Engineering Structures 33 (2011) 255–264

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Coupling of smoothed particle hydrodynamics and finite element method forimpact dynamics simulationZhichun Zhang ∗, Hongfu Qiang, Weiran GaoXi’an Hi-Tech Institute, Xi’an 710025, China

a r t i c l e i n f o

Article history:Received 4 February 2010Received in revised form9 October 2010Accepted 14 October 2010Available online 10 November 2010

Keywords:SPHFEMAttachmentContactBackground particleImpact dynamics

a b s t r a c t

This paper presents an alternativemethod for coupling smoothed particle hydrodynamics (SPH) and finiteelement method (FEM) in a Lagrangian framework. The attachment and contact between SPH particlesand finite elements are calculated. FE nodes are added to the SPH neighbor list for the attachment, andthe continuity of the interface is guaranteed. The contact force on SPH particles and FE nodes is calculatedwith the same approach used in SPH particle to particle contact algorithm, and the identification of thecontact surface and the surface normal is not required. Background particles are assigned in the positionof FE nodes to facilitate particle approximation. The perforation of a cylindrical Arne tool steel projectileimpacting a plate Weldox 460 E steel target is simulated in 3D to demonstrate the performance of theSPH–FEM coupling algorithm. The coupled computational model of viscoplasticity and ductile damageand Gruneisen EOS are used for the target plate. A particle-kill algorithm is used to invalidate the damageparticles. Good agreement between the numerical simulations and the experimental results is obtained,and the ballistic limit velocity obtained from the SPH–FEM coupling algorithm gives a deviation of 2%from the experimental data.

Crown Copyright© 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Finite element method (FEM) has been used extensively forthe analysis of computational solid mechanics, and it can pro-vide robust capabilities for a wide range of problems. For theproblem with extreme distortions, the accuracy of FEM is notalways adequate [1–3]. More recently, smoothed particle hydro-dynamics (SPH) has been developed and applied to solid me-chanics problems [4–6]. The formulation of SPH is not affectedby the arbitrariness of the particle distribution due to the adap-tive nature of the SPH approximation. And it can naturally han-dle problems with extremely large deformation. However, SPH isnot generally as good as FEM for dynamics problems with milddistortions, and worse still, SPH encounters several difficulties:(1) the tensile instability [7,8]; (2) Dirichlet boundary condi-tions [9]; (3) comparatively high computational cost [9]. Therefore,coupling SPH and FEM seems a reasonable approach to take advan-tage of the best properties of both methods, which can provide atool capable of modeling the interaction of bodies subjected to ex-treme deformation. This approach allows for the use of accurateand efficient finite elements in the lower distortion regions, andfor the use of SPH particles in the higher distortion regions. The at-tachment and contact between SPH particles and finite elementsshould be calculated according to the application field.

∗ Corresponding author. Tel.: +86 29 83348248; fax: +86 29 84744092.E-mail address: zzchun323@yahoo.com.cn (Z. Zhang).

0141-0296/$ – see front matter Crown Copyright© 2010 Published by Elsevier Ltd. Aldoi:10.1016/j.engstruct.2010.10.020

An important effort has been dedicated to the coupling of SPHand FEM in the past years. For the attachment between SPH par-ticles and finite elements, Johnson et al. [10,11] developed a rigidcouplingmethodwhere SPH particles were fixed to FE nodes in theinterface zone. The computation of the stresses at an interface par-ticle (node as seen from the FE region)was donewith contributionsfrom the interface particles and from other standard particles, butthere was no contribution from the neighboring FE nodes. Conse-quently, the continuity of the interface between SPH and FEM wasnot guaranteed with this approach [12]. Sonia Fernández-Méndezet al. [12] developed a transition region between SPH and FE do-mains, and shape function of the coupled SPH and FE interpolationswas used in this region. However, the interpolation shape functionis hard to get, and the computational cost is high. For the contactbetween SPH particles and finite elements, Attaway [13] and John-son et al. [10,11] developed a master–slave method to describethe contact between SPH particles and finite elements. In everytime step, they checkedwhether particles penetrate element faces.The calculated contact forces that prevent the interpenetration arealways normal to the corresponding element surface. Sliding be-tween particles and elements in tangential direction is allowed.However, this master–slave method needs to define the contactsurface and the surface normal in every time step, which is com-plex in 3D.

The aim of this paper is to present an alternative SPH–FEMcoupling algorithm to deal with the attachment and contactbetween SPH and FEM, and to simulate the perforation of a

l rights reserved.

