T样条的k-细分及其在等几何分析中的应用

In order to provide the possibility for integration problems between engineering design and analysis, iso-geometric analysis (IGA) uses the smooth spline basis to define the geometry basis for analysis. T-splines have emerged as an important technology for engineering design and IGA, which have been applied to solve many limitations inherent in the NURBS(non-uniform rational B-splines) representation. However, up to now, researchers have paid much attention to h-refinement (knot insertion) for T-splines based IGA, while there are few articles focus on p-refinement (degree elevation) and k-refinement (both h and p-refinement are preformed) for T-splines. This project will study k-refinement for T-splines and its application in IGA. Applying the existing local refinement algorithms for T-splines cannot ensure the produced T-splines have linearly independent blending functions. While applying the existing AS refinement can produce the T-splines with linearly independent blending functions, but we need a preprocessing of conversing general T-splines into AS T-splines. So we will present an optimized refinement algorithm for T-splines to make sure that the new T-splines after refinement have linearly independent blending functions. Besides local refinement, degree elevation is another useful technology in IGA. The meshes of NURBS are tensor-product meshes, while T-meshes are not. So it is more difficult to provide a degree elevation algorithm for T-splines. We will develop a recursive algorithm for general T-splines local degree elevation and develop an analysis-suitable local degree elevation algorithm. Then we will use these algorithms to develop k-refinement and apply it in IGA application.

为了实现工程设计和分析的相结合，研究者们提出了等几何分析，用样条函数同时作为几何基和分析基。而T样条在工程应用领域又有着独特的优势。本项目对基于T样条的k-细分（同时要求网格细分和升阶）及其在等几何分析中的应用进行研究。已有的T样条细分算法在细分之后得到的混合函数还存在可能的线性相关性，已有的AS细分算法需要增加一个步骤来保证最终得到线性无关的混合函数, 为解决这些问题，本项目首先提出优化的细分算法。升阶算法也是等几何分析中的重要技术，而T样条由于其网格不同于NURBS（非均匀有理B样条）的张量积网格，所以其升阶算法较NURBS更难提出，目前该问题在国内外的研究仍处于初级阶段，成果较少，本项目拟给出T样条的局部升阶算法。然后在这两个算法的基础上发展T样条的k-细分，并应用到等几何分析中。

等几何分析用相同的基函数来表示几何并且用于分析，有助于实现工程设计和分析的相结合。T样条在工程应用领域有着独特的优势。本项目针对细分算法及等几何分析等进行研究，主要研究了以下内容：.T样条已经被证明适合用来进行几何设计，如果对T样条做一些限制，又可以满足等几何分析所需要的性质。本项目针对T样条给出了一种有效的自适应的细分算法，解决了目前常用的细分算法存在的问题，保证了较好的局部性，同时保持了混合函数的线性无关性。将该算法应用到等几何分析中也取得了较好的结果。在几何设计中，对于任意给定的控制网格，提供了可以生成曲率连续的三次IPH (Indirect Pythagorean hodographs)曲线的算法，给出了理论证明，并且还给出了局部编辑的算法。利用特征多面体技术，调整了特征多面体中角度的取值，生成了非均匀G1连续Catmull-Clark细分曲面，相比较已有算法，通过改进以后的算法所生成的极限曲面更加光滑，品质更高。等几何分析与几何设计对实现工程设计，分析及两者之间的融合具有重要意义。在工程应用上，通过求解光波动方程，降低重构振幅与目标振幅之间的误差，利用平均算法提高重构振幅的均一性和信噪比，实现了较高效的飞秒激光图形化加工。.本项目为等几何分析的发展提供理论基础和技术支持，促进工程设计和分析的紧密结合。