局部与超限插值T样条的构造和应用研究

T-splines and splines over T-meshes have received an extensive attention in complex surface modeling, scientific data visualization and isogeometric analysis in the past years. Interpolation is an essential method in spline theory system, which is an important subject in the field of scientific computation. Up to now, most of the existed T-splines and splines over T-meshes belong to the type of approximation spline. By using these approximation splines to interpolate the data located at T-meshes, however, we should solve linear systems to calculate the control points. Even worse, when a data point is changed, we need to resolve the linear systems again so as to ensure interpolation, which brings many inconveniences to practical applications. In this project, we plan to systematically explore on the research of constructions and applications of local and transfinite interpolation T-splines. The project mainly includes the constructions of C2 interpolation T-splines with local tension parameters, C2 interpolation and approximation T-splines with a unified expression, Coons transfinite interpolation splines over T-meshes, and their applications in triangular mesh models parametric surfaces reconstruction, three-dimensional scattered data implicit surface reconstruction and isogeometric analysis. By the research of the project, we hope to provide highly efficient and stable local and transfinite interpolation T-splines novel schemes.

T样条和T网格上的样条近年来在复杂曲面建模、科学数据可视化和等几何分析等领域中受到广泛的研究关注。插值是样条理论体系中的基本方法，是科学计算领域的重要研究课题。目前已有T样条和T网格上的样条大多为逼近型样条，在对数据点集的插值应用中，逼近型样条需要求解一个与数据点及其数量有关的线性方程组进行反算控制点，当任一个数据点变动时，为保证插值必须重新求解线性方程组，不利于实际应用。本项目拟计划系统地开展局部与超限插值T样条的构造和应用研究，内容包括构造带局部张力参数C2插值T样条、C2插值和逼近统一表示T样条和T网格上Coons超限插值样条，及其在三角网格模型参数曲面重建，三维散乱数据隐式曲面重建和等几何分析方面的应用。本项目的研究将提供高效稳定的局部和超限插值T样条新方法。

项目负责人按计划开展了C2插值T样条的构造及其在参数曲面重建和等几何分析应用方面的研究工作。主要研究内容包括：C2插值T样条和等几何分析应用；单变量多项式再生插值逼近样条递推表示的一般理论框架；C2插值逼近结合径向基函数和隐式曲面重建应用；C∞插值逼近结合样条曲线曲面；保形插值样条曲线曲面；三角域和矩形域插值样条曲面；拟Bernstein基和广义拟B样条基；C2张力插值样条等。经过三年努力，项目负责人以第一或通讯作者在科学计算与计算机辅助几何设计领域顶级期刊Computer Methods in Applied Mechanics and Engineering，SIAM Journal on Scientific Computing和Computer Aided Geometric Design等发表了18篇SCI检索论文和4篇EI检索论文。其中5个代表性研究成果：C2插值T样条和等几何分析应用, 发表于Computer Methods in Applied Mechanics and Engineering；单变量多项式再生插值逼近样条递推表示的一般理论框架, 发表于SIAM Journal on Scientific Computing；C2插值逼近结合径向基函数和隐式曲面重建应用, 发表于Computer Aided Geometric Design；C∞插值逼近结合样条曲线曲面, 发表于Numerical Algorithms；C2上下约束有理插值样条曲面, 发表于Journal of Mathematical Imaging and Vision。本项目的研究成果丰富了计算机辅助几何设计和数值逼近等领域的样条理论体系，拓展了曲线曲面建模和等几何分析等工程应用。