Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems

We discuss the symmetric smoothed particle hydrodynamics (SSPH) method for generating basis functions for a meshless method. It admits a larger class of kernel functions than some other methods, including the smoothed particle hydrodynamics (SPH), the modified smoothed particle hydrodynamics (MSPH), the reproducing kernel particle method (RKPM), and the moving least squares (MLS) methods. For finding kernel estimates of derivatives of a function, the SSPH method does not use derivatives of the kernel function while other methods do, instead the SSPH method uses basis functions different from those employed to approximate the function. It is shown that the SSPH method and the RKPM give the same value of the kernel estimate of a function but give different values of kernel estimates of derivatives of the function. Results computed for a sine function defined on a one-dimensional domain reveal that the L ², the H ¹ and the H ² error norms of the kernel estimates of a function computed with the SSPH method are less than those found with the MSPH method. Whereas the L ² and the H ² norms of the error in the estimates computed with the SSPH method are less than those with the RKPM, the H ¹ norm of the error in the RKPM estimate is slightly less than that found with the SSPH method. The error norms for a sample problem computed with six kernel functions show that their rates of convergence with an increase in the number of uniformly distributed particles are the same and their magnitudes are determined by two coefficients related to the decay rate of the kernel function. The revised super Gauss function has the smallest error norm and is recommended as a kernel function in the SSPH method. We use the revised super Gauss kernel function to find the displacement field in a linear elastic rectangular plate with a circular hole at its centroid and subjected to tensile loads on two opposite edges. Results given by the SSPH and the MSPH methods agree very well with the analytic solution of the problem. However, results computed with the SSPH method have smaller error norms than those obtained from the MSPH method indicating that the former will give a better solution than the latter. The SSPH method is also applied to study wave propagation in a linear elastic bar.