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Variational formulation for the smooth

particle hydrodynamics (SPH) simulation of fluid

and solid problems

J. Bonet *, S. Kulasegaram, M.X. Rodriguez-Paz, M. Profit

Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea, Singleton Park,

Swansea SA2 8PP, UK

Received 5 July 2003; received in revised form 10 October 2003; accepted 4 December 2003

Abstract

The paper describes the variational formulation of smooth particle hydrodynamics for both fluids and solids

applications. The resulting equations treat the continuum as a Hamiltonian system of particles where the constitutive

equation of the continuum is represented via an internal energy term. For solids this internal energy is derived from the

deformation gradient of the mapping in terms of a hyperelastic strain energy function. In the case of fluids, the internal

energy term is a function of the density. Once the internal energy terms are established the equations of motion are

developed as equations of Lagrange, where the Lagrangian coordinates are the current positions of the particles. Since

the energy terms are independent of rigid body rotations and translations, this formulation ensures the preservation of

physical constants of the motion such as linear and angular momentum.

Ó 2004 Elsevier B.V. All rights reserved.

Keywords: Smooth particle hydrodynamics

1. Introduction

Smooth particle hydrodynamics (SPH) is a truly mesh-free technique initially developed by Lucy,

Gingold and others for the simulation of astrophysical problems [1–11]. The technique was later extended

to model free surface flows by Monaghan [11–13] and to simulate large strain solid mechanics problems byLibersky and others [14–16]. More recently several authors have made substantial progress on improving

the accuracy of the basic technique by introducing correction or re-normalisation terms [17–22]. In addi-

tion, the stability properties of the method have been studied in detail by various authors, specially for the

case of solids, where the presence of tension leads to the phenomenon of tension instability [23–27].

* Corresponding author. Tel.: +44-1797-295-689; fax: +44-1792-295-676.

E-mail address: [email protected] (J. Bonet).

0045-7825/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.12.018

Comput. Methods Appl. Mech. Engrg. 193 (2004) 1245–1256

www.elsevier.com/locate/cma

http://mail%20to:%[email protected]/

http://mail%20to:%[email protected]/



The aim of the paper is to present the corrected SPH technique from the point of view of a variational

formulation for both fluids and solids. In this way, the continuum is represented by a discrete set of

particles each with a given mass and velocity. The constitutive representation of the material is achieved via

an internal energy term, which for adiabatic reversible problems is simply a function of the particle posi-tions. In this way the continuum is modelled as a Hamiltonian system of particles and the motion of each

particle is given by the classical Lagrange equations. This procedure of deriving the governing equations

bypasses the standard differential equations of equilibrium and, more importantly, ensures that the con-

stants of the motion such as linear and angular momentum are preserved. The final equations for the

internal (stress based) forces can then be extended to more complex constitutive models without loss of

momentum preservation characteristics.

For the fluid case the internal energy term is derived from the traditional SPH equation for the density

and the paper will show that this is variationally consistent with the most commonly used expression for the

internal forces. The paper will then explain how to introduce dissipative effects in the form of viscosity into

the variational framework.

In the solid case, a total Lagrangian approach is employed, which has been shown not to suffer from

tension instability [27]. This leads to a set of equations for the internal forces based on the first Piola–

Kirchhoff tensor. The paper will finally present some examples to illustrate the techniques proposed.

2. Smooth particle hydrodynamics interpolation

In the smooth particle hydrodynamics (SPH) method, a given function f ðxÞ and its gradient r f are

approximated in terms of values of the function at a number of neighbouring particles and a kernel

function W ðx À xb; hbÞ ¼ W bðx; hbÞ as

f hðxÞ ¼ X M

b¼1

V b f bW bðx; hbÞ and r f hðxÞ ¼ X M

b¼1

f bgbðxÞ; ð1Þ

where h is the smoothing length and determines the support of the kernel (see Fig. 1); V b denotes a tributary

volume associated to particle b (typically evaluated as the particle mass dived by the density); and in the

standard SPH technique, the gradient vectors g are simply gb ¼ V brW b. More recently, however, re-nor-

malised or corrected SPH methods have been derived in which the gradient functions are amended to

Fig. 1. SPH interpolation.

