# Corrected higher order Laplacian for enhancement of pressure calculation by projection-based particle methods with applications in ocean engineering

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## Abstract

A corrected higher order Laplacian (CHL) scheme is proposed for enhancement of pressure calculation in projection-based particle methods. The CHL scheme is derived by meticulously taking divergence of a corrected SPH gradient model in a similar manner to derivation of higher order Laplacian (HL) scheme performed by Khayyer and Gotoh (Appl Ocean Res 32(1):124–131, 2010; Appl Ocean Res 37:120–126, 2012). Unlike the original SPH gradient model considered in derivation of HL, the (first-order) consistency of the corrected SPH gradient model is strictly guaranteed. The enhanced performance of CHL with respect to HL is shown by a set of numerical simulations corresponding to designed sinusoidal pressure oscillations, unperturbed/perturbed water jets impinging on a flat plate and a 2D diffusion problem. Hence, the CHL scheme is suggested to be applied in place of the HL one, especially for practical engineering applications including those encountered in ocean engineering.

## Keywords

Corrected higher order Laplacian Particle method Moving particle semi-implicit method Pressure calculation Consistency## 1 Introduction

Particle methods or Lagrangian gridless methods have been increasingly applied in a wide range of engineering fields including ocean and marine engineering. In particular, successful simulations of violent sloshing flows (e.g., Gotoh et al. 2014; Hwang et al. 2014; Delorme et al. 2009), slamming loads on ships (e.g., Veen and Gourlay 2012) and scouring of offshore structures (e.g., Ulrich et al. 2013) have been carried out by two well-known particle methods, namely, moving particle semi-implicit (MPS; Koshizuka and Oka 1996) and smoothed particle hydrodynamics (SPH; Gingold and Monaghan 1977) methods.

Despite their robustness and wide potential range of applicability, particle methods have been suffering from major shortcomings, that is, presence of unphysical pressure oscillations that results from local particle-based interpolations by incomplete/inconsistent differential operator models (Gotoh 2009; Gotoh et al. 2013). As a result of this shortcoming, particle methods have not been extensively applied for practical ocean and marine engineering applications, particularly those corresponding to pressure calculations (e.g., wave impact pressure). Considerable efforts, however, have been made to minimize such unphysical oscillations and enhance the accuracy of particle methods by deriving corrected (e.g., Bonet and Lok 1999; Khayyer et al. 2008), higher order (e.g., Colagrossi and Landrini 2003; Khayyer and Gotoh 2009a, b) differential operator models, error mitigating terms (e.g., Hu and Adams 2009; Khayyer and Gotoh 2011, 2013; Kondo and Koshizuka 2011), dynamic stabilizers (e.g., Tsuruta et al. 2013), particle shifting techniques (e.g., Lind et al. 2012) and enhanced boundary conditions (e.g., Adami et al. 2012; Tsuruta et al. 2015). In the context of explicit SPH methods, a so-called delta-SPH scheme (Antuono et al. 2010, 2012) has proven to substantially enhance the pressure calculation. In a comprehensive and rigorous work, Touzé et al. (2013) highlighted the significance of higher order interpolation schemes to improve the pressure field.

In an attempt to improve the pressure calculation by a projection-based particle method, namely, MPS method, Khayyer and Gotoh (2010) derived a higher order Laplacian model (abbreviated as HL) by meticulously taking the divergence of a commonly applied SPH gradient model (Monaghan 1992) for discretization of Laplacian of pressure in the Poisson pressure equation (PPE). This derivation was later extended to 3D with verified enhancing effects (Khayyer and Gotoh 2012). One numerical issue of the HL scheme corresponds to its derivation on the basis of a SPH gradient model without a guaranteed consistency for irregularly distributed particles and/or particles without a full compact support (e.g., at and in the vicinity of free surface) (Randles and Libersky 1996; Gotoh et al. 2013; Souto-Iglesias et al. 2013). A common approach to guarantee the consistency of gradient models in particle methods is to derive corrective matrices based on Taylor-series expansions of the considered physical field (e.g., Oger et al. 2007; Khayyer and Gotoh 2011).

In this paper, a corrected higher order Laplacian, hereafter abbreviated as CHL, is derived by considering a corrected SPH gradient model and by performing a careful and meticulous derivation similar to those performed by Khayyer and Gotoh (2010, (2012). The enhanced performance of the CHL scheme will be verified by a set of simulations comprising of designed sinusoidal and exponentially excited sinusoidal pressure variations (Khayyer and Gotoh 2012), unperturbed/perturbed jets impinging on a flat plate (Molteni and Colagrossi 2009) and a 2D diffusion problem (Young et al. 2005).

## 2 MPS-HS-HL-ECS-GC method

The MPS method is a macroscopic, deterministic and projection-based particle method, initially proposed for simulation of incompressible fluid flows by Koshizuka and Oka (1996). The method reproduces the flow field by solving the continuity and Navier–Stokes equations as the governing equations. Through the past years, refined numerical schemes have been proposed in order to enhance the stability and performance of MPS method. In this study, enhanced MPS methods benefitting from so-called HS, HL (or CHL), ECS and GC schemes are considered. In this section, concise descriptions of HS, HL, ECS and GC schemes are presented in precedent order. Detailed descriptions can be found in Khayyer and Gotoh (2009a; b, 2010, 2011) and Gotoh et al. (2013).

