显式粒子方法求解不可压缩纳维-斯托克斯方程的收敛性研究及最优权重函数
Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier--Stokes equations
AI总结：
本文研究了显式粒子方法在求解不可压缩纳维-斯托克斯方程中的收敛性及优化权重函数，通过分析方法的收敛性和截断误差，推导出收敛条件，并通过数值实验验证了优化权重函数的有效性。
摘要：
为了提高粒子方法在不可压缩粘性流问题中的模拟可靠性，本文对一种完全显式的粒子方法进行了收敛性研究和精度改进。该显式粒子方法基于惩罚问题，理论上收敛到不可压缩纳维-斯托克斯方程，并通过定义为比光滑粒子流体动力学（SPH）和移动粒子半隐式（MPS）方法更广泛类别的近似算子进行空间离散。通过考虑显式粒子方法的解析推导和近似算子的截断误差估计，推测出收敛的充分条件。在这些条件下，通过比较精确解与近似解之间的误差数值验证了显式粒子方法的收敛性。此外，通过关注近似算子的截断误差，推导出最优权重函数，以减少一般粒子分布上的截断误差。通过截断误差的数值结果和驱动腔流的实验，验证了具有最优权重函数的近似算子的有效性。作为自由表面流动问题的应用，该显式粒子方法被应用于水坝破裂流。
英文摘要：
To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier--Stokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible Navier--Stokes equations, and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured.Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the truncation errors of the generalized approximate operators, an optimal weight function is derived by reducing the truncation errors over general particle distributions. The effectiveness of the generalized approximate operators with the optimal weight functions is confirmed using numerical results of truncation errors and driven cavity flow. As an application for flow problems with free surface effects, the explicit particle method is applied to a dam break flow.
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