Numerical Simulations of Shock Problems With the Revised KDF-SPH Method
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摘要: 基于光滑粒子流体动力学(smooth particle hydrodynamics, SPH)方法中的光滑核近似和Taylor级数展开原理，利用核函数矩对KDF-SPH(kernel derivative free SPH)方法进行了修正.为了验证修正方法的适用性和可行性，将该方法应用于不同情况下一维激波管问题的数值模拟，并对模拟结果进行分析.结果表明，修正方法能很好地捕捉到激波和接触不连续的位置和强度，修正方法不要求核函数可导性，不计算核函数矩，计算量更小，计算效率更高.Abstract: Based on the smooth kernel approximation and the Taylor series expansion of the smooth particle hydrodynamics (SPH) method, the kernel function moment was used to revise the KDF-SPH (kernel derivative free SPH) method. To prove the applicability and feasibility of the proposed revised scheme, the scheme was applied to the numerical simulations of 1D shock tube problems under different conditions, and the simulation results were analyzed. The results show that, the revised method can well capture the positions and strengths of shock waves and contact discontinuities. The revised method does not require the derivability of the kernel function, does not calculate the kernel function moment, and has a smaller computation cost with a higher calculation efficiency.
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图 3 t=0.2 s时刻，Sod问题的SPH解、KDF-SPH解、修正解与精确解
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 3. The SPH solution, the KDF-SPH solution, the revised solution and the exact solution of the Sod problem at t=0.2 s
图 4 t=0.15 s时刻，Sjögreen测试的SPH解、KDF-SPH解、修正解与精确解
Figure 4. The SPH solution, the KDF-SPH solution, the revised solution and the exact solution of the Sjögreen test at t=0.15 s
图 5 t=0.012 s时刻，爆炸波问题的SPH解、KDF-SPH解、修正解与精确解
Figure 5. The SPH solution, the KDF-SPH solution, the revised solution and the exact solution of the blast wave problem at t=0.012 s
图 6 t=0.035 s时刻，强冲击碰撞问题的SPH解、KDF-SPH解、修正解与精确解
Figure 6. The SPH solution, the KDF-SPH solution, the revised solution and the exact solution of the strong shock problem at t=0.035 s
表 1 计算量对比
Table 1. Comparison of calculations
revised KDF-SPH KDF-SPH conventional SPH nuclear derivative needless needless needed kernel function moment needless needed needless 
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表 2 程序运行时长比较
Table 2. Comparison of program running time costs
case N t/s revised KDF-SPH KDF-SPH conventional SPH Sod problem 600 37.066 70 52.095 91 47.466 41 6 000 2 890.730 82 4 429.704 88 4 232.116 63 Sjögreen test 600 37.789 27 52.667 87 48.074 76 6 000 3 027.268 75 4 543.114 52 4 077.673 14 blast wave problem 600 46.298 74 63.743 67 57.654 41 6 000 2 974.784 74 4 603.566 58 4 017.884 46 strong shock problem 600 391.313 07 551.295 24 508.942 10 6 000 29 577.200 15 46 118.617 35 40 347.813 63
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[1] LUCY L B. A numerical approach to the testing of the fission hypothesis[J]. The Astronomical Journal, 1977, 82: 1013. doi: 10.1086/112164 [2] HE F, ZHANG H S, HUANG C, et al. Numerical investigation of the solitary wave breaking over a slope by using the finite particle method[J]. Coastal Engineering, 2020, 156: 103617. doi: 10.1016/j.coastaleng.2019.103617 [3] HE F, ZHANG H S, HUANG C, et al. A stable SPH model with large CFL numbers for multi-phase flows with large density ratios[J]. Journal of Computational Physics, 2022, 453: 110944. doi: 10.1016/j.jcp.2022.110944 [4] FENG D Y, IMIN R. A kernel derivative free SPH method[J]. Results in Applied Mathematics, 2023, 17: 100355. doi: 10.1016/j.rinam.2023.100355 [5] HUANG C, LEI J M, LIU M B, et al. A kernel gradient free (KGF) SPH method[J]. International Journal for Numerical Methods in Fluids, 2015, 78(11): 691-707. doi: 10.1002/fld.4037 [6] MAATOUK K. Third order derivative free SPH iterative method for solving nonlinear systems[J]. Applied Mathematics and Computation, 2015, 270: 557-566. doi: 10.1016/j.amc.2015.08.083 [7] IMIN R, IMINJAN A, HALIK A. A new revised scheme for SPH[J]. International Journal of Computational Methods, 2018, 15(5): 1-17. [8] IMIN R, WEI Y, IMINJAN A. New corrective scheme for DF-SPH[J]. Computational Particle