1246 J. Bonet et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1245–1256



ensure that the gradient of a general constant or linear function is correctly evaluated. This requirement

leads to two simple conditions for these gradient vectors, namely

X M

b¼1gbðxÞ ¼ 0 and

X M

b¼1xb gbðxÞ ¼ I: ð2Þ

A possible simple way of fulfilling these conditions is obtained by introducing a vector and tensor cor-

rection terms, e and L, respectively, to give

gbðxaÞ ¼ V bLa½rW bðxaÞ þ eadab: ð3Þ

Substituting this equation into the above requirement leads to explicit equations for the correction terms as

ea ¼ ÀX M a

b¼1

V brW bðxaÞ and La ¼X M a

b¼1

V bðxb

"À xaÞ rW bðxaÞ

#À1

: ð4Þ

With the help of these terms, the second expression in Eq. (1) will furnish the correct gradient of constant

and linear functions. This interpolation will be used later to evaluate strains inside a solid continuum.

3. Equations of motion

Consider now a continuum represented by a large set of particles as shown in Fig. 2. Each particle a is

described by a mass ma, a position vector xa, and a velocity vector va. In order to proceed with a variational

formulation of the equations of motion of the continuum, it is necessary to define the kinetic, internal and

external energy of the system. For instance, the total kinetic energy of the system can be simply approx-

imated as the sum of the kinetic energy of each particle:

K ¼

1

2Xamaðva Á vaÞ: ð5Þ

Similarly, for a common case where the external forces result form a gravitational field g, the total external

energy is

Pext ¼ ÀXa

maðxa Á gÞ; ð6Þ

The internal energy, on the other hand will incorporate the constitutive characteristics of the continuum. In

general, it is possible to express the total internal energy as the sum of the products of particle masses by the

amount of energy accumulated per unit mass p, that is

Fig. 2. Continuum represented by a set of particles.

J. Bonet et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1245–1256 1247



Pint ¼Xa

mapðq; . . .Þ; ð7Þ

where p will depend on the deformation, density or other constitutive parameters.

The equations of motion of the system of particles representing the continuum can now be evaluated

following the classical Lagrangian formalism to give

d

dt

oL

ova

ÀoL

oxa

¼ 0; Lðxa; vaÞ ¼ K ðvaÞ À PintðxaÞ À PextðxaÞ: ð8Þ

Substituting Eqs. (5)–(7) into the above expression leads to the standard Newton Õs second law for each

particle as

maaa ¼ Fa À Ta; ð9Þ

where aa is the acceleration of the particle; the external forces Fa, for the simple gravitational case, are

Fa ¼ À

oPext

oxa ¼ mag ð10Þ

and the internal constitutive forces are defined as

Ta ¼oPint

oxa

¼o

oxa

Xb

mbpðqb; . . .Þ: ð11Þ

The evaluation of this last term will inevitably require the constitutive definition of the material in question

and will therefore be carried out in the following sections, separately for solids or fluids. Note that provided

that the internal forces are evaluated in accordance with the above equation, the resulting expression will be

consistent with the preservation of linear and angular momentum.

In a computational context, it is now possible to update the velocities and positions of the particles by,

for instance, using a simple leap-frog integration scheme defined by

vnþ12

a ¼ vnÀ12

a þ Dt ana; xnþ1

2a ¼ xn

a þ Dt vnþ12

a : ð12Þ

This type of time integrator preserves the constant of the motion and is therefore highly suitable for

mechanical systems. However, it is well known that its explicit nature makes it conditionally stable and that

the time step size Dt must satisfy the condition:

Dt 6 ah

c; ð13Þ

where a is the Courant number and c the speed of the sound wave.

4. Solids––total Lagrangian formulation

Consider the case in which the continuum modelled as a system of particles represents an elastic solid.

The motion is then described by a mapping / (see Fig. 3), which relates the current to the initial position of

each continuum particle [28]

x ¼ /ðX; t Þ: ð14Þ

The internal energy per unit mass is given by the elastic strain energy which is a function of the deformation

gradient tensor F as

p ¼ pðFÞ; F ¼ox

oX: ð15Þ




The deformation gradient tensor at each particle can be evaluated with the help of Eq. (1) in terms of thecurrent nodal positions as

F ¼X N b¼1

xb g0bðXÞ; ð16Þ

where g0b denote the gradient vectors at the initial body configuration. It is now possible to obtain the

expression for the internal forces by differentiating the internal energy per unit mass with respect to nodal

positions to give

Ta ¼Xb

mb

q0b

Pbg0aðXbÞ; ð17Þ

where the first Piola–Kirchhoff vector is defined as

P ¼ q0b

op

oFð18Þ

and is related to the Cauchy stresses by [28]:

q0

qr ¼ PFT: ð19Þ

The above derivation has been carried out exclusively for the case of an elastic solid. However, it is now

possible to employ the final equation for the internal forces in the more general context of irreversible

constitutive models such as plasticity, viscoelasticity, etc. In such cases the internal forces may not be given

as the derivative of a potential with respect to nodal positions, but if Eq. (17) is used, they will still be

consistent with the preservation of momentum.

