保形约束曲面造型方法与应用

In computer aided design, data visualization or other areas, tensor product format with basis functions possessing shape parameters is commonly used to generate shape preserving surfaces. However, when the shape parameters of two adjacent patches are not equal, the splicing gap will be generated by this format. First, considering the boundary of the two adjacent patches is undoubtedly continuous, we construct a C1 continuous tensor product format by using Hadamard product. This format is the generalized version of the existing tensor product format. All the basis functions used by the old format can be directly used by the new one to generate C1 continuous surfaces, and does not require the shape parameters of two neighbor are equal. Next, the properties of the new format, such as the approximation, polynomial reproduction and local control capability, are analyzed, and a algorithm for shape preserving surface modeling is designed based on the genetic algorithm and Levenberg-Marquardt algorithm. Then, by referring to the way of constructing the high order continuous spline curves, we will construct the high order continuous tensor product surface method with only the point location information. The properties of the method is analyzed, and we use the method to generate the shape preserving surface. Then, these new continuous tensor product methods are applied to the shape preserving modeling on the scattered data. Finally, considering the more general constraints, design a software to generate the constrained surface for the mesh data and the scattered data.

基函数中带形状参数的张量积格式常用于计算机辅助设计或数据可视化等领域的保形约束曲面造型。然而，当相邻曲面片形状参数不相等时，这种格式曲面在拼接处将产生缝隙。考虑到两相邻曲面片在边界处必定是连续的，首先构造基于Hadamard积的C1连续张量积格式。该格式是现有张量积格式的推广，可以直接使用现有格式的任何基函数来生成C1连续曲面，且不需要相邻曲面片形状参数相等。接着，分析新格式的逼近性、多项式重生性和局部控制能力等性质，并设计基于遗传Levenberg–Marquardt的保形约束曲面造型算法。然后，参考高阶连续样条曲线的构造方式，设计只有数据点位置信息的高阶连续张量积曲面，分析所构造曲面的性质，并将其应用于保形约束曲面造型。再接着，将这些新的张量积方法应用于散乱数据的保形约束造型。最后，考虑在更一般约束条件下，设计一套用于网格数据和散乱数据的约束曲面造型软件。

基函数中带形状参数的张量积格式常用于计算机辅助设计或数据可视化等领域的保形约束曲面造型。然而，当相邻曲面片形状参数不相等时，这种格式曲面在拼接处将产生缝隙。考虑到两相邻曲面片在边界处必定是连续的，在形状参数样条曲线的基础上，分析了现有形状参数张量积样条曲面格式在局部形状参数不一致产生不连续的原因，构造了 连续张量积格式，并在选择一组具体基函数的基础上，分析新格式的逼近性、多项式重生性及局部控制能力。研究了保形约束曲面造型方法。探讨了高阶连续张量积曲面，分析所构造曲面的性质，并将其应用于C2连续的保形约束曲面造型。接着，研究了散乱网格数据的保形约束曲面造型。开发了一套处理网格数据和散乱数据的曲面造型程序。项目资助期间，课题组发表学术论文8篇、申报发明专利1项。在项目资助下，在研究曲面构造过程中使用最优化及系统动力学模型相关理论过程中，我们成功将与本项目研究相关理论知识应用于机械制造中的齿面设计、生态环境研究，为学科交叉做出了贡献。