On the Galerkin formulation of the Smoothed Particle

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng2000;00:1–6 Prepared usingnmeauth.cls [Version: 2000/01/19 v2.0]

On the Galerkin formulation of the Smoothed ParticleHydrodynamics method

L. Cueto-Felgueroso, I. Colominas∗, G. Mosqueira, F. Navarrina, M. Casteleiro

Group of Numerical Methods in Engineering, GMNIDept. of Applied Mathematics, Civil Engineering School

Universidad de La CorunaCampus de Elvina, 15192 La Coruna, SPAIN

SUMMARY

In this paper we propose a Galerkin based SPH formulation with moving least squares meshless approximation,applied to free surface flows. The Galerkin scheme provides a clear framework to analyze several procedureswidely used in the classical SPH literature, suggesting that some of them should be reformulated in order todevelop consistent algorithms. The performance of the methodology proposed is tested through various dynamicsimulations, demonstrating the attractive ability of particle methods to handle severe distortions and complexphenomena. Copyrightc© 2000 John Wiley & Sons, Ltd.

KEY WORDS: Computational Fluid Dynamics, Particle Methods, SPH, Free Surface Flows, Galerkin method.

1. INTRODUCTION

Meshless methods have experimented an intense development in the last decade. Their potential seemsto be such that many researchers feel to be living a period of pre-revolutionary activity in computationalmechanics (if Kuhn’s conception is applicable), even though it is not clear where the definitive fracturemay come from...

Particle methods are (too) frequently regarded asnumerical modelsrather thannumerical methods,well suited to give physically reasonablequalitative solutions in complex problems, but notquantitativelyaccurate solutions, even in simple situations. Much work has been devoted in recentyears to banish such a prejudice.

In the basis of meshless formulations we find specific interpolation techniques, such askernel estimates [1] or moving least-squares approximations [2]. However, meshless methods incomputational mechanics are not simply different interpolation schemes but constitute, indeed,a powerful and ambitious attempt to solve the equations of continuum mechanics without thecomputational workload associated to the explicit partition of the domain into certain non-overlappingcells. This essentially distinctive nature of meshless methods is, in turn, the origin of many well known

∗Correspondence to: E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de La Coruna, Campus de Elvina, 15192La Coruna, SPAIN. Email: [email protected]

Received YesterdayCopyright c© 2000 John Wiley & Sons, Ltd. Revised Today

2 L. CUETO-FELGUEROSO, I. COLOMINAS, G. MOSQUEIRA, F. NAVARRINA, M. CASTELEIRO

shortcomings. Unlike finite elements, the absence of a spatial framework complicates the application ofcertain numerical methodologies such as the weighted residuals (Galerkin) method. Nevertheless, wemust note that these techniques, nowadays the strongest basis to develop practical implementations ofmeshless methods, may be not as adequate for particle methods as they are for finite element methods.Certainly, the development of the mathematical knowledge would provide better techniques to obtainthe discrete equations in computational mechanics, taking out the whole power of the particle approach.

The Smoothed Particle Hydrodynamics (SPH) method was developed in the late 70’s to simulatefluid dynamics in astrophysics [3],[4]. The extension to solid mechanics was introduced by Libersky,Petschek et al. [5] and Randles [6]. Johnson and Beissel proposed a Normalized Smoothing Function(NSF) algorithm [7] and other corrected SPH methods have been developed by Bonet et al. [8],[9] andChen et al. [10]. More recently, Dilts has introduced Moving Least Squares (MLS) shape functionsinto SPH computations [11].

Early SPH formulations included both a new approximation scheme and certain characteristicdiscrete equations (the so-called SPH equations), which may look quite “esoteric” for those researcherswith some experience in methods with a higher degree of formalism such as finite elements. Theformulation described in this paper follows a different approach, and the discrete equations are obtainedusing a Galerkin weighted residuals scheme. This derivation may result somewhat disconcerting forthose accustomed to the classical SPH equations. However, we believe that Galerkin formulationsprovide a strong framework to develop consistent algorithms. Note that we use moving least-squaresshape functions and a Galerkin formulation: is this SPH or EFG? It is not our purpose to carry out a“taxonomic” study of meshless methods. Perhaps the correct question is: how many (really different)meshless methods there exist? Whatever the answer, we feel that our endeavour follows the spirit ofLucy, Gingold and Monaghan’s Smoothed Particle Hydrodynamics.

The outline of the paper is as follows. We begin with a brief review of standard SPH and moving leastsquares approximations. After introducing the model equations, their discrete counterpart is obtainedusing a Galerkin formulation, and various important computational issues, with special emphasis onnumerical integration, are addressed. Finally, the methodology is applied to the simulation of fluiddynamics and free surface flows.

2. MESHLESS APPROXIMANTS

2.1. Standard SPH shape functions.

The simplest way to construct meshless test and trial functions corresponds to the standard SPHapproximants, given by

Nj(xxxxxxxxxxxxxx) = VjWj(xxxxxxxxxxxxxx) = VjW (xxxxxxxxxxxxxx− xxxxxxxxxxxxxxj , h) (1)

In the above expression,W (xxxxxxxxxxxxxx−xxxxxxxxxxxxxxj , h) is a kernel (smoothing) function with compact support centeredat particlej andVj is the tributary or statistical “volume” associated to particlej. The parameterh,usually calledsmoothing lengthin the SPH literature ordilation parameterin the RKPM literature[12], is a certain characteristic measure of the size of the support ofWj (e.g. the radius in circularsupports). Exponential and spline funtions are most frequent kernels. The SPH aproximationu(xxxxxxxxxxxxxx) ofa given functionu(xxxxxxxxxxxxxx) can be posed in terms of the shape functions (1) and certain particle ornodalparametersuj as

Copyright c© 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng2000;00:1–6Prepared usingnmeauth.cls

ON THE GALERKIN FORMULATION OF THE SMOOTHED PARTICLE HYDRODYNAMICS METHOD 3

u(xxxxxxxxxxxxxx) =n∑

j=1

Nj(xxxxxxxxxxxxxx)uj =n∑

j=1

VjWj(xxxxxxxxxxxxxx)uj (2)

Using standard kernels, the aproximation given by (2) is poor near boundaries, and lacks even zerothorder completeness, i.e.

n∑

j=1

Nj(xxxxxxxxxxxxxx) 6= 1 (3)

Thus, the setNj(xxxxxxxxxxxxxx), j = 1, . . . , n, wheren is the total number of particles, does not constitute asigned partition of unity in the sense of Duarte [13]. The gradient ofu(xxxxxxxxxxxxxx) is evaluated as

∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxu(xxxxxxxxxxxxxx) =n∑

j=1

∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxx)uj =n∑

j=1

Vj∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxWj(xxxxxxxxxxxxxx)uj (4)

Alternative expressions are frequent in the SPH literature (see for example [14], [15]) to enforceconservation properties in the discrete equations, which are not assured by the approximation scheme.

2.2. Moving Least Squares based shape functions.

Other meshless interpolation schemes have been proposed. Although different in their formulation,kernel based approximants (Moving Least Squares, Reproducing Kernel Particle Method) can be seenas corrected SPH methods, and in practice they are very similar.

Within this approach both standard Moving Least Squares (MLS) [2] and Moving Least SquaresReproducing Kernel (MLSRKPM) [12] shape functions are analyzed. Let us consider a functionu(xxxxxxxxxxxxxx)defined in a bounded, or unbounded, domainΩ. The basic idea of the MLS approach is to approximateu(xxxxxxxxxxxxxx), at a given pointxxxxxxxxxxxxxx, through a polynomial least-squares fitting ofu(xxxxxxxxxxxxxx) in a neighbourhood ofxxxxxxxxxxxxxx as:

u(xxxxxxxxxxxxxx) ≈ u(xxxxxxxxxxxxxx) =m∑

i=1

pi(xxxxxxxxxxxxxx)αi(zzzzzzzzzzzzzz)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

= ppppppppppppppT (xxxxxxxxxxxxxx)αααααααααααααα(zzzzzzzzzzzzzz)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

(5)

where ppppppppppppppT (xxxxxxxxxxxxxx) is an m-dimensional polynomial basis andαααααααααααααα(zzzzzzzzzzzzzz)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

is a set of parameters to be

determined, such that they minimize the following error functional:

J(αααααααααααααα(zzzzzzzzzzzzzz)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

) =∫

yyyyyyyyyyyyyy∈ΩxxxxxxxxxxxxxxW (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)

∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

[u(yyyyyyyyyyyyyy)− ppppppppppppppT (yyyyyyyyyyyyyy)αααααααααααααα(zzzzzzzzzzzzzz)


]2

dΩxxxxxxxxxxxxxx (6)

beingW (zzzzzzzzzzzzzz−yyyyyyyyyyyyyy, h)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

a symmetric kernel with compact support (denoted byΩxxxxxxxxxxxxxx), frequently chosen

among the kernels used in standard SPH. As mentioned before,h is the smoothing length, whichmeasures the size ofΩxxxxxxxxxxxxxx. The stationary conditions ofJ with respect toαααααααααααααα lead to

∫

yyyyyyyyyyyyyy∈Ωxxxxxxxxxxxxxxpppppppppppppp(yyyyyyyyyyyyyy)W (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)


u(yyyyyyyyyyyyyy)dΩxxxxxxxxxxxxxx = MMMMMMMMMMMMMM(xxxxxxxxxxxxxx)αααααααααααααα(zzzzzzzzzzzzzz)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

