C A N A L D E E X P E R I E N C I A S H I D R O D I N Á M I C A S , E L P A R D O

Publicación núm. 194

A PARTICLE-BASED ‘LAGRANGIAN’ CFD TOOL FOR FREE-SURFACE SIMULATION

POR

D. MUÑOZ

V. GONZÁLEZ M. BLAIN J. VALLE

J. C. DÍAZ-CUADRA

Ministerio de Defensa

MADRID MARZO 2006

A PARTICLE-BASED ‘LAGRANGIAN’ CFD TOOL FOR FREE-SURFACE SIMULATION

POR

D. MUÑOZ

V. GONZÁLEZ M. BLAIN J. VALLE

J. C. DÍAZ-CUADRA

Trabajo presentado en la conferencia “World Maritime Technology Conference, WTMC 2006”. Londres, Reino Unido, Marzo 2006

A particle-based ‘Lagrangian’ CFD tool for free-surface simulation

D. Muñoz

Next Limit Technologies V. Gonzalez

Next Limit Technologies M. Blain

Next Limit Technologies J. Valle CEHIPAR

J C Díaz-Cuadra NAVANTIA

SYNOPSIS

Computation of fluid motion in the presence of free surfaces is essential in many engineering applications. Nowadays, however, it is difficult to find technologies capable of robustly handling this type of scenario. Traditional mesh-based CFD codes often struggle to provide meaningful solutions to problems involving large free surface deformation and complex interaction with moving boundaries. This paper presents a new tool for fluid dynamics simulation that focuses on complex fluid motion with large free surface deformation, ‘sloshing’ phenomena, breaking waves, floating bodies and fully coupled fluid-structure interaction.

INTRODUCTION

From the point of view of industry, a tool capable of accurately solving fluid dynamics problems where free surfaces are present is of great value. Providing this type of tool to the engineer is a must for determining the correct dimensions of engineering structures at the design stage. With technologies currently available, engineers are not able to correctly take into account loads due to free fluid mass motion within tanks partially filled and forced by an external excitation. Furthermore, under some circumstances, fluid motions with large free-surfaces inside containers could lead to important stability issues. Within the domain of naval architecture and marine engineering, the ‘sloshing problem’ is of particular relevance. The ‘sloshing’ phenomena is a highly nonlinear resonant effect that arises due to coupling between an external excitation that moves a tank partially filled with liquid and the internal forces generated by the fluid itself. ‘Sloshing’ arises in all marine structures containing liquids. The effects of ‘sloshing’ loads are of great importance when, for example, designing LNG and FPSO tankers. The importance of ‘sloshing’ effects in these kinds of ships is related to the dimensions of their tanks. Oil tankers may have two or three tanks along the breadth in order to reduce the effects of ‘sloshing’ loads. ‘Sloshing’ may also be a problem encountered by offshore cargo operations. In the case of LNG’s the sloshing loads are an important subject as ‘sloshing’ effects introduce severe limitations on the level to which tanks can be filled.

DISCUSSION

Framework Traditional software for fluid dynamics computation typically have at their heart the long-established finite element method. Most of these suites simply cannot simulate cases where moving boundaries arise. Those who cater for these kinds of problems, also usually have to make use of indirect approaches to model the moving boundaries effect (as introducing inertial forces).

