二维静电场问题的面片拼接等几何分析方法研究

Isogeometric Analysis for the Two-Dimensional Electrostatic Field Problem Based on the NURBS Patch Splicing Technique

等几何分析方法在求解静电场问题时,实现了几何模型和计算模型的统一以及自适应网格划分过程,然而受制于单个NURBS曲面片拓扑的局限性,单片等几何分析方法难以处理含角点非凸几何域静电场及多媒质静电场问题.本文基于面片拼接技术,将单片等几何分析方法扩展到多片,并用来求解二维含角点非凸几何域静电场及多媒质静电场问题,NURBS曲面片拼接处的控制点和网格细分前后要求必须匹配.由于NURBS基函数不满足插值性,在非齐次Dirichlet边界条件的处理上本文采用Lagrange乘子法进行处理.数值算例表明:修正后的多面片等几何分析方法可以很好地处理二维含角点非凸几何域静电场及多媒质静电场问题,且相比传统的有限元法,该方法具有自由度消耗小、精度高、收敛速度快等优点.

Isogeometric analysis （IGA） based on the non-uniform rational B-splines （NURBS） realizes the unification of the geometric model, the computational model and adaptive mesh generation process in solving electrostatic problems. However,IGA based on one single NURBS patch is difficult to deal with the non-convex electrostatic field with corners and inhomogeneous electrostatic field because of limitation of the NURBS patch topology. In this paper, IGA based on the patch splicing is used to solve two-dimensional electrostatic problems of this kind while IGA extends from one single NURBS patch to multipatches. The control points and meshes of different patches must coincide on the interface, even after refinement. Due to lack of interpolation properties for the NURBS basis functions, Lagrange multiplier method is adopted to deal with non-homogeneous Dirichlet boundary conditions. Numerical examples are presented to show that modified multi- patch IGA can solve two-dimensional electrostatic problems of this kind well and possess the advantages of better convergence on a per-degree-of-freedom and high accuracy.