Incompressible SPH method based on Rankine source solution for violent water wave simulation

Abstract

. Incompressible SPH method based on Rankine source solution for violent water wave simulation. Journal of Computational Physics, 276, pp. 291-314. doi: 10.1016Physics, 276, pp. 291-314. doi: 10. /j.jcp.2014 This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent AbstractWith wide applications, the smoothed particle hydrodynamics method (abbreviated as SPH) has become an important numerical tool for solving complex flows, in particular those with a rapidly moving free surface. For such problems, the incompressible Smoothed Particle Hydrodynamics (ISPH) has been shown to yield better and more stable pressure time histories than the traditional SPH by many papers in literature. However, the existing ISPH method directly approximates the second order derivatives of the functions to be solved by using the Poisson equation. The order of accuracy of the method becomes low, especially when particles are distributed in a disorderly manner, which generally happens for modelling violent water waves. This paper introduces a new formulation using the Rankine source solution. In the new approach to the ISPH, the Poisson equation is first transformed into another form that does not include any derivative of the functions to be solved, and as a result, does not need to numerically approximate derivatives. The advantage of the new approach without need of numerical approximation of derivatives is obvious, potentially leading to a more robust numerical method. The newly formulated method is tested by simulating various water waves, and its convergent behaviours are numerically studied in this paper. Its results are compared with experimental data in some cases and reasonably good agreement is achieved. More importantly, numerical results clearly show that the newly developed method does need less number of particles and so less computational costs to achieve the similar level of accuracy, or to produce more accurate results with the same number of particles compared with the traditional SPH and existing ISPH when it is applied to modelling water waves.

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

“…They particularly indicated that the relative error of the solution became larger with increased random perturbation: 1.5 % corresponding to (±0.1)S, but 17 % to (±0.5)S. They also demonstrated that the LP-SPH04 may lead to results with a convergent rate of 1.2-1.3 (less than 2 as shown for h = 0.268 √ S by Schwaiger 2008) with the particle shifting scheme to maintain the particle orderliness. Zheng et al (2014) performed similar tests, but used the function of f (x,y) = cos(4π x + 8π y) defined in the region of 2 ≤ x ≤ 3 and 2 ≤ y ≤ 3. This function is closer to the real pressure in water waves than x m +y m and x m y m .…”

“…From the figure, one can see that the average error of LP-SPH04 is consistently reduced with reduction of S. This trend is similar to the results of Schwaiger (2008) for a function of x m + y m obtained using h = 0.268 √ S, but different from those of Schwaiger (2008) obtained using h = 1.2S which is shown to be constant with the reduction of S in their papers. The reason is perhaps because the smooth length used in Zheng et al (2014) was larger, though it was still proportional to S. The average errors of LP-SPH03 can increase with the reduction of S, which is a divergent behaviour. Figure 1b demonstrates that the maximum error of LP-SPH04 consistently decreases until S = 0.0125 or Log(S) ≈ −1.9, but increase with increasing the resolution of the particles after that.…”

“…To demonstrate if there is a significant number of particles with a large error, Zheng et al (2014) plotted a figure similar to Fig. 3.…”

“…[20, 30 %], while the vertical axis shows the number of particles whose error Fig. 3 The number of particles with an error larger than a certain values for S = 0.01 and k = 1.2 (originally presented in Zheng et al 2014) lies in a range. For example, in the range of [20, 30 %], there are about 230 particles for the LP-SPH04.…”

“…Tamai et al (2016) carried out tests using the discrete Laplacians to estimate the values of Laplacian for a given function on a square domain 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. This is similar to the method used by Schwaiger (2008) and Zheng et al (2014) discussed above, but Tamai et al (2016) used a sum of four exponential functions. In their tests, the random distribution of particles was also achieved by randomly shifting the particle position on the basis of uniform distribution.…”