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The principal problem with the finite element approach is that it is necessary to compute a mesh that partitions the simulation space. Typically, for simple steady cases the mesh generation step occupies over eighty percent of the time of a fluid dynamics simulation run. Moreover, mesh quality is a critical factor determining the accuracy of the solution. In addition, when treating problems where free surfaces are present, the use of additional techniques for interface tracking is essential. Finite elements are definitely not naturally suited for free surface simulation. X There are two possible perspectives for handling mass transport problems in continuum mechanics: Eulerian is the most common and most intuitive approach in which ‘sensors’ that measure medium properties are placed at fixed locations throughout the simulation space. By properly combining these measures estimations of several mass transport related properties can be obtained. On the other hand in a Lagrangian approach ‘sensors’ follow the fluid mass and consider the variations which happen while tracking the flow. Mesh-based formulations of the incompressible Navier-Stokes equations usually need artificial numerical stabilization due to the way in which incompressibility is enforced (often a Poisson equation for the pressure). In addition, formulations that take the Eulerian approach also encounter problems due the presence of a convective term in the equations that which describes the transport of mass. Even for small deformations the distortion introduced in the mesh leads to inaccuracies and instabilities that are usually resolved by expensive remeshing procedures. Furthermore, attempting to capture compressibility effects with mesh-based techniques aggravates of these problems. Recently, great strides have been taken in the development of mesh-less techniques able to cope with mass transport equations. Lagrangian formulations do not require the discretization of a convective term as evolution laws for fluid magnitudes are expressed in material derivatives unlike mesh-based methods. The aim of mesh-free methods is to provide accurate and stable numerical solutions for integral and partial differential equations relying solely upon a set of arbitrarily distributed nodes without any kind of mesh framework. This new generation of computational mesh-free methods promises to be superior to the conventional grid-based finite differences and finite elements. In order to improve upon conventional mesh-based methods efforts have been principally focused on trying to solve problems for which these methods perform poorly or are difficult to apply such as problems with free surfaces, deformable boundaries, moving interfaces and large deformations that would require an extremely complex mesh generation procedure. Smoothed Particle Hydrodynamics (SPH) is one of the earliest mesh-free methods. It was presented as an approximation of the Monte Carlo method for the resolution of gas dynamics problems. It was first applied by Lucy (1977) for modelling astrophysical phenomena, and later widely extended for application to problems in continuum solid and fluid mechanics. The SPH method and its variants have been incorporated into many commercial codes. Smoothed Particle Hydrodynamics uses an integral representation for field function approximation and efficiently reproduces the behaviour of compressible and quasi-incompressible flow. The use of an integral representation of field functions transfers differentiation operations on the field function to the smoothing (weight) function. This relaxes requirements regarding the order of continuity of the approximated field function. As a result, SPH has been proved to be very stable for many problems with extremely large deformations. In order to solve fluid dynamics problems, the SPH formalism can easily be applied to the Navier-Stokes equations. The pure Lagrangian nature of SPH method handles fluid dynamics problems with extremely large free surface deformation effortlessly since particles implicitly delineate the free surface position. In contrast to finite element methods there is no need to explicitly capture and track the free surface with SPH. The classic SPH formulation provides very good conservation properties and has been proved very stable under highly demanding circumstances. On the other hand a simple analysis of the reproducing conditions of the SPH formulation reveals a lack of consistency when the nodes are not evenly distributed. The Navier-Stokes equations are second order partial differential equations and therefore linear consistency is a sufficient condition to ensure that in the limit the numerical model converges to the true equations. Linear consistency is always a desirable condition although it is not strictly necessary. Classic SPH with randomly distributed nodes does not constitute even a partition of unity and thus SPH cannot even guarantee zero-order consistency. It is known that the lack of consistency with SPH is a source of error and can lead to rapid deterioration in the solution accuracy. Recently, moving least squares corrections (MLS) have been applied to SPH in order to ensure n-order consistency.

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Particle methods usually also encounter difficulties when handling boundary conditions. A key aspect in order to ensure correct treatment of boundary conditions is to be aware of the non delta-kronecker property of SPH and MLS-SPH shape functions. In classic finite elements, information is easily interpolated between nodes, and the value of nodal properties match the values measured using the shape functions. However, in SPH or MLS-SPH, nodal properties do not match the measured field magnitudes approximated by the shape functions. This paper presents a new approach for solution of the viscous Navier-stokes equations that takes into account recent advances in mesh-free particle methods. This new technology exploits the powerful capabilities of mesh-less particle methods by using a Lagrangian formulation that handles compressibility effects in liquids, free and forced floating bodies, moving boundaries, multiphase flow, and complex coupled fluid-structure interaction. This technology is also highly applicable to the study of ‘sloshing’ phenomena.

