高阶曲线矢量有限元方法实现及关键问题

Implementation for higher order curvilinear vector FEM

针对传统的直线型高阶矢量基函数对曲边界模拟不好的缺点,研究了基于四面体叠层矢量元的高阶曲线建模技术。系统而显式地分析了三维高阶曲线矢量元的实现过程,探讨了实现过程中的一些关键问题。通过分析一个球形谐振腔,系统比较了各种直线或曲线形式的高、低阶矢量元的性能(如计算精度、条件数等),并将其用于分析不均匀腔体的谐振问题。计算结果表明:采用曲线元对曲线模型进行建模,在不影响计算效率的情况下,可极大地提高计算精度。

To overcome the shortcoming of typical rectilinear higher order vector elements which cannot exactly model curved boundaries, an efficient higher order geometrical mapping technology based on tetrahedral hierarchical vector elements is studied. The implementation process for 3D higher order curvilinear vector elements is researched, and some key issues are discussed. The results of a numerical experiment that investigates the resonant problem of a spherical cavity, compare the performance of different rectilinear and curvilinear vector elements systematically （such as calculated accuracy, condition numbers）, which are also applied to the eigen-selution of inhomogeneously-filled cavities. Numerical results demonstrate that for the same computational efficiency, curvilinear elements can provide more accurate results than rectilinear elements when higher order elements are used to such problems.