INVESTIGATION ON MULTIGRID FEATURES OF THE FINITE VOLUME METHOD WITH WALSH BASIS FUNCTION
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摘要: Walsh函数有限体积法(FVM-WBF)是一种能够在网格内部捕捉间断的新型数值方法. 持续增加Walsh基函数数目能够稳步提高FVM-WBF方法的求解分辨率, 但计算量暴发式增长和收敛速度下降的问题也会同步出现. 针对Walsh基函数数目增加而引起的计算效率问题, 本文分析了Walsh基函数及其系数所能影响的网格单元局部均值区域尺度, 发现其中隐含类似多重网格的尺度特征, 据此提出一种结合多重网格策略的FVM-WBF方法. 在定常流场计算中根据各级Walsh基函数影响尺度的不同, 对每级Walsh基函数设置满足其稳定性约束的时间步长, 在时间推进求解的过程中快速消除不同波长的数值误差, 实现多重网格的加速收敛效果. 选取NACA0012翼型和二维圆柱的定常无黏绕流问题作为算例, 对引入多重网格策略的FVM-WBF方法和不考虑多重网格策略的FVM-WBF方法进行对比测试. 数值结果证实: 新发展的FVM-WBF方法具备多重网格的关键特征, 在不增加任何特殊处理和计算量的情况下, 只需通过时间步长的调整, 就能够达到多重网格的加速效果, 显著提升计算效率.Abstract: The finite volume method with Walsh basis functions (FVM-WBF) is a novel numerical method with the ability to capture discontinuity inside grid. The numerical resolution of the FVM-WBF method can be effectively improved by increasing the number of basis functions, but the explosive growth of computation and the decrease of convergence speed will also appear simultaneously. To relieve the computational costs due to the increasing of the number of basis functions, the scales of the piecewise continuous mean value subdomains inside the grid cell, which are dominated by different levels of Walsh basis functions and their coefficients, have been analyzed. It is found that FVM-WBF method implicitly has scale characteristics similar to multigrid. Based on this finding, an FVM-WBF method combined with multigrid strategy is presented. In time integration stage of steady flow simulation, this newly developed FVM-WBF method defines the maximal time step for each level of Walsh basis function according to their influence scales and the corresponding numerical scheme stability constraint. As a result, the numerical error of different wavelengths in the process of time advancing is quickly eliminated and the convergence can be accelerated. Several test cases are selected to evaluate the multigrid features of the presented FVM-WBF method, including the low speed inviscid flow over two-dimensional cylinder and a set of inviscid steady flow with different Mach number around NACA0012 airfoil. The numerical results confirm that the newly developed FVM-WBF method has the key characteristics of multigrid, and convergence rate can be greatly enhanced only by adjusting the time step without any additional processing and computational costs.
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Key words:
- Walsh function /
- finite volume method /
- multigrid /
- discontinuity /
- convergence rate
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图 1 单位正方形上二维Walsh基函数示意图
Figure 1. Schematic diagram of the two-dimensional Walsh basis function on the unit square
图 2 FVM-WBF方法对单位正方形区域上的变量
$Q(x,y) = - {(x - 0.4)^2} - {(y - 0.6)^2} + 0.1 x\sin (2 y) + 1$ 拟合数值结果Figure 2. The numerical results of two-dimensional variable
$Q(x,y) = - {(x - 0.4)^2} - {(y - 0.6)^2} + 0.1 x\sin (2 y) + 1$ simulated by the FVM-WBF method on unit square area
图 3 四边形网格的Walsh基函数级数子区域划分示意图
Figure 3. Subdivision of a quadrilateral grid according to Walsh basis function series
图 4 不同级别的基函数所对应的时间步长
Figure 4. Correspondence between different levels of basis functions and time steps
图 8 FVM-WBF_MG方法在圆柱低速绕流计算中的加速效果
Figure 8. Effect of FVM-WBF_MG method in the calculation of the low speed flow over cylinder
图 10 NACA0012翼型低速绕流马赫数云图
Figure 10. Mach number contours of the low speed flow over NACA0012 airfoil
图 11 FVM-WBF_MG方法在NACA0012翼型低速绕流计算中的加速效果
Figure 11. Effect of FVM-WBF_MG method in the calculation of the low speed flow over NACA0012 airfoil
图 12 NACA0012翼型跨声速绕流密度云图
Figure 12. Density contours of the transonic flow over NACA0012 airfoil
图 13 NACA0012翼型表面压力系数对比图
Figure 13. Comparison of pressure coefficients on NACA0012 airfoil surface
图 14 FVM-WBF_MG方法在NACA0012翼型跨声速绕流计算中的加速效果
Figure 14. Effect of FVM-WBF_MG method in the calculation of the transonic flow over NACA0012 airfoil
图 15 NACA0012翼型超声速绕流密度云图
Figure 15. Density contours of the supersonic flow over NACA0012 airfoil
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