The smoothed particle hydrodynamics method via residual iteration

Abstract

In this paper we propose for the first time an iterative approach of the Smoothed Particle Hydrodynamics (SPH) method. The method is widespread in many areas of science and engineering and despite its extensive application it su↵ers from several drawbacks due to inaccurate approximation at boundaries and at irregular interior regions. The presented iterative process improves the accuracy of the standard method by updating the initial estimates iterating on the residuals. It is appealing preserving the matrix-free nature of the method and avoiding to modify the kernel function. Moreover the process refines the SPH estimates and it is not a↵ected by disordered data distribution. We discuss on the numerical scheme and experiments with a bivariate test function and di↵erent sets of data validate the adopted approach.

User-Friendly Expressions of the Coefficients of Some Exponentially Fitted Methods

“…Further developments of this research will be oriented to the reformulation, through C and S functions, of existing methods for ordinary differential equations [ 2 , 17 , 20 , 25 , 26 , 28 , 37 – 39 , 41 , 42 , 44 , 48 , 51 , 53 , 56 , 77 , 80 ], integral equations [ 5 – 8 , 10 , 11 , 24 , 29 , 32 , 34 , 55 , 71 ], stochastic problems [ 9 , 12 , 13 , 18 , 19 , 29 , 47 ], fractional equations [ 12 , 13 , 21 , 22 , 36 ], partial differential equations [ 57 – 59 , 66 , 74 ].…”

Enhancing the Iterative Smoothed Particle Hydrodynamics Method

“…By increasing p, the order of accuracy increases, requiring the solution of linear systems for each evaluation point [16,17]. In [15] an iterative matrix-free approach (ISPH) is introduced to improve the standard method. In [18] it is adopted refining the normalized approach (4) as starting values for the iterations.…”

“…The method provides a spatial approximation via a discrete kernel convolution. In [15] the standard procedure has been improved by means of an iterative approach refining the residuals. An approximant is generated by employing strictly definite positive kernel functions and the method, in convergence, improves the results without requiring evenly spaced data distribution.…”

A normalized iterative Smoothed Particle Hydrodynamics method

“…The envelope of strictly definite positive kernel functions is considered and the method, in convergence, can be successfully applied without any restriction on pair-wise distinct data sites. Moreover, ensuring the linear approximation order [13,17,30], the iterative procedure provides more accurate results than those obtained with the standard SPH as initial values for both evenly and irregularly data distribution [19]. The paper is organized as follows.…”