Comparison study of MPM and SPH in modeling hypervelocity impact problems

Abstract

Due to the high nonlinearities and extreme large deformation, the hypervelocity impact simulation is a challenging task for numerical methods. Meshfree particle methods, such as the smoothed particle hydrodynamics (SPH) and material point method (MPM), are promising for the simulation of hypervelocity impact problems. In this paper, the material point method is applied to the simulation of hypervelocity impact problems, and a three-dimensional MPM computer code, MPM3D, is developed. The Johnson–Cook material model and Mie–Grüneisen equation of state are implemented. Furthermore, the basic formulations of MPM are compared with SPH, and their performances are compared numerically by using MPM3D and LS-DYNA SPH module.This study shows that the material point method is an efficient and promising method for simulating the hypervelocity impact problems. MPM possesses many prominent features. The formulation of MPM is simple and similar to the traditional finite element method (FEM). Spatial derivatives are calculated based on a regular computational grid in MPM, so that the time consuming neighbor searching, which is compulsory in most meshfree methods, is not required. The approximation of field variables and their spatial derivatives is efficiently evaluated using the information of only 8 grid nodes in three-dimensional problem, and the shape functions exactly satisfy the constant and linear consistency. The boundary conditions can be applied in MPM as easily as in FEM, and contact algorithm can be efficiently implemented whose cost is linear in the number of bodies. Because the same regular computational grid can be used in all time steps, the time step size keeps constant in MPM simulation.

Full text

Comparison study of MPM and SPH in modeling
hypervelocity impact problems
q
S. Ma, X. Zhang
*
, X.M. Qiu
School of Aerospace, Tsinghua University, Beijing 100084, China
article info
Article history:
Received 28 June 2007
Received in revised form 28 May 2008
Accepted 2 July 2008
Available online 10 July 2008
Keywords:
Meshfree
Smoothed particle hydrodynamics
Material point method
Hypervelocity impact
abstract
Due to the high nonlinearities and extreme large deformation, the hypervelocity impact simulation is
a challenging task for numerical methods. Meshfree particle methods, such as the smoothed particle
hydrodynamics (SPH) and material point method (MPM), are promising for the simulation of hyperve-
locity impact problems. In this paper, the material point method is applied to the simulation of hyper-
velocity impact problems, and a three-dimensional MPM computer code, MPM3D, is developed. The
Johnson–Cook material model and Mie–Gru
¨neisen equation of state are implemented. Furthermore, the
basic formulations of MPM are compared with SPH, and their performances are compared numerically
by using MPM3D and LS-DYNA SPH module.
This study shows that the material point method is an efficient and promising method for simulating the
hypervelocity impact problems. MPM possesses many prominent features. The formulation of MPM is
simple and similar to the traditional finite element method (FEM). Spatial derivatives are calculated
based on a regular computational grid in MPM, so that the time consuming neighbor searching, which
is compulsory in most meshfree methods, is not required. The approximation of field variables and
their spatial derivatives is efficiently evaluated using the information of only 8 grid nodes in three-
dimensional problem, and the shape functions exactly satisfy the constant and linear consistency. The
boundary conditions can be applied in MPM as easily as in FEM, and contact algorithm can be efficiently
implemented whose cost is linear in the number of bodies. Because the same regular computational grid
can be used in all time steps, the time step size keeps constant in MPM simulation.
2008 Elsevier Ltd. All rights reserved.
1. Introduction
The numerical simulations of hypervelocity impact problems
are of great interest in many engineering applications, such as
shield design for spacecraft protection. Because extremely large
deformations and material fracture are involved in the process of
hypervelocity impact, they are challenging tasks for computer
simulation.
There are two fundamental frames for describing the motion of
material: the Eulerian description and the Lagrangian description.
In the Lagrangian description, the mesh is embedded in and
deforms with the material domain. It presents no convective effects
so that the boundary conditions at free surfaces, moving bound-
aries and material interfaces are automatically imposed. It is
ideal for history-dependent material. However, if the deformation
is very large, as in hypervelocity impact and metal forming,
mesh distortion and element entanglement become the signifi-
cant limitations of the Lagrangian description. Contrary to the
Lagrangian description, material flows through a grid fixed in space
in the Eulerian description. It completely avoids element distor-
tions, but there are still difficulties in tracking the deformation
history of material and dealing with the material interfaces. The
dissipation problem associated with mass flux between adjacent
elements also arises in the Eulerian method. There are also some
mixed methods to strengthen the advantages of these two
descriptions and to avoid their disadvantages, such as the arbitrary
Lagrangian–Eulerian (ALE) [1]. Unfortunately, the convection term
still exists in the ALE formulation, which may cause numerical
difficulties. In addition, it still remains a challenging task to design
an efficient and effective mesh moving algorithm to maintain mesh
regularity for three-dimensional complicated material domain.
Recently developed meshfree methods [2–5] use a set of
discrete points to construct trial functions, so that the difficulties
associated with mesh distortion can be avoided or alleviated.
Nevertheless, most of the meshfree methods suffer from higher
computational cost and the accuracy of some meshfree methods is
still dependent on the node regularities to some extent. Therefore,
only a few of them perform well in hypervelocity impact problems,
q
Supported by National Natural Science Foundation of China (10472052) and the
Science Foundation of Computational Physics, IAPCM, China.
*Corresponding author. Tel./fax: þ86 10 62782078.
E-mail address: xzhang@tsinghua.edu.cn (X. Zhang).
Contents lists available at ScienceDirect
International Journal of Impact Engineering
journal homepage: www.elsevier.com/locate/ijimpeng
0734-743X/$ – see front matter 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijimpeng.2008.07.001
International Journal of Impact Engineering 36 (2009) 272–282