5. Compressible and nearly incompressible fluids

In the case of a fluid the notion of a fixed reference state is not physically appropriate and it is therefore

necessary to define internal energy making reference only to the current nodal positions. Again, it is first

necessary to restrict the derivation to the reversible case, in which the internal energy is simply a function of

the density of the fluid. Moreover, the explicit nature of the time integration proposed above makes it

necessary to allow for some compressibility in the fluid. The incompressible case is then only achieved in a

penalty sense by increasing the bulk modulus of the material.

Fig. 3. Large strain solid deformation.




The evaluation of the density for a given distribution of particle masses is obtained by simply applying

the first part of Eq. (1) to the density to give

qðxaÞ ¼X M

b¼1mbW aðxb; haÞ; ð20Þ

which is the classical SPH approximation for the density of the fluid at each particle. The corresponding

internal forces are now evaluated by simple differentiation to give

Ta ¼o

oxa

Xb

mbpðqbÞ ¼Xb

mb

dp

dqb

oqboxa

: ð21Þ

The derivative of the internal energy for a reversible adiabatic process is related to the pressure by

p ¼ q2 dp

dq: ð22Þ

In addition, the derivative of the density with respect to nodal positions can obtained from Eq. (22). Itsevaluation, however, will depend on whether the smoothing length ha is kept constant throughout the

motion or allowed to vary. In general, for nearly incompressible flows, this value is kept constant, whilst it

is typically allowed to change when the compressibility is significant.

5.1. Constant h case

In this case the derivative of the density with respect to nodal positions is simply obtained as

oqboxa

¼ marW bðxaÞ À dabXc

mcrW aðxcÞ: ð23Þ

Substituting into Eq. (21) and noting that for the common case of spherical kernel functions with constantsmoothing length: rW bðxaÞ ¼ ÀrW aðxbÞ, yields the internal force expression as

Ta ¼Xb

mamb

p a

q2a

þp b

q2b

rW bðxaÞ; ð24Þ

which is, in fact, the classical equation used in SPH calculations [12]. Again, it is now important to

emphasize that the resulting equations for the internal forces can now be employed in more general irre-

versible processes.

5.2. Variable h case

Generally, if the flow is highly compressible it is necessary to change the value of the smoothing length inaccordance with the density. Otherwise, leaving a constant value may lead to particles with hardly any

neighbours in the case of expansion or with far too many in the case of compression. To avoid this, the

smoothing length is typically linked to the density via the equation:

qahd a ¼ qaðt 0Þhaðt 0Þ

d ¼ constant: ð25Þ

The derivation of the variationally consistent internal forces is now more complex and will be detailed in

a forthcoming publication. Suffice to say that the resulting expression for the internal forces is

Ta ¼Xb

mamb

p a

aaq2a

rW aðxb; haÞ

À

p b

abq2b

rW bðxa; hbÞ

; ð26Þ




where a represents a scalar correction term given by

aa ¼1

qad Xbmbr ab

dW a

dr ; r ab ¼ kxa À xbk: ð27Þ

5.3. Viscosity

In the context of the proposed variational formulation, viscosity can be introduced via a dissipative

potential. This leads to a new term in the Lagrange equations as

d

dt

oL

ova

ÀoL

oxa

¼ ÀoPdis

ova: ð28Þ

In general, the dissipative potential will be expressed as the sum of viscous potentials per unit mass w, which

in turn are functions of the rate of deformation tensor d, as

Pdis ¼ Xa

mawðdÞ; 2d ¼ rv þ rvT: ð29Þ

For instance, in the case of a Newtonian fluid with kinematic viscosity m ¼ l=q, the viscous potential is

defined as

wðdÞ ¼ mðd0: d0Þ; d0 ¼ d À 1

3ðtr dÞI ð30Þ

so that the viscous stresses are obtained by differentiation to give

rvis ¼ q

ow

od¼ 2ld0: ð31Þ

The gradient of the velocity at each continuum particle is obtained with the help of Eq. (1) as

rva ¼X M a

b¼1vb gbðxaÞ: ð32Þ

This enables the internal forces due to viscous effects to be evaluated after simple algebra as