(7)

where the moment matrixMMMMMMMMMMMMMM(xxxxxxxxxxxxxx) is



MMMMMMMMMMMMMM(xxxxxxxxxxxxxx) =∫

yyyyyyyyyyyyyy∈Ωxxxxxxxxxxxxxxpppppppppppppp(yyyyyyyyyyyyyy)W (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)


ppppppppppppppT (yyyyyyyyyyyyyy)dΩxxxxxxxxxxxxxx (8)

In numerical computations, the global domainΩ is discretized by a set ofn particles. We canthen evaluate the integrals in (7) and (8) using those particles insideΩxxxxxxxxxxxxxx as quadrature points (nodalintegration) to obtain, after rearranging,

αααααααααααααα(zzzzzzzzzzzzzz)∣∣∣zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

= MMMMMMMMMMMMMM−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx)uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx (9)

where the vectoruuuuuuuuuuuuuuΩxxxxxxxxxxxxxx contains certain nodal parameters of those particles inΩxxxxxxxxxxxxxx, the discrete versionof M is M(xxxxxxxxxxxxxx) = PΩxxxxxxxxxxxxxxWV(xxxxxxxxxxxxxx)PT

Ωxxxxxxxxxxxxxx , and matricesPΩxxxxxxxxxxxxxx andWV(xxxxxxxxxxxxxx) can be obtained as:

PΩxxxxxxxxxxxxxx =(pppppppppppppp(xxxxxxxxxxxxxx1) pppppppppppppp(xxxxxxxxxxxxxx2) · · · pppppppppppppp(xxxxxxxxxxxxxxnxxxxxxxxxxxxxx)

)(10)

WV(xxxxxxxxxxxxxx) = diag Wi(xxxxxxxxxxxxxx− xxxxxxxxxxxxxxi)Vi , i = 1, . . . , nxxxxxxxxxxxxxx (11)

Complete details can be found in [12]. In the above equations,nxxxxxxxxxxxxxx denotes the total number of particleswithin the neighbourhood of pointxxxxxxxxxxxxxx and Vi and xxxxxxxxxxxxxxi are, respectively, the tributary volume (usedas quadrature weight) and coordinates associated to particlei. Note that the tributary volumes ofneighbouring particles are included in matrixWV, obtaining an MLS version of the ReproducingKernel Particle Method (the so-called MLSRKPM) [16]. Otherwise, we can useW instead ofWV,

W(xxxxxxxxxxxxxx) = diag Wi(xxxxxxxxxxxxxx− xxxxxxxxxxxxxxi) , i = 1, . . . , nxxxxxxxxxxxxxx (12)

which corresponds to the classical MLS approximation (in the nodal integration of the functional(6), the same quadrature weight is associated to all particles). Introducing (9) in (5) the interpolationstructure can be identified as:

u(xxxxxxxxxxxxxx) = ppppppppppppppT (xxxxxxxxxxxxxx)M−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx)uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx = NNNNNNNNNNNNNNT (xxxxxxxxxxxxxx)uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx (13)

And, therefore, the MLS shape functions can be written as:

NNNNNNNNNNNNNNT (xxxxxxxxxxxxxx) = ppppppppppppppT (xxxxxxxxxxxxxx)M−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx) (14)

It is most frequent to use a scaled and locally defined polinomial basis, instead of the globally definedpppppppppppppp(yyyyyyyyyyyyyy). Thus, if a function is to be evaluated at pointxxxxxxxxxxxxxx, the basis would be of the formpppppppppppppp(yyyyyyyyyyyyyy−xxxxxxxxxxxxxx

h ). Theshape functions are, therefore, of the form

NNNNNNNNNNNNNNT (xxxxxxxxxxxxxx) = ppppppppppppppT (00000000000000)M−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx) (15)

In the 2D examples shown in this work, a linear polynomial basispppppppppppppp(yyyyyyyyyyyyyy−xxxxxxxxxxxxxxh ) =

(1, y1−x1

h , y2−x2h

)was

used, where(x1, x2) and (y1, y2) are, respectively, the cartesian coordinates ofxxxxxxxxxxxxxx andyyyyyyyyyyyyyy. This basisprovides linear completeness, i.e.

n∑

j=1

Nj(xxxxxxxxxxxxxx) = 1,

n∑

j=1

∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxx) = 00000000000000 (16)

n∑

j=1

xxxxxxxxxxxxxxjNj(xxxxxxxxxxxxxx) = xxxxxxxxxxxxxx,

n∑

j=1

xxxxxxxxxxxxxxj ⊗∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxx) = IIIIIIIIIIIIII (17)



2.3. The choice of kernel.

A wide variety of kernel functions appear in the literature, most of them being spline or exponentialfunctions. We have not found a general criterion for an optimal choice. The following cubic spline hasbeen extensively used [14]:

Wj(xxxxxxxxxxxxxx) = W (xxxxxxxxxxxxxx− xxxxxxxxxxxxxxj , h) =α

hν

1− 32s2 + 3

4s3 s ≤ 114 (2− s)3 1 < s ≤ 20 s > 2

(18)

wheres = ‖xxxxxxxxxxxxxx− xxxxxxxxxxxxxxj‖h

, ν is the number of dimensions andα takes the value23 ,107π or 1

π in one, two orthree dimensions, respectively. The coeficientα/hν is a scale factor neccesary only if non-correctedSPH interpolation is being used, to assure the normality property

∫WdV = 1. We do not use it in our

MLS computations.This approach corresponds to radial weights (i.e. the support of the kernel in two/three dimensions

is a circle/sphere with radius2h). However, 2D an 3D kernels can be constructed as tensor-productweights, where the weighting function in higher dimensions is computed as the product of one-dimensional kernels as

Wj(xxxxxxxxxxxxxx− xxxxxxxxxxxxxxj , h) =ν∏

n=1

Wnj (xn − xn

j , hn) (19)

wherexn is the n-th coordinate of particlexxxxxxxxxxxxxx. In the above expression we letWnj andhn (the one-

dimensional kernel function and its caracteristic smoothing length) be different for each dimension. Ifthe same weighting scheme is employed in all dimensions, then the support is a square/cube in 2D/3D.

3. CONTINUUM EQUATIONS

In finite deformation analysis two possible coordinate systems can be chosen to describe the continuumunder consideration [17],[18]:

• a certain reference configuration (usually an ”initial configuration”). This is called a Lagrangianor material description, and all relevant quantities are referred to an initial problem domain,Ω0;

• the current continuum configuration. This is called an Eulerian or spatial description; relevantquantities are referred to the current problem domain,Ω.

The former is most frequent in solid mechanics, whereas the latter is typical in fluid mechanics.These two descriptions will lead, in general, to non-equivalent discretizations in particle methods [19].

Let us assume that the behaviour of a continuum could be analyzed as if it was governed by thefollowing equations:

(a) Continuity equation. Conservation of mass can be written in a material form as an algebraicequation:

ρJ = ρ0 (20)



whereρ0 andρ are, respectively, the initial and current densities andJ is the determinant ofthe deformation gradient,J = det(FFFFFFFFFFFFFF ), FFFFFFFFFFFFFF = dxxxxxxxxxxxxxx

dXXXXXXXXXXXXXX. In the following, XXXXXXXXXXXXXX, xxxxxxxxxxxxxx = xxxxxxxxxxxxxx(XXXXXXXXXXXXXX), ∇∇∇∇∇∇∇∇∇∇∇∇∇∇XXXXXXXXXXXXXX

and∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxx denote coordinates and gradient operators in the reference and current configurations,respectively. Most SPH codes use an eulerian rate form for mass conservation,

dρ

dt= −ρ div(vvvvvvvvvvvvvv) (21)

where d·dt

denotes the material time derivative anddiv(vvvvvvvvvvvvvv) is computed in the currentconfiguration in terms of the velocity gradient tensorllllllllllllll as [17]:

div(vvvvvvvvvvvvvv) = tr(llllllllllllll), llllllllllllll =∂vvvvvvvvvvvvvv(xxxxxxxxxxxxxx, t)

∂xxxxxxxxxxxxxx= ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxvvvvvvvvvvvvvv (22)

(b) Momentum equation. In a Lagrangian description, conservation of linear momentum can bewritten as:

ρ0 dvvvvvvvvvvvvvv

dt= ∇∇∇∇∇∇∇∇∇∇∇∇∇∇XXXXXXXXXXXXXX · PPPPPPPPPPPPPP + bbbbbbbbbbbbbb (23)

wherebbbbbbbbbbbbbb is the body force per unit volume andPPPPPPPPPPPPPP is the first Piola-Kirchhoff stress tensor. ItsEulerian counterpart is

ρdvvvvvvvvvvvvvv

dt= ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxx · σσσσσσσσσσσσσσ + bbbbbbbbbbbbbb (24)

where stresses are now related to the Cauchy stress tensorσσσσσσσσσσσσσσ, andρ is the current density. Notethat, in fluid dynamics, the governing equations include a momentum equation which can bewritten in an arbitrary Lagrangian-Eulerian (ALE) form as [20]:

ρ(∂vvvvvvvvvvvvvv

∂t+ vvvvvvvvvvvvvv∗∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxvvvvvvvvvvvvvv

)= ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxx · σσσσσσσσσσσσσσ + bbbbbbbbbbbbbb (25)

wherevvvvvvvvvvvvvv∗ is the convective velocity. In finite element analysis,vvvvvvvvvvvvvv∗ is defined as the differencebetween the fluid velocity and the mesh velocity. Further details can be found in the excellentbook by Donea and Huerta [21] on finite element methods for flow problems. SPH-like particlemethods follow the movement of a set of particles, sovvvvvvvvvvvvvv∗ = 00000000000000 and the convective term in (25)vanishes. This is considered a Lagrangian description of the movement. When we say that (24) isposed in eulerian form we mean that relevant quatities are referred to the current configuration,although the description is Lagrangian in the aforementioned sense.