such as the smoothed particle hydrodynamics (SPH) [6–8], material
point method (MPM) [9,10], hybrid particle-element method
[11–13], conversion of distorted elements into particles method
[14–16].
Meshfree methods and particle methods can be classified into
the same group of methods. Particle methods in nature are well
suited for modeling extremely large deformation problems with
failure and fracture, so they are superior to the Lagrangian grid-
based methods in the simulation of hypervelocity impact problems.
Smoothed particle hydrodynamics (SPH) [6–8] is one of the
earliest meshfree Lagrangian particle methods. SPH was first
proposed by Lucy [17] and Gingold and Monaghan [18] in 1977 to
solve astrophysical problems in three-dimensional open space and
has been extensively studied and extended to dynamics response
with material strength as well as dynamic fluid flows with large
deformations. SPH and its improved versions have been success-
fully applied to the hypervelocity impact simulations and become
one of the most popular and powerful meshfree methods in this
area. Because of its good performance, several commercial soft-
wares, such as AUTODYN [19], PAM-SHOCK [20] and LS-DYNA [21],
have incorporated SPH into their solvers.
Material point method (MPM) [9,10], which is also a particle
method, is an extension of the FLIP particle-in-cell [22] method to
solid mechanics. In MPM, material domain is discretized by a group
of points, termed as particles or material points. These Lagrangian
particles carry all material information and track the deformation
history. The momentum equations are solved on a predefined
background grid, which can be fixed in space or arbitrarily defined,
and provide a Eulerian description of the material domain. With
characteristic of both Lagrangian description and Eulerian
description, MPM is a good choice for modeling hypervelocity
impact problems which involve extreme large deformation.
Although MPM is an extension of FLIP particle-in-cell method, it
is closely related to the Lagrangian finite element method (FEM).
MPM can be viewed as a special Lagrangian FEM in which the
particles rather than the Gauss points serve as the quadrature
points. For small deformation problems, the MPM is equivalent to
the FEM using Gauss points at the same locations as those of
material points in each cell. Bear this in mind, Zhang et al. [23]
proposed an explicit material point finite element method, which
provides an uniform formulation for the MPM and FEM, and
extended it to the hypervelocity impact simulation.
In this paper, the material point method is applied to the
simulation of hypervelocity impact simulation, and a three-
dimensional material point method code, MPM3D, is developed
with FORTRAN 90 to simulate various hypervelocity impact
problems. The Johnson–Cook material model and Mie–Gru
¨neisen
equation of state are implemented. The basic formulation and
features of MPM are compared with SPH from following aspects:
neighbor searching, approximation functions, consistency of shape
functions, tensile instability, time integration, boundary conditions
and contact algorithm. The performance of MPM is compared
numerically with SPH by using the MPM3D and LS-DYNA SPH
module.
The remaining part of the paper is organized as follows. The
governing equations are briefly summarized in Section 2. The basic
formulations of MPM and SPH in modeling hypervelocity problems
are presented in Sections 3and 4, respectively. The basic formu-
lation and features of MPM and SPH are compared in detail in
Section 5, and their performances are numerically investigated in
Section 6. Section 7draws some conclusions.
2. Governing equations
In the updated Lagrangian description, the material is governed
by the momentum equations [21]
s
ij;j
þ
r
f
i
¼
r
€
u
i
cx
i
˛V(1)
subject to the traction boundary conditions
s
ij
n
j
¼t
i
ðtÞcx
i
˛
G
t
(2)
and the displacement boundary conditions
u
i
ðX
a
;tÞ¼d
i
ðtÞcx
i
˛
G
d
(3)
where Vis the current material domain,
G
t
and
G
d
are, respectively,
the boundary portions of Vprescribed with traction and displace-
ment,
s
ij
is the Cauchy stress,
r
is the current density, f
i
is the body
force density, €
u
i
is the acceleration, the comma denotes covariant
differentiation and n
j
is the unit outward normal to the boundary.
The conservation equation of mass is stated as
_
r
þ
rn
i;i
¼0 (4)
The energy equation is given by
_
E¼J
s
ij
_
3
ij
¼Js
ij
_
3
ij
Jp_
3
kk
(5)
where Jis the determinant of the deformation gradient matrix
F
ij
¼vx
i
=vX
j
,Eis the energy per unit initial volume. _
3
ij
is the strain
rate, s
ij
and prepresent the deviatoric stresses and pressure, i.e.
s
ij
¼
s
ij
þp
d
ij
.
The stresses at time t
kþ1
can be obtained by
s
kþ1
ij
¼
s
k
ij
þ_
s
k
ij
D
t(6)
where _
s
ij
is the material time derivative of the stress, and given by
_
s
ij
¼
s
V
ij
þ
s
jl
u
il
þ
s
il
u
jl
(7)
in which
u
ij
¼ð1=2Þð_
x
i;j
_
x
j;i
Þis the spin tensor, and
s
V
ij
is the
Jaumann stress rate and determined from the strain rate
_
3
ij
¼ð1=2Þð_
x
i;j
þ_
x
j;i
Þby a constitution model. Johnson–Cook
material model [24] is implemented in MPM3D code, in which the
equivalent tensile flow stress is expressed as
s
y
¼ðAþB
3
n
Þ1þCln_
3
*
1T
*m
(8)
where
3
is equivalent plastic strain, _
3
*
¼_
3
=_
3
0
is the dimensionless
plastic strain rate for _
3
0
¼1:0s
1
and T
*
¼(TT
room
)/(T
melt
T-
room
)˛[0, 1] is the homologous temperature. A,B,n,Cand mare the
material constants. Temperature is computed by assuming that the
90% of the energy dissipated due to plastic work goes to raise the
temperature.
The pressure is updated by an equation of state. The Mie–
Gru
¨neisen equation of state [25] is implemented in MPM3D, in
which the pressure is updated by
p¼p
H
1
gm
2þ
gr
E(9)
p
H
¼a
0
m
þb
0
m
2
þc
0
m
3
m
>0
a
0
mm
<0(10)
The subscript Hrefers to the Hugoniot curve,
m
¼
r
=
r
0
1is
used to represent the compression and
g
is the Gru
¨neisen param-
eter. The constants a
0
,b
0
, and c
0
in Eq. (10) are related to Cand Sin
shock wave velocity and particle velocity relation U
s
¼CþSU
p
through a Taylor’s expansion of the Hugoniot curve.
a
0
¼
r
0
C
2
;b
0
¼a
0
½1þ2ðS1Þ;c
0
¼a
0
½2ðS1Þþ3ðS1Þ
2
.
3. Material point method
The material domain is discretized by a finite number of parti-
cles in MPM, as illustrated in Fig. 1. Each particle carries all of
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282 273

