参数曲面/体的理论及其在等几何分析中的应用研究

参数曲面与参数体的理论研究不仅对计算几何与计算机辅助几何设计的理论发展有重要意义，而且对曲面与实体造型技术在计算机科学与工程中的应用有重要的指导意义。本项目拟对一些尚未发现的参数曲面/体的理论进行研究，同时拟将所研究的部分理论应用于面向等几何分析的计算域参数化中。本项目拟开展的主要研究内容包括：三元toric-Bernstein基函数的升阶与递推公式；GT-Bézier曲面/体与toric体的升阶公式与de Casteljau算法；toric体、有理Bézier体、NURBS曲面/体的单值性条件与判定算法及其在计算域参数化中的应用；toric曲面/体的分解算法及其在计算域分解与单元细化中的应用；计算域参数化中权因子的作用。通过本项目的研究，将数学理论与实际应用相结合，在丰富与完善参数曲面/体的理论、性质与算法的同时，拓展这些理论在等几何分析等方向中的应用。

The study on theory of parametric surfaces and volumes are not only important for the improvement of computational geometry and computer aided geometric design, but also important for applications of surface and solid modeling in computer science and engineering. This research project aims to study on the theory of parametric surfaces/volumes, and its application for parametrization of computational domain in isogeometric analysis. This project involves researches in the following topics: the degree elevation and recursive formulas of trivariate toric-Bernstein basis functions; degree elevations and de Casteljau algorithms of GT-Bézier surfaces/volumes and toric volumes; the injectivity conditions and algorithms for toric volumes, rational Bézier volumes, NURBS surfaces/volumes, and their application in parametrization of computational domain; the decomposition algorithm of toric surfaces/volumes and its application in parametrization of computational domain and element refinement; application of weights in parametrization of computational domain. We believe the results of this project will not only enrich and improve the theory, properties and algorithms of parametric surfaces and volumes, but also extend their applications in isogeometric analysis and related fields.