(c) Angular Momentum Conservation. We consider neither mass distributions of polar momenta normagnetizable media.

(d) Energy equation. Conservation of energy may also be considered in processes involving heattransfer or other related phenomena:

ρdU

dt= σσσσσσσσσσσσσσ : dddddddddddddd− div(qqqqqqqqqqqqqq) + ρQ (26)

whereU is the internal energy per unit mass,qqqqqqqqqqqqqq is the energy flux,Q a thermal source (energyper unit time and mass) anddddddddddddddd is the deformation gradient tensor, defined as

dddddddddddddd =12(∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxvvvvvvvvvvvvvv +∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxvvvvvvvvvvvvvvT ) (27)



We confine our study to problems governed by equations (20)– (24). The extension of themethodology is straightforward.

4. DISCRETE EQUATIONS

4.1. Weighted residuals. Test and trial functions.

The meshless discrete equations can be derived using a weighted residuals formulation. The discretecounterpart of the Galerkin weak form is almost equivalent to that obtained from kernel estimates[22] such as classical SPH formulations. Furthermore, such an equivalence indicates that SPH can bestudied in the context of Galerkin methods. The global weak (integral) form of the spatial momentumequation can be written as:

∫

Ω

ρdvvvvvvvvvvvvvv

dt· δvvvvvvvvvvvvvv dΩ = −

∫

Ω

σσσσσσσσσσσσσσ : δllllllllllllll dΩ +∫

Ω

bbbbbbbbbbbbbb · δvvvvvvvvvvvvvv dΩ +∫

Γ

σσσσσσσσσσσσσσnnnnnnnnnnnnnn · δvvvvvvvvvvvvvv dΓ (28)

beingΩ the problem domain,Γ its boundary andnnnnnnnnnnnnnn the outward unit normal to the boundary. Ifδvvvvvvvvvvvvvv andvvvvvvvvvvvvvv are approximated by certain test and trial functionsδvvvvvvvvvvvvvv andvvvvvvvvvvvvvv,

∫

Ω

ρdvvvvvvvvvvvvvv

dt· δvvvvvvvvvvvvvv dΩ = −

∫

Ω

σσσσσσσσσσσσσσ : δllllllllllllll dΩ +∫

Ω

bbbbbbbbbbbbbb · δvvvvvvvvvvvvvv dΩ +∫

Γ

σσσσσσσσσσσσσσnnnnnnnnnnnnnn · δvvvvvvvvvvvvvv dΓ (29)

The spatially discretized equations are obtained after introducing meshless test and trial functions andtheir gradients in (29) as

δvvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

i=1

δvvvvvvvvvvvvvviN∗i (xxxxxxxxxxxxxx), ∇∇∇∇∇∇∇∇∇∇∇∇∇∇δvvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =

n∑

i=1

δvvvvvvvvvvvvvvi ⊗∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxx) (30)

vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxx), ∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvvj ⊗∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxx) (31)

to yield,

n∑

i=1

δvvvvvvvvvvvvvvi· n∑

j=1

∫

Ω

ρN∗i (xxxxxxxxxxxxxx)Nj(xxxxxxxxxxxxxx)

dvvvvvvvvvvvvvvj

dtdΩ +

∫

Ω

σσσσσσσσσσσσσσ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxx)dΩ−

−∫

Ω

N∗i (xxxxxxxxxxxxxx)bbbbbbbbbbbbbb dΩ−

∫

Γ

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓ

= 0 (32)

Thus, for each particlei the following identity must hold:

n∑

j=1

∫

Ω

ρN∗i (xxxxxxxxxxxxxx)Nj(xxxxxxxxxxxxxx)

dvvvvvvvvvvvvvvj

dtdΩ = −

∫

Ω

σσσσσσσσσσσσσσ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxx)dΩ +

∫

Ω

N∗i (xxxxxxxxxxxxxx)bbbbbbbbbbbbbb dΩ +

∫

Γ

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓ (33)

In this paper we follow a Bubnov Galerkin approach and, therefore,N∗j = Nj . The Lagrangian

counterpart of (33) (i.e. with quantities referred to the initial configurationΩ0 and Lagrangian shapefunctions) can be written as:



n∑

j=1

∫

Ω0ρ0N∗

i (XXXXXXXXXXXXXX)Nj(XXXXXXXXXXXXXX)dvvvvvvvvvvvvvvj

dtdΩ0 = −

∫

Ω0PPPPPPPPPPPPPP ∇∇∇∇∇∇∇∇∇∇∇∇∇∇XXXXXXXXXXXXXXN∗

i (XXXXXXXXXXXXXX)dΩ0+

+∫

Ω0N∗

i (XXXXXXXXXXXXXX)bbbbbbbbbbbbbb dΩ0 +∫

Γ0N∗

i (XXXXXXXXXXXXXX)PPPPPPPPPPPPPPnnnnnnnnnnnnnn dΓ0 (34)

whereXXXXXXXXXXXXXX denote particle coordinates in the reference configuration. For convenience, we can write (33)in a compact form:

MMMMMMMMMMMMMMaaaaaaaaaaaaaa = FFFFFFFFFFFFFF int + FFFFFFFFFFFFFF ext (35)

where the mass matrixMMMMMMMMMMMMMM = mij, internal forcesFFFFFFFFFFFFFF int = ffffffffffffff inti and external forcesFFFFFFFFFFFFFF ext = ffffffffffffffext

i are respectively defined by:

mij =∫

Ω

ρN∗i (xxxxxxxxxxxxxx)Nj(xxxxxxxxxxxxxx)dΩ (36)

ffffffffffffff inti = −

∫

Ω

σσσσσσσσσσσσσσ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxx)dΩ (37)

ffffffffffffffexti =

∫

Ω


∫

Γ

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓ (38)

A completely analogous expression could be derived for the Lagrangian version (34). The internalforces will be related to the field variables through the nominal stress tensor,σσσσσσσσσσσσσσ or PPPPPPPPPPPPPP , and thecorresponding constitutive equations.

The MLS shape functions do not vanish on essential boundaries and, therefore, the boundary integralin (38) can be decomposed as:

∫

Γ

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓ =

∫

Γu

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓu +

∫

Γn

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓn (39)

whereΓu andΓn are, respectively, the parts of the boundary where essential and natural boundaryconditions are prescribed andΓ = Γu ∪ Γn. This feature will be revisited later in section 5.3.2.

If expression (21) is used for mass conservation, its Galerkin weak form is equivalent to a pointcollocation scheme and, thus, the continuity equation must be enforced at each particlei,

dρi

dt= −ρidiv(vvvvvvvvvvvvvv)i = −ρi

n∑

j=1

vvvvvvvvvvvvvvj · ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxxi) (40)

where expression (31) for∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvvi has been used.

4.2. Numerical integration.

4.2.1. Introduction. The final step to obtain a set of discrete equations corresponds to the numericalintegration of the weak form. This is a most important issue in meshless methods and is the source ofwell known inaccuracies and instabilities [9],[22],[23],[24]. Some aspects must be considered whenchoosing (or designing) a numerical quadrature for particle methods.



• The method should provide reasonable accuracy.• In Lagrangian hydrodynamics applications (at least), the numerical quadrature should retain the

meshless character of the method.• It should be computationally efficient.

The matter of numerical integration concerns the nature itself of meshless methods, and has receivedmuch attention in Galerkin-based meshless formulations such as the Element-Free Galerkin (EFG)method [2]. However, numerical integration has not been explicitly studied in SPH, probably becausenodal integration lies in the basis of its early formulations and the method was considered a collocationmethod. In the context of SPH, the use of alternative numerical quadratures appeared implicitly withinthe concept of “stress-points”. More recently, Belytschko and coworkers [22] have reinterpreted SPHas a nodally integrated Galerkin method. Following a similar approach, we believe that, in the contextof Galerkin methods, the question about SPH and numerical integration gains full sense, providing aclear framework to analyze the use of the aforementioned “stress-points”.

4.2.2. Particle methods and continuum equations.There are important differences between meshlessmethods and mesh-based methods such as the finite element method, concerning numerical integration,when a global Galerkin weak form is defined over the entire problem domain in a continuum mechanicsproblem:

• The complexity of the shape functions.• The absence of a spatial framework to define the integration points and their corresponding

weights.

Moving least squares shape functions and their derivatives are complex functions that, in general,cannot be integrated exactly using numerical quadratures [24]. Moreover, the actual integrationdomains in the globally defined Galerkin weak form correspond to the intersection between nodalsupports. Given an arbitrary set of particles, the definition of quadrature subdomains on the basis ofsupport intersections may constitute a formidable task in dynamic problems, requiring the generationof a complex integration mesh each time step.

However, the second difference is much more important and recalls the question about the “nature”of meshless methods. Mesh-based methods perform a partition of the domain into certain nonoverlapping “elements”, which areexplicitly representative of a piece of the domain. In domainsundergoing very high deformations, these individual elements may suffer from severe distortions,with a dramatic loss of accuracy in the computations. In turn, the mesh of elements providesa natural spatial framework to integrate the globally defined Galerkin weak form, which is splitinto assembled “elemental contributions” (calculated element-by-element). Moreover, the discreteequations are stablished in terms of equilibrium between “regions” (elements) of the domain, which isconsistent with the derivation of the continuum mechanics model.