Tvisa ¼

oPdis

ova¼Xb

mb

rvisb

qb

gaðxbÞ: ð33Þ

Note that in contrast with the equations derived in the previous section for the solid case, the gradient

vectors are now in the current configuration. It has been shown that this type of equation can lead to

instabilities in the presence of negative pressures (tension). In general, however, fluids will remain in

compression, at least in absolute terms, and this problem will not arise.

6. Examples

6.1. Taylor impact bar

This section presents numerical results from the simulation of a small cylindrical copper bar against a

rigid plane wall. The bar has an initial length of 0.0324 m and initial radius 0.0032 m. The Initial velocity of

the bar is 227 m/s and the termination time of the problem is 80 ls. Von Mises plasticity with linear iso-

tropic hardening is employed for the numerical computation. Material properties used for copper are given

in Table 1. This is a classical dynamic test example and the results obtained match closely those achieved

using a standard FE formulation [29].




Fig. 4 above shows the deformed shape and the distribution of equivalent plastic strain at various stages

of the numerical simulation. The results show that the mesh-less method yields a slightly larger maximal

equivalent plastic strain than the finite element solution reported in [29], but the discrepancy is small.

6.2. Oscillating cylinder

This example involves a nearly incompressible neo-Hookean [28] cylinder travelling with initial velocity

of 1.88 m/s to the right which is suddenly fixed at its base. The initial radius is 0.32 m and the length 3.24 m.

The shear modulus is 0.3571 MN/m2 and the bulk modulus is 1.67 MN/m2. The shapes obtained at different

times are shown in Fig. 5. The same example has been run using a standard dynamic FE code with identical

initial nodal positions and tri-linear eight noded cube elements. The SPH and FE solution for the centreline

of the cylinder at three different times are compared in Fig. 6, where the agreement can be seen to be

excellent.

Table 1

Copper bar material properties

Elastic modulus E (GN/m2) 117

PoissonÕs ratio m 0.35

Yield stress r y (GN/m2) 0.4

Hardening modulus H (GN/m2) 0.1

Density q (kg/m3) 8930

Fig. 4. High speed impact test. Contours of plastic strain.




6.3. Dam breaking flow past a cylinder

This final example involves the collapse of a rectangular 3-D dam in the presence of a rigid cylinder. The

constitutive equation use for the fluid ignores viscosity and allows a small amount of incompressibility by

using an equation of state of the form [12]:

p ¼ jq

q0

cÀ 1

; ð34Þ

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2

FE solution

CSPH solution

y - p o s ( m )

x-pos (m)

Fig. 6. Oscillating cylinder––FE comparison.

Fig. 5. Oscillating cylinder.




where c ¼ 7 and the bulk modulus employed is derived so that the Mach number of the flow is small(typically 0.1–0.01). This is achieved by setting [12]:

k ¼ c2maxq=c; where cmax ¼ kvkmax=m; ð35Þ

where m is the desired Mach number. Fig. 7 shows various stages of the solution.

7. Concluding remarks

The paper has presented a unified variational formulation covering solids, perfect fluids and viscous

fluids. The main advantages of using such a procedure to derive the equations of motion is that the resulting

Fig. 7. Dam breaking flow past a cylinder.




discrete equations are guaranteed to preserve the constant of the motion such as linear and angular

momentum. The procedure proposed works well for both fluids and solids with the possible exception of

solids where as a result of plasticity very large deformations will take place. In such cases the solid will start

to behave more like a fluid and the viscous formulation proposed may be more appropriate than mappingthe deformation to an initial reference state. The alternative would be to repeatedly update the reference

configuration as deformation progresses. This type of updated Lagrangian formulation is currently under

consideration.

Acknowledgements

This work was partly funded by research contract GR/M84312 from EPSRC. This financial support is

gratefully acknowledged. The third author wishes to thank CONACYT (The Mexican Council of Science

and Technology) and ORS for the support given to this research project.