256 Z. Zhang et al. / Engineering Structures 33 (2011) 255–264

cylindrical Arne tool steel projectile impacting a plate Weldox 460E steel target in 3D. The center part of the target plate wherethere will be large deformation is modeled with SPH particles,and the other part with finite elements. The SPH–FEM attachmentalgorithm is used in the interface between SPH particles andFE elements, and the continuity of the interface is enhanced.The SPH–FEM contact algorithm is used between the projectileand the target, and no contact surface and normal are required.In both the algorithms, background particles are assigned inthe position of FE nodes to facilitate particle approximation.The coupled computational model of viscoplasticity and ductiledamage is adopted for the target plate,with aGruneisen EOS for thevolumetric response. A particle-kill algorithm is used to removethe damaged particle from the model. Good agreement betweennumerical simulations and experimental observations is obtained.

Section 2 recalls some basic concepts on SPH method. Inparticular, Section 3.1 gives the weak form of linear momentumbalance of SPH, initially proposed in [13], providing the foundationfor the coupling of SPH–FEM. Sections 3.2 and 3.3 presentthe SPH–FEM attachment and contact algorithm, respectively.Section 3.4 describes the time step control to make sure that thecodes of SPH and FEM use the same time integration scheme.Finally some numerical examples are presented to demonstratethe applicability of the method.

2. Basic equations of SPH

The formulation of SPH is often divided into two key steps[14,15]. The first step is the integral representation or the so-calledkernel approximation of field functions. The second one is theparticle approximation. In the first step, the integral representationof a function f (x) in the volume Ω is

⟨f (x)⟩ =

∫Ω

f (x′)W (x − x′, h)dx′. (1)

And in the second step, the function is given by

⟨f (xi)⟩ =

N−j=1

mj

ρjf (xj)Wij (2)

wheremj and ρj are themass and density of particle j, respectively.N is the number of particles within the support domain of particlei. The angle bracket ⟨⟩ marks the kernel approximation operator.The smoothing kernel function W should satisfy a number ofconditions as follows:∫

Ω

W (x − x′, h)dx′= 1 (3)

limh→0

W (x − x′, h) = δ(x − x′) (4)

W (x − x′, h) = 0,x − x′

> kh (5)

where x is the particle position, and h is smoothing length, and k isa constant defining the support domain of the smoothing function.δ(x) is the Dirac delta function.

The SPH formulation for hydrodynamics withmaterial strengthis obtained as

dρi

dt=

N−j=1

mj

v

β

i − vβ

j

∂Wij

∂xβ

i

(6)

dvαi

dt= +

N−j=1

mj

σ

αβ

i

ρ2i

+σ

αβ

j

ρ2j

− Πij

∂Wij

∂xβ

i

(7)

deidt

=12

N−j=1

mj

v

β

j − vβ

i

σαβ

i

ρ2i

+σ

αβ

j

ρ2j

− Πij

∂Wij

∂xβ

i

(8)

where the sum calculation is for all the SPH particles in the supportdomain. σ is stress, and v is velocity. Π is the artificial viscosity,avoiding unphysical oscillations. e is the internal energy.

3. SPH–FEM coupling algorithm

3.1. Momentum balance in SPH approximation

The differential form of the linear momentum balance can beexpressed in a weak form by multiplying by a weighting functionδu [13]:∫

Ω

δu(∇ · σ + ρu + fb)dV = 0. (9)

For the case of the body force fb is zero, integration by parts ofEq. (9) gives∫

Ω

δuρudV −

∫Ω

σ · ∇δudV +

∫∂Ω

tδudS = 0 (10)

where t = σ · n is the traction on the surface with normal n. Toobtain a weighted residual statement, replace f (x) with displace-ment u, and substitute Eq. (2) into Eq. (10), i.e.

Ru =

∫Ω

ρu−

δumj

ρjWij

dV −

∫Ω

σ ·

−δu

mj

ρj∇Wij

dV

+

∫∂Ω

t−

δumj

ρjWij

dS. (11)

The residual at each particle can be minimized by imposing

∂Ru

∂δu= 0. (12)

For particle i, Eq. (12) gives∫Ω

ρuWijmj

ρjdV −

∫Ω

σ · ∇Wijmj

ρjdV

+

∫∂Ω

tWijmj

ρjdS = 0. (13)

Eq. (13) can be approximated using particles as−j∈hi

ρjWijmi

ρi

mj

ρju =

−j∈hi

σj∇Wijmi

ρi

mj

ρj−

−j∈hi

tWijmi

ρi1Sj (14)

where particle j ∈ hi, i.e. those particles that are within thesmoothing length hi of particle i. Eq. (14) can be assembled to givea mass matrix and force vector i.e.

Mu = F ext− F int. (15)

For dynamic problems, Eq. (15) can be integrated forward in timeusing a central difference scheme.