Unlike elements, particles bear certain masses and volumes of which they areimplicitlyrepresentative (the volumes areconcentratedat particles regardless of their actual shape). This is avery powerful computational advantage of particle methods, but the spatial framework is no longerpreserved. The resulting discrete equations can be seen as force equilibrium between interactingparticles, which does not correspond to the fundamentals of continuum mechanics: in this sense,the particle philosophy is not conceptually consistent with the continuum mechanics model. Thiscontradiction is partially eliminated by considering the particles as representative of a certainregion; note that, however, Cauchy’s fundamental axiom of continuum mechanics stablishes that the



interactions between regions occur in the form of boundary force densities, but not forces betweencentroids. Nevertheless, the spatial “uncertainty” introduced by particles suffices to turn the simpleintegration of the Galerkin weak form into a challenging numerical problem.

In the following, we present a brief review of various integration techniques widely used byresearchers and some comments about our own experience in their practical implementation. We mustnote that this question is far from being closed and is likely to produce fundamental modifications inparticle formulations.

4.2.3. Nodal integration. Nodal integration has been used, at least implicitly, in all SPH formulations,and lies, indeed, in the basis of its early formulation. Obviously, this is the cheapest option andthe resulting scheme is truly meshless (no background mesh is needed). The particles are used asquadrature points and the corresponding integration weights are their tributary volumes. Recalling theweak form derived in the previous section, the discrete eulerian momentum equation can be written as:

MMMMMMMMMMMMMMaaaaaaaaaaaaaa = FFFFFFFFFFFFFF int + FFFFFFFFFFFFFF ext (41)

where

mij =n∑

k=1

ρkN∗i (xxxxxxxxxxxxxxk)Nj(xxxxxxxxxxxxxxk)Vk (42)


n∑

k=1

σσσσσσσσσσσσσσk∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxxk)Vk (43)


n∑

k=1

N∗i (xxxxxxxxxxxxxxk)bbbbbbbbbbbbbbkVk +

n∑

k=1

N∗i (xxxxxxxxxxxxxxk)σσσσσσσσσσσσσσknnnnnnnnnnnnnnAk (44)

In the above,Vk represents the tributary volume associated to particlek. Usual techniques to determinesuch volumes vary from simple domain partitions to Voronoi diagrams. In the most frequent approachin SPH simulations, the particles are set up with certain initial densities, volumes and, therefore,masses. ThesephysicalmassesMk remain constant during the simulation and densities are fieldvariables updated using the continuity equation. Thus, particle volumes are obtained each time stepasVk = Mk

ρk. Note that, in our formulation, thereal or physicalparticle massesMk are different, in

general, from thenumericalmassesmij given by (42), and derived in the Galerkin scheme.To enforce natural boundary conditions, according to (44), we must firstlocate the boundary, in fact

an approximated boundary formed by certain particles, and then determine the boundary weightsAk,associated to each boundary particle and neccesary to compute (44).

Nodal integration is the origin of well known instabilities in meshless methods. Beissel andBelytschko [23], and Bonet and Kulasegaram [9], have analyzed the performance of this quadraturein the context of Element-Free Galerkin (EFG) and Corrected SPH (CSPH) methods, and proposedmodified variational principles based on least-squares stabilizations, requiring second derivatives ofthe shape functions. Chen and coworkers developed a stabilized conforming nodal integration [25],based on a strain smoothing, which requires the Voronoi diagram of the cloud of particles, at a highcomputational cost.



4.2.4. Background integration mesh.The most frequent approach in EFG and RKPM is the definitionof a background integration mesh, composed by non overlapping cells covering the whole domain,where high order Gauss quadratures are defined (Fig. 1)[26]. In general, these cells do not matchintegration domains; however, thespatial frameworkrequired by the Galerkin method is recovered (atthe cost of the generation of an integration mesh). The mass matrix and force vectors are obtained as:

mij =ninte∑

k=1

ρkN∗i (xxxxxxxxxxxxxxk)Nj(xxxxxxxxxxxxxxk)Wk (45)


ninte∑

k=1

σσσσσσσσσσσσσσk∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxxk)Wk (46)


ninte∑

k=1

N∗i (xxxxxxxxxxxxxxk)bbbbbbbbbbbbbbkWk +

ninteB∑

k=1

N∗i (xxxxxxxxxxxxxxk)σσσσσσσσσσσσσσknnnnnnnnnnnnnnWB

k (47)

whereninte is the number of integration points (in actual computations only a few points would beconsidered, according to nodal supports), andWk is the quadrature weight of pointk. The numberof boundary integration points isninteB , andWB

k is the boundary weight of boundary particlek.This technique, which has been succesfully applied to a wide variety of problems in computationalmechanics [26], is not appropriate for lagrangian SPH simulations, where the domain is continuouslychanging. Instead, a similar but somewhat relaxed scheme has been used in the SPH literature: theso-called “stress-point” approach.

PARTICLES QUADRATURE POINTS QUADRATURE CELLS

Figure 1. Background integration mesh.



4.2.5. Stress points.The concept of “stress points” was introduced by Dyka, Randles and Ingel [27],as an attempt to eliminate tensile instabilities [28] in SPH. The basic idea, which still remains inthe stress-point SPH literature, is to “calculate stresses away from the centroids (particles)”. This isequivalent to “use a quadrature other than nodal integration” in the Galerkin weak form. Therefore,the discrete equations are completely analogous to (45)–(47), but now the integration points arecalledstress points, which aremoving integration points spread among the cloud of particles, withno reference to any background mesh (Fig. 2). Stress points are set up in certain positions and theirmovement is completely determined by the movement of the particles, as

vvvvvvvvvvvvvvsi =

n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxxsi ),

dxxxxxxxxxxxxxxsi

dt= vvvvvvvvvvvvvvs

i (48)

where the superscripts was used in reference to the stress points. In this context, some authors [22]refer to stress points and particles as “slave” nodes and “master” nodes, respectively.

In our implementation, and for consistency with the definition of the test functions, densities arecomputed at particles through the continuity equation and then interpolated at stress points, as

ρsi =

n∑

j=1

ρjNj(xxxxxxxxxxxxxxsi ) (49)

Note that in the above we assumeρj = ρj ; that is, we use asdensity nodal parametersρj thereal nodal valuesρj , obtained from the continuity equation. We expect the errors introduced by thisassumption to be negligible, particularly in the case of nearly incompressible fluids. Vignjevic et al. [29]have proposed a different implementation, where the continuity equation is enforced at stress points.Considering the resulting discrete equations we can readily understand that displacement, velocity andacceleration are tracked at particles, whereas other field variables such as stress are required only atquadrature (stress) points [6], [29].

With this technique, quadrature points move continuosly in time: this allows the integration “mesh”to adapt the moving domain, avoiding the rigid background mesh. We have the positions of quadraturepoints but, which are their weights? The answer is that the two sets of points are uncoupled. Initially,both represent the total computational domain and particles and integration points are set up withmasses, densities and volumes such that

n∑

ip=1

Vip = V,

ns∑

is=1

Vis = V (50)

n∑

ip=1

Mip = M,

ns∑

is=1

Mis = M (51)

where M ,V ,n,ns are the initial mass and volume of the body under study, number of particlesand number of quadrature (stress) points, respectively. With time, both sets result in two differentcomputational domains and volumes: particles representpositionsandmovementsof the body, whereasthe set of quadrature points constitutes theactual computational domain. Obviously, this domainduplicity must be kept under control to preserve the accuracy of the method.

Belytschko and coworkers [22] have proposed an alternative implementation, where both particlesand stress points are used as quadrature points (the stress points are, therefore,additionalquadraturepoints) (Fig. 3). The internal forces, without boundary tractions, result:



PARTICLES

STRESS POINTS

Figure 2. Particles and stress points (double grid).

PARTICLES

STRESS POINTS

Figure 3. Particles, stress points and Voronoi cells.




n∑

ip=1

σσσσσσσσσσσσσσip∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxxip)V

pip −

ns∑

is=1

σσσσσσσσσσσσσσis∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxxis)V s

is (52)

The weightsV pip andV s

is (different from “real” volumes) are such that

n∑

ip=1

V pip +

ns∑

is=1

V sis = V (53)

The determination of such weights is the most important drawback of this scheme for eulerian kernels.The Voronoi diagram of the cloud of particles and stress points must be computed at each time step,with an important computational cost. However, if Lagrangian kernels are employed the diagram iscomputed only in the initial configuration, and the method results quite efficient.

Finally, we would like to propose another implementation of stress points, which is in some sense a2D extension of the 1D algorithm by Dyka, Randles and Ingel [27], where an “element” was associatedto each SPH particle and stresses computed using two integration (stress) points inside each “element”.In a 2D version of this approach, certain region is associated to each particle and severalrepresentativepoints within that region are used as quadrature points. Unlike the “double grid approach”, stress pointsare nowassociatedto particles and represent certain portion of the nodal volume. We would also liketo avoid theexplicitdetermination of such nodal associated volumes (computing the Voronoi diagram),so anassumednodal region is used instead. Stress points are defined in such regions (Fig.4), and givenquadrature weightsV s

ik, such that

nsi∑

k=1

V sik = Vi (54)

wherensi is the number of stress points associated to particlei, V sik is the weight of stress pointk

associated to particlei andVi is the volume of particlei. We assume that the relationV sik/Vi remains

constant throughout the simulation. Particles are usually set up in regular lattices, so the set up of stresspoints (the determination of initial assumed nodal associated regions) should be easy. If stress pointsare moved with the same velocity as their associated particle, the shape of the assumed nodal regionsremains constant. This approach may not be adequate under high distortions, so it is better to movethe stress points using (48). With this implementation, the movement of stress points provides certainmeasure of the distortion of the regions associated to particles.