Standard SPH Formulations

The key idea of SPH is that any field function ( )rf can be approximated by ( )rf defined as:

(1)

where W , the kernel or smoothing function, is a scalar function of the distance between particles and h is referred to as the smoothing length. The kernel must satisfy the following conditions:

(2)

(3)

A Gaussian smoothing function normalized for 2D computations would have the following form

(4)

The Gaussian is a ∞C function that fulfils all the kernel requirements. From a computational point of view it is more efficient to define a kernel with compact support that vanishes for particle separations 2hr >

(5)

Fig 1: Shape of the kernel for 2D simulations

Equation (1) can be approximated by the following expression:

(6)

( ) ( )

( ) ( ) ( ) ( ) ( )i i i j1Ω

f rf r W r -r ,h r dr r -r ,h

r

Nj

jj j

mf r Wρ

ρ ρ=

′′ ′ ′= ≈

′ ∑∫

( ) ( ) ( )r,hΩ

f r f r W r dr′ ′ ′=∫

( ) ( )ij ijh 0lim W r ,h δ r→

=

( )ij jΩ

W r ,h dr 1=∫

( )2

2rh

2

1W r,h eh π

−=

( ) ( )( ) ( )

( )

22

2

222

2

r( 4)1 h for 0 r 2hr4h π 3 log 16W r,h ( 4)h

0 for r 2h

⎧−⎪

≤ <⎪−= ⎨ +⎪

⎪ ≥⎩

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This can used to measure field properties at any location within the simulation domain by just using the nodal information stored with each particle. Writing equation (1) for the gradient of a function ( )rf and integrating by parts gives:

(7)

Where Ω∂ is the boundary of the fluid domainΩ . The surface integral is usually neglected because the function or the kernel is supposed to vanish at the boundary of the domain. However, care must be taken in the vicinity of solid boundaries. Standard SPH is generally used to solve the Euler equation which is limited to inviscid flows since it is not easy to obtain expressions for the second derivatives in a straightforward manner using SPH formalism. However, the SPH formalism may be applied to the full Navier-Stokes equations in order to simulate general-purpose flow simulations. The continuity equation in Lagrangian form is: (8)

Similarly, the momentum equation is:

(9) where the superscripts βα, denote the coordinate directions, v is the velocity and F is the external body force per unit mass and σ is the full internal stress tensor:

(10) For Newtonian fluids, the viscous shear stress is proportional to the shear denoted by the strain tensor ε by a factor known as the dynamic viscosity µ :

(11) where:

(12)

Finally the material derivative for the internal energy may be expressed as the combination of two parts. The first term is the isotropic pressure contribution multiplied by the volumetric strain whereas the second term represents the energy dissipation due to viscous shear forces.

(13)

Using the equality ( ) ρ∇+ρ−∇=∇ρ− vvv and applying SPH formalism we can express the material density derivative for each particle as follows:

(14)

Alternatively, mass conservation may be exactly enforced by directly applying equation [3] for the density:

(15)

( ) ( ) ( ) ( ) ( ) ( ) ( ) rΩ Ω Ω

f r f r W r-r ,h dr f r W r-r ,h dr f r W r-r ,h n ds′∂

′ ′ ′ ′ ′ ′ ′ ′∇ = ∇ = ∇ +∫ ∫ ∫r

D vDtρ ρ= − ∇

1Dv FDt x

α αβα

β

σρ∂

= +∂

pαβ αβ αβσ δ τ=− +

αβ αβτ ε µ=

( )23

v v vx x

β ααβ αβ

α βε δ∂ ∂= + − ∇⋅∂ ∂

2De p vDt

αβ αβµ ε ερ ρ

= − ∇⋅ +

( ) ( )i j ijW r -r ,hj i j iji

D m v vDtρ

= − ∇∑

( )i j ijW r -r ,hjij

mρ =∑

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After some manipulation the SPH approximation of the momentum equation for any given particle can be written in the following discretized form

(16)

Using equation (12) SPH approximation for ε we finally obtain:

(17)

The energy equation can be obtained by following a similar procedure to that of the momentum equation. The discretized energy equation has the final form:

(18)

Particle positions are easily updated by using following evolution law: (19)

