches, on which the Gram determinant of the mapping admits a Bernstein polynomial representation. This enables a coefficient-based analysis of local regularity by exploiting the convex hull and positivity properties of the Bernstein basis. Based on this formulation, we derive a sufficient condition for bijectivity from the nonnegativity of Bernstein coefficients, together with a necessary condition based on the sign consistency of corner coefficients. For cases where these conditions are inconclusive, we introduce a hierarchical subdivision strategy that progressively localizes ambiguous regions and resolves them through refinement. The proposed method provides a certified and adaptive procedure for bijectivity verification that avoids dense numerical sampling and remains computationally efficient. Numerical experiments on complex T-spline geometries demonstrate that the approach accurately detects both valid and near-degenerate configurations, while scaling effectively to large models with thousands of rational Bézier patches. The framework is fully compatible with standard isogeometric analysis workflows.
Comments28 pages, 8 figures
