On criterions for smoothed particle hydrodynamics kernels in stable field

Smoothed particle hydrodynamics (SPH) particle approximation equations for functions and their derivatives are analyzed in stable field. Three criterions for determining suitable kernels are proposed to evaluate the accuracy of computation. The first criterion is used to evaluate the estimation errors of the functions. The second and third criterions are for the first derivatives of the functions. The second criterion requires that the first derivative of a kernel should be zero when the position of the neighbor particle is approaching the estimated one. The third criterion is so defined that the particle estimation of the first derivatives in stable field should be zero. Ten SPH kernels with different orders of function are selected to demonstrate the application of the criterions. The effects of the position of the estimated particles and the smoothing length on behaviors of the kernels are analyzed. To verify the feasibility of the three criterions in dynamic field, one dimensional shock tube problem is simulated with four deliberately chosen kernels. The simulation results, including profiles of density, pressure, velocity and energy, are compared with the exact solutions. Through analyses, it is found that the three criterions proposed in this study are feasible to evaluate the properties of kernels. Of the three criterions, the first one is more critical than the other two. In terms of computational accuracy, Gaussian and Q-spline kernels can be regarded as the best kernels of the ten proposed kernels in this study.