material variables, such as the mass, position, velocity, strain and
stress. Instead of evaluating the momentum equations on particles
as in the SPH, the momentum equations are evaluated on the
predefined background grid nodes. The solution process of MPM is
divided into two phases. In the first phase of solution, particles are
rigidly attached to the background grid and they deform with the
grid. After obtaining the kinematic solution on the grid nodes, they
are mapped back to the particles to update their positions and
velocities. The deformed grid is discarded in the subsequent time
step and a new regular grid is used to avoid mesh distortion.
Generally speaking, the same fixed regular grid can be used in all
time steps.
The fundamental formulation of MPM can be obtained from the
weak form of momentum equations and traction boundary condi-
tion as:
dP
¼Z
V
r
€
u
i
d
u
i
dVþZ
V
rs
s
ij
d
u
i;j
dVZ
V
r
f
i
d
u
i
dVZ
G
t
t
i
d
u
i
d
G
¼0
(11)
where
s
s
ij
¼
s
ij
=
r
is the specific stress and
d
u
i
is the virtual
displacement. The displacement boundary conditions have been
assumed to be satisfied a priori.
Since the movement of particles represents the deformation of
physical domain and mass is carried by particles, the mass
conservation is automatically satisfied in MPM.
Because the particles are rigidly attached to the computational
grid, the particle displacement u
pi
and its derivatives u
pi,j
can be
obtained by mapping their grid point values u
gi
and u
gi,j
to the
particle using the standard finite element shape functions of the
grid, namely
u
pi
¼X
8
g¼1
N
pg
u
gi
(12)
u
pi;j
¼X
8
g¼1
N
pg;j
u
gi
(13)
where N
pg
¼N
g
(x
pi
). In this paper, 8-point hexahedron grid is used
as the background grid so that the shape function is given by
N
g
¼1
81þ
xx
g
1þ
hh
g
1þ
zz
g
;g¼1;2;/8 (14)
if the particle (
x
,
h
,
z
) is inside the hexahedron, where
x
g
,
h
g
and
z
g
take on their nodal values (1, 1, 1) at the grid node g. If the
particle is outside the hexahedron, N
g
¼0.
The material mass is lumped at particles, hence the density
r
at
point x
i
can be approximated as
r
ðx
i
Þ¼ X
n
p
p¼1
M
p
d
x
i
x
pi
(15)
where x
pi
denotes the coordinate of pth particle in ith
direction.
Substituting Eqs. (12), (13) and (15) into the weak form (11), and
invoking the arbitrariness of
d
u
hi
yields
_
p
hi
¼f
int
hi
þf
ext
hi
;h¼1;2;/;n
g
(16)
where n
g
is the number of grid nodes,
p
hi
¼X
n
g
g¼1
m
hg
_
u
gi
(17)
is the momentum of hth grid point in the idirection,
m
hg
¼P
n
p
p¼1
M
p
N
ph
N
pg
is the mass matrix. If the lumped mass
matrix m
h
¼P
n
p
p¼1
M
p
N
ph
is used, the momentum p
hi
can be
simplified as
p
hi
¼m
h
_
u
hi
(18)
Moreover,
f
int
hi
¼
X
n
p
p¼1
V
p
N
ph;j
s
pij
(19)
and
f
ext
hi
¼X
n
p
p¼1
M
p
N
ph
f
pi
þX
n
p
p¼1
M
p
N
ph
t
s
pi
h
1
(20)
are the internal force and external force, t
s
pi
¼t
pi
=
r
is the specific
traction, his the thickness of the boundary layer,
s
pij
¼
s
ij
ðx
p
Þand
f
pi
¼f
i
ðx
p
Þ.
The Lagrangian formulation is used in MPM so that the
acceleration in Eq. (16) does not contain the convection term
which can cause significant numerical error in purely Eulerian
approaches.
Using the central difference scheme, the momentum equations
on gride nodes can be integrated as
p
kþ1
hi
¼p
k
hi
þf
int
hi
þf
ext
hi

