SOLUTION INTERPOLATION FOR HIGH-ORDER ACCURATE ADAPTIVE FLOW SIMULATION
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摘要: 网格自适应技术和高阶精度数值方法是提升计算流体力学复杂问题适应能力的有效技术途径. 将这两项技术结合需要解决一系列技术难题, 其中之一是高阶精度流场插值. 针对高阶精度自适应流动计算, 提出一类高精度流场插值方法, 实现将前一迭代步网格中流场数值解插值到当前迭代步网格中, 以延续前一迭代步中的计算状态. 为实现流场插值过程中物理量守恒, 该方法先计算新旧网格的重叠区域, 然后将物理量从重叠区域的旧网格中转移到新网格中. 为满足高阶精度要求, 先采用k-exact最小二乘方法对旧网格上的数值解进行重构, 获得描述物理量分布的高阶多项式, 随后采用高阶精度高斯数值积分实现物理量精确地转移到新网格单元上. 最后, 通过一个具有精确解的数值算例和一个高阶精度自适应流动计算算例验证了本文算法的有效性. 第一个算例结果表明当网格规模固定不变时, 插值精度阶数越高, 插值误差越小; 第二个算例显示本文方法可以有效缩短高精度自适应流动计算的迭代收敛时间.Abstract: Mesh adaptation and high order numerical methods are regarded as effective techniques to improve the adaptability of computational fluid dynamics (CFD) to complex problems. The combination of these two techniques requires solving a series of technical challenges, one of which is the flow field interpolation for high order numerical methods among different adaptation steps. A high-order accurate solution interpolation method is proposed for the high-order accurate adaptive flow simulation. In this method, it interpolates the numerical flow solution from the mesh in the previous iteration step into the mesh of the current iteration step, to allow the simulation to be restarted from the previous state. To realize the conservation of physical quantities in the process of flow field interpolation, the method first computes the overlapping regions of the new and old meshes and then transfers the physical quantities from the old mesh to the new mesh in the overlapping regions. To achieve high-order accuracy, the k-exact least-squares method is first used to reconstruct the numerical solution on the old mesh, and as a result, a polynomial with the required order that represents the distribution of the physical quantity is obtained over each element of the background mesh. Then Gaussian numerical integration is used to integrate the physical quantities over each element of the new mesh, which accurately transfers the physical quantities from the background mesh to each element of the new mesh. Finally, the effectiveness of the proposed algorithm is verified by a numerical example with an exact solution and an example of high-order accurate adaptive flow simulation. The results of the first example show that a smaller interpolation error exists when higher-order accurate interpolation is adopted, and the second example shows that the method in this paper can effectively shorten the iterative convergence time of high order accurate flow simulation.
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图 1 二维物理量守恒插值示意图, 背景网格和当前网格分别用黑边和红边表示
Figure 1. Schematic diagram of two-dimensional conservation of physical quantity interpolation, the background mesh and the current mes are represented by red and black edges respectively
图 4 翼型计算域在不同自适应计算迭代步中的网格
Figure 4. Meshes of the airfoil computational domain in different adaptive computation iteration steps
图 5 自适应计算收敛过程中气动系数随网格规模变化的变化
Figure 5. Convergence of lift and drag coefficients against degrees of freedom (vertices)
图 6 自适应迭代收敛后的马赫数分布图
Figure 6. The distribution of Mach number after adaptive solution convergences
表 1 不同阶数精度对应的积分点信息
Table 1. Information of integration points corresponding to different order
Order Points Point location Weight 2 1 $ \left(\dfrac{1}{3},\dfrac{1}{3},\dfrac{1}{3}\right) $ 1 3 3 $ \left(\dfrac{1}{2},\dfrac{1}{2}, 0\right) , \left(\dfrac{1}{2},0, \dfrac{1}{2}\right) ,\left(0,\dfrac{1}{2},\dfrac{1}{2}\right) $ $ \dfrac{1}{3} , \dfrac{1}{3} , \dfrac{1}{3} $ 4 4 $ \left(\dfrac{1}{3},\dfrac{1}{3},\dfrac{1}{3}\right) ,\left(\dfrac{1}{5},\dfrac{1}{5},\dfrac{3}{5}\right) , \left(\dfrac{1}{5},\dfrac{3}{5},\dfrac{1}{5}\right) , \left(\dfrac{3}{5},\dfrac{1}{5},\dfrac{1}{5}\right) $ $ {{ - }}\dfrac{9}{{16}} ,\dfrac{{25}}{{48}} ,\dfrac{{25}}{{48}} , \dfrac{{25}}{{48}} $ 
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表 2 两组不同规模的网格数据
Table 2. Two groups of meshes with different scales
Fist group of meshes Second group of meshes Background mesh Current mesh Background mesh Current mesh elements 1043 1225 3363 4944 points 580 679 1792 2597
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表 3 在不同网格规模和不同插值精度下的流场插值误差
Table 3. Interpolation errors of flow field under different mesh scales and different orders of interpolation accuracy
Order Physical quantity First group of meshes Second group of meshes L1 norm L2 norm L∞ norm L1 norm L2 norm L∞ norm 2 ρ 3.05 × 10−4 5.25 × 10−4 4.25 × 10−3 1.12 × 10−4 1.73 × 10−4 1.34 × 10−3 u 2.07 × 10−4 3.60 × 10−4 3.21 × 10−3 7.41 × 10−5 1.21 × 10−4 1.35 × 10−3 v 2.03 × 10−4 3.57 × 10−4 2.57 × 10−3 7.48 × 10−5 1.22 × 10−4 1.18 × 10−3 P 2.86 × 10−4 4.70 × 10−4 3.52 × 10−3 1.01 × 10−4 1.58 × 10−4 1.32 × 10−3 3 ρ 1.65 × 10−5 3.08 × 10−5 3.13 × 10−4 3.15 × 10−6 5.98 × 10−6 6.39 × 10−5 u 1.21 × 10−5 2.37 × 10−5 2.72 × 10−4 2.41 × 10−6 6.05 × 10−6 1.52 × 10−4 v 1.11 × 10−5 2.22 × 10−5 2.78 × 10−4 2.44 × 10−6 6.11 × 10−6 1.13 × 10−4 P 3.23 × 10−5 6.03 × 10−5 5.92 × 10−4 6.82 × 10−6 1.30 × 10−5 1.24 × 10−4 4 ρ 2.55 × 10−6 5.34 × 10−6 4.19 × 10−5 3.11 × 10−7 7.11 × 10−7 7.65 × 10−6 u 1.16 × 10−6 2.56 × 10−6 3.51 × 10−5 1.43 × 10−7 4.50 × 10−7 1.26 × 10−5 v 1.17 × 10−6 2.30 × 10−6 2.26 × 10−5 1.41 × 10−7 4.32 × 10−7 1.32 × 10−5 P 4.40 × 10−6 8.73 × 10−6 6.53 × 10−5 5.08 × 10−7 1.05 × 10−6 9.43 × 10−6
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表 4 有无流场插值功能时求解收敛情况
Table 4. Convergence of solution with or without the flow field interpolation
Steps of adaptation Elements Vertices With flow field interpolation Without flow field interpolation Convergence iterations Convergence time/s Convergence iterations Convergence time/s initial 7790 3966 16 34.7 − − 1 9235 4693 11 33.0 15 40.1 2 12 517 6364 8 35.2 13 49.6 3 16 719 8477 8 47.5 14 69.9 4 21 290 10 773 10 70.7 23 138.8 5 27 100 13 708 10 91.0 22 170.9 6 34 823 17 593 10 125.3 29 294.6
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