Accuracy analysis of different Laplacian models of incompressible SPH method improved by using Voronoi diagram

The purpose of this paper is to investigate the effect of employment of Voronoi diagram drawing as an alternative approach to estimate the particle or nodal volume on accuracy of three well-known Laplacian operator models of incompressible SPH method (Hu and Adams in J Comput Phys 227:264–278, 2007; Schwaiger in Int J Numer Methods Eng 75:647–671, 2008; Shao and Lo in Adv Water Resour 26:787–800, 2003). In addition to this, the numerical performance of these modified Laplacian models is compared with that of another newly developed higher-order Laplacian model proposed by Shobeyri (Iran J Sci Technol Trans Civil Eng https://doi.org/10.1007/s40996-018-0226-9, 2019). The numerical errors of the above Laplacian models for solving different 2-D elliptic partial differential equations are analyzed on a unit square computational domain discretized with highly irregular node distributions. In addition to this, the patch test for an arbitrary function is conducted to evaluate numerical behavior of the Laplacian models. The numerical results indicate that the three Laplacian models coupled with Voronoi diagram technique can get smaller errors compared with the standard models. The most significant finding of this study is that the Laplacian model proposed by Shobeyri (Iran J Sci Technol Trans Civil Eng https://doi.org/10.1007/s40996-018-0226-9, 2019) produces lower numerical errors even in comparison with the improved models. Voronoi diagram technique with simple implementation can significantly improve numerical accuracy of the presented higher-order Laplacian models for arbitrary and highly irregular node configurations, and it is expected that this strategy is applicable for other SPH Laplacian models to obtain higher-accurate results.