可局部加细样条中若干基本问题的研究

With the rapid development of complicated surface modeling and isogeometric analysis, the local refinable splines with adaptive refinement came into being. In recent years, domestic and foreign scholars have put forward a series of local refinable splines, from the T-spline and the PHT spline defined over the quadrilateral mesh, to the hierarchical box spline on the triangulation, and to the truncated hierarchical subdivision surfaces defined over arbitrary topological partition. However, the technique of local refinable splines is far from mature. Some basic problems such as efficient evaluation algorithm and efficient numerical integral need further study. The underlying mesh of local refinable splines is no longer a tensor product mesh, so some well-established techniques of NURBS can not be directly extended. In this project, we study three typical splines in local refinable splines. First, the de-Boor-like algorithm and quasi-interpolation operator for AST spline will be considered. Then, we study how to modify the PHT spline so that it has one-to-one correspondence control net and how to generalize to the T-mesh with singular points. Finally, the efficient quadrature scheme of hierarchical subdivision surfaces will be addressed. The research on these issues provides the necessary theoretical support for the development of local refinable splines, which further promotes the application of AST splines, PHT splines and subdivision surfaces in geometric modeling and isogeometric analysis.

随着复杂曲面造型和等几何分析的快速发展，具有自适应加细功能的可局部加细样条应运而生。近十几年来可局部加细样条得到了快速发展，从四边形网格上的T样条和PHT样条，拓展到了三角剖分上的层次箱样条，直至任意拓扑剖分的可局部加细的细分曲面。但是可局部加细样条技术远没有成熟，存在的一些基本问题如高效的求值算法、高效的积分格式等问题还需要深入研究。定义可局部加细样条的网格不再是一个张量积网格，因此NURBS一些成熟的技术无法进行直接推广。本项目针对可局部加细样条中三种典型的样条展开研究:研究AST样条的类de Boor求值算法和拟插值算子;研究如何修正PHT样条使其具有一一对应的控制网和如何推广到带有奇异点的T网格上;可局部加细的细分曲面上高效的数值积分格式。这些问题的研究对促进可局部加细样条技术走向成熟提供必要的理论支持，进而推动AST样条、PHT样条和细分曲面在几何造型和等几何分析中的进一步应用。

随着复杂曲面造型和等几何分析的快速发展，具有自适应加细功能的可局部加细样条应运而生。近十几年来可局部加细样条得到了快速发展，但是可局部加细样条技术远没有成熟。本项目计划系统地研究可局部加细样条中存在的一些基本问题，诸如AST样条的高效求值算法和拟插值算法，PHT样条曲面的进一步推广和可局部加细细分曲面的高效积分格式等问题，为局部可加细样条走向成熟提供有力的保障。项目按照计划执行，在AST样条的高效求值和PHT样条曲面的进一步推广问题上取得了一些成果，在细分曲面（可局部加细细分曲面）和AST样条的拟插值相关问题上取得了一些进展，实现了预定研究目标。另外，我们还在基于层次B样条的等几何分析的经济型数值解和B样条曲线曲面拟合中的节点计算领域取得了一系列的研究成果。这些成果进一步促进了可局部加细样条在几何造型和等几何分析中的应用。三年来在国内外重要学术期刊上发表SCI论文4篇，主要工作发表在专业核心权威期刊Computer-Aided Design，Computer Aided Geometric Design和Communications in Mathematics and Statistics。