SOLUTION INTERPOLATION FOR HIGH-ORDER ACCURATE ADAPTIVE FLOW SIMULATION
-
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.
-
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}} $ 
下载: 导出CSV
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
下载: 导出CSV
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
下载: 导出CSV
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
下载: 导出CSV
-
[1] 张来平, 常兴华, 赵钟等. 计算流体力学网格生成技术. 北京: 科学出版社, 2017Zhang Laiping, Chang Xinghua, Zhao Zhong, et al. Mesh Generation Techniques in Computational Fluid Dynamics. Beijing: Science Press, 2017 (in Chinese) [2] Park MA, Loseille A, Krakos J, et al. Unstructured grid adaptation: status, potential impacts, and recommended investments towards CFD 2030//46th AIAA Fluid Dynamics Conference, 2016, 2016-3323 [3] Caplan PC, Haimes R, Darmofal DL, et al. Extension of local cavity operators to 3D + t space-time mesh adaptation//AIAA Scitech 2019 Forum, 2019, 2019-1992 [4] Caplan PC, Haimes R, Darmofal DL, et al. Four-dimensional anisotropic mesh adaptation. Computer-Aided Design, 2020, 129: 102915 doi: 10.1016/j.cad.2020.102915 [5] Loseille A, Alauzet F, Menier V. Unique cavity-based operator and hierarchical domain partitioning for fast parallel generation of anisotropic meshes. Computer-Aided Design, 2017, 85: 53-67 doi: 10.1016/j.cad.2016.09.008 [6] Xiao ZF, Ollivier-Gooch C, Vazquez JDZ. Anisotropic tetrahedral mesh adaptation with improved metric alignment and orthogonality. Computer-Aided Design, 2022, 143: 103136 doi: 10.1016/j.cad.2021.103136 [7] Sharbatdar M, Ollivier-Gooch C. Mesh adaptation using C1 interpolation of the solution in an unstructured finite volume solver. International Journal for Numerical Methods in Fluids, 2018, 86(10): 637-654 doi: 10.1002/fld.4471 [8] Coulaud O, Loseille A. Very high order anisotropic metric-based mesh adaptation in 3D. Procedia Engineering, 2016, 163: 353-365 doi: 10.1016/j.proeng.2016.11.071 [9] Wang ZJ, Fidkowski K, Abgrall R, et al. High‐order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids, 2013, 72(8): 811-845 doi: 10.1002/fld.3767 [10] Hoshyari S, Mirzaee E, Ollivier-Gooch C. Efficient convergence for a higher-order unstructured finite volume solver for compressible flows. AIAA Journal, 2020, 58(4): 1490-1505 doi: 10.2514/1.J058537 [11] 廖飞. 高阶精度数值方法及其在复杂流动中的应用. [博士论文]. 西安: 西北工业大学, 2018Liao Fei. Efficient high-order high-resolution methods and the applications. [PhD Thesis]. Changsha: Northwestern Polytechnical University, 2018 (in Chinese) [12] Pan JH, Ren YX, Sun YT. High order sub-cell finite volume schemes for solving hyperbolic conservation laws. II: Extension to two-dimensional systems on unstructured grids. Journal of Computational Physics, 2017, 338: 165-198 doi: 10.1016/j.jcp.2017.02.052 [13] 雷国东, 李万爱, 任玉新. 求解可压缩流的高精度非结构网格WENO有限体积法. 计算物理, 2011, 28(5): 633-640 (Lei Guodong, Li Wanai, Ren Yuxin. A high-order unstructured-grid WENO FVM for compressible flow computation. Chinese Journal of Computational Physics, 2011, 28(5): 633-640 (in Chinese)Lei Guodong, Li Wanai, Ren Yuxin, A high-order unstructured-grid WENO FVM for compressible flow computation. Chinese Journal of Computational Physics, 2011, 28(5): 633-640 (in Chinese)) [14] Xu Z, Cambier L, Alonso JJ, et al. Towards a scalable hierarchical high-order CFD solver//AIAA Scitech 2021 Forum, 2021, 2021-0494 [15] Alauzet F, Mehrenberger M. P1-conservative solution interpolation on unstructured triangular meshes. International Journal for Numerical Methods in Engineering, 2010, 84(13): 1552-1588 doi: 10.1002/nme.2951 [16] 王瑞利. 散乱物理量逼近的插值重映算法. 计算物理, 2005, 22(4): 299-305 (Wang Ruili. An interpolated remapping algorithm for scattered physics quantities. Chinese Journal of Computational Physics, 2005, 22(4): 299-305 (in Chinese) doi: 10.3969/j.issn.1001-246X.2005.04.003Wang Ruili. An interpolated remapping algorithm for scattered physics quantities. Chinese Journal of Computational Physics, 2005, 22(04): 299-305 (in Chinese)) doi: 10.3969/j.issn.1001-246X.2005.04.003 [17] 王永健, 赵宁. 