修正变光滑长度SPH方法及其应用

Modified SPH method considering full variable smoothing lengths effects and its applications

提出了修正的变光滑长度SPH方法及其算法实现。与传统变光滑长度SPH法相比,新方程组基于对称形式核函数近似,并从根本上考虑了变光滑长度影响。方程组中,密度演化方程与变光滑长度方程隐式关联;而SPH动量方程和能量方程在Springel提出的全守恒SPH方程组基础上,改进其分散核近似形式为对称核近似形式得到。由于新方程组密度演化方程和变光滑长度方程隐式耦合,采用迭代求解密度演化方程和变光滑长度方程,显式求解SPH动量方程和能量方程的办法,迭代过程增加的计算量相对很少。给出了2个一维激波管算例和二维Sedov算例来验证方法的有效性。结果表明,新方法很好地解决了传统SPH法中变光滑长度影响,特别是在模拟二维Sedov问题时新方法能得到比Springel方法更准确的压强峰值位置,中心压强也更接近理论值。特别适合模拟爆炸与冲击、大变形大扭曲等密度和光滑长度变化剧烈的问题。

Modified SPH equations including fully variable smoothing length aspects and its implementation were proposed. Unlike the existing adaptive kernel SPH method, the fully variable smoothing lengths effects have been considered essentially in the scheme based on the adaptive symmetrical kernel estimation. Among the new equations, the evolution equation of density was derived, which is implicitly coupled with the variable smoothing length equation and the momentum equation and energy equation were derived from Springel＇s fully conservative formulation SPH using the symmetrical kernel estimation instead of the scattered kernel estimation algorithm. Because the new SPH density evolution equation was implicitly coupled with the variable smoothing length equation, an additional iteration process was employed necessarily to solve the evolution equations of density and the variable smoothing lengths equation, and the SPH momentum equation and the energy equation ware solved, which little cost in the iteration algorithm. The new equations and its algorithm are tested via two 1D shock-tube problems and a 2D Sedov problem, which shows that the new algorithm corrects the variable smoothing lengths effect fairly well, especially in the 2D Sedov problem, and the pressure peak is captured by the presented method more accurately than that of Springel＇s scheme, and the accuracy of the pressure at the center in Sedov problem is also improved. The new method can deal well with the large density gradient and large smoothing length gradient problems, such as large deformation and serious distortion problems in high velocity impact and blasting phenomenon.