Study on Particle Inconsistency Problem in Smoothed Particle Hydrodynamics

Abstract

Abstract. In the smoothed particle hydrodynamics (SPH) method, the particle inconsistency problem significantly influences the calculation accuracy. In the present study, we investigate primarily the influence of the particle inconsistency on the first derivative of field functions and discuss the behavior of several methods of addressing this problem. In addition, we propose a new approach by which to compensate for this problem, especially for functions having a non-zero second derivative, that is less computational demanding, as compared to the finite particle method (FPM). A series of numerical studies have been carried out to verify the performance of the new approach. IntroductionSmoothed particle hydrodynamics (SPH) [1] is a mesh-free or particle-based method for solving physical phenomena expressed by partial differential equations and has been widely used in various spheres. However, SPH has some numerical problems. One of these problems is the particle inconsistency that results from the particle approximation process, which leads to low calculation accuracy. Concretely speaking, because of the particle approximation features of SPH, the particle inconsistency problem associated with boundary particles, irregular distributed particles, or the choice of the smoothing length will occur. An appropriate countermeasure to address this problem is very important, because these cases frequently appear in actual analyses.In order to address this problem, several effective techniques, such as the corrective smoothed particle method (CSPM), have been proposed [2]. The finite particle method (FPM) has also been proposed by Liu and Liu [3]. In their paper, the accuracy of approximating field functions in such cases is investigated using conventional SPH, the CSPM, and the FPM, which is shown to be extremely accurate.However, they discuss only the case for approximating field functions, and only limited a constant field function and a linear field function are investigated. In fact, evaluation of the influence on the accuracy resulting from the particle inconsistency is indispensable not only for a field function but also for derivatives of the field function.From this point of view, we have developed the extended CSPM, which is based on the conventional CSPM, in order to approximate the derivatives of a field function more accurately with a smaller matrix than in the FPM, considering that the calculation in a particle method will be repeated for each particle and each time step. Several calculations in the one-dimensional problem are carried out in order to investigate the effect of the new approach.

Smoothed particle hydrodynamics method applied to oral region: A narrative review

“…Examples of the applications of the SPH method include crack-propagation simulation of a solder ball 21) , simulation of the forging process 22) , dam break analysis 23) , and fluid mechanics of splashing bow waves on ships 24) . However, this method has some disadvantages, such as tensile instability, unclear material boundary, and particle inconsitency [25][26][27] . Furthermore, for small deformations, the FEM proves to be more efficient than the SPH method, whereas the SPH method surpasses the FEM in efficiency for large deformations 28) .…”