### 2.1 The HS scheme

### 2.2 The HL scheme

### 2.3 The ECS scheme

### 2.4 The GC scheme

## 3 Corrected higher order Laplacian (CHL) scheme

## 4 Verification tests

The enhancing performance of the CHL scheme with respect to the HL scheme will be shown by simulations of designed sinusoidal and exponentially excited sinusoidal pressure variations (Khayyer and Gotoh 2012), unperturbed/perturbed jets impinging on a flat plate (Molteni and Colagrossi 2009) and a 2D diffusion problem (Young et al. 2005). The simulations are performed by an improved version of the MPS method, namely, MPS-HS-ECS-GC method (Gotoh et al. 2013; Khayyer and Gotoh 2011), briefly described in Sect. 2, incorporated with either HL or CHL schemes (MPS-HS-HL-ECS-GC or MPS-HS-CHL-ECS-GC).

### 4.1 Designed sinusoidal pressure variations (Khayyer and Gotoh 2012)

### 4.2 Designed exponentially excited sinusoidal pressure variations (Khayyer and Gotoh 2012)

RMSE (root mean square error) corresponding to HL and CHL schemes—exponentially excited sinusoidal pressure variations

\(d_{0}\) | 10 mm | 5 mm | 2.5 mm | |||
---|---|---|---|---|---|---|

Scheme | HL | CHL | HL | CHL | HL | CHL |

Irregular | 234.5438 | 19.2850 | 162.1980 | 30.8089 | 150.9107 | 24.2292 |

### 4.3 Jet impingement on a flat plate (Molteni and Colagrossi 2009)

The impingement of a water jet on a flat plate has been considered as a benchmark test to illustrate the enhanced pressure calculations by improved versions of both SPH (e.g., Molteni and Colagrossi 2009; Antuono et al. 2010) and MPS (e.g., Khayyer and Gotoh 2011) methods. A two-dimensional inviscid water jet impinges on a horizontal rigid plate. After the impact of jet and release of shock pressure, the flow regime becomes steady and the pressure at the stagnation point is that obtained from the Bernoulli equation.

### 4.4 Perturbed jet impingement on a flat plate

The superiority of CHL with respect to HL in simulation of a perturbed jet impingement suggests that this scheme should be preferred to the HL scheme in simulations of violent fluid flows that often contain irregularities and fragmentations.

### 4.5 Two-dimensional diffusion problem

RMSE (root mean square error) corresponding to HL and CHL schemes—a 2D diffusion problem

\(L/D=10\) | \(L/D=100\) | ||
---|---|---|---|

HL | CHL | HL | CHL |

0.00748421 | 0.00649334 | 0.00125827 | 0.00105165 |

## 5 Concluding remarks

The paper presents a corrected higher order Laplacian (CHL) model for enhancement of pressure calculation by moving particle semi-implicit (MPS) method. The proposed CHL scheme is derived by meticulously taking the divergence of a corrected SPH gradient model in a similar manner to derivation of higher order Laplacian (HL) scheme conducted by Khayyer and Gotoh (2010, (2012). The considered corrected SPH gradient is characterized by a corrective matrix to assure the first-order consistency of pressure gradient approximations.

The enhanced performance of CHL with respect to HL is shown by performing designed sinusoidal and exponentially excited sinusoidal pressure variations (Khayyer and Gotoh 2012), unperturbed/perturbed jets impinging on a flat plate (Molteni and Colagrossi 2009) and a 2D diffusion problem (Young et al. 2005) through both qualitative and quantitative comparisons. The superiority of CHL with respect to HL is found to be more realizable in presence of irregularities in particle distributions or highly accelerated flow fields, often encountered in simulation of violent fluid flows. Thus, despite relative complexity in formulation and coding, the CHL scheme should be preferred to the HL one, in simulations related to practical engineering applications, including those related to ocean engineering. More rigorous studies on the accuracy and convergence of both HL and CHL schemes are scheduled to be conducted by the authors.

Although in this paper, derivation and application of CHL is considered for the MPS method, similar developments can be easily made for another well-known projection-based particle method, i.e., incompressible SPH method (e.g., Shao and Lo 2003).

Development of refined differential operator models, such as CHL, will be also helpful to achieve a more accurate and more reliable reproduction of SPS (sub-particle scale) turbulence (Gotoh et al. 2001) in hydrodynamic fluid flows where presence of unphysical pressure oscillations remains to be a challenging difficulty (Gotoh and Sakai 2006). Upon achieving an accurate and fully reliable MPS-based solver, real-time fluid flow simulations are expected to be obtained via high-performance GPU (graphics processing unit)-based computations (e.g., Hori et al. 2011). Gotoh (2009) and Koshizuka (2011) provide comprehensive reviews on key issues for extension of particle methods for practical engineering applications.

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