D
t(21)
where the superscripts ‘‘k’’ and ‘‘kþ1’’ denote the values at time t
k
and t
kþ1
, respectively. The particle velocities and positions are
updated by
x
kþ1
pi
¼x
k
pi
þX
n
g
h¼1
p
kþ1
hi
m
k
h
N
k
ph
D
t(22)
n
kþ1
pi
¼
n
k
pi
þX
n
g
h¼1
f
k
hi
m
k
h
N
k
ph
D
t(23)
where f
k
hi
¼f
int
hi
þf
ext
hi
.
In order to update stress, the strain rate _
3
pij
is required to be
calculated at the particles using updated velocity as
_
3
pij
¼1
2X
8
h¼1
v
kþ1
hi
N
k
ph;j
þ
n
kþ1
hj
N
k
ph;i
(24)
Fig. 1. Solid line denotes the boundary of material domain, solid dot denotes material
point and dash lines are background computational mesh.
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282274

where velocities at grid nodes are mapped back from the updated
particle velocities by
n
kþ1
hi
¼P
n
p
p¼1
M
p
n
kþ1
pi
N
k
ph
=m
k
h
.
In MPM, there is no interaction between material particles
separated by grid cells. Fracture can be simulated roughly without
special treatment [23]. In order to obtain more precise simulation,
fracture model can be added into the code. One of the failure
models in LS-DYNA [21] was implemented in the MPM3D in which
particles will fail if tensile pressure or plastic strain exceed the limit,
i.e. p<p
min
or
3
p
>
3
p
max
. Once failure has occurred, pressure may
never be negative and the components of deviatoric stress are set
zero for all the time so that the failed particle can only carry
compression loads.
4. Smoothed particle hydrodynamics
Similar to MPM the material domain Vin SPH is also represented
by a finite number of particles that carry individual mass m
J
and
occupy individual space
D
V
J
¼m
J
/
r
J
, in which
r
J
is the density of Jth
particle. The function u
i
(x) is approximated by the kernel approx-
imation (integral function representation) as
u
i
ðxÞxu
h
i
ðxÞ¼Z
V
u
i
ðxÞWðxx;hÞdV
x
(25)
where Wis the kernel function, which should be normalized and
compactly supported, his the smoothing length defining the
influence domain of the kernel function W.Wshould become Delta
function when happroaches zero.
The kernel approximation for the divergence u
i,i
(x) is given by
u
i;i
ðxÞxu
h
i;i
ðxÞ¼Z
V
u
i;i
ðxÞWðxx;hÞdV
x
(26)
where the subscript ‘i’, denotes differentiation with respect to x
i
,
namely, u
i;i
¼P
3
i¼1
vu
i
=vx
i
. Integrating by parts and using the
divergence theorem, Eq. (26) can be reduced to
u
h
i;i
ðxÞ¼Z
V
u
i
ðxÞW
;i
ðxx;hÞdV
x
¼Z
V
u
i
ðxÞW
;i
ðxx;hÞdV
x
(27)
for those points whose support domain is inside the problem
domain. In Eq. (27),W
;i
ðxx;hÞ¼W
;i
ðxx;hÞ.
After discretizing the material domain Vby a set of particles, Eqs.
(25) and (27) can be evaluated by
u
h
i
ðxÞ¼X
N
J¼1
Wxx
J
;hm
J
r
J
u
iJ
(28)
u
h
i;i
ðxÞ¼X
N
J¼1
W
;i
xx
J
;hm
J
r
J
u
iJ
(29)
where the subscript Jdenotes the quantities associated with the Jth
particle, u
iJ
¼u
i
ðx
J
Þis the displacement of the Jth particle in x
i
direction, and Nis the number of particles within the support
domain of the point x.
According to Eqs. (28) and (29), the function u
i
(x) and its
divergence u
i,i
(x) can be approximated at Ith particle as
u
h
i
ðx
I
Þ¼X
N
J¼1
W
IJ
m
J
r
J
u
iJ
(30)
u
h
i;i
ðx
I
Þ¼X
N
J¼1
W
IJ;i
m
J
r
J
u
iJ
(31)
where
W
IJ
¼Wx
I
x
J
(32)
W
IJ;i
¼W
;i
xx
J
j
x¼x
I
(33)
The derivatives of the field functions can also be approximated
as [6]:
u
h
i;i
ðx
I
Þ¼1
r
I
2
4X
N
J¼1
m
J
u
iJ
u
iI
W
IJ;i
3
5(34)
u
h
i;i
ðx
I
Þ¼
r
I
2
4X
N
J¼1
m
J
u
iJ
r
2
J
þu
iI
r
2
I
!W
IJ;i
3
5(35)
In the above two equations, the field functions appear in the
term of paired particles.
In SPH method, the momentum Eq. (1), continuity Eq. (4) and
energy Eq. (5) are evaluated on particles. Applying SPH approxi-
mation to these equations will result in the SPH formulations [6–8]:
€
u
iI
¼X
N
J¼1
m
J
s
Iij
r
2
I
þ
s
Jij
r
2
J
!W
IJ;i
(36)
_
r
I
¼X
N
J¼1
m
J

