First of first, SPH inherently handles well problems with free surface, deformable boundary, moving surface as well as extremely large deformation and distortion. The latter capability is of most significance. FEM is notorious for dealing with distortion and extremely large deformation due to its grid/mesh based characteristics. Mesh distortion is nontrivial to resolve and thus bring instability to calculation. Different techniques are proposed to combat mesh distortion. However, none of them has achieved the publicly acceptable level. FE analysis engineer thus find out more than 50 percent of their overall work is fighting mesh issues. As is said by Dr. Gui-Rong Liu and Dr. M. B. Liu, they themselves times and again are frustrated by FEM in their explosion related research.

SPH methods as a meshfree, Lagrangian particle method,is characterized by its adaptivity nature. However, whether this adaptivity allows a arbitrarily distributed of particles is still in doubt. Actually, a uniformly distributed particles are recommend based on imperial experience by Dr. Monaghan. This is somehow contradictory with what has been claimed By G.R.Liu and M.B.Liu[1]. Furthermore, as a meshfree method, particles in SPH has more physical meaning than its counterpart in FEM. Material properties like density/mass are carried on these SPH particles besides acting as interpolation points. That being said, SPH particle functions as both approximation points and material component.

Furthermore, SPH does not require a mesh to calculate derivative. And accounting of the well known fact that integration is replaced by summation over neighboring nodes and thus background mesh is unnecessary in SPH, it’s safe to conclude that SPH is a true meshless method.

Several drawbacks are identified including tensile instability, imposition of essential boundary conditions, accuracy near the boundary and particle inconsistency. Numerous improvements has been made. However, in a overall sense, numerical analysis of SPH method is barely conducted so far. For example, disordered particles in high speed impact has an unknown relationship with the final result. Nevertheless, SPH has already find its great success in fluid calculation comparing its embarrassing situation faced in solid mechanics.

SPH 的发展, 引用 Gui-Rong Liu, M. B. Liu 两位教授话说,已经接近或正在快速于成熟.不过,本人实在是不敢苟同. 毋 庸置疑, Gui-Rong Liu, M. B.Liu在无网格领域有着巨大的成绩,然而就事论事,SPH只能说是在高速碰撞的领域有了长足的进步和发展应用. 众所周知, 固体在高速以及超高速的背景之下, 非常接近于流体的行为表征. SPH 本身在流体力学领域的长时间发展为这一应用打下了良好基础在静态, 但是 在准静态 以及 低速的情况下, 固体的表现和流体就有显著的差别, 单举一例, 边界条件在流体力学中对最终解的作用远远不如其在固体力学的计算中影响巨大, 所以边界条件的施加仍然是一个值得商榷的问题.显而易见的道理就是, 偏微分方程的解依赖于边界条件.如果不能保证边界条件准确有效的施加,那 么所算出来的结果就没有那么可信了.虽然不少商业软件(例如,Ls-Dyna, Ansys-Autodyn)已经有了SPH的求解器, 但是其功能远不能说是强大. 总而言之, SPH 在固体力学领域的应用仍有很多问题尚未解决.

【参考文献】 Reference