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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