n
iI

n
iJ
W
IJ;i
(37)
_
e
I
¼1
2X
N
J¼1
m
J
p
I
r
2
I
þp
J
r
2
J
!
n
iI

n
iJ
W
IJ;i
þ1
r
I
s
Iij
_
3
Iij
(38)
for body force free case, where
s
Iij
¼
s
ij
ðx
I
Þ;s
Iij
¼s
Iij
ðx
I
Þ;
3
Iij
¼
3
ij
ðx
I
Þ;and e¼E=
r
Jis the internal energy per unit mass.
Libersky [25,26] extended SPH to treat the dynamics response of
solids in 1990. Because of the Lagrangian and particlenature in SPH,
it is easily to handle large deformations and is well suited in the
simulation of hypervelocity impact problems [27,28].
After SPH was extended to solid dynamics, tensile instability
[29,30] was exposed. It is more frequently encountered in solid
because there are usually no tensile stress in astrophysical or
general fluid applications. If the materials are under tensile stress in
SPH calculation, particles are tend to clumped together and voids
may be formed. Sometimes, this instability could cause numerical
fracture which is unreal. Several remedies have been proposed to
avoid the tensile instability.
Nowadays, SPH and its improved versions are among the most
popular and powerful methods in hypervelocity impact
simulations.
5. Comparison between MPM and SPH
SPH and MPM are both meshfree particle methods, and are both
well suited for the simulation of hypervelocity impact problems.
However, they possess different advantages and disadvantages. In
this section, MPM and SPH are compared in detail.
5.1. Neighbor searching
All numerical methods for solving partial differential equations
(PDEs) require a nodal connectivity to evaluate the spatial deriva-
tives of the field variables. The nodal connectivity is constructed
based on finite elements in FEM, and constructed by searching
neighbor nodes in meshfree methods. Although the term ‘meshless
methods’ has been widely used to name the methods in which the
approximation functions are constructed based on a collection of
discrete points instead of on a predefined mesh/grid, they are not
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282 275

truly meshless. In these methods, nodal connectivity is still
essential for calculating the spatial derivatives of the approxima-
tion functions. However, the nodal connectivity in these methods
might be changed at each time step, rather than fixed as in the grid-
based method like the finite element method. Therefore, the term
‘meshfree methods’ is more appropriate to name these methods.
Similar to the most meshfree methods, the approximation of
field variables and their spatial derivatives in SPH is calculated
based on a current local set of arbitrarily distributed particles, see
Eqs. (28) and (29). Therefore, it is required in SPH to search the
nearest neighbor particles of a particle to define nodal connectivity.
Neighbor particles for a given particle may change with time in
hypervelocity impact problems due to the large deformation, so
that the neighbor searching has to be carried out in every time step.
Several efficient search schemes, such as tree-like method [31–33]
and box algorithm [25,34,35], are available, usually the complexity
of searching algorithms can reach order of O(Nlog N). Neighbor
searching in SPH is still time consuming, especially for larger scale
problems.
On the contrary, the nodal connectivity in MPM is defined based
on the computational grid. The same regular computational grid is
usually used in all time steps in MPM, so that the nodal connectivity
does not change with time. It can be seen from Eq. (13) that the
derivatives of the displacement of the pth particle are calculated
from their nodal values at the grid nodes of the hexahedron in
which the particle is located. It is trivial to determine the index of
hexahedron and its grid nodes. The index of the hexahedron in
which the pth particle is located can be obtained efficiently from
the following FORTRAN statements
NumCellx ¼int((SpanX(2)-SpanX(1))/DCell þ0.5)
NumCelly ¼int((SpanY(2)-SpanY(1))/DCell þ0.5)
NumCellxy ¼NumCellx * NumCelly
ix ¼ceiling((xp-SpanX(1))/DCell)
iy ¼ceiling((yp-SpanY(1))/DCell)
iz ¼ceiling((zp-SpanZ(1))/DCell)
InWhichCell ¼(iz 1) * NumCellxy
þ(iy 1) * NumCellx þix
where xp,yp and zp are the coordinates of the pth particle, DCell
is the size of the hexahedron, SpanX,SpanY and SpanZ are the
minimum and maximum x,yand zcoordinates of the
computational grid, InWhichCell is the index of the hexahedron.
After obtaining the index of the hexahedron, the grid nodes used
to calculate the derivatives of the field variables can be retrieved
from a array without any extra costs. Therefore, the computational
cost required in SPH to define the nodal connectivity is much
higher than that required in MPM.
5.2. Approximation functions
One of the most significant advantages of SPH is that it does not
require a pre-defined mesh or grid for the purpose of both
approximation construction and integration calculation. The
continuous integral representations of a function and its deriva-
tives, Eqs. (25) and (27), are converted to the discretized summa-
tions over all particles within the support domain of the point x,as
shown in Eqs. (28) and (29) or (30) and (31). To ensure the accuracy
and the numerical stability, the support domain should be suffi-
ciently large, so that sufficient particles (sampling points for inte-
gration) are included in the summation.
Compared with SPH, the traditional finite element approxima-
tion is used in MPM to evaluate the approximation of field variables
and their derivatives. As shown in Eqs. (12) and (13), the approxi-
mation functions are constructed on grid nodes and particle values
are calculated by using the information on the grid nodes of the
hexahedron in which the particle is located. The summation in Eqs.
(12) and (13) is taken over 8 grid nodes, while the summation in
Eqs. (30) and (31) has to be taken over Nneighbors of a particle. In
three-dimensional hypervelocity impact analysis, Nis usually
greater than 8. As a consequence, the computational cost required
to evaluate the approximation of field functions and their deriva-
tives in SPH is higher than that in MPM.
Bardenhagen and Kober [36] proposed a generalization of MPM
named generalized interpolation material point (GIMP) method, in
which the approximation function could have continuous deriva-
tive (in contrast to C
0
continuous finite element shape function
used in MPM). This method increases the computational cost of
MPM, but benefits much in precision and stability as discussed in
the following sections. Fig. 2 illustrates the typical approximation
functions of SPH, MPM and GIMP.
Because the same regular computational grid can be used in all
time steps, the mesh distortion and element entanglement asso-
ciated with the traditional finite element method are avoided in
MPM.
It worth noting that density approximation
r
ðx
i
Þ¼X
N
J¼1
m
J
W
J
ðx
i
Þ(39)
is usually preferred in SPH simulation [27]. In this way, the density
field is smoothed by the kernel function. The particle in both SPH and
MPM carries mass. The difference is that the kernel function W
J
(x
i
)in
Eq. (39) is replaced by
d
function in MPM as shown in Eq. (15).
5.3. Consistency of shape functions
For a numerical method to converge, the shape function
employed must satisfy a required degree of consistency. Unfortu-
nately, due to the unbalanced particles contributing to the
summation in Eq. (28), the constant consistency condition
X
N
J¼1
Wxx
J
;hm
J
r
J
¼1 (40)
and the linear consistency condition
0
0.5
1
SPH MPM GIMP
Fig. 2. Typical approximation functions of SPH, MPM and GIMP in one dimension. White circles indicate the particles.
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282276

