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基于拟共形映射的三角Bézier参数化

Triangular Bézier Patch Based Parameterization via Quasi-conformal Mapping

  • 摘要: 区域参数化作为等几何分析方法中非常关键的一步, 近年来已得到广泛的关注与研究. 针对现有参数化方法大都将参数域限定为正方形(体), 对于一些类似三角形状的区域无法获得高质量参数化映射的问题, 以三角Bézier面片为表示形式, 以拟共形映射为计算框架, 提出一种有效的平面区域参数化方法. 给定参数域(单位直角三角形)与计算域之间的边界对应; 构建参数域到计算域具有双射性和低扭曲的拟共形映射问题可以建模为一优化模型, 其目标函数为映射的共形扭曲和光滑性, 约束条件为映射的单射特性; 然后通过交替求解两个二次优化问题, 计算拟共形映射; 最后, 将本文方法与基于张量积B样条表示的参数化方法作比较. 实验结果表明, 所提方法可以得到更高质量的映射.

     

    Abstract: Domain parameterization, which is an essential step in isogeometric analysis, has been extensively studied in recent years. To address the problem that most of the existing parameterization methods restrict the parameter domain to the square (cube) and cannot obtain high-quality mapping for triangle-like domains, we propose an effective parametrization method for planar domains by adopting the triangular Bézier patches as the representation and the quasi-conformal mapping as the computational framework. Given the boundary correspondence between the parametric domain (unit right triangle) and the physical domain, the problem of constructing a quasi-conformal mapping with low distortion can be modeled as an optimization model, in which the objective function is the conformal distortion and smoothness of the mapping with the injectivity of the mapping as a constraint. Then the optimization model is addressed by solving two quadratic optimization problems alternatively. Finally, we compare our method with the tensor-product B-spline based parameterization method, and the experimental results demonstrate that the proposed approach can obtain parametrizations with higher quality.

     

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