In standard SPH for compressible flows, the particle motion is driven by the pressure gradient, where the particle pressure is calculated using the local particle density and internal energy in a “state equation”. For incompressible flows, however, there is no equation of state for pressure. Moreover, the actual equation of state for the fluid will lead to prohibitively small timesteps. Although it is possible to include a constant density constraint into the SPH formulation the resultant equations are too cumbersome to be solved. Usually an artificial compressibility approach is used. It is based in the fact that every incompressible fluid is theoretically compressible and therefore it is feasible to use a quasi-incompressible equation of state to model the incompressible flow. The purpose of introducing the artificial compressibility is to produce a time derivative of pressure. The following artificial equation of state is widely used

(20) where sc the isentropic speed of sound and oρ is a nominal reference density.

Improving consistency of SPH

Although the traditional SPH formulation presents very good conservation properties and has been proved stable is widely known that this formulation loses its consistency as soon as the nodes are not uniformly distributed. This rapidly deteriorates the solution quality. MLS shape functions can be applied in order to impose linear order consistency. This improves stability and accuracy of standard SPH algorithms. For evenly distributed nodes, classic kernel approximations reproduce constant and linear scalar fields provided a sufficient number of neighbours are present. As soon as the nodes are not placed uniformly within the domain we have

(21)

i ip j ij i j j ijj j i

j ji j i i j ii

p W WDv m m FDt x x

αβ αβα

α β

µ ε µ ερ ρ ρ ρ+ ∂ + ∂

= − + +∂ ∂∑ ∑

ji ji2v v3

j ij ijji i iji

j j i i

m W Wv W

x xαβ β α αβ

α βε δρ

∂ ∂⎛ ⎞= + − ⋅∇⎜ ⎟∂ ∂⎝ ⎠∑

i ip1

2 2j

j ij i ij i iji i j i

pDe m v WDt

αβ αβµ ε ερ ρ ρ+

= ⋅∇ +∑

ii

Dr vDt

=

2

1i s ii

o

cPγ

ρ ργ ρ

⎛ ⎞⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( )n

jj 1

W x 1=

≠∑- 5 -

This means that SPH approximations are not able to reproduce even a constant scalar field. This problem is magnified at the boundaries where the lack of information always introduces severe errors in kernel approximations even for uniformly distributed nodes. Recent SPH corrections resolve this situation making use of moving least squares (first successfully applied by Dilts [3]). MLS corrections have made it possible to impose an arbitrary order of consistency in shape function approximation given sufficient information (nodes). MLS exactly reproduces an arbitrary functional base

(22)

The procedure consists of modifying the kernel weight function in the following way

(23) where ( )xβ

ρρ is a vector of coefficients that depends on the spatial distribution of Lagrangian nodes.

When the coefficients have been computed, field functions may be approximated by:

(24) For example, in order to impose linear consistency the following base must be adopted

(25)

Thus, the MLS shape function takes the following form

(26)

The coefficients vector ( )xβ

ρρ must be computed in the following manner

(27)

(28)

(29)

As soon as n-order consistency has been imposed via MLS corrections the reproducing conditions of MLS kernels ensure that any function expressed as a linear combination of the base functions will be exactly reproduced. The figure below shows the shape functions of standard SPH (Wo) with Gaussian kernel function in a

unidimensional bounded domain attempting to reproduce a scalar field of the form 4

x3)x(f += (grey

line) with evenly distributed nodes. Following this the MLS-0 and MLS-1 results are shown.

2 2p 1,x,y,z,x ,y ,...=r

( ) ( ) ( ) ( )( )Mlsi jN x N x p x β x,x= ⋅

rrr r r r r

( )MLSi j j i

ja a N x=∑ r%

p 1,x,y,z=r

( ) ( ) ( )( ) ( )( ) ( )( )( )Mlsj i 0 i x i i j y i i j z i i j ijN x β x β x x x β x y y β x z z N= + − + − + −

r r r r r

0

1

( ) 1( ) 0

( ) ( )( ) 0( ) 0

i

x ii i

y i

z i

xx

x A xxx

ββ

βββ

−

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

rrr r rrr

( )( )i ij j ij

A x A N x=∑r r%

2

2

2

1 ( ) ( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( ) ( )( )( ) ( )( ) ( )( ) ( )

i j i j i j

i j i j i j i j i j i jij

i j i j i j i j i j i j

i j i j i j i j i j i j

x x y y z zx x x x x x y y x x z z

Ay y x x y y y y y y z zz z x x z z y y z z z z

− − −⎡ ⎤⎢ ⎥− − − − − −⎢ ⎥=⎢ ⎥− − − − − −⎢ ⎥− − − − − −⎢ ⎥⎣ ⎦

%

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Fig 2: SPH and MLS kernels for evenly distributed nodes