X
N
J¼1
xx
J
!Wxx
J
;hm
J
r
J
¼0 (41)
are not satisfied in SPH for the following two cases:
– Particles at or near the boundary of the problem domain so that
the support domain intersects with the boundary, even for
regular particle distribution.
– Irregularly distributed particles, even for the interior particles
whose support domains are not truncated.
Although there are different ways [27,37] to restore the
consistency condition, they require much more additional
computational cost. Furthermore, they may lead to some problems
in simulating hydrodynamic problems [7], such as causing negative
density and negative energy that can result in a breakdown of the
entire computation.
With the use of density approximation (39), the approximation
function is reformed to Shepard function. The constant consistency
is satisfied and simulation results can be improved, but the linear
consistency condition remains unsatisfied under the two cases
mentioned above.
The constant and linear consistency conditions are satisfied
exactly in MPM because the traditional finite element shape func-
tions are used. As for GIMP, the shaped function has similar char-
acteristics as kernel function. Usually evenly spaced grid node is
used, so the constant and linear consistency are satisfied in most of
the computational domain except a thin layer near the boundaryas
discussed above.
5.4. Instability
SPH converts the integration over the support domain into
a summation only over a finite number of particles. Similar to the
node integration for the element-free Galerkin method (EFG) [38],
insufficient sampling points for integration may result in numerical
instability. When particles are under tensile stress state, the motion
of the particles become unstable, which may lead to particle
clumping or complete blowup in the computation [29]. Several
means have been proposed to improve the tensile instability, such
as the special smoothing functions [39], the corrective smoothed
particle method (CSPH) [40], the artificial force [41] and stress
points [42,43].
Different from SPH, the grid nodes in MPM serve as field nodes
to construct the approximation functions of the field variables,
whereas the material points serve as sampling points for integra-
tion. Usually, the number of material points is much greater than
that of the grid nodes, so that the numerical instability arisen from
insufficient sampling points for integration is avoided in MPM.
In some MPM simulations, numerical noises, especially in stress
or velocity results, can be observed when the material points move
across the cell boundaries during deformation. The noise was
identified as cell crossing noise. This noise may be due to the abrupt
change of node mass when particle cross the cell boundary or C
0
continuous feature of the finite element shape function. To
suppress the noise, Bardenhagen and Kober [36] proposed
a general mathematical framework named the generalized inter-
polation material point method (GIMP), which is a good alternative
for those problems in which the cell crossing noise bothers much.
5.5. Time integration
Explicit integration is used in both SPH and MPM, so the critical
time step which assure the stability can be obtained from the
Courant–Friedrichs–Lewy (CFL) condition as
D
t
cr
¼h
c(42)
where cis the wave speed of material, his a characteristic length of
element. The critical time step is determined by the smallest
smoothing length [42] in SPH, and determined by the background
mesh size [44] in MPM. Since the same background mesh is used in
all time steps, the constant time step can be used in the whole
process of computation in MPM. However, the smoothing length h
in SPH varies with particle distances which decrease with the
material compression. Therefore, the critical time step may
decrease significantly in the process of computation in SPH, espe-
cially in extremely large deformation problems. Consequently, SPH
requires much more time steps than MPM for the simulation of the
same hypervelocity impact problem.
5.6. Boundary conditions
It should be noted that Eq. (27) is not valid for points near or on
the boundary because at which the residual boundary term
Z
S
u
i
ðxÞWðxx;hÞn
i
dS
x
(43)
in the integrating by parts does not vanish. Therefore, the
enforcement of boundary conditions is not trivial as that in the
traditional finite element method. The early fluid dynamic uses of
SPH either did not require boundary conditions or only required
simple conditions. Meanwhile, the approximations in SPH do not
have the property of strict interpolants so that they are not equal to
Pressure
Pressure
XX
ab
Fig. 3. Pressure profile of analytical solution (solid line) and MPM simulation (dots) at time t¼0.143. (a) 200 cells, (b) 1000 cells.
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282 277

the particle value of the dependent variable. This complicates the
enforcement of the essential boundary conditions.
Recently some improvements have been proposed to treat the
boundary conditions. Campbell [45] addressed the SPH boundary
condition problem for equations involving gradients, and treated
the boundary conditions by including the residual boundary term,
Eq. (43), in the integration by parts when evaluating the original
kernel integral involving gradients. An alternative way to treat the
boundary conditions is to use virtual particles [46] or ghost parti-
cles [25,27,47].
The MPM formulation is derived from the weak form (11), hence
the boundary conditions at the free surface are satisfied automat-
ically without any special treatments. The implementation of
traction boundary conditions requires choosing a suitable set of
material points to represent the boundary layer, see Eq. (20).
Because the standard finite element shape functions are used in
MPM, the essential boundary conditions can be easily applied on
the background grid nodes in the same way as that in the
traditional finite element method. However, as for GIMP method,
which has smoother shaped function and larger influence region
for each grid node, enforcement of the essential boundary condi-
tions also became a problem to be dealt with. Analogous to SPH,
ghost grid nodes may be used to ensure exact satisfaction of
essential boundary conditions.
5.7. Contact algorithm
SPH can be used in impact problems without any special
treatments to contact boundary condition. Contact was handled
automatically by treating particles from both contact body equally
when finding neighbor particles. Since SPH does not require the
velocity field to be single-valued, a degree of penetration and
mixing may occur at the contact surface. To solve this problem,
Campbell et al. [48] developed a particle-to-particle contact algo-
rithm by using penalty method.
Unlike SPH, the velocity field is single-valued in MPM, so that
the interpenetration of material is precluded without any iteration
and no-slip contact condition between impinging bodies is auto-
matically satisfied at no additional cost. To release the inherent no-
slip contact constraint in MPM while avoiding interpenetration,
contact/sliding/separation schemes were proposed [49–51]. The
idea is that if the bodies are moving toward one another, the
material points are moved in the usual center-of-mass velocity field
Table 1
Material properties
r
(kg/m
3
)E(GPa)
n
A(MPa) B(MPa) nC
8930 117 0.35 157 425 1.0 0.0
Table 2
Numerical results obtained by SPH and MPM in Taylor bar impact simulation
L(mm) D(mm) W(mm)
D
Test 16.2 13.5 10.1 –
SPH1 15.4 15.6 9.9 0.075
SPH2 15.5 14.7 10.0 0.047
MPM 16.3 13.0 9.6 0.031
FEM 16.3 13.2 10.1 0.009
Table 3
Variation of time step size in Taylor bar impact simulation
D
t
max
(
m
s)
D
t
min
(
m
s) Steps
SPH1 0.120 0.118 672
SPH2 0.140 0.138 576
MPM 0.133 0.133 604
FEM 0.0398 0.0123 5486
X
X
X
Density
Pressure
Velocity
Energy
X
ab
d
c
Fig. 4. Profiles of density, velocity, pressure and energy obtained by analytical solution (solid line) and GIMP simulation using 1000 cells (dots).
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282278

which automatically enforces the no-penetration condition. If the
bodies are moving away from one another, they move in their own
velocity field which allows the separation to occur. Therefore, if the
bodies are determined to be in contact, the equation of motion (16)
is solved, where the internal force f
hi
int
results from all material
points even if they are from different bodies. If the bodies are
releasing from one another, the equation of motion is solved for
a separate body, namely
_
p
I
hi
¼f
I;int
hi
þf
I;ext
hi
;h¼1;2;/;n
g
(44)
where the internal force f
hi
I, int
results only from body I. The cost of
these contact algorithms is linear in the number of bodies because it
can be applied to all bodies with one sweep over the computational
grid without iteration, so they are able to account for the interactions
between a large number of bodies, such as grains [50,52].
6. Numerical examples
To compare the accuracy and efficiencyof MPM and SPH, several
numerical examples are presented. In the following examples, the
SPH simulations are carried out by using LS-DYNA version 970
while the MPM simulations are carried out by using our MPM3D
code.
6.1. Shock tube problem
Shock tube problem is often used to test the capability of a code
on simulating compressible fluid. Sod’s model problem [53] is
regarded as a standard test. This problem consists of a shock tube
where a diaphragm separates two regions which have different
densities and pressures. The fluid in two regions are initially at rest.
The initial values are:
r
1
¼1.0, p
1
¼1.0,
r
2
¼0.125, and p
2
¼0.1. At
time t>0 the diaphragm is broken. Then the shock and the contact
interface travel at different speed. The results usually plotted at
t¼0.143 when shock travels a distance of about 0.25.
SPH has been used to simulate shock tube problem in its early
age [54]. Monaghan [55] and Inutsuka [56] improve the SPH shock
simulation by introducing Riemann solver. Sigalotti et al. [57]
capture the shock by using adaptive kernel estimation and get
relatively good result. Usually SPH gets blurred or smooth shock
profiles, but Monaghan [58] argued that real shock fronts is only
a few molecular mean free path. No current method gives the width
of a shock front accurately. Therefore it is more important to get
pre- and post-shock values of physical variables as far as evaluating
SPH is concerned.
The same argument holds for MPM too. York et al. [59] simu-
lated shock tube problem by using MPM. Oscillation, which might
be caused by cell crossing noise, was observed in their results.
MPM3D can be used to simulate compressible fluid by simply
ignoring deviatoric part of stress and strain and using perfect gas
EOS to update pressure. For thisone dimension problem, MPM3D is
used by setting the solution variables constant in yand zdirections.
Two cases are analyzed, in which background meshes of 200 cells
and 1000 cells are used, respectively. In both cases, four particles
are initially placed in each cell for left high density region and two
particles per cell for right low density region, so that 600 particles
and 3000 particles are used, respectively. All data are plotted atgrid
Fig. 5. Final configuration of the bar. (a) SPH1, (b) SPH2, (c) MPM.
Fig. 6. Final configuration of the bar (top view). (a) SPH1, (b) SPH2, (c) MPM.
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282 279

