Subdivision Methods for Computer-Aided Design and Engineering

曲线形状的表示是几何造型与设计中的一个基本问题。目前存在多种建模曲线形状的方法，其中最流行的是边界表示（B-rep），其表面是修剪的NURBS（非均匀有理b样条）。不幸的是，建立一个基于修剪NURBS的建模器是非常困难的，至少有两个原因。首先，修剪两个NURBS补丁以使其沿共同边缘一致通常涉及复杂的曲面/曲面相交算法和大量近似。其次，即使在没有锋利边缘的情况下，也经常需要修补，因为单个NURBS修补只能对拓扑上等同于平面、圆柱体和环面的表面进行建模。其他拓扑类型的曲面建模（即使是像球体一样简单的曲面）需要将曲面分解成单个NURBS补丁的绗缝。然后必须施加复杂的连续性条件以保证相邻的贴片平滑地连接。细分为NURBS提供了一个有希望的替代方案。通过细分，一个多面体被映射到一个更密集的多面体序列，其极限是一个弯曲的对象。每个新的多面体都是通过将其每个面分割成四个新面并使用固定的平均掩码定位其新顶点而获得的。不同的掩模会在限制对象中产生诸如角、曲线边缘或光滑面之类的表面特征。该方法简单，只需要多面体建模。该方法是通用的，生产的对象表面光滑，边缘弯曲锋利，不需要修剪或修补。最后，细分提供了一种分层表示，允许设计高效的多分辨率处理算法。本研究解决了理论和应用问题的结合，这些问题的解决是必不可少的，如果细分是作为一种实用的方法来建模曲线形状。具体发展为：曲率连续细分方法；局部细化和引入形状和边缘的细分方法；不规则三角剖分的功能细分方法（FEA所必需的）和算法：与NURBS等其他表示的转换，细分曲面的编辑和显示，细分曲面的有限元分析，计算细分曲面的交集，并集或差异。

Representing curved shape is a fundamental problem in geometric modeling and design. A variety of approaches currently exist for modeling curved shape, by far the most popular of which is a boundary representation (B-rep) whose surfaces are trimmed NURBS (non-uniform rational B-splines). Unfortunately, building a modeler based on trimmed NURBS is exceedingly difficult for at least two reasons. First, trimming two NURBS patches to agree along a common edge usually involves complex surface/surface intersection algorithms and substantial approximation. Second, even in the absence of sharp edges, patching is often required since a single NURBS patch is capable of modeling only surfaces topologically equivalent to planes, cylinders, and tori. Modeling surfaces of other topological types (even those as simple as a sphere) require the surface be decomposed into a quiltwork of individual NURBS patches. Complicated continuity conditions must then be imposed to guarantee that adjacent patches join smoothly. Subdivision provides a promising alternative to NURBS. Using subdivision, a polyhedron is mapped to a sequence of denser polyhedra whose limit is a curved object. Each new polyhedron is derived from the previous polyhedron by splitting each of its faces into four new faces and positioning its new vertices using a fixed averaging mask. Different masks yield surface features such as corners, curved edges, or smooth faces in the limit object. The method is simple, requiring only polyhedral modeling. The method is general, producing objects with smooth faces and sharp curved edges without trimming or patching. Finally, subdivision provides a hierarchical representation that allows the design of efficient multi-resolution processing algorithms. This research addresses a combination of theoretical and applied problems whose solution is essential if subdivision is to be of use as a practical method for modeling curved shape. Specifically it develops: Curvature continuous subdivision methods; Subdivision methods for local refinement and introduction of shapes and edges; Functional subdivision methods over irregular triangulations (necessary for FEA) and algorithms for: Conversion to and from other representations such as NURBS, Editing and display of subdividion surfaces, Finite element analysis over subdivision surfaces, Computing the intersection, union, or difference of subdivision surfaces.