The figure clearly illustrates that, for evenly distributed nodes, the classic SPH approximation reproduces the linear function everywhere except at the boundaries due to lack of information. The figure also shows that MLS-1 exactly reproduces the linear scalar function across the whole domain. The figure below shows the shape functions of standard SPH (Wo), MLS-0 and MLS-1 reconstructing the same field with nodes at randomly distributed positions.

Fig 3: SPH and MLS kernels for randomly distributed nodes

It is clearly evident that the classic SPH approximation leads to severe errors while the MLS-1 shape functions exactly reproduce the linear scalar field across the entire domain.

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Implementation

The new work carried out by the authors improves the existing mesh-less methods solving most of the outstanding issues related to the treatment of boundary conditions. This technology also exploits the improved consistency fully Lagrangian formulation provided by the use of MLS-like shape functions. New approach takes special care in boundary conditions treatment. Boundary conditions play a decisive role on the solution of a fluid dynamics simulation. In finite elements, imposing these conditions is not very difficult. In most commercial pre-processor, users can easily specify these conditions either to the geometric entities or directly to the mesh. However, simulate accurately boundary conditions with actual engineering systems requires experience, knowledge, time and accurate engineering decision. Although all the procedures developed in finite elements are applicable with some modifications to mesh-less methods, boundary conditions implementation usually requires special care. In classic SPH formulations, there is no direct boundary condition. Nodes near the solid boundaries lack of information as there is no particles outside the fluid domain. Indirect approaches as ‘boundary particles’ which exert central forces on fluid particles are the common choice for modelling the solid boundary. But the deficit of information near the walls usually leads to incorrect solutions. Finally, even using MLS approximations for constructing shape functions, special techniques are required to impose essential boundary conditions, because the shape functions created do not satisfy the kroneker delta conditions. The technology developed makes use of in-house special techniques that physically imposes boundary conditions taking into account these subtle issues. This has been implemented in both 2 and 3 dimensions and solves the full viscous Navier-Stokes equations with compressibility effects. It provides a promising solution to easily cope with complex problems which include free surfaces, multiple phases, arbitrary moving solid boundaries and coupled fluid-structure interaction. The authors have decided to call this method XPH (Extended Particle Hydrodynamics) due to the similitude with SPH and related work. XPH has been tested in several scenarios related to the ‘sloshing’ phenomena. In the following figure, the simulation (bright water) has been superimposed over the real video footage of the tank (dark water). The figure shows how the simulated results match almost perfectly the profile of the real water.

Fig 4: simulation with XPH

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VALIDATION

The new method has been extensively compared against both theoretical and experimental data in several scenarios. Our first tests focus on the capabilities of the method to simulate ‘sloshing’ phenomena.

Sloshing tests

In order to test the method’s capabilities for solving the ‘sloshing’ problem we start with a simple parallelepiped tank. This tank is subjected to several rolling motions of various amplitudes and frequencies whilst filled to various levels. The goal is to capture for each water level the magnitude and frequency of the resonant ‘sloshing’ effects. In this paper the tank dimensions are (length x width x height) 800mm x 400mm x 250mm. The following sequence shows some snapshots from a simulation with filling level of 50mm subjected to an excitation of amplitude 10mm with frequency 5 rad·s-1. The numerical simulation has been superimposed over the experimental video data to illustrate the remarkable correspondence between the numerical and experimental free-surfaces.

Fig 5: The experimental tank video and the overlaid simulation visualization show remarkable

correspondence.