一类基于 ENO 插值的守恒重映算法. 计算物理, 2004, 21(4): 329-334 (Wang Yongjian, Zhao Ning. A kind of rezoning(remapping) algorithms based on ENO Interpolation. Chinese Journal of Computational Physics, 2004, 21(4): 329-334 (in Chinese) doi: 10.3969/j.issn.1001-246X.2004.04.008Wang Yongjian, Zhao Ning. A kind of rezoning(remapping) algorithms based on ENO Interpolation. Chinese Journal of Computational Physics, 2004, 21(4): 329-334 (in Chinese)) doi: 10.3969/j.issn.1001-246X.2004.04.008 [18] 赵小杰, 赵宁, 王东红. 一类基于RBF插值的守恒重映算法. 计算物理, 2012, 29(1): 10-16 (Zhao Xiaojie, Zhao Ning, Wang Donghong. A kind of conservative remapping algorithms based on RBF interpolations. Chinese Journal of Computational Physics, 2012, 29(1): 10-16 (in Chinese) doi: 10.3969/j.issn.1001-246X.2012.01.002Zhao Xiaojie, Zhao Ning, Wang Donghong. A kind of conservative remapping algorithms based on RBF interpolations. Chinese Journal of Computational Physics, 2012, 29(1): 10-16 (in Chinese)) doi: 10.3969/j.issn.1001-246X.2012.01.002 [19] 徐喜华, 刘娜, 陈艺冰. 非结构多面体二阶局部保界全局重映算法. 计算物理, 2018, 35(1): 22-28 (Xu Xihua, Liu Na, Chen Yibing. Second-order local-bound-preserving conservative remapping on unstructured polyhedral meshes. Chinese Journal of Computational Physics, 2018, 35(1): 22-28 (in Chinese)Xu Xihua, Liu Na, ChenYibing. Second-order local-bound-preserving conservative remapping on unstructured polyhedral meshes. Chinese Journal of Computational Physics, 2018, 35(1): 22-28 (in Chinese)) [20] Farrell PE, Piggott MD, Pain CC, et al. Conservative interpolation between unstructured meshes via supermesh construction. Computer Methods in Applied Mechanics and Engineering, 2009, 198(33-36): 2632-2642 doi: 10.1016/j.cma.2009.03.004 [21] Slattery SR. Mesh-free data transfer algorithms for partitioned multiphysics problems: conservation, accuracy, and parallelism. Journal of Computational Physics, 2016, 307: 164-188 doi: 10.1016/j.jcp.2015.11.055 [22] Alauzet F. A parallel matrix-free conservative solution interpolation on unstructured tetrahedral meshes. Computer Methods in Applied Mechanics and Engineering, 2016, 299: 116-142 doi: 10.1016/j.cma.2015.10.012 [23] Cheng J, Shu CW. A high order accurate conservative remapping method on staggered meshes. Applied Numerical Mathematics, 2008, 58(7): 1042-1060 doi: 10.1016/j.apnum.2007.04.015 [24] Lei N, Cheng J, Shu CW. A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113497 [25] Zhang M, Huang WZ, Qiu JX. High-order conservative positivity-preserving DG-interpolation for deforming meshes and application to moving mesh DG simulation of radiative transfer. SIAM Journal on Scientific Computing, 2020, 42(5): A3109-A3135 doi: 10.1137/19M1297907 [26] Ollivier-Gooch C, Nejat A, Michalak K. Obtaining and verifying high-order unstructured finite volume solutions to the Euler equations. AIAA Journal, 2009, 47(9): 2105-2120 doi: 10.2514/1.40585 [27] Lo SH, Wang WX. A fast robust algorithm for the intersection of triangulated surfaces. Engineering with Computers. 2004, 20(1): 11-21 [28] Campen M, Kobbelt L. Exact and robust (self-) intersections for polygonal meshes//Computer Graphics Forum, 2010: 397-406 [29] McLaurin D, Marcum D, Remotigue M, et al. Repairing unstructured triangular mesh intersections. International Journal for Numerical Methods in Engineering. 2013, 93(3): 266-275 [30] Stroud AH, Secrest D. Gaussian Quadrature Formulas. Englewood Cliffs: Prentice-Hall, 1966 [31] Pagnutti D, Ollivier-Gooch C. A generalized framework for high order anisotropic mesh adaptation. Computers & Structures, 2009, 87(11-12): 670-679 [32] Xiao ZF, Ollivier-Gooch C. Smooth gradation of anisotropic meshes using log-euclidean metrics. AIAA Journal, 2021, 59(10): 4105-4122 doi: 10.2514/1.J059864 [33] Malik S, Ollivier Gooch CF. Mesh adaptation for wakes via surface insertion//AIAA Scitech 2019 Forum, 2019, 2019-1996 -
下载:
点击查看大图
图(6) / 表(4)
计量
- 文章访问数: 376
- HTML全文浏览量: 110
- PDF下载量: 82
- 被引次数: 0
