# 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.

## References

- Adami S, Hu XY, Adams NA (2012) A generalized wall boundary condition for smoothed particle hydrodynamics. J Comput Phys 231(21):7057–7075CrossRefMathSciNetGoogle Scholar
- Antuono M, Colagrossi A, Marrone S, Molteni D (2010) Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Comput Phys Commun 181(3):532–549CrossRefMathSciNetzbMATHGoogle Scholar
- Antuono M, Colagrossi A, Marrone S (2012) Numerical diffusive terms in weakly-compressible SPH schemes. Comput Phys Commun 183(12):2570–2580CrossRefMathSciNetGoogle Scholar
- Antuono M, Bouscasse B, Colagrossi A, Marrone S (2014) A measure of spatial disorder in particle methods. Comput Phys Commun 185(10):2609–2621CrossRefGoogle Scholar
- Bonet J, Lok TS (1999) Variational and momentum preservation aspects of smooth particle hydrodynamic formulation. Comput Methods Appl Mech Eng 180:97–115CrossRefMathSciNetzbMATHGoogle Scholar
- Chen JK, Beraun JE, Jih CJ (1999) An improvement for tensile instability in smoothed particle hydrodynamics. Comput Mech 23:279–287CrossRefzbMATHGoogle Scholar
- Colagrossi A, Landrini M (2003) Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J Comput Phys 191(2):448–475CrossRefzbMATHGoogle Scholar
- Delorme L, Colagrossi A, Souto-Iglesias A, Zamora-Rodriguez R, Botia-Vera E (2009) A set of canonical problems in sloshing, part I: pressure field in forced roll-comparison between experimental results and SPH. Ocean Eng 36(2):168–178CrossRefGoogle Scholar
- Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–89CrossRefzbMATHGoogle Scholar
- Gotoh H, Shibahara T, Sakai T (2001) Sub-particle-scale turbulence model for the mps method—Lagrangian flow model for hydraulic engineering. Comput Fluid Dyn J 9(4):339–347Google Scholar
- Gotoh H, Sakai T (2006) Key issues in the particle method for computation of wave breaking. Coast Eng 53:171–179CrossRefGoogle Scholar
- Gotoh H (2009) Lagrangian particle method as advanced technology for numerical wave flume. Int J Offshore Polar Eng 19(3):161–167Google Scholar
- Gotoh H, Khayyer A, Ikari H, Arikawa T, Shimosako K (2014) On enhancement of Incompressible SPH method for simulation of violent sloshing flows. Appl Ocean Res 46:104–115CrossRefGoogle Scholar
- Gotoh H, Okayasu A, Watanabe Y (2013) Computational wave dynamics. World Scientific Publishing Co, SingaporeCrossRefzbMATHGoogle Scholar
- Hori C, Gotoh H, Ikari H, Khayyer A (2011) GPU-acceleration for moving particle semi-implicit method. Comput Fluids 51(1):174–183CrossRefzbMATHGoogle Scholar
- Hu XY, Adams NA (2009) A constant-density approach for incompressible multi-phase SPH. J Comput Phys 228(6):2082–2091CrossRefMathSciNetzbMATHGoogle Scholar
- Hwang SC, Khayyer A, Gotoh H, Park JC (2014) Development of a fully Lagrangian MPS-based coupled method for simulation of fluid-structure interaction problems. J Fluids Struct 50:497–511CrossRefGoogle Scholar
- Khayyer A, Gotoh H, Shao SD (2008) Corrected Incompressible SPH method for accurate water-surface tracking in breaking waves. Coast Eng 55(3):236–250CrossRefGoogle Scholar
- Khayyer A, Gotoh H (2009a) Modified moving particle semi-implicit methods for the prediction of 2D wave impact pressure. Coast Eng 56:419–440CrossRefGoogle Scholar
- Khayyer A, Gotoh H (2009b) Wave impact pressure calculations by improved SPH methods. Int J Offshore Polar Eng 19(4):300–307Google Scholar
- Khayyer A, Gotoh H (2010) A higher order Laplacian model for enhancement and stabilization of pressure calculation by the MPS method. Appl Ocean Res 32(1):124–131CrossRefGoogle Scholar
- Khayyer A, Gotoh H (2011) Enhancement of stability and accuracy of the moving particle semi-implicit method. J Comput Phys 230:3093–3118CrossRefMathSciNetzbMATHGoogle Scholar
- Khayyer A, Gotoh H (2012) A 3D higher order Laplacian model for enhancement and stabilization of pressure calculation in 3D MPS-based simulations. Appl Ocean Res 37:120–126CrossRefGoogle Scholar
- Khayyer A, Gotoh H (2013) Enhancement of performance and stability of MPS meshfree particle method for multiphase flows characterized by high density ratios. J Comput Phys 242:211–233CrossRefMathSciNetzbMATHGoogle Scholar
- Kondo M, Koshizuka S (2011) Improvement of stability in moving particle semi-implicit method. Int J Numer Methods Fluids 65:638–654Google Scholar
- Koshizuka S, Oka Y (1996) Moving particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421–434Google Scholar
- Koshizuka S (2011) Current achievements and future perspectives on particle simulation technologies for fluid dynamics and heat transfer. J Nucl Sci Technol 48(2):155–168CrossRefGoogle Scholar
- Lind S, Xu R, Stansby P, Rogers B (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J Comput Phys 231(4):1499–1523CrossRefMathSciNetzbMATHGoogle Scholar
- Liu WK, Adee J, Jun S (1993) Reproducing kernel and wavelets particle methods for elastic and plastic problems. Adv Comput Methods Mater Model 180(268):175–190Google Scholar
- Molteni D, Colagrossi A (2009) A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Comput Phys Commun 180(6):861–872CrossRefMathSciNetzbMATHGoogle Scholar
- Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30:543–574CrossRefGoogle Scholar
- Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method: towards higher order convergence. J Comput Phys 225(2):1472–1492CrossRefMathSciNetzbMATHGoogle Scholar
- Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Meth Appl Mech Eng 139:375–408CrossRefMathSciNetzbMATHGoogle Scholar
- Shao SD, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26(7):787–800CrossRefGoogle Scholar
- Souto-Iglesias A, Macià F, González LM, Cercos-Pita JL (2013) On the consistency of MPS. Comput Phys Commun 184(3):732–745CrossRefMathSciNetGoogle Scholar
- Touzé DL, Colagrossi A, Colicchio G, Greco M (2013) A critical investigation of smoothed particle hydrodynamics applied to problems with free surfaces. Int J Numer Methods Fluids 73:660–691Google Scholar
- Tsuruta N, Khayyer A, Gotoh H (2013) A short note on dynamic stabilization of moving particle semi-implicit method. Comput Fluids 82:158–164CrossRefMathSciNetzbMATHGoogle Scholar
- Tsuruta N, Khayyer A, Gotoh H (2015) Space potential particles to enhance the stability of projection-based particle methods. Int J Comput Fluid Dyn. doi: 10.1080/10618562.2015.1006130
- Ulrich C, Leonardi M, Rung T (2013) Multi-physics SPH simulation of complex marine-engineering hydrodynamic problems. Ocean Eng 64:109–121Google Scholar
- Veen D, Gourlay T (2012) A combined strip theory and smoothed particle hydrodynamics approach for estimating slamming loads on a ship in head seas. Ocean Eng 43:64–71Google Scholar
- Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4:389–396CrossRefMathSciNetzbMATHGoogle Scholar
- Young DL, Chen KH, Lee CW (2005) Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 209:290–321CrossRefzbMATHGoogle Scholar