In order to produce the forced motion of the tank a movable platform has been constructed. This platform restricts all degrees of freedom except the angular rotation about an axis normal to the longest side of the tank. A computer controlled linear actuator generates the forced motion and also measures the angle of oscillation and the magnitude of the overall rolling momentum on the tank due to the motion of the free fluid mass. Below we present the numerical and experimental results for a complete range of frequencies for amplitude 10º and filling level 100mm. The numerical simulations are fully 3 dimensional. The figures show the amplitude of the rolling momentum due to water motion and the phase offset between excitation and response of the rolling momentum. The results for rolling momentum confirm that the numerical model closely matches the experimental data and that the resonant frequency is captured both qualitatively and quantitatively.

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Fig 6: amplitude of water momentum (experimental versus numerical)

Fig 5: phase offset (experimental versus numerical)

Fig 7: experimental tank versus simulation

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Wave generation

Linear wave generation has been tested and compared with analytical models. Good agreement with linear wave theory has been observed. A bi-dimensional numerical wave tank 36 meters long filled with particles to a depth of 1.25 meters has been used for wave generation simulations. The waves are generated by using a moving wall. A numerical beach dissipates the waves as approach the channel end. Linear wave theory states that the relation between wave number, channel depth and wave-maker frequency is as follows:

(30)

Fig 8: wave generation test The above image shows a wave generated by a piston with a 0.2m stroke and a period of 1.25 seconds. Equation (30) predicts a wavelength of 2.4 m that closely matches the simulated wavelength of 2.6m.

Fig 9: wave generation test In this case the period is 2.13 seconds and the wave-maker stroke is 0.4 m. The wavelength predicted by linear wave theory is 6.1 m while the simulated wave length is 6m.

( )2

ss

ωTanh h k =-k g

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Ship sunk The method can easily deal with the behaviour of free floating bodies as they interact with free surfaces. Below is a visualization of a bi-dimensional simulation of a 150 meters long tanker with a breach just aft of the bow. The simulation contains more that 800,000 particles and is run at full-scale using the true speed of sound for water. The image focuses on a subset of the numerical domain which is three times longer than the ship.

Fig 10: sink simulation The sequence shows below several snapshots from the simulation. Pressure waves can be observed radiating from the ship.

Fig 11: sink simulation sequence

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Multiphase tests This sequence shows a cross-section from a full 3 dimensional computation of the evolution of a bubble fully immersed in a denser medium. A high density ratio was used without surface tension effects.

Fig 12: bubble simulation Below we show another cross-section, this time of a 3D Rayleigh-Taylor instability simulation. Once again the model neglects surface tension effects.

Fig 13: Rayleigh-Taylor instability

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ACKNOWLEDGEMENTS

The new methodology presented in this paper has been developed by the authors as part of an official Spanish research and development program supported by the Ministry of Industry, the Ministry of Science and Technology and the Ministry of Defence. Next Limit Technologies has provided the expertise for the development of the simulation and visualization tools. Commercial applications of this technology will be supported by Next Limit Technologies.

The experimental validation of the ‘sloshing’ cases has been performed in collaboration with CEHIPAR (El Pardo Model Basin). CEHIPAR is an autonomous organization belonging to the Spanish Ministry of Defence. Its activities include ship and propeller design and model tests and research and development in hydrodynamics and Computational Fluid Dynamics (CFD) technologies all focused on obtaining of optimal ship hydrodynamic behaviour.

NAVANTIA is the leading Spanish company in the military shipbuilding sector. From the perspective of size and technological capability it occupies a leading position in European and Worldwide military shipbuilding. NAVANTIA’s engineers have provided information resources to the research and development team including specific technical discussions regarding applications in the marine industry.

REFERENCES

1. J.J. Monaghan, Simulating Free Surface Flows with SPH. Annu. Rev. Astron. Astrophys., 30, 543-574; 1992 2. P.W. Randles, L.D. Libersky, SPH: Some recent improvements and applications. Comput., Methods in Appl. Mech. And Engtg., 139, 375-408; 1996 3. G.A. Dilts, Moving-Least-Squares-Particle Hydrodynamics. Int. J. Numer. Meth. Engng. Part I 44, 1115-1155(1999), Part II 48, 1503 (2000)

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