Mechanics, 2020, 7(3): 471-478. doi: 10.1007/s40571-019-00273-w [9] HUANG C, LONG T, LI S M, et al. A kernel gradient-free SPH method with iterative particle shifting technology for modeling low-Reynolds flows around airfoils[J]. Engineering Analysis With Boundary Elements, 2019, 106: 571-587. doi: 10.1016/j.enganabound.2019.06.010 [10] GAROOSI F, SHAKIBAEINIA A. Numerical simulation of free-surface flow and convection heat transfer using a modified weakly compressible smoothed particle hydrodynamics (WCSPH) method[J]. International Journal of Mechanical Sciences, 2020, 188: 105940. doi: 10.1016/j.ijmecsci.2020.105940 [11] HUANG C, LEI J M, LIU M B, et al. An improved KGF-SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows[J]. International Journal for Numerical Methods in Fluids, 2016, 81(6): 377-396. doi: 10.1002/fld.4191 [12] 王建玲, 李小纲, 汪文帅. 一个改进的三阶WENO-Z型格式[J]. 应用数学和力学, 2021, 42(4): 394-404. doi: 10.21656/1000-0887.410203WANG Jianling, LI Xiaogang, WANG Wenshuai. An improved 3rd-order WENO-Z type scheme[J]. Applied Mathematics and Mechanics, 2021, 42(4): 394-404. (in Chinese) doi: 10.21656/1000-0887.410203 [13] 张成治, 郑素佩, 陈雪, 等. 求解理想磁流体方程的四阶WENO型熵稳定格式[J]. 应用数学和力学, 2023, 44(11): 1398-1412. doi: 10.21656/1000-0887.440178ZHANG Chengzhi, ZHENG Supei, CHEN Xue, et al. A 4th-order WENO-type entropy stable scheme for ideal magnetohydrodynamic equations[J]. Applied Mathematics and Mechanics, 2023, 44(11): 1398-1412. (in Chinese) doi: 10.21656/1000-0887.440178 [14] MONAGHAN J J, GINGOLD R A. Shock simulation by the particle method SPH[J]. Journal of Computational Physics, 1983, 52(2): 374-389. doi: 10.1016/0021-9991(83)90036-0 [15] LI M K, ZHANG A M, PENG Y X, et al. An improved model for compressible multiphase flows based on smoothed particle hydrodynamics with enhanced particle regeneration technique[J]. Journal of Computational Physics, 2022, 458: 111106. doi: 10.1016/j.jcp.2022.111106 [16] MENG Z F, ZHANG A M, WANG P P, et al. A shock-capturing scheme with a novel limiter for compressible flows solved by smoothed particle hydrodynamics[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 386: 114082. doi: 10.1016/j.cma.2021.114082 [17] WANG P P, ZHANG A M, MENG Z F, et al. A new type of WENO scheme in SPH for compressible flows with discontinuities[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 381: 113770. doi: 10.1016/j.cma.2021.113770 [18] SIROTKIN F V, YOH J J. A smoothed particle hydrodynamics method with approximate Riemann solvers for simulation of strong explosions[J]. Computers & Fluids, 2013, 88: 418-429. [19] 徐建于, 黄生洪. 圆柱形汇聚激波诱导Richtmyer-Meshkov不稳定的SPH模拟[J]. 力学学报, 2019, 51(4): 998-1011.XU Jianyu, HUANG Shenghong. Numerical simulation of cylindrical converging shock induced Richtmyer-Meshkov instability with SPH[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(4): 998-1011. (in Chinese) [20] FULK D A, QUINN D W. An analysis of 1-D smoothed particle hydrodynamics kernels[J]. Journal of Computational Physics, 1996, 126(1): 165-180. doi: 10.1006/jcph.1996.0128 [21] LIU G R, LIU M B. Smoothed Particle Hydrodynamics: a Meshfree Particle Method[M]. Singapore: World Scientific Publishing, 2003. [22] SIGALOTTI L D G, LÓPEZ H, TRUJILLO L. An adaptive SPH method for strong shocks[J]. Journal of Computational Physics, 2009, 228(16): 5888-5907. doi: 10.1016/j.jcp.2009.04.041 [23] DANAILA I, JOLY P, KABER S M, POSTEL M. An Introduction to Scientific Computing[M]. New York: Springer-Verlag, 2007. [24] SOD G A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J]. Journal of Computational Physics, 1978, 27(1): 1-31. doi: 10.1016/0021-9991(78)90023-2 [25] MONAGHAN J. SPH and Riemann solvers[J]. Journal of Computational Physics, 1997, 136(2): 298-307. doi: 10.1006/jcph.1997.5732 [26] WOODWARD P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics, 1984, 54(1): 115-173. doi: 10.1016/0021-9991(84)90142-6 [27] TORO E F. Riemann Solvers and Numerical Methods for Fluid Dynamics[M]. Berlin: Springer-Verlag, 2009. -
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