多胞形样条的理论及其应用研究

Polytope splines are piecewise defined over polygonal tessellations or polyhedral tessellations. They have received much attention in the fields of computer graphics and finite element analysis, due to their inherently excellent properties. Driven by different applications, various kinds of polytope splines have been proposed. However, all of them have their own limitations, including low approximation orders, low continuity orders across tessellation edges and restrictions on tessellations. Besides, efficient polygonal/polyhedral mesh generation methods for producing high-quality tessellations are lacking. To maximize the potential of polytope splines, we are inspired to develop a novel construction method of polytope splines which guarantees high approximation orders and continuity orders and is applicable to general polygonal/polyhedral tessellations, and tailor efficient methods for generating high-quality polygonal/polyhedral tessellations. In this project, we will focus on developing a simple and easy-to-use method for constructing new polytope splines that inherit the excellent properties of existing polytope splines while overcoming their drawbacks. Furthermore, we will investigate the theory of the proposed polytope splines and develop efficient evaluation algorithms and high-quality polygonal/polyhedral mesh generation methods. The theoretical and algorithmic results will be applied to the applications in traditional and emerging research fields, such as computer graphics, computer aided geometric design, finite element analysis, etc. In particular, we will apply our results to geometric modeling, data approximation and interpolation, image/video warping, solving equations and so on.

多胞形样条指的是在多边形、多面体剖分上定义的样条函数，它固有的许多优良性质备受计算机图形学、有限元分析等领域的青睐。在应用需求的驱动下，各类多胞形样条应运而生，但它们还存在两方面的缺陷：一是样条函数逼近精度低、跨越剖分边界连续性低、对剖分适应性差；二是缺乏为多胞形样条设计的高质、高效的多边形、多面体剖分生成方法。给出在一般多边形、多面体剖分上定义的具有高逼近阶和连续阶的多胞形样条的构造方法，同时给出相应的剖分生成方法，将有望打破多胞形样条的应用瓶颈，使其优良性质被最大限度地利用。本项目主要研究多胞形样条简单易用的构造方法，使其在继承现有多胞形样条优良性质的同时能克服其不足，同时研究其理论性质并提供配套的高效计算方法及剖分生成方法。研究成果将用于计算机图形学、计算机辅助几何设计、有限元分析、等几何分析等传统或新兴的研究领域，具体应用包括几何造型、插值与逼近、图像视频变形、微分方程求解等方面。

本项目主要研究在一般多边形、多面体剖分上定义的具有高逼近阶和连续阶的多胞形样条的简单易用的构造方法，使其在继承现有多胞形样条优良性质的同时能克服其不足，同时研究其理论性质并提供配套的高效计算方法和剖分生成方法。特别地，针对传统多边形元形函数大多只具有一阶逼近阶或只能在凸多边形上定义的局限性，我们给出了在一般复杂多边形（包括凹多边形）上构造具有二阶逼近阶型函数的方法，研究了其逼近性质并进行了误差估计。根据函数的理论性质，我们制定了多边形单元的生成标准，并给出了系列高质量多边形、多面体网格生成算法。相关研究成果进一步应用于方程求解、图形图像逼近、图形变形，并提供配套的应用算法框架。我们研究并完善了基于高阶多边形剖分的TCB样条的理论性质，针对复杂 CAD 模型一般需大量的张量积曲面拼接，存在数据冗余和非水密，导致后续分析优化存在困难的问题，实现了在单张曲面中同时描述跳跃、C0、C1 等不同连续阶变换的复杂几何构造方法，并初步应用于矢量图的实时交互编辑。本项目也将所研究的多胞形样条引入等几何分析领域，构建了基于多胞形样条的等几何分析框架，首次给出了无需优化的显示高质量参数化计算方法。此外，本项目的理论研究成果还应用于复杂几何处理、图像逼近与变形等计算机图形学、计算机辅助几何设计领域的应用中。通过本项目的研究，有望打破多胞形样条的应用瓶颈，使其固有的许多优良性质被最大限度地利用，为后续的复杂几何模型的建模分析一体化研究提供必要的理论及算法支持。