Review of Development of the Smooth Particle Hydrodynamics (SPH) Method

Abstract

The paper gives an overview of developments of the SPH method. Especial attention is given to the main shortcomings of the original form of the method namely consistency, tensile instability and zero energy modes. An example of derivation of correction necessary to assure first order consistency is given. The origin of the tensile instability and few proposed solutions to this problem are described. Similar consideration is given with respect to the zero energy modes typical for the collocational SPH method.

Consistent SPH Simulations of Protostellar Collapse and Fragmentation

“…As a final remark, it has been demonstrated that C 0 particle consistency for both the estimates of a function and its gradient implies preservation of the homogeneity and isotropy of the discrete space, which have as consequences conservation of the linear and angular momentum, respectively (Vignjevic & Campbell 2009;Sigalotti et al 2016). Therefore, we may expect that local linear and angular momentum are well conserved in our consistent collapse calculations.…”

“…In particular, satisfaction of relation (5) means that the homogeneity of space is not affected by the SPH kernel approximation, which has as a consequence the conservation of linear momentum. On the other hand, fulfillment of relation (11) expresses that the isotropy of space is preserved by the kernel approximation, and therefore angular momentum is locally conserved (Vignjevic & Campbell 2009;Sigalotti et al 2016). …”

“…In particular, restoring C 0 particle consistency implies that the homogeneity and isotropy of the discrete space are preserved, which has as a consequence the conservation of linear and angular momentum in practical calculations (Vignjevic & Campbell 2009;Sigalotti et al 2016). The goal here is to track the quality of the particle consistency relations in a true hydrodynamic evolution involving large density and pressure gradients as well as large spatial and temporal variations of the smoothing length.…”

Braiding Simulation and Prediction of Mechanical Properties

“…The following brief overview is extracted from [11], and details some main points of the method. Spatial discretisation of the state variables is provided at the SPH particles using interpolation of the Kernel function.…”

“…4a. The The set of neighbouring particles [11] contributions of each particle to a property are weighted according to their distance from the particle of interest and their density. The Kernel function has the following properties:…”