The SPH equations are derived using an approach that parallelsthe derivation of the classical displacement-based finite elementmethod. From this perspective, the SPHmethod can be viewed as aspecial case of the finite element method, where the connectivityof the element is constructed by a search for the nearest neighbors.The SPH method can be easily embedded within existing finiteelement code architecture, if the particles are viewed as elementswhose connectivity must be determined for each time step.

3.2. SPH–FEM attachment algorithm

Fig. 1 shows how SPH particles can be attached to standardfinite elements in the interface. The smaller real circle representsSPH particle; the larger broken circle around particle i represents

Z. Zhang et al. / Engineering Structures 33 (2011) 255–264 257

Fig. 1. SPH particles’ attachment to finite elements.

the support domain of the SPH particle; the smaller brokencircle around FE node represents background particle. Backgroundparticle has SPH particle property, and the particle variables areconsistent with those of the corresponding FE nodes, such asthe particle mass, position, velocity and stress. The SPH–FEMattachment algorithm present in this paper adds FE nodes to SPHneighbor list in the mode of background particles. The particleapproximation of density, momentum and energy for particle iis done with contributions from particles n1, n2, . . . , n5 and FEnodes n6, n7, n8. The support domain of the SPH particle near theinterface does not interact with the problem domain. Therefore,the smoothing function W is not truncated by the boundary,avoiding boundary effects [14] (see in Fig. 2). The solutionprocedure is shown in Fig. 3. At the start of each time step, allthe relative data to SPH approximation are transferred from FEnodes to the corresponding background particles. The backgroundparticles only increase the particle number in the support domain,and they are only passively searched by other SPH particle. At theend of each time step, the relative data are transferred from SPHparticles to the corresponding FE nodes in the interface, updatingthe FE data. For the FE part, the transferring of data from theinterface particles to the corresponding FE nodes is just like loadingboundary condition, and this is feasible because the shape functionof FE has the property of Kronecker delta tensor. For the SPHpart, FE nodes are added to the neighbor list of particles, and theboundary effects are avoided. As a result, the continuity betweenSPH particles and finite elements is guaranteed.

The principle of the attachment between SPH and FEM is theextension of the SPH summation on all parts of the FE discretizationinside the approximation radius near the attachment interface.Different approximation methods are used according to theposition and the discrete mode. For the SPH particle far from thecoupling interface, pure SPH approximation is used. For the SPHparticles near the coupling interface, the FE nodes in the supportdomain are added to SPH summation in the mode of backgroundparticles. For the finite elements, pure FEM approximation is used.And the approximation formulation for the whole model is givenby

⟨f (xi)⟩ =

N−j=1

mj

ρjf (xj)Wij SPH only (16)

Fig. 2. The support domain of the smoothing functionW and problem domain.

Fig. 3. Solution procedure for SPH–FEM attachment algorithm.

⟨f (xi)⟩ =

N−j=1

mj

ρjf (xj)Wij +

Nb−j=1

mbj

ρbjf (xj)Wij

coupled SPH–FEM (17)

⟨f (x)⟩ =

−i

Ni(x)f (xi) FEM only (18)

where Nb is the number of background particle in the supportdomain of SPH particle i; mbj and ρbj are the mass and density ofparticle j, respectively; Ni(x) is the FE shape function.

3.3. SPH–FEM contact algorithm

Fig. 4 shows how SPH particles contact with neighboring finiteelements. The smaller real circle, smaller broken circle and largebroken circle represent SPH particle, background particle andsupport domain of particle i, respectively. The contact between SPHparticles and finite elements occurswhen the penetration becomespositive, i.e. the distance between the particles and nodes smallerthan two times the smoothing length (the supporting radius is twotimes the smoothing length). The contact force vector on particlesand nodes is calculated in the same way as the SPH particle toparticle contact algorithm [16]. Any particle or node that is closeenough to the node or particle being considered for contact will beadded to its contact neighbor list. The solution procedure is shownin Fig. 5.

258 Z. Zhang et al. / Engineering Structures 33 (2011) 255–264

Fig. 4. SPH particles’ contact with finite elements.

The starting point for the contact algorithm is the definition ofcontact potential φ(xi)

φ(xi) =

NCONT−j

mj

ρjK

W (rij)W (1pavg)

n

(19)

where NCONT defines the list of neighbor particles that belongto a different body to particle i. W is SPH kernel function, andW (xA − xB) = 0 if xA and xB belong to the same body. rij is thedistance between particles, and 1pavg is the average value of thesmoothing length. In Fig. 4, for particle i,NCONT = 3, the contactforce is calculated with the contribution from nodes n6, n7, n8. Kand n are user defined parameters, and K presents contact stiffnesspenalty.

