固体介质中SPH方法的拉伸不稳定性问题研究进展

光滑粒子流体动力学法(smoothed particle hydrodynamics, SPH)是一种基于核估计的无网格Lagrange数值方法.它用粒子方程 离散流体动力学的连续方程, 既可以处理有限元难于处理的大变形和严重扭曲问题, 又可以 处理有限差分法不易处理的自由边界和材料界面的问题, 在固体力学中的冲击、爆炸和裂纹 模拟中具有广阔的发展前景.但是, 该算法的拉伸不稳定性(tensile instability)问题是它在固体力学领域中应用的最大障碍.对SPH稳定性分析表明, 算法不 稳定性的条件仅与应力状态和核函数的2阶导数有关.目前, 应力点法(stress points)、Lagrange核函数法、人工应力法(artificial stress)、修正光滑粒子法(corrective smoothed particle method, CSPM)和守恒光滑法(conservative smoothing)以及其他一些方法成功地改善了SPH的拉伸不稳定性, 但是每一种方法都不能彻 底解决SPH的拉伸不稳定性问题.本文介绍了SPH法的方程和Von Neumann稳定性分析的思想, 以及国内外在这几个方面的研究成果及其最新进展, 同时指出目前研究中存在的问题和研究的方向.

ADVANCES IN THE TENSILE INSTABILITY OF SMOOTHED PARTICLE HYDRODYNAMICS APPLIED TO SOLID DYNAMICS

SPH(smoothed particle hydrodynamics)is a gridless Lagrangian numerical method based on kernel approximation.In this method,the continuum equations of fluid dynamics are replaced by particle equations. SPH can deal with large deformation and extensive tangling,which are difficult to be handled in FEM,as well as free surface and material interface,which are tricky problems in FDM.SPH is robust in the simulation of impact,explosion and crack.However,the tensile instability is its biggest drawback in the applications to solid dynamics.Von Neumann stability analysis shows that the criterion for stability or instability can be expressed in terms of the stress state and the second derivative of the kernel function.At present,SPH tensile instability is alleviated in the stress point method,artificial stress method,CSPM(corrected smoothed particle hydrodynamics method),conservative smoothing method and other methods,but they can not completely overcome SPH tensile instability.In this paper,the theory of SPH and the idea of Von Neumann stability analysis are introduced,and the research results and recent advances in SPH tensile instability are analyzed. The existing problems and future trends in these fields are discussed.