4.3. Mass lumping.

As it is known in FEM analysis, it is not efficient to use the complete mass matrix in practicalapplications, and lumped (diagonal) mass matrices are most frequently used. A simple lumpingtechnique corresponds to a row-sum mass matrix. Thus, the lumped massMi associated to particlei is

Mi =n∑

j=1

mij =n∑

j=1

∫

Ω

ρN∗i (xxxxxxxxxxxxxx)Nj(xxxxxxxxxxxxxx)dΩ =

=∫

Ω

ρN∗i (xxxxxxxxxxxxxx)

( n∑

j=1

Nj(xxxxxxxxxxxxxx))

dΩ =∫

Ω

ρN∗i (xxxxxxxxxxxxxx)dΩ (55)



PARTICLESASSOCIATED STRESS POINTS

Figure 4. Particles and associated stress points.

provided that trial functions are, at least, zeroth order complete. The discrete counterpart of (55),regardless the particular integration method,

Mi =ninte∑

k=1

ρkN∗i (xxxxxxxxxxxxxxk)Wk (56)

whereWk is the weight of quadrature pointk. Note that, if test functions are also zeroth order complete,this lumping verifies

n∑

i=1

Mi =n∑

i=1

ninte∑

k=1

ρkN∗i (xxxxxxxxxxxxxxk)Wk =

ninte∑

k=1

ρk

( n∑

i=1

N∗i (xxxxxxxxxxxxxxk)

)Wk =

ninte∑

k=1

ρkWk = M (57)

whereM is the total mass of the body. Note the importance of a correct choice of quadrature weightsand densities for the integration points in this scheme.

However, (56) is not a common expression in the SPH literature. The most extended practicecorresponds to using thereal particle masses asnumericallumped masses, as

Mi = Mi (58)

verifying, obviously,

n∑

i=1

Mi = M (59)



5. APPLICATIONS. FLUID DYNAMICS.

5.1. Stress tensor.

We assume a compressible newtonian fluid and eulerian kernels (derivatives and quantities referred tothe current configuration). Thus, the internal forces are related to the Cauchy stress tensor, given by:

σσσσσσσσσσσσσσ = −pIIIIIIIIIIIIII + 2µdddddddddddddd′ (60)

wherep is the pressure,µ the viscosity anddddddddddddddd′ the deviatoric part of the rate of deformation tensordddddddddddddd,given by

dddddddddddddd′ = dddddddddddddd− 13tr(dddddddddddddd)IIIIIIIIIIIIII, dddddddddddddd =

12(∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv +∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvvT ) (61)

The discrete velocity gradient∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) and rate of deformation tensordddddddddddddd are obtained as

∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvvj ⊗∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxx), dddddddddddddd =12(∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv +∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvvT ) (62)

Thus, the velocity gradient at a given pointxxxxxxxxxxxxxx (in actual computations a particle or quadrature point) iscomputed in terms of particle velocity parametersvvvvvvvvvvvvvvj. The velocity divergence is given by,

∇∇∇∇∇∇∇∇∇∇∇∇∇∇ · vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) = tr(∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv(xxxxxxxxxxxxxx)) =n∑

j=1

vvvvvvvvvvvvvvj · ∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxx) (63)

We use an equation of state of the form [30]:

p = κ

[(ρ

ρ0

)γ

−1]

(64)

where typicallyγ = 7 andκ is chosen such that the fluid is nearly incompressible.In gravity flows the initial particle densities are adjusted to obtain the correct hydrostatic pressure

computed as (64) [30]:

ρ = ρ0

(1 +

ρ0g(H − z)κ

)1/γ

(65)

whereH is the total depth andg = 9.81 m/s2.

5.2. Discrete equations.

We can now write the complete spatially discretized set of equations. We assume a Bubnov-Galerkinscheme, where both test and trial functions are chosen from the same space.

• Momentum equation.

Midvvvvvvvvvvvvvvi

dt= ffffffffffffff int

i + ffffffffffffffexti (66)



whereMi is the lumped mass of particlei andffffffffffffff inti andffffffffffffffext

i are respectively the internal andexternal forces, given by:


ninte∑

k=1

σσσσσσσσσσσσσσk∇∇∇∇∇∇∇∇∇∇∇∇∇∇Ni(xxxxxxxxxxxxxxk)Wk (67)


ninte∑

k=1

Ni(xxxxxxxxxxxxxxk)bbbbbbbbbbbbbbkWk +ninteB∑

k=1

Ni(xxxxxxxxxxxxxxk)σσσσσσσσσσσσσσknnnnnnnnnnnnnnWBk (68)

We prefer this general approach, whereninte is the total number of quadrature points, regardlessthe particular integration technique chosen. Note that appropriate weights,Wk andWB

k , mustbe defined for interior and boundary quadrature points. The stress tensor must be computed ateach quadrature point,

σσσσσσσσσσσσσσk = −pkIIIIIIIIIIIIII + 2µkdddddddddddddd′k (69)

anddddddddddddddd′k is related to the velocity gradient tensor as expressed in (62).

• Continuity equation.

dρi

dt= −ρi div(vvvvvvvvvvvvvv)i = −ρi

n∑

j=1

vvvvvvvvvvvvvvj · ∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxxi) (70)

5.3. Consequences of the lack of interpolation property.

5.3.1. Moving the particles. Recall the meshless approximationu(xxxxxxxxxxxxxx) of a functionu(xxxxxxxxxxxxxx), computedin terms of the shape functions as

u(xxxxxxxxxxxxxx) =n∑

j=1

ujNj(xxxxxxxxxxxxxx) (71)

wheren is the total number of particles anduj is a set of nodal parameters. In this context, theinterpolated velocity at a given pointxxxxxxxxxxxxxx can be obtained from neighbour particles as

vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxx) (72)

Note that, unlike finite element interpolants, standard SPH and moving least squares shape functionsdo not verify the interpolation property, i.e.

Nj(xxxxxxxxxxxxxxi) 6= δij (73)

Thus, in general,

vvvvvvvvvvvvvvi = vvvvvvvvvvvvvv(xxxxxxxxxxxxxxi) =n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxxi) 6= vvvvvvvvvvvvvvi (74)

and nodal parameters do not necessarily coincide with interpolated values. We suspect this propertyhas introduced some confusion in many SPH formulations where the fact is disregarded. In order



to derive a consistent algorithm, the discrete equations must be treated carefully, as they are not asdirectly meaningful as those obtained with shape functions that bear the interpolation property. Thediscrete momentum and continuity equations are written in terms of velocities, velocity gradientsand accelerations, which are computed using the nodal velocity parametersvvvvvvvvvvvvvvj, not the ”real”(interpolated) nodal velocitiesvvvvvvvvvvvvvvj. However, before moving the particles, we must compute suchinterpolatedvelocities

vvvvvvvvvvvvvv(xxxxxxxxxxxxxxi) =n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxxi) (75)

and usevvvvvvvvvvvvvvj to move the particles according to

dxxxxxxxxxxxxxx

dt= vvvvvvvvvvvvvv (76)

We must note that moving the particles withvvvvvvvvvvvvvvj is not a correction, but thecorrect scheme,consistent with the meshless approximation method used. Many SPH practitioners have been usinga somewhat modified version of this approach. We refer to the so-called XSPH correction, proposed byMonaghan [30]. To illustrate this point, let us consider MLS shape functionsNj(xxxxxxxxxxxxxx). After computingthe nodal parametersvvvvvvvvvvvvvvj, we move particlei with the velocity

vvvvvvvvvvvvvvi =n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxxi) = vvvvvvvvvvvvvviNi(xxxxxxxxxxxxxxi) +∑

j 6=i


Even MLS shape functions with constant basis form a partition of unity and, therefore, the aboveexpression can be written as

vvvvvvvvvvvvvvi = vvvvvvvvvvvvvvi

(1−

∑

j 6=i

Nj(xxxxxxxxxxxxxxi))+

∑

j 6=i


which can be rearranged, to yield

vvvvvvvvvvvvvvi = vvvvvvvvvvvvvvi +∑

j 6=i

(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)Nj(xxxxxxxxxxxxxxi) = vvvvvvvvvvvvvvi +n∑

j=1

(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)Nj(xxxxxxxxxxxxxxi) (79)

Now consider standard SPH shape functions, given byNj(xxxxxxxxxxxxxx) = mjρj

Wj(xxxxxxxxxxxxxx). The above derivation is

not correct for such functions, since they do not form a partition of unity,

n∑

j=1

mj

ρjWj(xxxxxxxxxxxxxx) 6= 1 (80)

Let us writeNi(xxxxxxxxxxxxxx) as

Ni(xxxxxxxxxxxxxx) =mi

ρiWi(xxxxxxxxxxxxxx) = 1− α

∑

j 6=i

mj

ρjWj(xxxxxxxxxxxxxx) (81)

whereα is a certain parameter which may depend on the nodal arrangement, kernel function, etc.Introducing this expression in (77) and rearranging,



vvvvvvvvvvvvvvi = vvvvvvvvvvvvvvi + α

n∑

j=1

mj

ρj(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)Wj(xxxxxxxxxxxxxxi) + (1− α)

∑

j 6=i

mj

ρjvvvvvvvvvvvvvvjWj(xxxxxxxxxxxxxxi) (82)

Using a simplified version of (82) we can write,

vvvvvvvvvvvvvvi = vvvvvvvvvvvvvvi + ε

n∑

j=1

mj

ρj(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)Wj(xxxxxxxxxxxxxxi) (83)

where ε is another (unknown) parameter. This expression is completely analogous to the XSPHcorrection, defined as [30]

vvvvvvvvvvvvvvi = vvvvvvvvvvvvvvi + ε

n∑

j=1

mj

ρij(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)Wj(xxxxxxxxxxxxxxi) (84)

whereρij = 0.5(ρi + ρj). We believe this “correction” is indeed aheuristicversion of the consistentform (75). In the XSPH correction, the continuity equation is solved using the corrected velocitiesgiven by (84). Note that this procedure is not consistent with our formulation, since the continuityequation is posed in terms ofvvvvvvvvvvvvvvj.

