基于拟共形映射的三角Bézier参数化  

Triangular Bézier Patch Based Parameterization via Quasi-Conformal Mapping

作　　者：张珊珊 潘茂东 陈发来[1] Zhang Shanshan;Pan Maodong;Chen Falai(School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026;School of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing 211106)

机构地区：[1]中国科学技术大学数学科学学院,合肥230026 [2]南京航空航天大学数学学院,南京211106

出　　处：《计算机辅助设计与图形学学报》2024年第12期1863-1869,共7页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家重点研发计划青年科学家项目(2022YFB3302900);国家自然科学基金(12471359,12101308);江苏省自然科学基金(BK20210268).

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),对于一些特殊(类似三角形状)的区域无法获得高质量的映射的问题,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.首先给定参数域(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调性;然后通过交替求解2个二次优化问题,计算拟共形映射;最后,通过计算若干区域的扭曲值,将参数化结果与基于拟共形映射的张量积B样条参数化结果作比较.实验结果表明,该方法可以得到扭曲值更低的映射,所提方法是有效的.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 some special(trian-gle-like)domains,we propose an effective parametrization method for planar domains by adopting the tri-angular Bézier patches as the representation and the quasi-conformal mapping as the computational frame-work.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,and the constraint is the injectivity of the mapping.Then the optimization model is addressed by solving two quadratic optimization problems alternatively.Finally,by calculating the distortion of several domains,the parameterized results are compared with the tensor-product B-spline parameterized results based on quasi-conformal mapping.The experimental results show that this method can obtain mappings with lower distortion,and the proposed method is effective.

关 键 词：区域参数化 等几何分析 三角Bézier面片 拟共形映射 交替优化方法 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]