瞬态Navier-Stokes方程的最小二乘有限元降阶计算方法

Reduced-order Method for the Transient Navier-Stokes Equation Based on Least-squares Finite Element Method

为了解决在应用数值算法计算瞬态流体方程以分析流体运动特征时,方程的自由度过大,所耗用计算资源较多的问题,提出了一种求解瞬态不可压缩流体问题的降阶计算方法。该方法首先应用最小二乘有限元法计算部分时刻流场的解构成瞬像矩阵,然后对瞬像矩阵进行特征正交分解(Proper Orthogonal Decomposition,POD)提取流场的特征函数,最后结合Galerkin正交投影方法构建求解瞬态Navier-Stokes方程的降阶计算模型。应用降阶计算方法对顶盖驱动流模型进行分析,计算结果表明:前9阶POD正交基可以捕获流场超过99%的广义能量信息,能够比较精确地描述流场的主要物理特征,所构造的降阶模型方程的自由度远小于全阶模型方程的自由度;降阶模型的计算结果与全阶模型的计算结果符合得很好,验证了降阶计算方法的准确性。

When studying the features of the fluid flow by using the numerical method, the degree of equation is very large, thus consuming large amount of computing resources. In order to improve the computational efficiency, this pa- per proposed a reduced-order method to solve the transient incompressible fluid problems. The method firstly achieves a snapshot matrix consisting of the results of fluid field on some discrete time-in-point which computed by applying the least-squares finite element method. Then, the eigenvectors of the snapshot matrix are obtained through proper orthogo- nal decomposition （POD）. Finally, basing on Galerkin orthogonal projection, the reduced-order model is constructed to solve the transient Navier-Stokes equation. The obtained reduced-order model is applied to solve the fluid problem of lid-driven cavity model. The results show that the largest nine POD orthogonal basis can capture over 99 percent of the total general energy of the flow field, which means the main features of the flow field can be described rather accurately by these POD orthogonal basis. The degree of the reduced-order model finite element equation constructed by the POD basis is much smaller than the full-order model. Meanwhile, the flow field computation results with POD reduction method is in good agreement with the full-order least-squares finite element method, and the accuracy of the proposed POD reduction method is verified.