And the body force is calculated as the gradient of the potential

b(xi) = ∇φ(xi) =

NCONT−j

mj

ρjKn

W (rij)n−1

W (1pavg)n∇xiW (rij). (20)

Then the contact force is calculated as

Q (xi) =

NCONT−j

mj

ρj

mi

ρiKn

W (rij)n−1

W (1pavg)n∇xiW (rij). (21)

And the direction of the contact force is determined by the SPHapproximation of the gradient of the contact potential.

For SPH particles, the contact force is added in the momentumequation

dvαi

dt= +

N−j=1

mj

σ

αβ

i

ρ2i

+σ

αβ

j

ρ2j

− Πij

∂Wij

∂xβ

i

−Q (xi)mi

. (22)

And for FE nodes, the contact force is added in the dynamicequation as the external force

Mu + Cu + Ku = Q (xi) (23)

where u, u, u are the acceleration, velocity and displacement of FEnode, respectively. M, C, K are the mass matrix, damping matrix,stiffness matrix of the system, respectively.

The contact force is applied to boundary particles and nodesonce they get within two times the smoothing length from eachother. The SPH–FEM contact algorithm facilitates the calculationof direction of contact forces and avoids the problems relatedto non-uniqueness of the surface normal at particles. Themethod is consistent with SPH approximation in general andits implementation in 3D is not complex. Moreover, the contactalgorithm makes it easy for the surface to come together andseparate in a physically correct manner.

3.4. Time step control

In the SPH–FEM coupling algorithm, the SPH equations areintegrated with leap-frog scheme, which can be described in thefollowing way.

Fig. 5. Solution procedure for SPH–FEM contact algorithm.

Z. Zhang et al. / Engineering Structures 33 (2011) 255–264 259

(1) At the end of the first time step

ui

t0 +

1t2

= ui(t0) +

1t2

dui(t0)dt

(24)

ri(t0 + 1t) = ri(t0) + 1tvi

t0 +

1t2

. (25)

(2) At the start of each subsequent time step

ui(t) = ui

t −

1t2

+

1t2

dui(t − 1t)dt

. (26)

(3) At the end of each time step

ui

t +

1t2

= ui

t −

1t2

+ 1t

dui(t)dt

(27)

ri(t + 1t) = ri(t) + 1tvi

t +

1t2

(28)

where ui can be the density, velocity and energy of particle i,and ri is the particle position.

The FEM equations are integrated with central differencescheme, and the nodal acceleration, velocity and displacement arecalculated as

un+1 = −M−1· (Rinn+1 − Rcon+1 + Rex) (29)

un+1 = un +1t2

(un + un+1) (30)

un+1 = un + 1tun +121t2un (31)

where Rin, Rco, Rex are the internal force, cohesive force andexternal force, respectively.

The same time step must be calculated for the equations ofSPH and FEM to make sure that the integration of SPH and FEMis synchronous within a Lagrangian framework. The time stepcalculation is described as

1tSPH–FEM = min1tSPH, 1tFEM (32)

1tSPH = αhC

(33)

1tFEM ≤ 1tcr =LC

(34)

where L is the smallest element length; C is the speed of sound; αis the time scale factor.

4. Numerical examples

This section presents three numerical examples to demonstratethe applicability and performance of the SPH–FEM couplingalgorithm. In all three examples, 8-node continuumelementswereused for FE mesh. The XSPH technique proposed by Monaghan [9]was adopted to make the SPH particle move with a velocity closerto the average velocity of the neighboring particles. The artificialstress proposed by Monaghan [7] was adopted to remove thetensile stability in SPH formulation. The fully variable smoothinglengths algorithm proposed by Qiang and Gao [17] was adopted inSPH formulation to avoid the variable smoothing lengths’ effect.

4.1. Bar pressed at one end

The first test to validate the SPH–FEM attachment algorithmis a cubic bar pressed at one end. The dimension of the bar is0.005×0.005×0.1m3, and a linear elastic material model is usedwith a density of ρ = 7.83 × 103 kg/m3, an elastic modulus of

Fig. 6. Model of a bar pressed at one end.

Fig. 7. Plot for velocity_Z .

E = 207 × 109 Pa and a Poisson’s ratio of υ = 0.3. One half of thebar is modeled with 1250 hexahedron elements, while the otherhalf with 1836 SPH particles (see in Fig. 6). Uniform pressure P =

40×109 Pa is loaded at the end of the FE part in the time 0–10−6 s,and the total calculation time is 14 × 10−6 s. A second simulationwas performed to give a comparison for the SPH–FEM attachmentalgorithm, and the bar ismodeledwith 2500hexahedron elements.

In the first simulationwith the SPH–FEMattachment algorithm,SPH particles 750, 800, 1050, 1100 and FE nodes 909, 910, 1215,1216 are chosen for the test points, while in the second simulationwith FEM only, FE nodes 1521, 1522, 2127, 2128 are chosen (thesamepositionswith SPHparticles 750, 800, 1050, 1100). Figs. 7 and8 give the velocity and stress curves for all the test points, wherethe real lines represent the first simulation, and the broken linesrepresent the second one. The velocity and stress curves of the fourparticles and four nodes are all coincident because the position ofthe particles and the nodes are all symmetric, indicating that thesimulation resultsmeet the symmetric demand. At time t = 10.5×10−6 s and t = 11.0 × 10−6 s, the four particles and four nodesreach the highest velocity vz = −582 m/s and vz = −539 m/s,while the stress reaches the highest value σz = −27.3 × 109 Paand σz = −27.9 × 109 Pa. The comparisons show that the changetendency of the velocity and stress of the SPH particles, on theone hand, is in good agreement with that of the nodes in theSPH–FEM attachment algorithm, and on the other hand, coincideswith the nodes in the second simulation. All of these showthat the calculation of the SPH–FEM attachment algorithm meets

260 Z. Zhang et al. / Engineering Structures 33 (2011) 255–264

Fig. 8. Plot for stress_zz.

the continuity in the interface between SPH particles and FEelements.

4.2. Bar impact

Two bar impacts are simulated to validate the performance ofthe SPH–FEM contact algorithm. Two bars of 0.05× 0.05× 0.2 m3

are initially separate with the distance 0.035m, and they are mov-ing with an equal and opposite velocity of 200 m/s. The materialparameters are: density ρ = 7.83 × 103 kg/m3, elastic modulusE = 207×109 Pa and Poisson’s ratio υ = 0.3. The simulation stopsat the termination time of 200 × 10−6 s. Two simulations wereperformed, one simulation where one bar was modeled with 500SPH particles, and the other with 500 hexahedron elements (seein Fig. 9), andwith the solution of SPH–FEM contact algorithm. Thesecond simulationwas performedwith LS-DYNA, and both the barswere modeled with 500 hexahedron elements.

Figs. 10 and 11 give the velocity and stress curves for particle81 and node 631. The velocity in the Z direction is compared withthat obtained from LS-DYNA. The velocity tendency is in goodagreement, but there is more obviously oscillation in the velocityof finite element, as in SPH solution, artificial viscosity is used,and the serious oscillation is avoided. The velocity obtained fromLS-DYNA lags behind that obtained from the SPH–FEM contactalgorithm 25 × 10−6 s, that is because the contact force is appliedto SPH particles and finite elements once they get within two

Fig. 10. Plot for velocity_Z compared with LS-DYNA.

times the smoothing length (h = 0.01 m) from each other, whilein LS-DYNA, the contact distance is much smaller. The terminationvelocity of particle 81 and node 631 differ by 12.3% and 7.0% fromthe corresponding value of the element and node obtained in LS-DYNA, respectively.

4.3. Cylindrical projectile impacting plate target

Ballistic impact is a complicated dynamic problem, andmaterialnonlinearity, geometric nonlinearity and contact nonlinearityare all involved. The characterization of material involves notonly the stress–strain response at large strains, different strainrates and temperatures, but also the accumulation of damageand the mode of failure. Some research has been dedicated toimpact computation and mechanics of materials and structures.Delhomme et al. [18] simulated a block impacting a reinforcedconcrete slab with a finite element model and a mass–springsystem. Dolce et al. [19] described the conceptual design, theengineering process and the implementation of an impact resistingstructure for volcanic shelter. Morquio and Riera [20] gaveexperimental and theoretical evidence on the joint influence of sizeand strain rate on the mechanical properties of steel.

In this section the perforation of a cylindrical Arne tool steelprojectile impacting a plate Weldox 460 E steel target wassimulated in 3D to demonstrate the performance of the SPH–FEMcoupling algorithm for this type of problem. The dimensions of themodel are shown in Fig. 12. The center of the target ismodeledwith21853 SPH particles, and the particle spacing is 10−3 m. The otherpart of the model is modeled with 23744 hexahedral elements,

Fig. 9. Model of two bars’ impact.

Z. Zhang et al. / Engineering Structures 33 (2011) 255–264 261

Fig. 11. Plot for stress_zz.

and the element length near the coupling interface is 10−3 m. Themesh is somewhat coarsened towards the boundary to reduce thecomputational time. The initial separation distance between theprojectile and the target is 2 × 10−3 m. Fixed end support is usedfor boundary condition of the target. The total calculation time is300 × 10−6 s.

There is large deformation in the target during the perforation.The coupled computational model of viscoplasticity and ductiledamage presented by Børvik et al. [21] was used for the targetmaterial ofWeldox 460 E steel, and theGruneisen EOSwas adoptedfor the volumetric response. For vonMises plasticity, the correcteddeviatoric stress Sαβ

est is calculated as

Sαβest = Sαβ

σ 2Y

3J2

0.5

(35)

where J2 is the second invariant of the deviatoric stress.The model includes linear thermoelasticity, the von Mises

yield criterion, the associated flow rule, nonlinear isotropic strainhardening, strain rate hardening, temperature softening dueto adiabatic heating, isotropic ductile damage and failure. Theequivalent von Mises stress σeq is given as

σeq = (1 − D)[A + Brn][1 + C ln r∗][1 − T ∗m

] (36)

where A, B, C, n,m are material constants; r is the damageaccumulated plastic strain given as r = (1−D)p and r∗

= (1−D)p∗; p∗

= p/p0 is a dimensionless strain rate, where p is the plasticstrain rate, and p0 is a user defined reference strain rate; T ∗

=

Fig. 13. Particle-kill algorithm in SPH.

(T − T0)/(Tm − T0) is the homologous temperature, where T isthe actual temperature, T0 is the room temperature, and Tm is themelting temperature. D is the damage variable, and it takes valuesbetween 0 and 1, where D = 0 for an undamaged material andD = 1 for a fully brokenmaterial. However, the critical value ofD atwhich amacrocrack occurs is less than 1, and the damage evolutionrule is

D =

0 p < pdDC

pf − pdp p ≥ pd

(37)

where DC is the critical damage, pd is the damage threshold and pfis a fracture strain depending on stress triaxiality, strain rate andtemperature. Johnson and Cook [22] proposed an expression forthe fracture strain as

pf = [D1 + D2 exp(D3σ∗)][1 + D4 ln p∗

][1 + D5T ∗] (38)

where D1 − D5 are material constants, σ ∗= σm/σeq is the stress

triaxiality ratio, and σm = (σx + σy + σz)/3 is the mean stress.In Eq. (38), the first bracket gives the material degradation due todamage; the second one describes the yield and strain hardening;the third bracket formulates the effect of strain rate hardening, andthe effect of temperature softening on the equivalent stress is givenin the last bracket.

The deformation of the projectile is comparatively small, andan elastic material model is used for the projectile material of Arnetool steel. Thematerial parameters for the target and the projectileare shown in Tables 1 and 2, respectively.

The particle-kill algorithm [23] was adopted to describethe evolution of damage for SPH particles in the perforation

Fig. 12. Model size and discretization mode of impact problem (unit: m).

262 Z. Zhang et al. / Engineering Structures 33 (2011) 255–264

Table 1Material parameters of the target.

Elastic constants and density Yield stress and strain hardening Strain rate hardening Damage evolutionE (GPa) v ρ (kg/m3) A (MPa) B (MPa) n p0(s−1) C DC pd

200 0.33 7850 490 807 0.73 5 × 10−4 0.012 0.3 0

Adiabatic heating and temperature softening Fracture strain constantsCp (J/kg K) α Tm (K) T0 (K) m D1 D2 D3 D4 D5

452 0.9 1800 293 0.94 0.0705 1.732 −0.54 −0.0123 0

Fig. 14. Plots for the simulation of impact process.

course (see Fig. 13). And the damaged particles will be removedfrom the model when the damage variable reaches the damagecritical value. However, the damaged particles are still alive incompression state, because the material still has loadability in thecompression state even if damage occurs. In Eq. (2), the damageparticle j is removed from the integration of particle i, even if it is

in the support domain of particle i. But the mass and momentumof particle j are not removed, and the conservation of mass andmomentum is guaranteed.

The simulation results are shown in Fig. 14 for the initialprojectile velocity v0 = 210 m/s, and the experimental resultsfor the projectile and the plug are shown in Fig. 15 [21]. The

Z. Zhang et al. / Engineering Structures 33 (2011) 255–264 263

Fig. 15. Projectile and plug after impact.

Table 2Material parameters of the projectile.

E (GPa) v ρ (kg/m3)

200 0.33 7838

shape of the projectile and the plug are close to the experimentalobservations. The contact between the projectile and the targetoccurs at t = 7 × 10−6 s. After that, the target mass in front of theprojectile is accelerated, and the deformation mechanism changesinto a highly localized shear zone surrounding the projectile nose.In the shear zone, damage develops rapidly due to the severeplastic straining of the particles closest to the projectile–targetinterface. The particles start to erode after the critical damagevalue is reached, and the crack propagates towards the rear sideof the target. In the final stage, the failure mode is combined shearand tensile failure, and the plug is fully fractured before completeseparation from the target. The deformation and the damage of theprojectile are comparatively small, so the use of elastic materialmodel for the projectile is acceptable.

Figs. 16 and 17 give the numerical simulations and experimen-tal results [21] for some typical cross-sections of penetrated tar-get plates at increasing initial projectile velocity, respectively. Thetarget deformation consists of a combination of localized bulgingand global dishing. The deformation is mainly global dishing in thestatic loading case, while in the dynamic case, it is mainly localizedbulging. The target plate is not totally perforation when the initialprojectile velocity is less than the ballistic limit, and the bulging atthe rear side of the target plate increases with the initial projectilevelocity. The perforation occurs when the initial projectile velocity

is well above the ballistic limit, and the permanent target deforma-tion decreases with the initial projectile velocity. In the test withthe initial velocity close to the ballistic limit, a larger part of theplate is activated, involving a larger global target deformation. Themaximum deformation of the target is less than half of the targetplate thickness, showing that bending stress dominates the globalresponse during impact.

Table 3 gives the comparisons between the numerical andexperimental results, and the numerical results include threesimulations: (1) coupled SPH–FEM; (2) SPH only; (3) LS-DYNA. Thesimulations (2) and (3) were performed to provide comparisonsfor the performance of the SPH–FEM coupling algorithm. vpr is theresidual velocity of the projectile, and vtr is the residual velocityof the plug. Good agreement is obtained between the numericalsimulations and the experimental observations, and the simulationof coupling the SPH–FEM algorithm gives higher spatial accuracythan that of SPH only and LS-DYNA. In test 9, perforation is notobtained. In order to define the ballistic limit velocity, tests 10and 11 are conducted. The ballistic limit velocity is found tobe 186 m/s, and this gives a deviation from the experimentallyobtained ballistic limit of 2%. Considering the complexity of theproblem, the ballistic limit velocity obtained from the SPH–FEMalgorithm is acceptable.

Fig. 18 gives the comparisons of projectile velocity–time curvesbetween different initial velocities for the coupled SPH–FEMsimulations. From the slope of the curves, one can conclude that theprojectile impacts the plug several times during the perforation.After the first impact between the projectile and the plug, theplug is accelerated to a higher velocity. However, the restrainingby the shear zone prevents the plug from leaving the target,and the plug velocity lowers to some degree. Then the projectilecatches up with the plug, giving it a new impact, and the courseof impact and separation repeats for several times. Until very closeto fracture, the last impact becomes a collision between two freebodies, and from then on, the projectile and the plug keep theirown residual velocities. It can be seen that the perforation timedecreases drastically with the initial projectile velocity, on theother hand, the kinetic energy loss of the projectile increases withthe initial projectile velocity.

Fig. 16. Cross-section of the target after impact (Numerical).

Fig. 17. Cross-section of the target after impact (Experimental).

264 Z. Zhang et al. / Engineering Structures 33 (2011) 255–264

Table 3Comparisons between numerical and experimental results.

Test v0 (m/s) Numerical ExperimentalSPH–FEM SPH LS-DYNA vpr (m/s) vtr (m/s)vpr (m/s) vtr (m/s) vpr (m/s) vtr (m/s) vpr (m/s) vtr (m/s)

1 303.5 200.5 265.2 186.5 235.1 206 265 199.7 242.32 285.4 169.0 229.6 164.4 233.6 – – 181.1 224.73 244.2 139.3 176.6 99.4 131.9 109 142 132.6 187.74 224.7 110.0 160.5 65.4 106.8 94 123 113.7 169.05 210.0 90.6 133.5 26.5 40.9 38 47 – –6 205.0 71.7 117.8 0 0 0 0 – –7 200.4 68.4 105.3 0 0 0 0 71 1048 189.6 41.0 58.8 0 0 0 0 42 629 181.5 0 0 0 0 0 0 0 0

10 187.0 5.03 18.2 – – – – – –11 185.0 0 0 – – – – – –

Fig. 18. Velocity–time curves for different initial velocities.

5. Conclusions

An alternative SPH–FEMcoupling algorithmhas beenpresentedfor the dynamic impact problem. This approach adopts backgroundparticles in the position of FE nodes. The attachment and contactbetween SPH particles and finite elements are considered. Theexamples demonstrated the capability to link SPH particles withfinite elements such that highly distorted flows can be combinedwith structural response in a single computation. The ballisticlimit velocity deviation of 2% between the simulation result andthe experimental observation showed that the SPH–FEM couplingalgorithm could effectively treat the problem of a hard projectileimpacting a soft target. On the other hand, the use of a layer ofFEs along the boundary, coupled with the SPH particles, facilitatedthe enforcement of essential boundary conditions. This algorithmcan be applied to the coupling of finite element and othermeshlessmethods, such as EFG and RKPM.

