二维不可压缩Navier-Stokes方程的并行谱有限元法求解

Solution of two dimensional incompressible Navier-Stokes equation by parallel spectral finite element method

针对不可压缩Navier-Stokes(N-S)方程求解过程中的有限元法存在计算网格量大、收敛速度慢的缺点,提出了基于面积坐标的三角网格剖分谱有限元法(TSFEM)并进一步给出了利用OpenMP对其并行化的方法。该算法结合谱方法和有限元法思想,选取具有无限光滑特性的指数函数取代传统有限元法中的多项式函数作为基函数,能够有效减少计算网格数量,提高算法的精度和收敛速度;利用面积坐标便于三角形单元计算的特点,选取三角单元作为计算单元,增强了适用性;在顶盖方腔驱动流问题上对该算法进行验证。实验结果表明,TSFEM较传统有限元法(FEM)无论是收敛速度还是计算效率都有了显著提高。

Due to a large number of computational grids and slow convergence existed in the numerical simulation of Navier-Stokes （N-S） equation, Triangular mesh Spectral Finite Element Method based on area coordinate （TSFEM） was proposed. And further, TSFEM was paralleled with OpenMP. Spectral method was combined with finite element method, and the exponential function with infinite smoothness was selected as the basis function to replace the polynomial function in the traditional finite element method, which can efficiently reduce the amount of computational grids as well as improve the convergence and accuracy of the proposed algorithm. Because area coordinates can facilitate the calculation of triangular units, which were selected as the computing units to enhance the applicability of the algorithm. The lid-driven cavity flow was used to verify the TSFEM. The experimental results show that, compared with the traditional Finite Element Method （FEM）, the TSFEM greatly improves the convergence rate and the calculation efficiency.