分级T网格上样条空间的理论与应用

Spline spaces over T-meshes are a new type of multivariate splines proposed by the applicant and his collaborators. The new spline extends the partition fashion of spline domains and has important applications in some areas including adaptive surface modeling and isogeometric analysis. The proposed project intends to research high smootherness splines and its variants over a special type of T-meshes (i.e., hierarchical T-meshes), such that we can make a new attemp to avoid the problem of dimension instability of spline spaces over general T-meshes. The research topics of the project include basis funnction construction of C1 biquadratic spline spaces, dimension formulae and basis function construction of C2 bicubic spline spaces and the spline spaces with higher degree and higher smootherness, and variants of spline spaces over hierarchical T-meshes. We will also discuss their theories and applications in some areas including surface modeling and isogeometric analysis. The main theoretical techniques include linear space embedding and direct sum decomposition proposed in our previous publications. We have obtained some research accumulation about these research topics, and hence it is reasonable to expect that we will reach the research objects, build a systemeric theory (similar to univariate B-splines) over hierarchical T-meshes, and setup a solid foundation for adaptive surface modeling and analyatical computations.

T网格上样条是由申请人与合作者提出的一种新型多变量样条，拓展了多元样条中的定义域剖分方式。这种样条在自适应曲面造型和等几何分析等领域中具有重要应用。本项目拟研究一类特殊的T网格（即分级T网格）上的高光滑阶样条以及其它各种类型样条,这为避开T网格上的样条面临的维数不稳定性等问题提供了新的尝试。本项目主要研究分级T网格上的C1双二次样条空间的基函数构造，C2双三次样条空间和更高次更高光滑度样条空间的维数和基函数构造，以及分级T网格上的其它类型样条，并且讨论这些样条的理论和它们在曲面造型与等几何分析等领域中的应用。理论工作方面主要技术手段是申请人在前期工作中提出的线性空间嵌入和直和分解方法。相关工作已有了一定的工作积累，有望完成研究目标，从而在分级T网格上建立起类似于单变量B样条的系统理论，为自适应曲面造型和分析计算奠定基础。

T 网格上样条是由项目负责人与合作者提出的一种新型多变量样条，拓展了多元样条中的定义域剖分方式。这种样条在自适应曲面造型和等几何分析等领域中具有重要应用。本项目研究一类特殊的T 网格（即分级T 网格）上的高光滑阶样条以及其它各种类型样条, 这为避开T 网格上的样条面临的维数不稳定性等问题提供了新的尝试。主要研究成果是：分级T网格上多项式样条的维数分析和基函数构造，等几何分析中参数化的构造，以及其它与几何造型相关主题的研究。这些研究成果有助于为自适应曲面造型和分析计算奠定理论基础。经过四年的研究，本项目发表有标注论文23篇，其中SCI/EI收录20篇，主要代表作发表在ACM Transactions on Graphics，Computer Methods in Applied Mechanics and Engineering等一、二区期刊和Computer Aided Design, Computer Aided Geometric Design等专业顶级期刊上。培养毕业博士8名，其中一人获得中科院院长特别奖。