Constructing smoothing functions in smoothed particle hydrodynamics with applications

This paper presents a general approach to construct analytical smoothing functions for the meshfree, Lagrangian and particle method of smoothed particle hydrodynamics. The approach uses integral form of function representation and applies Taylor series expansion to the SPH function and derivative approximations. The constructing conditions are derived systematically, which not only interpret the consistency condition of the method, but also describe the compact supportness requirement of the smoothing function. Examples of SPH smoothing function are constructed including some existing ones. With this approach, a new quartic smoothing function with some advantages is constructed, and is applied to the one dimensional shock problem and a one dimensional TNT detonation problem. The good agreement between the SPH results and those from other sources shows the effectiveness of the approach and the newly constructed smoothing function in numerical simulations.