多重网格在黏性流动最小二乘等几何模拟中的应用

An Application of Multigrid in Viscous Flow Simulations with Least Squares Isogeometric Method

针对最小二乘等几何方法模拟黏性流动时条件数大、迭代法收敛速度慢的问题,提出了基于多重网格技术的加速方法。计算中自动生成一系列疏密不同的网格,在最密网格上用最小二乘等几何方法将Navier-Stokes方程离散为代数方程组,用多重网格方法作为独立求解器或共轭梯度法的预处理器迭代求解所得到的代数方程组。对雷诺数为100、400、1 000和2 500的顶盖驱动流进行了数值模拟,计算中进行23次迭代可使方程组的余量降低10个数量级,流动特征量的计算误差在1%以内。计算结果表明,通过多重网格技术加速迭代,提高了最小二乘等几何方法模拟黏性流动的计算效率。

A multigrid technique to accelerate iteration is proposed to solve the problem of low convergence rate due to large condition number in simulating viscous flow with the least squares isogeometric method. A series of grids with different element sizes are automatically generated in analysis. The least squares isogeometric method is used to discretize the Navier-Stokes equations into a system of linear equations on the densest grid, and the system is iteratively solved using either the multigrid method as standard solver or the preconditioner of the conjugate gradient method. Numerical simulations are performed on lid driven cavity flow with Re= 100, 400, 1 000 and 2 500. The residual of algebraic equations is reduced about 10 order of magnitude after 23 iterations, and the calculation error on characteristic parameters of the cavity flow is less than 1%. The results show that the performance of the least squares isogeometrie method for viscous flow is improved when the multigrid acceleration technique is used.