In practice, nodal velocity parameters are used to solve the discrete equations and interpolated nodalvelocities are used to move the particles. We would like to emphasize that this approach isneithera smoothing nor a correctionof the velocities, but the procedure consistent with an approximationscheme which does not bear the interpolation property. Note that the continuity equation is posed interms of “real” densities and, therefore, the so obtained densities can be introduced directly in theequation of state to compute pressures.

5.3.2. Enforcement of the essential boundary conditions.The imposition of boundary conditions isa most important and problematic issue in SPH, since they cannot be enforced as directly as in finiteelements. Natural boundary conditions are included in the weak form through the boundary tractionterm. The most important difficulty at this point “reduces” to the determination of the boundaryparticles and their weights. In this study, boundary tractions correspond to free surface boundariesand thus, the natural boundary integrals vanish (surface tension is neglected). Unfortunately, the testfunctions do not vanish on essential boundaries, and a term including tractions on essential boundariesremains in the problem weak form. The external forces in the momentum equation of a fluid particleiare, thus:


∫

Ω


∫

Γu

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓu (85)

The discrete external forces, in the case of nodal integration, result


n∑

j=1

N∗i (xxxxxxxxxxxxxxj)bbbbbbbbbbbbbbjVj +

∑

j∈BN∗

i (xxxxxxxxxxxxxxj)σσσσσσσσσσσσσσjnnnnnnnnnnnnnnAj (86)

whereB is the set of particles from the fluid located on the essential boundaries (located “sufficiently”close to the boundary). Note that the second sum in (86) involves those neighbours of particleibelonging toB.

In addition, given that the MLS shape functions do not bear the interpolation property, velocitiescannot be imposed directly. In the SPH literature, essential boundary conditions have been rather



treated as fluid-structure contact and specific techniques such as mirror particles and boundary forceshave been extensively used [14]. In the classical boundary force approach, certain forces are appliedto those particles that approach the boundary (those particles inB). The contact with solid boundariesis detected by checking the distance between fluid particles and certainboundary particles, which arefixed at the solid boundary and are not included in the general computations. Consider, for instance, afluid particlei ∈ B. The external force vector for particlei:


n∑

j=1


∑

j∈BN∗

i (xxxxxxxxxxxxxxj)σσσσσσσσσσσσσσjnnnnnnnnnnnnnnAj +∑

j∈SFFFFFFFFFFFFFF j

i (87)

whereS is the set of solid boundary particles andFFFFFFFFFFFFFF ji is the boundary force exerted by boundary particle

j on i (which may depend on the distance betweeni andj, their relative velocity, etc.). In most SPHformulations the second term on the right hand side of (87) (tractions on essential boundaries) is simplydropped, to yield


n∑

j=1


∑

j∈SFFFFFFFFFFFFFF j

i (88)

whereas for interior particles (i /∈ B),


n∑

j=1

N∗i (xxxxxxxxxxxxxxj)bbbbbbbbbbbbbbjVj (89)

This technique is reformulated here by means of a penalization technique similar to that usedin implicit meshless formulations [9]. Thus, the essential boundary conditions are enforced using apenalty boundary potentialΠbp given by

Πbp =η

2

∫

Γu

(vvvvvvvvvvvvvv − vvvvvvvvvvvvvvB

)2dΓu (90)

whereη is a high penalty value andvvvvvvvvvvvvvvB is the prescribed velocity at boundaryΓu. The first variation of(90) is

δΠbp = η

∫

Γu

δvvvvvvvvvvvvvv · (vvvvvvvvvvvvvv − vvvvvvvvvvvvvvB)dΓu (91)

and this expression should be added to the Galerkin weak form (28). Using nodal integration, thefollowing expression is obtained for the forceffffffffffffff bp

i over particlei due to the boundary potential (90):

ffffffffffffff bpi = η

∑

j∈B

(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvB

j

)N∗

i (xxxxxxxxxxxxxxj)Aj (92)

The expression

FFFFFFFFFFFFFF j = η(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvB

j

)Aj (93)

can be interpreted as aforce due to the fact thatj is close to the boundary(i.e. thatj ∈ B), and (92)rewritten as



ffffffffffffff bpi =

∑

j∈BFFFFFFFFFFFFFF jN

∗i (xxxxxxxxxxxxxxj) (94)

The above expression suggests a new implementation of the boundary force approach for solidboundaries. Thus, the “forces”FFFFFFFFFFFFFF j are computed in a fashion similar to that exposed above for theclassical boundary force scheme, as

FFFFFFFFFFFFFF j =∑

k∈SFFFFFFFFFFFFFFk

j (95)

where a general dependence forFFFFFFFFFFFFFFkj could be of the form

FFFFFFFFFFFFFFkj = FFFFFFFFFFFFFFk

j (∆vvvvvvvvvvvvvvjk, ∆xxxxxxxxxxxxxxjk) (96)

being∆vvvvvvvvvvvvvvjk = vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvBk and∆xxxxxxxxxxxxxxjk = xxxxxxxxxxxxxxj −xxxxxxxxxxxxxxk, wherevvvvvvvvvvvvvvB

k is the velocity prescribed at boundary particlek. The particular expression forFFFFFFFFFFFFFFk

j used in this study will be presented in section 6. Using (94) and(95), the external force vector, for a given particlei, is


n∑j=1


∑j∈B

N∗i (xxxxxxxxxxxxxxj)σσσσσσσσσσσσσσjnnnnnnnnnnnnnnAj +

∑j∈B

N∗i (xxxxxxxxxxxxxxj)FFFFFFFFFFFFFFj =

=n∑

j=1


∑j∈B

(FFFFFFFFFFFFFF j + σσσσσσσσσσσσσσjnnnnnnnnnnnnnnAj

)N∗

i (xxxxxxxxxxxxxxj) (97)

Note that, following this approach, boundary forces are not only “felt” by those particles locatedinmediatly close to the boundary, but also by their neighbours. Furthermore, note that the termcorresponding to the integral over essential boundaries has been incorporated in the formulation. Giventhat the penalty forcesFFFFFFFFFFFFFF j are much greater than the termsσσσσσσσσσσσσσσjnnnnnnnnnnnnnnAj we can assume that


n∑

j=1


∑

j∈BFFFFFFFFFFFFFFjN

∗i (xxxxxxxxxxxxxxj) (98)

5.3.3. Initialization of the field variables.Finally, special attention must be paid to the initializationof the field variables, in particular the initial velocity field, known in terms of nodal velocitiesvvvvvvvvvvvvvv0

j.Once more, in our computations we need the nodal velocityparametersvvvvvvvvvvvvvv0

j, such that,

vvvvvvvvvvvvvv0i =

n∑

j=1

vvvvvvvvvvvvvv0jNj(xxxxxxxxxxxxxxi) (99)

and a linear system of equations should be solved forvvvvvvvvvvvvvv0j. Fortunately, the linear completeness of

the MLS shape functions chosen simplifies certain initial conditions. Consider, for instance, a constantinitial velocity field,vvvvvvvvvvvvvv0(xxxxxxxxxxxxxx) = vvvvvvvvvvvvvv0. Then, we can choosevvvvvvvvvvvvvv0

j = vvvvvvvvvvvvvv0, ∀j = 1, . . . , n because

vvvvvvvvvvvvvv0(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvv0jNj(xxxxxxxxxxxxxxi) = vvvvvvvvvvvvvv0

n∑

j=1

Nj(xxxxxxxxxxxxxx) = vvvvvvvvvvvvvv0 (100)

Similarly, given a linear fieldvvvvvvvvvvvvvv0(xxxxxxxxxxxxxx) = CCCCCCCCCCCCCCxxxxxxxxxxxxxx, whereCCCCCCCCCCCCCC is a constant2× 2 matrix,



vvvvvvvvvvvvvv0(xxxxxxxxxxxxxx) =n∑

j=1

CCCCCCCCCCCCCCxxxxxxxxxxxxxxjNj(xxxxxxxxxxxxxxi) = CCCCCCCCCCCCCC

n∑

j=1

xxxxxxxxxxxxxxjNj(xxxxxxxxxxxxxx) = CCCCCCCCCCCCCCxxxxxxxxxxxxxx (101)

Therefore,vvvvvvvvvvvvvv0j = CCCCCCCCCCCCCCxxxxxxxxxxxxxxj , ∀j = 1, . . . , n. The above is true provided that linearly complete shape

functions are used. However, even with standard SPH shape functions (which lack zeroth ordercompleteness), it is a common practice to takevvvvvvvvvvvvvv0

j = vvvvvvvvvvvvvv0j . We suspect that, indeed, most SPH

formulations assume thatvvvvvvvvvvvvvvj ≡ vvvvvvvvvvvvvvj throughout the computations. This practice, as we would like todemonstrate in this paper, is not correct and causes important errors in the computations.

