基于层次T网格的分片孔斯曲面重构

本文提出一种基于任意层次T网格的多项式(PHT)样条空间$S(3,3,1,1,T)$的一个新的曲面重构算法.该算法由分片插值于层次T网格上每个小矩形单元对应4个顶点的16个参数的孔斯曲面形式给出.对于一个给定的T网格和相应基点处的几何信息(函数值,两个一阶偏导数和混合导数值),可得到与$S(3,3,1,1,T)$的PHT样条曲面相同的结果,且曲面表达形式更简单,同时,在离散数据点的曲面拟合中,我们给出了自适应的曲面加细算法.数值算例显示,该自适应算法能够有效的拟合离散数据点.     

Adaptive finite element methods for elliptic equations over hierarchical T-meshes

Isogeometric analysis using NURBS (Non-Uniform Rational B-Splines) as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement. In this paper, we use the polynomial splines over hierarchical T-meshes (PHT-splines) to construct basis functions which not only share the nice smoothness properties as the B-splines, but also allow us to effectively refine meshes locally. We develop a residual-based a posteriori error estimate for the finite element discretization of elliptic equations using PHT-splines basis functions and study their approximation properties. In addition, we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution.