heuristic acceleration correction algorithm for use in sph computations in impact mechanics

Comput. Methods Appl. Mech. Engrg. 198 (2009) 3962–3974

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Comput. Methods Appl. Mech. Engrg.

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Heuristic acceleration correction algorithm for use in SPH computationsin impact mechanics

Amit Shaw a,*, S.R. Reid b

a Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, Indiab School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK

a r t i c l e i n f o

Article history:Received 24 June 2009Received in revised form 31 August 2009Accepted 3 September 2009Available online 10 September 2009

Keywords:SPHArtificial viscosityImpact mechanicsEnergy loss

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.09.006

* Corresponding author.E-mail address: [email protected] (A. Sh

a b s t r a c t

Despite developments over the past 30 years, SPH and other mesh-free computational methods are notyet in general use as standard tools in dynamic structural mechanics. One possible reason for this is theuse of features such as artificial viscosity, to stabilize the numerical computations, which can result inphysically unreal phenomena. The effect of artificial viscosity in SPH computations is examined and aheuristic acceleration correction algorithm is proposed in this paper. The purpose is to improve the mod-elling of physically real effects and thereby make SPH a more attractive modelling option, particularly forstructural impact problems.

The essence of the proposed method is to calculate the change in the acceleration due to the artificialviscosity term and then correct the computed acceleration by subtracting a kernel approximation of itsartificial counterpart. The energy equation is also modified accordingly. By this means, the excessive dis-sipation is removed, while retaining the computational stabilizing effect of the artificial viscosity. Forillustrative purposes, the proposed method is applied to several classical elastic and elastic–plasticimpact problems and the results are compared with those available in the literature. In the process,the improved performance of the proposed algorithm vis-à-vis the standard SPH procedures is discussedas are the outstanding mathematical issues which require resolution to make the approach truly rigorous.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Smooth Particle Hydrodynamics [16,6,18,20] is one of the oldestand probably the simplest of mesh-free methods. It is a grid-lessLagrangian technique, developed initially to deal with mathemati-cal models of astrophysical and cosmological phenomena. SPH hasbeen successfully applied to a wide range of problems in computa-tional fluid dynamics [3] and has been extended to problems insolid mechanics by Libersky and Petschek [14] and Libersky et al.[15]. In demonstrating the value of the acceleration algorithmherein, we have followed the developments of Libersky and Pet-schek [14] closely.

Recently SPH has found considerable appeal amongst research-ers for the numerical modelling of high velocity impact and pene-tration problems (e.g. [10,17]). Due to its particle nature, SPH isparticularly very effective for modelling fragmentation and mate-rial separation caused by the formation of cracks and crack systemsand the coalescence of a multitude of small crack-like flaws [1],which are major physical phenomena encountered in several areasof impact mechanics. These are not dealt with easily by othermethods, e.g. FEM.

ll rights reserved.

aw).

However, the method is not yet in general use as a standard toolin dynamic structural mechanics due to some inherent computa-tional difficulties. These include the use of the artificial viscosity[22] to promote numerical stability, perhaps one of the majorproblems. In artificial viscosity formulations, an ‘artificial pressure’term is added into the momentum and energy equations wheneverthe system experiences any shock compression. The most basicphysical property that artificial viscosity generates is dissipation,i.e. it converts kinetic energy to internal energy. However, onehas to be careful when choosing the artificial viscosity parametersthat it does not induce any false pressure, which may lead to anexcessive loss of kinetic energy, making the system over-dissipa-tive and the predictions correspondingly physically unreal. Theseare of particular concern in impact mechanics problems.

Unfortunately, there is no standard procedure for choosing theartificial viscosity parameters which works for a range of problems.Most often, the SPH computations are performed with some gener-ally arbitrarily-prescribed values of these parameters. Johnson [11]examined the effect of artificial viscosity in impact computationsand showed that the accuracy of SPH computations can be signifi-cantly affected by the choice of artificial viscosity parameters. Oneway out of this difficulty is to perform numerical experiments(running the simulation with different values of the artificialviscosity parameters) in order to find the ‘optimum’ values of these

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A. Shaw, S.R. Reid / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3962–3974 3963

parameters for a given problem. This is a formidable task when thesize of the problem is large. Therefore an algorithm is neededwhich can bypass the requirement of user-defined parametersand yet controls the effect of artificial viscosity in order to preventthe system from being over dissipative, whilst still providing thenecessary numerical stability.

In the present paper, an attempt has been made to accomplishthe above objective by developing an acceleration correction algo-rithm. Herein, the acceleration equations are amended by subtract-ing a correction term. The correction term is taken as a kernelapproximation of that part of the rate of momentum producedby the artificial viscosity, using the same kernel function by whichthe governing equations are discretized. The energy equation isalso modified accordingly. For illustration, the proposed methodis applied here to two classical elastic and elastic–plastic impactproblems, the collinear impact of two elastic rods and a 2D repre-sentation of the Taylor bullet impact test. The results are comparedwith those available in the literature. The relative numericaladvantages of the new algorithm are brought out in the process.A third classical problem in structural impact, the Parkes cantileverproblem, is then given to illustrate its potential benefit for a widerange of structural impact problems.

The formulation of the acceleration correction algorithm pre-sented below is heuristic, being based on intuition and numericalobservations. Notwithstanding this, we have applied the algorithmto a series of problems related to impact mechanics and the resultsindeed are very encouraging. Although a detailed theoretical anal-ysis of the proposed algorithm is yet to be produced, its numericaladvantages vis-à-vis SPH models with standard artificial viscosityshows that the method has the potential to address the issues asso-ciated with using artificial viscosity in SPH computations.

The paper is organized as follows. In Section 2, the equations ofSPH, as applied to elastic and elastic–plastic dynamics problems,are outlined. The notion of artificial viscosity is described in Section3. The proposed correction algorithm is discussed in Section 4. Thethree test cases are provided in Section 5 to demonstrate the effi-cacy of the proposed method. As noted above, one of these is theParkes cantilever problem [25], a classical, structural plasticityproblem described in several textbooks and publications in the lit-erature (e.g. [12,30,28,29,13]). Conclusions are drawn in Section 6.

2. SPH – A brief overview

In smooth particle hydrodynamics (SPH), first the entire do-main is discretized by defining a set of particles. These particlesinteract with each other through a kernel function (sometimesalso called the window or weight function) such that at every par-ticle the conservation equations are satisfied. There exists exten-sive literature on SPH addressing different theoretical as well asnumerical aspects [16,18,20]. The objective of this section is tooutline the steps involved in SPH applied to solid mechanics prob-lems so that a potential reader can transfer the theory into a com-putational code without further reference to other literature.However for more comprehensive information, readers are recom-mended to refer to the review paper by Monaghan [21] and thereferences therein.

2.1. Conservation equations

The conservation equations for continuum mechanics are,

dqdt¼ �q

@vb

@xb; ð1Þ

dva

dt¼ � 1

q@rab

@xb; ð2Þ

dedt¼ �rab

q@va

@xb; and ð3Þ

dxa

dt¼ va; ð4Þ

where, for any material point, q denotes its mass density, e is thespecific internal energy, va and rab are respectively the elementsof velocity and Cauchy stress tensor, xa is the spatial coordinateand d

dt is the time derivative taken in the moving Lagrangian frame.In Eqs. (1)–(4), the effect of heat conduction is neglected assumingthat the deformation process is locally adiabatic.

2.2. Constitutive model

The stress component in Eqs. (2) and (3) may be written interms of hydrostatic and deviatoric stresses as Libesky and Pet-schek [14],

rab ¼ Pdab � Sab; ð5Þ

where P and Sab are respectively the pressure and the componentsof the traceless symmetric deviatoric stress tensor. The equationsinvolved in calculating P and Sab are given in the following sub-sections.

2.2.1. PressureThe pressure in Eq. (5) may be calculated through an equation

of state (EOS), which is generally a functional form of two or morethermodynamical properties (such as temperature, pressure,volume or internal energy) associated with the physics of the prob-lem. For solids there is no general EOS that is appropriate for allmaterials and circumstances. The Mie–Gruneisen EOS is widelyused in almost all hydrocodes [33]. For a comprehensive descrip-tion of EOS for solids and other materials one may refer to Elizeret al. [4]. In this paper two EOS are used.

For elastic problem, pressure is assumed to vary linearly withcompression ratio as,

PðqÞ ¼ Kqq0� 1

� �; ð6Þ

where, K is the bulk modulus and q0 is the initial mass density.For the elastic–plastic problem, the Mie–Gruneisen EOS [15],

described below, is used.

Pðq; eÞ ¼ 1� 12

Cg� �

PH þ Cqe; ð7Þ

where

PH ¼ a0gþ b0g2 þ c0g3 for g > 0 and PH ¼ a0g3; g < 0; ð8Þ

g ¼ qq0� 1

� �; ð9Þ

a0 ¼ q0C2; ð10Þb0 ¼ a0½1þ 2ðS� 1Þ�; ð11Þc0 ¼ a0½2ðS� 1Þ þ 3ðS� 1Þ2�: ð12Þ

Here, S and C, respectively, denote the linear shock-velocity and theparticle-velocity parameters to describe the Hugoniot fit and C isthe Gruneisen parameter.

2.2.2. Deviatoric stressThe deviatoric part of the Cauchy stress rate tensor may be writ-

ten as,

_Sab ¼ l _eab � 13

dab _ecc� �

; ð13Þ

Table 1Flowchart for SPH computation with the acceleration correction algorithm.

Input file

Read particle distribution fxigNi¼1

n owhere N is the total number of particles

Read xai ; va

i ; Sabi ; qi and ei 8i 2 ½1;N�

Read material properties E; l; ry (if plasticity is accounted for)End Input fileLoop 1 (time integration ðt0 6 t 6 tf Þ through predictor–corrector scheme) Operations in Loop1 may be different in other time-

integration scheme (e.g., Verlet algorithm, see Verlet, 1967for details)

Initialize time t ¼ t0

Predictor (predicts the response at t þ Dt=2ÞCall subroutine RATES to calculate rates at time tLoop 2 8i 2 ½1;N�

xai

� �tþDt=2 ¼ xa

i

� �t þ 0:5Dt _xa

i

� �t ; va

i

� �tþDt=2 ¼ va

i

� �t þ 0:5Dt _va

i

� �t

Sabi

� �tþDt=2

¼ Sabi

� �tþ 0:5Dt _Sab

i

� �t; ðqiÞtþDt=2 ¼ ðqiÞt þ 0:5Dtð _qiÞt

ðeiÞtþDt=2 ¼ ðeiÞt þ 0:5Dtð _eiÞt ; ðPÞtþDt=2 ¼ f ½ðqiÞtþDt=2; ðEiÞtþDt=2�Check for Von Mises yield criteria (if plasticity is accounted for) as, Other yield criteria may be used.

fi ¼min ry

rei;1

n o; re

i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3SabSab

p; Sab

i ¼ fSabi

End of Loop 2End of PredictorCorrector (Correct the response at t þ Dt=2)Call subroutine RATES to calculate rates at time t þ Dt=2Loop 3 8i 2 ½1;N�

xai

� �tþDt=2 ¼ xa

i

� �t þ 0:5Dt _xa

i

� �tþDt=2; va

i

� �tþDt=2 ¼ va

i

� �t þ 0:5Dt _va

i

� �tþDt=2

Sabi

� �tþDt=2

¼ Sabi

� �tþ 0:5Dt _Sab

i

� �tþDt=2

; ðqiÞtþDt=2 ¼ ðqiÞt þ 0:5Dt _qið ÞtþDt=2

ðeiÞtþDt=2 ¼ ðeiÞt þ 0:5Dtð _eiÞtþDt=2; ðPÞtþDt=2 ¼ f bðqiÞtþDt=2; ðEiÞtþDt=2cCheck for Von Mises yield criteria (if plasticity is accounted for) as, Other yield criteria may be used.

fi ¼min ry

rei;1

n o; re

i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 SabSab

q; Sab

i ¼ fSabi

End of Loop 3End of CorrectorLoop 4 8i 2 ½1;N� (Final response at t þ Dt)

xai

� �tþDt ¼ 2 xa

i

� �tþDt=2 � xa

i

� �t ; va

i

� �tþDt ¼ 2 va

i

� �tþDt=2 � va

i

� �t

Sabi

� �tþDt¼ 2 Sab

i

� �tþDt=2

� Sabi

� �t; ðqiÞtþDt ¼ 2ðqiÞtþDt=2 � ðqiÞt

ðeiÞtþDt ¼ 2ðeiÞtþDt=2 � ðeiÞt ; ðPÞtþDt ¼ f ðqiÞtþDt ; ðEiÞtþDt

End of Loop 4Loop 5 8i 2 ½1;N� (Initialization of variables for next time step)

xai

� �t ¼ xa

i

� �tþDt ; va

i

� �t ¼ va

i

� �tþDt ; Sab

i

� �t¼ Sab

i

� �tþDt

ðqiÞt ¼ ðqiÞtþDt ; ðeiÞt ¼ ðeiÞtþDt ; ðPÞt ¼ ðPÞtþDt

End of Loop 5Update time as t ¼ t þ Dt

End of Loop 1 (end of time integration)SUBROUTINE RATESLoop 6 8i 2 ½1;N�

Calculate rate equations as,Continuity equationdqidt ¼ qi

Pj

mj

qjvb

i � vbj

� �Wij;b

Deviatoric stress rate equationdSab

idt ¼

l2

Pj

mj

qjva

i � vaj

� �Wij;b þ vb

i � vbj

� �Wij;a � 1

3 vci � vc

j

� �Wij;c

h iMomentum equation

dvai

dt ¼ �P

jmjrab

iq2

iþ rab

j

q2j

� �Wij;b �

PjmjP

ð1;1Þij Wij;b þ 1

2

PjP

kmk Pð1;1Þik Wik;b þPð1;1Þjk Wjk;b

� �� Wij

Energy equation

deidt ¼

Pjmj va

i � vaj

� �rab

iq2

i

� �Wij;b þ 1

2

Pjmj va

i � vaj

� �Pð1;1Þij Wij;b � 1

2

Pj va

i � vaj

� � PkmkP

ð1;1Þjk Wjk;bWik

� �Positiondxa

idt ¼ va

i

where, For AV, use Pða1 ;a2Þij . An ‘‘optimum” ða1;a2Þ may be found by

repeating Loop1 until the computation becomes stable.However AC does not need such experiment as it always takeða1 ¼ 1;a2 ¼ 1Þ

Pð1;1Þij ¼ �cijlijþl2ij

qijif rij � v ij < 0 otherwise Pð1;1Þij ¼ 0

cij ¼ 0:5ðci þ cjÞ; qij ¼ 0:5ðqi þ qjÞ; rij ¼ xi � xj ; v ij ¼ v i � v j

End of Loop 6 8i 2 ½1;N�END OF SUBROUTINE RATES

3964 A. Shaw, S.R. Reid / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3962–3974


where l is the shear modulus, _eab is the component of the strainrate tensor and dab is the Kronecker delta. Since Eq. (13) is not mate-rial–frame-indifferent, the Jaumann stress rate, given below iswidely used,

_Sab ¼ l _eab � 13

dab _ecc� �

þ Sac _Rbc þ Scb _Rac; ð14Þ

where l is the shear modulus. The component _eab of the strain ratetensor and _Rab of the spin tensor may be obtained as,

_eab ¼ 12

@va

@xbþ @v

b

@xa

� �; ð15Þ

_Rab ¼ 12

@va

@xb� @v

b

@xa

� �: ð16Þ

When the plastic part of behaviour can be described as perfectly-plastic, following Libersky and Petschek [14] and Libersky et al.[15], the flow régime is determined by the Von Mises yield criterion.At every time step the second stress invariant J2 ¼ SabSab is checkedand if

ffiffiffiffiJ2

pexceeds the yield stress ry=

ffiffiffi3p

, where ryis the uniaxialyield stress, the individual stress components are brought back tothe yield surface using,

Sab ! fSab; ð17Þ

where f ¼minryffiffiffiffiffiffiffi3J2

p ;1

( ): ð18Þ

2.3. Discretization of governing equations

Let the computational domain X ¼ X [ @X � Rn be discretized

by a set of particles positioned at fxigNi¼1

n o, such that, for any suf-

ficiently smooth function uðxÞ defined on X, one can define a set ofdiscrete function values fui ¼ uðxiÞgN

i¼1. Then the discrete kernelapproximation of the function uðxÞ at any particle point xi maybe written as,

ui ¼X

j

ujWijDVj; ð19Þ

where Wij ¼Wðxi � xj;hÞ is the kernel function (also called thesmoothing or window function) centered at xi and DVj is thelumped volume (nodal volume) associated with particle j. The ker-nel function Wðx;hÞ is required to satisfy the following properties.

(i) Compact support: Wðx;hÞ is compactly supported with sup-port size h (also called the smoothing length). This propertyallows any particle to interact only with its neighbouringparticles and create a localized continuous field.

(ii) Partition of unity:R

X Wðx� y;hÞdy ¼ 1 enabling the kernelexpansion to reproduce a constant.

(iii) No local maxima or minima: Wðx;hÞ is a monotonicallydecreasing function away from its maximum. Thereforethe effect of one particle to another (and vice-versa)decreases as their inter-particle distance increases.

(iv) Wðx;hÞ is a finite representation of the Dirac delta distribu-tion and Wðx;hÞ !

h!0dðxÞ.

There are many possible choices for the kernel function depend-ing on its shape and the value of Dx=h, whereDx ¼ kxi � xjk. Formore details about the criterion for choosing kernel functions inSPH computations readers are referred to Fluk and Quinn [5] orHongbin and Xin [9]. In this paper, for illustration, the cubic splinekernel function, given below is used

Wðq;hÞ ¼ aD

1� 32 q2 þ 3

4 q3; 0 6 q 6 1;14 ð2� qÞ3; 1 6 q 6 2;0; q P 2;

8><>: ð20Þ

where aD ¼ 23h in 1D, aD ¼ 10

7ph2 in 2D.Similarly, derivatives of uðxÞ with respect to any spatial coordi-

nate xb may be written as,

@ui

@xb¼X

j

ujWij;bDVj; ð21Þ

where Wij;b ¼@Wij

@xb .Now each particle, say the ith particle, is associated with mass

mi, density qi, velocity component vai , internal energy ei, pressure

Pi, deviatoric stress component Sabi . Following Libersky and Pet-

schek [14], the semi-discrete form of the conservation Eqs. (1)–(3) and stress rate Eq. (14) may be written as,

dqi

dt¼ qi

Xj

mj

qjvb

i � vbj

� �Wij;b; ð22Þ

dvai

dt¼ �

Xj

mjrab

i

q2i

þrab

j

q2j

!Wij;b; ð23Þ

dei

dt¼X

j

mj vai � va

j

� � rabi

q2i

!Wij;b; and ð24Þ

dSabi

dt¼ l

2

Xj

mj

qjva

i � vaj

� �Wij;b þ vb

i � vbj

� �Wij;a

h

�13

vci � vc

j

� �Wij;c

�: ð25Þ

Once the continuum equations (Eqs. (1)–(3) and (14)) are discret-ized, the discrete SPH equations (Eqs. (22)–(25)) are updated (seeTable 1 in Section 4 for details) in time by a standard Predictor–Cor-rector scheme [19]. The time step size Dt is determined based onthe CFL (Courant–Fredrich–Levy) condition as,

Dt ¼mini

cshi

ci þ jv ij

� ; ð26Þ

where ciand hi are respectively the elastic wave speed and thesmoothing length associated with the ith particle. The elastic bar-wave speed ci may be calculated as ci ¼

ffiffiffiffiffiffiffiffiffiffiE=qi

p, where E is Young’s

modulus of the material. In Eq. (26) cs is the Courant number, pres-ently taken as 0.3 [19].

3. Artificial viscosity

In many impact mechanics computations e.g. projectile impact,high explosive detonation process, etc., discontinuous initial valuesfor physical quantities often leads to the propagation of ‘shockwaves’. The material experiences a sudden jump in the stress(and other physical quantities such as density, temperature, etc.)across a shock ‘front’. The inability of a numerical technique toapproximate these sudden jumps (and differentiate them) whilesatisfying the conservation Eqs. (22)–(25) across the ‘front’, can re-sult in the growth of unphysical, high frequency oscillation nearsuch sharp transitions of the physical quantities. Smoothening ofthese variations across the thickness of the shock wave is thereforerequired for a stable computation.

In the context of finite difference method, VonNeumann andRichtmyer [32] introduced the concept of artificial viscosity. Mona-ghan and Gingold [22] extended the concept of artificial viscosityto SPH computations. They showed via a one-dimensional shocktube problem that artificial viscosity smears the shock jump over


a few resolution lengths (or inter-particle distances) and stabilizesthe numerical computation in the presence of a shock. With theartificial viscosity term, Eq. (23) becomes,

dvai

dt¼ �

Xj

mjrab

i

q2i

þrab

j

q2j

þPij

!Wij;b: ð27Þ

Since artificial viscosity converts kinetic energy into internal energyof the system, the energy equation (24) is also modified to,

dei

dt¼X

j

mj vai � va

j

� � rabi

q2i

þ 12

Pij

!Wij;b: ð28Þ

There are many forms of the artificial viscosity Pij provided in theSPH literature but the form of the artificial viscosity proposed by[20] is perhaps the most widely used stabilizing mechanism inSPH computations. This will be taken as the exemplar for the algo-rithm described below. In this form for the artificial viscosity, Pij inEqs. (23) and (24) is given by,

Pij ¼�a1cijlij þ a2l2

ij

qijif rij � v ij < 0 otherwise Pij ¼ 0; ð29Þ

where

lij ¼hijrij � v ij

r2ij þ eh2

ij

; ð30Þ

cij ¼ ðci þ cjÞ=2; hij ¼ ðhi þ hjÞ=2; rij ¼ xi � xj;

v ij ¼ v i � v j and rij ¼ krijk:

Here, ða1;a2Þ are the artificial viscosity parameters. Henceforth, thenotation Pða1 ;a2Þ

ij will be used to imply the artificial viscosity withparameters ða1;a2Þ. In Eq. (30), e is introduced to prevent a singular-ity when rij ¼ 0 and is generally taken as 0.01 [20].

The major difficulty with this form of numerical dissipation isthe ‘‘optimum” choice of the artificial viscosity parametersða1;a2Þ. The choice of these parameters may affect the accuracyof the solution significantly as pointed out by Johnson [11]. Theartificial viscosity should not be so small that it cannot preventthe growth of unphysical oscillations near the shock nor shouldit be so large that it leads to a spuriously exaggerated smoothness,which makes the system over-dissipative. Unfortunately, as notedby Johnson [11], currently there is no standard set of parametersthat work for a wide range of problems. The ‘‘ideal” situationwould be when the form of artificial viscosity does not need anyuser specified parameters and yet stabilizes the numerical compu-tation without adding excess dissipation into the system. Towardsthis objective, an acceleration correction algorithm is proposedbelow.

4. Acceleration correction algorithm

A heuristic acceleration correction algorithm is proposed in thissection. The aim is to exemplify the use of this method to accom-plish two things. It reduces the unwanted effect of using the artifi-cial viscosity, whilst ensuring numerical stability. Having outlinedthe method, it is applied to some illustrative, classical problems. Inthe rest of the paper, the acceleration correction algorithm and thestandard artificial viscosity will be referred respectively to AC andAV.

As mentioned in Section 3 (and as shown in [11]), by taking dif-ferent combinations of a1 and a2 one can change the strength (theamount of dissipation it provides) of the artificial viscosity. In AC,the parameters a1 and a2 in Eq. (29) are both set equal to 1. Con-sequently, the form of artificial viscosity becomes,

Pð1;1Þij ¼�cijlij þ l2

ij

qijif rij � v ij < 0 otherwise Pð1;1Þij ¼ 0: ð31Þ

Eq. (27) may be rewritten as,

aai ¼ �aa

i þuai ; ð32Þ

where aai ¼

dvai

dt is the computed acceleration, �aai ¼ �

Pjmj

rabi

q2iþ

rabj

q2j

� �Wij;b without the artificial viscosity term and ua

i ¼ �P

jmjPð1;1Þij Wij;b

is the explicit contribution from the artificial viscosity, Pð1;1Þij beinggiven by Eq. (31). The addition of the artificial viscosity term, ua

i

into the momentum equation imparts dissipation into the systemand consequently some amount of kinetic energy is transformedinto internal energy.

Before proceeding, consider two important observations relatedto the use of artificial viscosity in SPH computations in solidmechanics. These form the basis of the development to follow.

(i) The amount of dissipation artificial viscosity generates,whilst removing the unphysical oscillation behind the shockfront, depends mainly on the choice of a1. a2 has itself asmall effect on the overall dissipation [2,11].

(ii) SPH computations in solid mechanics are generallyperformed with a1 6 1. Some commonly used (though arbi-trarily ascribed) values of ða1;a2Þ are ð0:5;0:5Þ [14], ð0:2 6a1 6 0:5;0:5 6 a2 6 4:0Þ [10], ð1:0;2:0Þ [17], ð1:0;1:0Þ [31],etc. Some authors have also used ð2:5;2:5Þ [15,31,17], how-ever nothing is stated about the reason for choosing this largevalue of a1. It is observed from the current study that an SPHcomputation in solid mechanics can be stabilized witha1 6 1.

Following the observations (i) and (ii), it is reasonable to con-sider that Eq. (31) is, generally, an overestimation of the artificialviscosity and may nevertheless produce excessive dissipation intothe system. Here our objective is to reduce this by subtracting acorrection term in Eq. (32) through,

aai ¼ �aa

i þuai � �ua

i : ð33Þ

The ideal situation would be when �uai ¼ ua

i . However, one cannothave �ua

i ¼ uai since it nullifies the numerical effect of artificial vis-

cosity and allows the numerical oscillations to grow near the shockfront.

However, consider taking �uai in the form

�uai ¼

12

Xj

uai þua

j

� �Wij; ð34Þ

where Wij ¼Wðxi � xj;�hÞ is the kernel function given by Eq. (20)

and �h is the smoothing length for the correction term. The criterionfor choosing h and �h are given in Section 4.1. Now from Eqs. (27),(31)–(34) the corrected momentum equation may be written as,

dvai

dt¼ �

Xj

mjrab

i

q2i

þrab

j

q2j

þPð1;1Þij

!Wij;b

þ 12

Xj

Xk

mk Pð1;1Þik Wik;b þPð1;1Þjk Wjk;b

� � !Wij; ð35Þ

) dvai

dt¼ �

Xj

mjrab

i

q2i

þrab

j

q2j

!Wij;b �

XjmjP

ð1;1Þij Wij;b|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Dissipation term

þ 12

Xj

Xkmk Pð1;1Þik Wik;b þPð1;1Þjk Wjk;b

� �� Wij|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Correction term

: ð36Þ

Note that the correction term is symmetric and its addition into themomentum equation (Eq. (36)) does not violate the Galilean invari-


ance (e.g. Newton’s laws hold in all inertial frames) of the formula-tion. Similarly the corrected energy equation may be written as,

dei

dt¼X

j

mj vai � va

j

� � rabi

q2i

!Wij;b

þ 12

Xjmj va

i � vaj

� �Pð1;1Þij Wij;b|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Dissipation term

� 12

Xj

vai � va

j

� � XkmkP

ð1;1Þjk Wjk;bWik

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Correction term

: ð37Þ

Having computed the correction terms for momentum and energy,the corrected momentum and energy equations (Eqs. (36) and (37),respectively) along with the continuity Eq. (22) and the stress Eq.(25) are updated in time. This is unlike SPH with AV which makesuse of Eqs. (22) and (25) respectively for momentum and energycalculations. For a better understanding, a flowchart indicatingthe different steps involved in SPH computation with AC is givenbelow in Table 1. While translating this flowchart into a computa-tional code one may need to follow an appropriate variable-storing,neighbour-search (e.g. see [23]) and a parallelization technique foran efficient computation. The advantages of using AC are given inSection 4.2.

4.1. Calculation of smoothing lengths ðhi and �hÞ

The accuracy of SPH computations greatly depends on thechoice of the smoothing length hi. Some SPH computations assumethat hi is the same for all particles, while others prefer to use a spa-tially and temporally variable hi [8,24]. Both approaches have theirown merits and demerits. As far as problems related to solidmechanics are concerned, one major advantage is the normallystrong cohesion between two adjacent/neighbouring particles.Therefore, even in very large deformation processes, one particlecannot move too far from its neighbouring particle unless thereis a fracture in or fragmentation of the material. Consequently, ifthe initial particle distribution is quasi-uniform, it is reasonableto take a constant hi ¼ h for all particles. We observed that approx-imately 25–30 neighbours per particle yields reasonably good re-sults (in the context of TREESPH, Hernquist and Katz [8]suggested 30–40 neighbours per particle).

h

h

Fig. 1. Smoothing lengths ðhi and �hÞ.

The smoothing length �h for the correction term is chosen suchthat each particle interacts only with its nearest neighbours asshown in Fig. 1. For the cubic kernel given by Eq. (20), �h is takenas h=2.

4.2. SPH with AV vs. SPH with AC

The main difference between the SPH with AV and SPH with theAC lies in the treatment of the artificial viscosity in the momentumand energy equations (see Eqs. (27) and (28) and (36) and (37)). InAC, terms related to the artificial viscosity are split into two parts.The first part is the dissipation term, which is the same as the arti-ficial viscosity term with ða1 ¼ 1;a2 ¼ 1Þ used in SPH computa-tions with AV. It provides dissipation into the system so thatstability of the numerical solution is ensured. The second term isthe correction term, which reduces the ‘loss’ of kinetic energy tothe benefit of the physicality of the solution. Salient properties ofAC vis-à-vis AV are given bellow.

1. AC does not involve any user-specified parameters unlike AVsince ða1;a2Þ is always taken throughout this paper as (1,1).Therefore the problematic (and unknown/uncontrolled) issueof spurious entropy production (loss of kinetic energy) due toalternative choices of these parameters is not relevant to theproposed algorithm.

2. In AV, the excessive loss of kinetic energy may be avoided, orsignificantly reduced, by choosing ‘‘optimum” artificial viscosityparameters. However in order to find the ‘‘optimum” artificialviscosity parameters for a particular problem, one has to per-form several, rigorous numerical experiments. This constitutesa formidable task when the size of the problem is large. Onthe other hand the proposed AC does not require any such numer-ical experiments and yet, as will be demonstrated, yields similaraccuracy.

3. AC requires some extra CPU time to compute the correctionterm. However it is observed that the extra time required tocompute the correction term is negligible compared to the totalCPU time. Moreover, since the AC does not require any numer-ical experiments (see Sections 5.1 and 5.3) for improvement,potentially huge computational effort is saved. ConsequentlyAC is usually more cost-effective than AV.

5. Numerical examples

The present section, is for illustrative purposes, and is focusedon a limited numerical exploration of the proposed method forthree, simple elastic and elastic–perfectly-plastic impact problemsof classical interest in dynamic structural mechanics. In Examples1 and 3 several simulations (a numerical experiment) were per-formed with different values of ða1;a2Þ in order to find the ‘‘opti-mum” value of ða1;a2Þ, required to model successfully the givenproblem (keeping the particle distribution, choice of kernel func-tion, etc. the same). Within the limits of this demonstration, the re-sults obtained with this ‘‘optimum” ða1;a2Þ are considered to bethe best predictions that can be obtained via SPH with artificial vis-cosity used as the stabilizing mechanism. In Example 2, the Taylorbullet problem, has been treated using SPH already by Libersky andPetschek [14] and there the main aim is to compare the uncor-rected with the corrected versions of SPH directly.

Determination of the ‘‘optimum” value of ða1;a2Þ through sucha numerical experiment can require of a huge computational effort.In many SPH computations in the literature, this extra numericaleffort is avoided by performing the computations with an arbi-trarily chosen ða1;a2Þ. However the artificial viscosity is oftenover-estimated, which consequently leads to an increase in


spurious entropy and thus the predictions can become unphysical.A basic numerical demonstration of the effect of arbitrarily chosenartificial viscosity parameters can be made by simulations run withsome commonly used (though arbitrarily ascribed) sets of ða1;a2Þavailable in the literature. Three sets of ða1;a2Þ, viz. (1,1), (1,2)

L L

VV

A

Fig. 2. Collinear collision of two bar with equal and opposite velocity (parametersare given in Table 2).

Table 2Geometric and material properties for Example 1.

Length (L) 20 mmVelocity (V) 50 m/sDensity 7900 kg/m3

Young’s modulus 215 GPaYield strength 1160 MPaPoisson’s ratio 0.3

-0.025 -0.02 -0.015 -0.01 -0.005 0-70

-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

-0.025 -0.02 -0.015 -0.01 -0.005 0-70

-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

a b

c d

Fig. 3. Particle velocity along the length of the left bar at 5.75 ls for different values of a2

is shown by dotted line.

and (2.5,2.5), hereafter referred respectively to as AV1, AV2 andAV3, and are used for this purpose for all examples.

The computed AV results for these examples are then comparedwith those obtained with the proposed AC and its efficacy is dis-cussed by comparing the results. In Examples 1 and 3 the ‘‘opti-mum” ða1;a2Þ for the AV solution is obtained through a limitednumerical experiment, deducing the ‘‘optimum” value. The prob-lem is then also solved using the AV models AV1, AV2 and AV3to show the range of potentially problematic SPH solutions. Asnoted above, in Example 2, as well as using the same modelsAV1-3, the previous SPH solution produced by Libersky and Pet-schek [14] using ða1;a2Þ ¼ ð0:5;0:5Þ is the basis of the comparisonwith AC.

5.1. Example 1: elastic collision of two identical bars having equalmass and opposite velocity

Consider the collinear elastic collision between two identicalhigh-carbon steel bars, moving towards each other axially withequal speeds, V, as shown in Fig. 2. The geometric and materialproperties are given in Table 2. One hundred and one particlesare placed along the length of each bar. The smoothing length his taken as 1:2Dx where Dx ¼ L=100.

From 1D elastic bar wave theory (see [12]), the bars reboundwith equal and opposite velocities as unstressed bodies at a time

-0.025 -0.02 -0.015 -0.01 -0.005 0-70

-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

-0.025 -0.02 -0.015 -0.01 -0.005 0-70

-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

keeping a1 ¼ 0; (a) a2 ¼ 0:5 (b) a2 ¼ 1:0 (c) a2 ¼ 1:5 (d) a2 ¼ 2:0. Theoretical result


tc ¼ 2L=CL ¼ 7:67 ls (where CL is the longitudinal bar-wave speedgiven by CL ¼

ffiffiffiffiffiffiffiffiffiE=q

pÞ after initial contact. The total kinetic energy

of each bar is conserved.First, consider the effect of artificial viscosity per se. The velocity

profiles along the length of the bar on the left near to the unloading

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

a b

c d

e

Fig. 4. Particle velocity along the length of the left bar at 5:75 ls for different values of(0.3,0.3). Theoretical result is shown by dotted line.

wave front at t ¼ 3tc=4 ls ¼ 5:75 ls obtained via SPH with differ-ent values of a2, keeping a1 ¼ 0 are shown in Fig. 3. It can be seenfrom Fig. 3 that a2 alone cannot remove the numerical noise. At-tempts to use non-zero a1 has a dramatic effect on the stabilityof the computation. Fig. 4 shows the velocity profiles at 5:75 ls

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)P

artic

le v

eloc

ity (

m/s

)

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

f

ða1;a2Þ (a) (0.05,0.05); (b) (0.1,0.1); (c) (0.15,0.15); (d) (0.2,0.2); (e) (0.25,0.25); (f)


for different values of a1, keeping a2 ¼ a1. It can be seen from Fig. 4that, for a1 P 0:3, oscillations in the velocity profile vanish and thesolution becomes stable. Therefore it is reasonable, in this numer-ical experiment, to take (0.3,0.3) as the ‘‘optimum” values forða1;a2Þ for the given problem and the prediction withða1 ¼ 0:3;a2 ¼ 0:3Þ are considered as the reference in order todemonstrate the efficacy of the proposed acceleration correctionalgorithm.

Next, in order to examine the effects produced by arbitrarilychosen artificial viscosity parameters, simulations were performedwith models AV1, AV2 and AV3. Velocity profiles at 5:75 ls andcomputed kinetic energies are compared respectively in Figs. 5and 6. It can be seen readily from Figs. 5 and 6 that computationwith the models AV1, AV2 and AV3 yield notably different resultthan those obtained with the ‘‘optimum” ða1;a2Þ (=(0.3,0.3) in thiscase). This discrepancy may be explained by the excessive smooth-ness (see the velocity profile in Fig. 5) provided by the artificial vis-cosity which manifest itself by a significant loss of kinetic energy(see Fig. 6). Note that the kinetic energy loss increases with in-

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m

/s)

AV1 & AV2

AV3(0.3,0.3)

Theoretical

Fig. 5. Comparison of particle velocity along the length of the left bar at 5:75 lsobtained via SPH with AV1, AV2, AV3 and (0.3,0.3). Theoretical result is shown bydotted line.

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Time (micro-sec)

KE

(J)

AV3

AV1 & AV2

(0.3,0.3)

Fig. 6. Computed total kinetic energy obtained via SPH with AV1, AV2, AV3 and(0.3,0.3).

crease in a1. Predictions for AV1 and AV2 are almost the sameand their negligible difference reconfirms that a2 has an insignifi-cant effect on the overall energy dissipation. A similar characteris-tic was observed by [2,11].

The simulation is then performed with the proposed AC as ex-plained in Section 4. The velocity profile at 5:75 ls obtained viaAV with artificial viscosity parameter taken as (0.3,0.3) and ACare shown in Fig. 7. Fig. 8 shows the velocity–time history at themid-section (section-A in Fig. 2) of the left bar. The computed totalkinetic energy is plotted in Fig. 9. It can readily be seen in Figs. 7–9that the performance of AC is in good agreement with AV using the‘‘optimum” value of ða1;a2Þ ¼ ð0:3;0:3Þ, which has been generatedthrough a series of rigorous numerical experiments. Presumablyany remaining difference with the ‘exact’ solution could be re-duced by refining the number of particles, etc. but the main pointof this paper about the effectiveness of AC is clear in Figs. 7–9.

5.2. Example 2: Taylor impact problem

Example 2 is aimed at demonstrating the performance of the ACin an impact computation for an elastic–plastic problem. Fig. 10

-0.025 -0.02 -0.015 -0.01 -0.005 0-60

-50

-40

-30

-20

-10

0

10

Distance (m)

Par

ticle

vel

ocity

(m/s

)TheoreticalAV(0.3,0.3)AC

Fig. 7. Particle velocity along the length of the left bar at 5:75 ls obtained via AV)with (0.3,0.3) and AC.

0 5 10 15 20 25 30 35 40 45 50-60

-40

-20

0

20

40

60

Time (micro-sec)

Par

ticle

vel

ocity

(m

/sec

)

AC

AV (0.3, 0.3)Theoretical

Fig. 8. Velocity–time history of mid-particle of the left bar obtained via AV withða1;a2Þ ¼ ð0:3; 0:3Þ and AC.

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Time (micro-sec)

KE

(J)

AV (0.3, 0.3)

AC

Fig. 9. Computed kinetic energy in time obtained via AV with ða1;a2Þ ¼ ð0:3;0:3Þand corrected SPH, AC.

V L

B

Fig. 10. Iron rod impact impacting a rigid surface normally (Example 2). Parametersare given in Table 3.

-0.01 -0.005 00

0.004

0.008

0.012

0.016

0.02

0.024

AV1 & AV2

AV3

EPIC-2

Fig. 11. Deformation of the iron rod at 50 ls as computed by SPH with AV1, AV2and AV3.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7x 10

4

Time (micro-sec)

Kin

etic

ene

rgy

(J)

AV1 & AV2

AV3AV (0.5,0.5)

AC

Fig. 12. Comparisons of computed kinetic energy obtained via SPH with AV1, AV2,AV3 and AC.


shows a 2D plane-strain model of an iron rod travelling at 200 m/s,striking normally a rigid surface. This problem was considered byLibersky and Petschek [14]. In their computation, the rod was dis-cretized by 21 � 67 = 1407 particles and the smoothing length hwas taken as 0.0076 m. The rigid wall was modelled by ghost par-ticles [27], placed outside the wall. The artificial viscosity parame-ters were taken as (0.5,0.5). The same problem is re-visited here. Inorder to permit a valid comparison, all relevant data (geometric,material and numerical) were taken the same as those given in Lib-ersky and Petschek [14].

Libersky and Petschek [14] showed that SPH predicted lessbulging at the impact end than that predicted by the EPIC-2 code.They also showed that the final shape of the impact end did notchange significantly even with an increased number of particles.In order to investigate whether this discrepancy is due to the arti-ficial viscosity parameters generally, first simulations were per-formed with AV1, AV2 and AV3. It is to be noted that, in thisexample no numerical experiment is performed by Libersky andPetschek in order to find the ‘‘optimum”ða1;a2Þ. The EPIC-2 coderesults in Figs. 11and 13 were digitized from Libersky and Petschek[14], and are considered as the reference solution.

The deformed shapes of the rod at 50 ls are compared inFig. 11. It can be seen that the impact (proximal) end of the rodexperiences less bulging as ða1;a2Þ increases. This may be ascribedto the artificial viscosity which acts as an energy sink and producesspurious entropy into the system. This is also evident in the com-puted kinetic energy shown in Fig. 12.

However running the program with the AC (N.B. starting withða1;a2Þ ¼ ð1;1ÞÞ, the deformed shapes of the rod at 50ls obtainedvia SPH and the EPIC-2 code are compared in Fig. 13. It can be seen

-0.01 -0.005 0 0.005 0.010

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

-0.01 -0.005 0 0.005 0.010

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022a b

Fig. 13. Deformation of the iron rod at 50 ls as computed by (a) SPH with standard artificial viscosity ða1 ¼ 0:5;a2 ¼ 0:5Þ and (b) SPH with acceleration correction algorithm(ða1;a2Þ are taken as (1,1)). Results obtained via EPIC-2 Code [14] are shown by bold line.


from this figure that SPH with ða1 ¼ 0:5;a2 ¼ 0:5Þ shows less bulg-ing than that predicted by the EPIC-2 Code as previously observedby Libersky and Petschek [14]. However the computed deforma-tion using the AC version of SPH is in very close agreement withthe EPIC-2 code result.

G

V

x

y

z

Fig. 14. Cantilever beam with tip mass.

Table 3Parameters used in the Taylor impact test simulations.

Parameter Values

E 2:08� 105 MPal 80� 103 MPary 600 MPaL 0:0254 mB 0:0076 mV 200 m=sC (in Eq. (10)) 0:36 cm=lsS (in Eq. (11)) 1.80C (in Eq. (7)) 1.80

Table 4Material and geometrical properties of the cantilever beam shown in Fig. 14.

Parameters Value

E ðN=mm2Þ 2:069� 105

r0 ðN=mm2Þ 344M0 ðNmÞ 24.8q ðkg=m3Þ 7493L ðmmÞ 304.8b ðmmÞ 6.6h ðmmÞ 6.6G ðkgÞ 0.0023b ¼ qLbd=2G 21.96V0 ðm=sÞ 481.6K0 ðJÞ 137.4

5.3. Example 3: cantilever beam subjected to impulsive at its tip

Finally we consider the application of AC for an elastic–per-fectly-plastic cantilever beam carrying an impulsively loaded tipmass as shown in Fig. 14 (often referred to as the Parkes cantileverproblem [25]). This problem is the exemplar for many structuralimpact problems in terms of the significance of the phenomenonof travelling plastic hinges etc.. It was shown in Symonds andFleming [30] that the transient behaviour of the beam significantlydepends on the energy ratio which is defined as the ratio betweenthe initial kinetic energy and the maximum elastic strain energy ofthe beam. For given geometric and material properties, the final tip

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

-0.2

-0.15

-0.1

-0.05

0

(1,1)

(2.5,2.5)

(1,2)

Fig. 15. Deflected positions of the beam at 15.8 ms obtained via AV1, AV2 and AV3.

0 0.05 0.1-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.05 0.1-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.05 0.1-0.25

-0.2

-0.15

-0.1

-0.05

0

Fig. 16. Maximum plastic deformation (at t = 15.80 ms) obtained with (a) a1 ¼ 0:05; a2 ¼ 0; (b) a1 ¼ 0:075; a2 ¼ 0 and (c) a1 ¼ 0:1; a2 ¼ 0.

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

AV (0.1,0.1)

AC

FEM (Reid and Gui 1987)

Fig. 17. Comparison of the deflected positions of the beam at 15.8 ms obtained viaAV (0.1,0.1), AC and FEM [27].


deflection and the amount of plastic work done during the beamdeformation increase with increase in the initial kinetic energy ofthe beam. The presence of artificial viscosity in an SPH computa-tion can result in the reduction of effective kinetic energy andcan thus affect the transient behaviour significantly.