5.4. Alternative discrete equations.

It is frequent in the SPH literature to calculate the velocity gradient as,

∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvvi =n∑

j=1

(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)⊗∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxxi) (102)

With linearly complete MLS trial functions,

n∑

j=1

(vvvvvvvvvvvvvvj − vvvvvvvvvvvvvvi)⊗∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxxi) =n∑

j=1

vvvvvvvvvvvvvvj ⊗∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxxi)− vvvvvvvvvvvvvvi ⊗n∑

j=1

∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxxi) =n∑

j=1

vvvvvvvvvvvvvvj ⊗∇∇∇∇∇∇∇∇∇∇∇∇∇∇Nj(xxxxxxxxxxxxxxi) (103)

and (102) is equivalent to (31) (except for round-off errors). Even though in the case of constant velocityfields (102) may eliminate such round-off errors [11], we have not found significant advantages ingeneral problems. If standard SPH interpolation is used, (103) is not true, and both formulations are notequivalent. This symmetrization has been widely used to assure correct velocity gradients for constantvelocity fields. However, according to our analysis, this corresponds to a different election of the trialfunctions, which should be considered, for consistency, wherever trial functions may appear in theformulation.

A similar reasoning could be applied to the case of the internal forces. An expression widely used inthe SPH literature is (in the case of nodal integration):


n∑

k=1

(σσσσσσσσσσσσσσk ± σσσσσσσσσσσσσσi)∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗i (xxxxxxxxxxxxxxk)Vk (104)

most frequently


n∑

k=1

(σσσσσσσσσσσσσσk + σσσσσσσσσσσσσσi)∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗i (xxxxxxxxxxxxxxk)Vk (105)

Standard SPH test functions verify∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗i (xxxxxxxxxxxxxxk)Vk = −∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗

k (xxxxxxxxxxxxxxi)Vi and, therefore, the use of (105)assures local conservation of linear momentum. Angular momentum will be also preserved in theabsence of shear stresses [8]. Note that, in general, using MLS approximation,

n∑

k=1

∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗i (xxxxxxxxxxxxxxk) 6= 00000000000000 (106)



and

∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗i (xxxxxxxxxxxxxxk)Vk 6= −∇∇∇∇∇∇∇∇∇∇∇∇∇∇N∗

k (xxxxxxxxxxxxxxi)Vi (107)

Therefore, (105) is neither skew-symmetric nor consistent with our definiton of the test functions (30)and, consequently, we do not use this expression for internal forces.

5.5. Time integration.

We use explicit time integration to update the field variables. One of the most widely used algorithmsis the leap-frog scheme, involving the following sequence of updates:• Compute velocities at stepk + 1

2 :

vvvvvvvvvvvvvvk+ 1

2i = vvvvvvvvvvvvvv

k− 12

i + 0.5(∆tk + ∆tk+1)aaaaaaaaaaaaaaki (108)

• Update densities and positions:

ρk+1i = ρk

i + ∆tk+1Di(vvvvvvvvvvvvvvk+ 12 ) (109)

xxxxxxxxxxxxxxk+1i = xxxxxxxxxxxxxxk

i + ∆tk+1vvvvvvvvvvvvvvk+ 1

2i (110)

In the above expressions,aaaaaaaaaaaaaaki = dvvvvvvvvvvvvvvk

i

dt is the acceleration nodal parameter of particlei (computed using

the momentum equation with variables at stepk) andDi(vvvvvvvvvvvvvvk+ 12 ) is the density ratedρi

dt , computed

with positions at stepk and intermediate velocitiesvvvvvvvvvvvvvvk+ 1

2i . With vvvvvvvvvvvvvvi we denote the interpolated nodal

velocities, computed as (72).We have also used the following second order predictor-corrector scheme, as exposed in [14]. The

sequence of updates to obtain the field variables at stepk + 1 is:• Predictions:

vvvvvvvvvvvvvvPi = vvvvvvvvvvvvvvk

i +∆t

2aaaaaaaaaaaaaak

i (111)

ρPi = ρk

i +∆t

2D(xxxxxxxxxxxxxxk, vvvvvvvvvvvvvvk) (112)

xxxxxxxxxxxxxxPi = xxxxxxxxxxxxxxk

i +∆t

2vvvvvvvvvvvvvvk

i (113)

• Corrections:

vvvvvvvvvvvvvvCi = vvvvvvvvvvvvvvk

i +∆t

2aaaaaaaaaaaaaa1k

i (vvvvvvvvvvvvvvP , ρρρρρρρρρρρρρρP , xxxxxxxxxxxxxxP ) (114)

ρCi = ρk

i +∆t

2D(xxxxxxxxxxxxxxP , vvvvvvvvvvvvvvC) (115)

xxxxxxxxxxxxxxCi = xxxxxxxxxxxxxxk

i +∆t

2vvvvvvvvvvvvvvC

i (116)

• Finally, the variables at stepk + 1 result,



vvvvvvvvvvvvvvk+1i = 2vvvvvvvvvvvvvvC

i − vvvvvvvvvvvvvvki (117)

ρk+1i = 2ρC

i − ρki (118)

xxxxxxxxxxxxxxk+1i = 2xxxxxxxxxxxxxxC

i − xxxxxxxxxxxxxxki (119)

where the superscriptsP andC have been used in reference to the prediction and correction phase,respectively. In practice, we takeaaaaaaaaaaaaaak ≈ aaaaaaaaaaaaaa2(k−1), with a lower cost without changing the order of themethod [14]. In both schemes the time step is limited by the Courant-Friedrichs-Lewy (CFL) stabilitycondition. Following Bonet and Lok [8],

∆t = CFLhmin

max(ci + ‖vvvvvvvvvvvvvvi‖) (120)

whereCFL is the Courant number (0 ≤ CFL ≤ 1) andci is the wave celerity at pointi,

ci =√

γκ/ρi (121)

beingγ andκ the same material properties as in (64).

6. IMPLEMENTATION

In this section we present a schematic flowchart and several remarks about the practical implementationof the methodology exposed above.

Searching for particle neighbours is a key issue in SPH and it is very important to choose anappropriately efficient algorithm. In this work we use a classical cell-partition algorithm, similar tothat exposed in [14].

In the examples shown below constant smoothing length is used throughout the computations; avalue ofh = 1.5 d, whered is the typical initial distance between particles, seems to be adequate,although we have not found a rigorous criterion for the choice ofh. Future versions of the codeshould allow variable smoothing lengths, but much care must be taken to the consequences of suchmodification, in order to retain the consistency of the formulation.

The construction of MLS shape functions with linear basis requires the inversion of the momentmatrix in (13). This matrix will be singular if there is not enough neighbours or they are aligned (in 2D).So, eventually, the computation of linear MLS shape functions at certain particles may be impossible. Insuch cases, we use MLS shape functions with constant basis (Shepard functions), which can always beconstructed. A similar variable-rank procedure is used in [11]. Note that, even with enough neigboursavailable, the moment matrix can become highly ill-conditioned and, in that case, the algorithm maybecome unstable. This latter effect can be alleviated by using adaptable time-stepping. In practice, wecompute for each particle the determinant or the condition number of the moment matrix and, if it islower than a certain tolerance, we use constant basis. We have not explored the errors introduced bythis implementation, but, given that the number of particles involved (if any) is very small, the effectsare expected to be negligible.



6.1. Main algorithm.

Let us consider the leap-frog time integration scheme. The outline of the main steps is as follows:

• Initialize geometry, cell conectivity and field variables:µ,κ,vvvvvvvvvvvvvv,V ,ρ,M ,p,c, etc.• At each time step:

I. Find neighbours.II. Compute shape functions and their derivatives at quadrature points, according to (14).

III. Compute stresses at quadrature points, according to (69) and (62).IV. Compute internal and external forces, according to (67) and (68). Compute nodal

acceleration parameters,aaaaaaaaaaaaaa.V. Update nodal velocity parameters, asvvvvvvvvvvvvvv∗ = vvvvvvvvvvvvvv−

12 + aaaaaaaaaaaaaa∆t.

VI. Compute boundary forces,ffffffffffffff b.VII. Update nodal velocity parameters, asvvvvvvvvvvvvvv

12 = vvvvvvvvvvvvvv∗ + ffffffffffffff b∆t.

VIII. Compute density rates, according to (70).IX. Compute the interpolated nodal velocitiesvvvvvvvvvvvvvv, used to move the particles, as (72).X. Update positions and other field variables.

• End loop.

6.2. Boundary conditions.

Throughout the examples shown in this paper, the essential boundary conditions were enforced usingthe boundary force approach exposed above. Solid boundaries are formed by particles fixed at theboundary, separated one third of the initial distance between domain particles. At each time step theboundary forces are computed as follows:

• Initialize boundary forcesffffffffffffff bi = 00000000000000, ∀i = 1, . . . , n.

• Determine the setB of particles located “sufficiently close” to the boundary. In practice, particlej belongs toB if, and only if, there exists a boundary particlek such that the distance betweenjandk is less than a certain specified valuer0; in this paper the valuer0 = 0.7h was used. Theset of boundary particlesk associated in the aforementioned sense to particlej is denoted bySj .