References

[1] L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astron. J. 82 (1977) 1013.

[2] R.A. Gingold, J.J. Monaghan, Smooth particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R.

Astron. Soc. 181 (1977) 375.

[3] M. Schussler, D. Schmitt, Comments on smoothed particle hydrodynamics, Astron. Astrophys. 97 (1981) 373.

[4] R.A. Gingold, J.J. Monaghan, Kernel estimates as a basis for general particle methods in hydrodynamics, J. Comput. Phys. 46

(1982) 429.

[5] J.J. Monaghan, An introduction to SPH, Comput. Phys. Commun. 48 (1988) 89–96.

[6] J.J. Monaghan, Why particle methods work, SIAM J. Sci. & Stat. Comput. 3 (1982) 422.

[7] J.J. Monaghan, Particle methods for hydrodynamics, Comput. Phys. Rep. 3 (1985) 71.

[8] J.J. Monaghan, An introduction to SPH, Comput. Phys. Commun. 48 (1988) 89.

[9] J.J. Monaghan, On the problem of penetration in particle methods, J. Comput. Phys. 82 (1989) 1.

[10] J.J. Monaghan, Smoothed particle hydrodynamics, Ann. Rev. Astron. Astrophys. 30 (1992) 543.

[11] H. Takeda, S.M. Miyama, M. Sekiya, Numerical simulation of viscous flow by smoothed particle hydrodynamics, Prog. Theor.

Phys. 92 (1994) 939.

[12] J.J. Monaghan, Simulating free surface flows with SPH, J. Comput. Phys. 110 (1994) 399.

[13] J.J. Monaghan, A. Kocharyan, SPH simulation of multi-phase flow, Comput. Phys. Commun. 87 (1995) 225.

[14] L.D. Libersky, A.G. Petschek, T.C. Carney, J.R. Hipp, F.A. Allahadi, High strain lagrangian hydrodynamics, J.Comput. Phys.

109 (1993) 67.

[15] W. Benz, E. Asphaug, Simulations of brittle solids using smooth particle hydrodynamics, Comput. Phys. Commun. 87 (1995) 253.

[16] G.R. Johnson, R.A. Stryk, S.R. Beissel, SPH for high velocity impact computations, Comput. Methods Appl. Mech. Engrg. 139

(1996) 347.

[17] W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Method Engrg. 20 (1995).

[18] G.R. Johnson, S.R. Beissel, Normalised smoothing functions for SPH impact computations, Int. J. Numer. Method Engrg. 39(1996) 2725.

[19] P.W. Randles, L.D. Libersky, Smoothed particle hydrodynamics: some recent improvements and applications, Comput. Methods

Appl. Mech. Engrg. 139 (1996) 375.

[20] J. Bonet, t.-s.L. Lok, Variational and momentum preservation aspects of smooth particle hydrodynamics formulations, Comput.

Methods Appl. Mech. Engrg. 180 (1999) 97–115.

[21] J. Bonet, S. Kulasegaram, Correction and stabilization of smooth particle hydrodynamics methods with applications in metal

forming simulations, Int. J. Numer. Method Engrg. (2000).

[22] G.A. Dilts, Moving least squares particle hydrodynamics––consistency and stability, Int. J. Numer. Method Engrg. 44 (2000)

1115–1155.

[23] J.W. Swegle, D.L. Hicks, S.W. Attaway, Smooth particle hydrodynamics stability analysis, J. Comput. Phys. 116 (1995).

[24] C.T. Dyka, R.P. Ingel, An approach for tension instability in smooth particle hydrodynamics (SPH), Comput. Struct. 57 (1995).

[25] C.T. Dyka, R.P. Ingel, Stress points for tension instability in SPH, Int. J. Numer. Method Engrg. 40 (1997).




[26] T. Belytschko, Y. Guo, W.K. Liu, S.P. Xiao, A unified stability analysis of meshless methods, Int. J. Numer. Method Engrg. 48

(2000) 1359–1400.

[27] J. Bonet, S. Kulasegaram, Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics

(CSPH) methods, Int. J. Numer. Method Engrg. 52 (11) (2001) 1203–1220.

[28] J. Bonet, R.D. Wood, Non-linear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge,UK, 1997.

[29] J. Bonet, A.J. Burton, A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic

explicit application, Commun. Numer. Method Eng. 14 (5) (1998) 437–449.


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