In this section the Parkes cantilever beam problem is briefly re-visited via SPH in order to study the effect of artificial viscosity andthe efficacy of the acceleration correction algorithm. Results arecompared with those given in Reid and Gui [28]. The geometryand the material properties of the beam as considered in (Table1, Example 2 in Reid and Gui [28]) are given in Table 4. Valuesfor parameters C, S and C are taken same as given in the previousexample (see Table 3) are taken here.

The beam is discretized by 201 � 5 particles. The tip mass ismodelled as an extra mass distributed over 5 � 5 = 25 particlesnear the tip of the cantilever beam. When the tip mass is set intomotion so too are the particles over which the tip mass is distrib-uted. Therefore following the conservation of momentum the ini-tial velocity of 5 � 5 = 25 particles near the tip is taken asV0 ¼ V0G=M ¼ 241:1 m=s where M is the total mass of the particlesover which the tip mass is distributed. A uniform smoothing lengthh ¼ 1:3Dx (corresponds to 30 particles on average within the sup-port of the B-spline kernel function given by Eq. (22)) is taken.

First simulations are performed with AV1, AV2 and AV3. Fig. 15shows the computed deflection at 15.80 ms for different values ofða1;a2Þ. It can be seen that the predicted deflection via AV3 is muchlesser than that of predicted via AV1 and AV2 (which are very sim-ilar and thus indistinguishable). Next a numerical experiment wasperformed with different values of ða1;a2Þ. Fig. 16 shows the de-formed shape of the cantilever beam at 15.80 ms obtained with dif-ferent ða1;a2Þ. It can be seen that for a1 < 0:1, the beamexperiences transverse cracking near its tip. This is not an actualcrack and but may be ascribed to the artificial viscosity which isnot adequate to provide enough dissipation into the system to re-move the ‘‘tensile instability”. With this observation, it is reasonableto assume that (0.1,0.1) is the optimum choice for ða1;a2Þ for theproblem under consideration.

Deflection of the beam obtained via the AV (with a1 ¼ 0:1;a2 ¼ 0:1Þ and AC are compared with the FEM solution [28] inFig. 17. It can readily be seen that although the SPH with the ‘‘opti-mum” choice of artificial viscosity parameter does not experienceany instability (or numerical fragmentation), the deflections aresignificantly smaller than those obtained via FEM. Whereas the

deformed shape obtained via the AC is in very good agreementwith the FEM results. However deformation of the impact end isnot as curled as in the FEM solution. This may be explained bythe effect of plastic shearing under the projectile, which was nottaken into consideration in FEM analysis [28]. This is being ex-plored further as explained in Section 5.3 below.

5.4. Discussion of examples

The above examples demonstrate that SPH computations usingartificial viscosity as a stabilizing mechanism present two choices.


First, the simulation can be performed with some arbitrarily chosenparameters equivalent to, say, AV1, AV2 or AV3. This may affect theaccuracy of the prediction in an uncontrolled fashion. Alternatively,one could choose the ‘‘optimum” values of the artificial viscosityparameters through a series of numerical experiments. Eitherway, there is uncertainty (and potentially heavily increase comput-ing time) remaining regarding the solution. By contrast with stan-dard SPH, the proposed heuristic method, AC, could be used. Thisis free from any user-specified parameters and yet has been shownto yield similar or better accuracy than that obtained through theuse of artificial viscosity parameters in the standard SPH.

Additionally, Example 3 provides a brief introduction to the use ofthe corrected SPH in structural impact problems. The accurate treat-ment of transient and large deformations is clear. However futurepublications using SPH will permit the examination of the basicassumptions used in previous models and the examination of the ef-fect of cracks (vide Petroski [26] and Stronge and Yu [29]), a strengthof SPH. Furthermore the effects of other features of projectile loading(e.g. the definition of ballistic limit) could also be examined there.

6. Closure

An acceleration correction algorithm for SPH has been proposedand illustrated through the re-analysis of three classical elasticand elastic–plastic impact problems. In the original form of SPH,an artificial viscosity term must be added to the momentum equa-tion in order to simulate shock phenomena and produce numericalstability. The form of the artificial viscosity generally used in anSPH computation (of which that described in Section 3 is justone example), requires the definition of a few user-defined param-eters. However, there is neither any mathematical basis for choos-ing such parameters nor does there exist any standard set ofparameters that work for a range of problems.

Most often, the SPH computations are performed with some‘‘standard” (often arbitrarily ascribed) values of artificial viscosityparameters. However, if caution is not exercised when choosingthese parameters, the artificial viscosity term may act as an energysink and affect the accuracy/reality of the solution and, as pointedout by [11], SPH computations with so-called ‘‘standard” artificialviscosity parameters could give misleading results. It is thereforeusually necessary to conduct numerical experiments (as illustratedin Examples 1 and 3) in order to determine the most suitable arti-ficial viscosity parameters for a particular problem. In doing this,one has to bear the potentially huge computational overhead ofsuch experiments, especially if the size of the problem is large.

In the present paper, an attempt has been made to reduce thesedifficulties by using the proposed correction algorithm. The issue ofuser-specified parameters is removed by setting the artificial vis-cosity parameters (in the case illustrated) both equal to 1. Theaccelerations are corrected by subtracting a kernel approximationof its artificial counterpart. The energy equation is also modifiedaccordingly.

The prime purpose of this paper is to introduce this algorithm.The question of whether the method always guarantees to yieldsimilar accuracy to that obtained through the ‘‘optimum” artificialviscosity parameters is yet to be investigated. However the numer-ical results obtained through the present form of the algorithm isvery encouraging. We anticipate that the method would benefitfrom a detailed mathematical analysis, including an estimation oferrors, in order to better control the effect of the artificial viscosity.

The effects of other parameters, such as the choice of the kernelfunction, h=Dx ratio and the randomness in the particle distribu-tion on the performance, as studied by [7], also require to be ex-plored. Extensions to other forms of artificial viscosity (e.g. [8])used in SPH could also be investigated.

Finally, it is relevant to note that the effect of artificial viscosityin the numerical examples provided here is probably also coupledwith the effects of other parameters such as particle distribution,choice of kernel function and smoothing length. The aim of this pa-per has not been to solve any classical impact problems ab initio,rather, at this stage, to demonstrate the efficacy of the proposedalgorithm in controlling the adverse effects of artificial viscosityin SPH. Other structural impact problems (e.g. the elasto-plasticimpact loading of structures, as in Example 3) can be treated sim-ilarly and will be the subject of future papers.

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heuristic acceleration correction algorithm for use in sph computations in impact mechanics

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