• For each particlej ∈ B:

• For each particlek ∈ Sj :

• Compute the boundary forceFFFFFFFFFFFFFFkj exerted by boundary particlek on j. In the

examples shown below, zero normal velocities were imposed. The expression usedfor boundary forces:

FFFFFFFFFFFFFFkj = f1(∆vvvvvvvvvvvvvvjk) f2(∆xxxxxxxxxxxxxxjk) (122)

where∆vvvvvvvvvvvvvvjk and∆xxxxxxxxxxxxxxjk are defined as in (96) and

f1(∆vvvvvvvvvvvvvvjk) =

1, ∆vvvvvvvvvvvvvvjk · nnnnnnnnnnnnnnk < 00, ∆vvvvvvvvvvvvvvjk · nnnnnnnnnnnnnnk ≥ 0

(123)

f2(∆xxxxxxxxxxxxxxjk) =A

r

((r0

r

)4

−(r0

r

)2)

nnnnnnnnnnnnnnk (124)



wherer = ‖∆xxxxxxxxxxxxxxjk‖ = ‖xxxxxxxxxxxxxxj−xxxxxxxxxxxxxxk‖, nnnnnnnnnnnnnnk is the unit normal to the boundary associated toboundary particlek andA is a certain constant. Note that (124) is a modified versionof the Lennard-Jones forces [30] (modified in the sense that we considernormalandnot radial forces).

• AddFFFFFFFFFFFFFFkj N∗

i (xxxxxxxxxxxxxxj) to the boundary forceffffffffffffff bi applied to each neighbouri of particlej.

• End loop.

• End loop.

Although the proposed implementation fairly outperforms our previous work with classical boundaryforce approaches in the case of simple non-penetrating boundaries [16, 31, 32, 33], we must note thatthe correct enforcement of more general boundary conditions is still an open problem in SPH.

Figure 5. Breaking dam flow: Scheme of the initial configuration.

7. EXAMPLES

In this section several fluid flow simulations are presented. In all the examples shown, nodal integrationand the leap-frog time integration have been used, in order to demonstrate that good results canbe obtained even with the simplest implementation. The proposed methodology to enforce non-penetrating boundaries was employed withA = 10 in all cases except example 7.5, where a lowervalue ofA = 4 was used.

7.1. Breaking dam problems.

The first example corresponds to a classical breaking dam simulation, with fluid (initially confinedbetween two walls) descending a ramp of slope 0.5 (Figure 5). The total number of particles involvedin the simulation is 2500, representing a square water column of side0.1 m. Particles are set upwith densities computed according to (65) andρ0 = 1000 kg/m3. The fluid viscosity isµ =0.5 kg m−1s−1. This a slow flow and, therefore, the use of the “real” water bulk modulus would



time=0.096 s time=0.120 s



Figure 6. Breaking dam flow: Simulation at various stages.

lead to very long computations, due to the CFL stability condition. Instead, an artificial bulk modulusis usually employed, computed as [8],[14],[30]:

κ =100ρ(2gH)

γ(125)

whereH is the height of the dam. Unfortunately, in long simulations this artificial (artificially low)bulk modulus results in an excessively compressible flow, demonstrating the convenience of thedevelopment of a fully incompressible algorithm. The evolution of the simulation is shown in Figure 6and the particle distribution att = 0.152 s is detailed in Figure 7.



Figure 7. Breaking dam flow: Simulation att = 0.152 s with µ = 0.5.

Figure 8. Bore: Scheme of the initial configuration.

7.2. Solitary waves.

The propagation of long gravity waves has been widely studied in last years, given its great interestin hydraulic and coastal engineering. Tsunamis are examples of such waves and shallow water modelsare often used to perform numerical simulations. No consideration of the vertical dimension is asimplification that reduces the complexity of the problem under study, but that is no longer valid incertain situations [34]. A scheme of the initial configuration of the example presented in this section isshown in Figure 8. The tank contains water at rest in two different levels, and the bore evolves rapidlyafter the beginning of the simulation (Figure 9). The density of water isρ0 = 1000 kg/m3 and theviscosity employed is againµ = 0.5 kg m−1s−1. The fluid was modelled using3980 particles.

7.3. Gravity currents.

A gravity current is the flow of a fluid into another fluid of different density [35]. In this example(Figures 10 and 11), we consider a tank with fluid of densityρ1

0 = 1000 kg/m3 (left) separatedby a lock gate from another fluid of densityρ2

0 = 1500 kg/m3 (right). The lock is rapidly(“instantaneously”) removed and fluid 2 flows under fluid 1. An experimental measurement of thevelocity of the head of the current was proposed by Rottman and Simpson (1983, cited by [35]):



time=0.060 s

time=0.076 s

time=0.092 s

time=0.108 s

Figure 9. Bore: Simulation at various stages.

vh ∼ 0.4(

gD∆ρ

ρ0

)1/2

(126)

whereD is the depth of the tank. In this example,D = 0.06 m, g = 9.81 m/s2, ρ0 = 1000 kg/m3

and∆ρ = 500 kg/m3. Thus, (126) predicts a head velocity ofvh = 0.217 m/s. In our numericalexperiments we have obtainedvh ∼ 0.2 m/s, which is in good agreement with the experimental value.The simulation involved5100 particles.



Figure 10. Horizontal gravity current: Scheme of the initial configuration.




Figure 11. Horizontal gravity current: Simulation at various stages.



Figure 12. Fluid-Fluid Impact: Scheme of the initial configuration.

time=0.0018 s time=0.0099 s

time=0.0297 s time=0.0477 s

time=0.0657 s time=0.0837 s

time=0.1017 s time=0.1197 s

time=0.1377 s time=0.1557 s

Figure 13. Fluid-Fluid Impact: Simulation at various stages.



Figure 14. Fluid-Fluid Impact: Simulation att = 0.0396 s (detail).

Figure 15. Fluid-Structure Impact: Scheme of the initial configuration.

Figure 16. Fluid-Structure Impact: Simulation att = 0.051 s.



time=0.015 s

time=0.051 s

time=0.087 s

time=0.033 s

time=0.069 s

time=0.105 s

Figure 17. Fluid-Structure Impact: Simulation at various stages (detail).

7.4. Impacts.

In the first example a circular drop of water falls vertically at2 m/s on a mass of water initially atrest (Figures 12 and 13). This example demonstrates the exceptional performance of the method in theabsence of boundary distortions (Figure 14). The fluid density and viscosity areρ0 = 1000 kg/m3 and



µ = 0.5 kg m−1s−1, respectively. The total number of particles is5539.An example of fluid-structure interaction is shown in figures 15, 16 and 17. The total number of fluid

particles is4470.

7.5. Mould filling.

If previous examples showed the good performance away from the boundaries, the followingdemonstrates that the formulation proposed is particularly sensitive to the correct (better, to theincorrect) enforcement of boundary conditions, which is still a challenging and open problem.

The simulation corresponds to the filling of a circular mould with core (Figure 18). The velocityof the jet at the gate is18 m/s and the viscosity isµ = 0.01 kg m−1s−1. The bulk modulusκ waschosen such that the wave celerity is1000 m/s and the total number of particles is14314. Severalinstants of the simulation are shown in Figure 19, with times referred to the impact between the jetand the core. The overall shape of the two jets passing the core looks quite satisfactory and agree withprevious results [36]. In spite of using the consistent formulation ofboundary forces, we have foundexcessive distortion near the boundaries, compared with the flow away from their influence (see Figure20). This effect could be caused by the “particle based” boundary approach. We expect to developbetter algorithms in the future. Figure 21 shows a comparison between the solution computed and theexperiments carried out by Schmid and Klein [37].

8. CONCLUSION AND FUTURE DEVELOPMENTS

In this paper we explored the application of a Galerkin based SPH formulation to the simulation ofviscous free surface flows. This approach provides a clear framework to interpret various techniquestraditionally used by SPH practitioners such as “stress-points”, boundary forces, special gradientcomputations, etc.

The start point of our research included an overwhelming amount of questions about the natureand application of meshless methods. And now we feel inmensely happy to leave them perfectlyunresolved. Much work is still to be done until a complete knowledge of meshless methods couldtake out their whole power. The set of unresolved questions include: development of an efficient fullyincompressible algorithm, adequate enforcement of boundary conditions, error estimations, efficientalgorithms with variable smoothing length, numerical integration of the weak form, development ofefficient local formulations, etc.

The results obtained so far are not less encouraging in view of all these “questions” for we keepa “logical” hope as, if Wittgenstein was right, if a question can at least be formulated, then it can beanswered.

ACKNOWLEDGEMENTS

This work has been partially supported by the SGPICT of the “Ministerio de Ciencia y Tecnologıa” of theSpanish Government (Grant DPI# 2001-0556), the “Xunta de Galicia” (Grants # PGDIT01PXI11802PR andPGIDIT03PXIC118002PN) and the University of La Coruna.

Mr. Cueto-Felgueroso gratefully acknowledges the support received from “Fundacion de la Ingenierıa Civil deGalicia” and “Colegio de Ingenieros de Caminos, Canales y Puertos”.



Figure 18. Mould filling: Dimensions of the mould.

time=5.76 ms time=8.64 ms

time=12.24 ms time=15.60 ms

Figure 19. Mould filling: Simulation at various stages.



Figure 20. Mould filling: Simulation att = 5.76 ms.

Figure 21. Mould filling: Experimental (left) and numerical (right) results.



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Documents

On the Galerkin formulation of the Smoothed Particle