However, in the SPH–FEM coupling algorithm, contact occursonce the distance between two bodies is smaller than two timesthe smoothing length, which is not in agreement with reality. Andthis error could be attenuated if we minimize particle spacing,because the smoothing length is in proportion to particle spacing.

The plastic deformation of the projectile may become severewith higher impact velocity, and this absorbs a lot of the initialkinetic energy. Unfortunately, the simple material model usedin these simulations for the projectile is not able to describethe behavior accurately. In a coming work, suitable constitutivemodels will be chosen for the projectile material with higherimpact velocity.

Acknowledgements

The authors would like to thank the New Century ExcellentTalents in University (NCET) in China for the financial support ofthe research. And the authors are very grateful to the anonymousreferees for the valuable suggestions on the improvement of themanuscript.

References

[1] Benson DJ. Computational methods in Lagrangian and Eulerian hydrocodes.Comput Methods Appl Mech Engrg 1992;99:235–394.

[2] Belytschko T, Liu WK, Moran B. Nonlinear finite elements for continua andstructures. New York: John Wiley and Sons; 2000.

[3] Zienkiewicz OC, Taylor RL. The finite element method volume 2: solidmechanics. Butterworth-Heinemann; 2000.

[4] Libersky LD, Petschek AG. Smooth particle hydrodynamics with strength ofmaterials. Trease H, Fritts J, Crowley W, editors. Proceedings of the next freeLagrange conference, vol. 395. New York: Springer-Verlag; 1991. p. 248–57.

[5] Benz W, Asphaug E. Simulations of brittle solids using smoothed particlehydrodynamics. Comput Phys Comm 1995;87:253–65.

[6] Libersky LD, Randles PW, Carney TC, et al. Recent improvements in SPHmodeling of hypervelocity impact. Int J Impact Eng 1997;20:525–32.

[7] Monaghan JJ. SPH without a tensile instability. J Comput Phys 2000;159:290–311.

[8] Gray JP, Monaghan JJ, Swift RP. SPH elastic dynamics. Comput Methods ApplMech Engrg 2001;190:6641–62.

[9] Monaghan JJ. Smoothed particle hydrodynamics. Ann Rev Astron Astrophys1992;30:543–74.

[10] Johnson GR. Linking of Lagrangian particle methods to standard finite elementmethods for high velocity impact computations. Nucl Eng Des 1994;150:265–74.

[11] Johnson GR, Stryk RA, Beissel SR. SPH for high velocity impact computations.Comput Methods Appl Mech Engrg 1996;139:347–73.

[12] Fernández-Méndez Sonia, Bonet Javier, Huerta Antonio. Continuous blendingof SPH with finite elements. Comput Struct 2005;83:1448–58.

[13] Attaway SW. Coupling of smoothed particle hydrodynamics with the finiteelement method. Nucl Eng Des 1994;150:199–205.

[14] Liu GR, Liu MB. Smoothed particle hydrodynamics: a meshfree particlemethod. World Scientific; 2004.

[15] Monaghan JJ. Smoothed particle hydrodynamics. Rep Progr Phys 2005;68:1703–59.

[16] Vignjevic R, De Vuyst T, Campbell JC. A frictionless contact algorithm formeshless methods. In: ICCES. vol. 3. 2007. p. 107–12.

[17] Qiang Hongfu, GaoWeiran. SPHmethodwith fully variable smoothing lengthsand implementation. Chinese J Comput Phys 2008;25:569–75.

[18] Delhomme F, Mommessin M, Mougin JP, et al. Simulation of a block impactinga reinforced concrete slab with a finite element model and a mass-springsystem. Eng Struct 2007;29:2844–52.

[19] Dolce M, Cardone D, Marnetto R. Structural design and analysis of an impactresisting structure for volcanic shelters. Eng Struct 2006;28:1634–49.

[20] Morquio Atilio, Riera Jorge D. Size and strain rate effects in steel structures.Eng Struct 2004;26:669–79.

[21] Børvik T, LangsethM, Hopperstad OS, et al. Ballistic penetration of steel plates.Int J Impact Eng 1999;22:855–86.

[22] Johnson GR, Cook WH. Fracture characteristics of three metals subjected tovarious strains, strain rates, temperatures, and pressures. Eng Fract Mech1985;21:31–48.

[23] Fu Xuejin. Research and engineering application of tensile instability insmoothed particle hydrodynamics. Ph.D. thesis. China: Xi’anHi-Tech Institute;2008.