nodes. The pressure results of MPM with different meshs are
illustrated in Fig. 3. Cell crossing noise could become an obstacle
when trying to get better results by refining the mesh. Pressure
profile result would deteriorate when finer mesh is used because
there are more cell crossing. This problem can be well solved by
using GIMP method as observed in Fig. 3. Interpolation function of
GIMP is taken from Pan’s article [52]. The profiles of density, velocity,
pressure and specific internal energy are shown in Fig. 4 for back-
ground mesh of 1000 cells. They agree well with analytical results.
Artificial bulk viscosity is important in shock simulation. We
adopted the same type viscosity with LS-DYNA [21]:
q¼c
0
r
l
2
e
_
3
kk

2
c
1
r
l
e
c_
3
kk
_
3
kk
<0
0_
3
kk
0(45)
where l
e
is characteristic length, _
3
kk
is bulk strain rate, cis sound
speed, and c
0
and c
1
are dimensionless constants. The default value
of c
0
and c
1
in LS-DYNA is 1.5 and 0.06. However, they are raised to
6.0 and 0.6, respectively, in this simulation to suppress the oscil-
lation. The peak in energy profile at contact surface as shown in
Fig. 4d might be caused by the artificial bulk viscosity.
Boundary condition can be applied to boundary grid nodes at
two ends of shock tube directly in MPM. As for GIMP, we treat two
layers of grid node as boundary nodes to ensure the exact satis-
faction of boundary condition.
6.2. Taylor bar problem
Taylor bar problem, which is a cylinder metal bar normally
impacted against a rigid wall, is often used to validate the consti-
tutive models in codes. There are ample experimental data which
can be referred to. In this example, SPH and MPM are used to
simulate the impact of a copper cylinder to a rigid wall with impact
velocity of 190 m/s to compare their accuracy and efficiency. In SPH
simulations, constant associated with smoothing length was set 1.2
and 1.4, respectively. The value of 1.2 is the default value used in LS-
DYNA, and larger value will increase the computational time but
may improve the result with more neighbors for each particle. In
the following tables and figures, these two cases were referred to as
SPH1 and SPH2. This problem is also simulated by using an explicit
finite element code with one point Gaussian quadrature developed
by us for comparison.
The initial length of the cylinder is L
0
¼25.4 mm and the initial
diameter is D
0
¼7.6 mm. The particles are evenly located with
initial interval of 0.38 mm. A total of 21172 particles are used in the
simulation. The Johnson–Cook model is used and material prop-
erties are taken from Ref. [24] as list in Table 1.
The simulation time is 80
m
s, when the kinetic energy reaches
zero. Table 2 compares the numerical results obtained by SPH and
MPM with the experimental results, which shows that MPM
possesses better accuracy than SPH in this example and SPH with
larger smoothing length results in better accuracy. The average
error
D
introduced by Johnson [24] is defined as:
D
¼1
3j
D
Lj
Lþj
D
Dj
Dþj
D
Wj
W(46)
Table 3 compares the maximum time step
D
t
max
, the minimum
time step
D
t
min
, time steps used in SPH and MPM simulation. In this
example, time step factor
a
is set 0.8 for all the case.
Figs. 5 and 6 compare the final configurations of the bar
obtained by using the two methods. It can be observed from Fig. 6
that the SPH algorithm suffers from numerical fracture due to
tensile instability. Enlarging the smoothing length can alleviate the
numerical fracture, but particle clumps may still exist. Furthermore,
enlarging the smoothing length enlarges the time step size but
increases the one step computational cost significantly. The total
cost is also increased as a consequence.
6.3. Debris cloud problem
Debris cloud could result from the typical hypervelocity impact
of a penetrator to a thin plate. Beissel et al. [16] have simulated
Table 4
Material properties for the aluminum
r
(kg/m
3
)E(GPa)
ns
y
(MPa) E
t
(MPa)
3
p
max pmin (MPa)
Sphere 2790 72.4 0.33 276 200 2.5 5000
Target 2700 68.9 0.33 276 200 2.0 4000
Fig. 7. Debris cloud obtained by SPH (top left), MPM (top right) and experiment [16] (bottom) at 8
m
s after impact.
S. Ma et al. / International Journal of Impact Engineering 36 (2009) 272–282280

hypervelocity impact of aluminum spheres on aluminum thin plate
using their coupled method in which distorted finite element was
converted to meshfree particle. An Al–Al impact is simulated here
using SPH and MPM. The sphere is of the 16 mm diameter. The
target plate has the thickness of 0.8 mm and the diameter of
60 mm. The size data have slight deviation from original data but
that deviation makes almost no difference to the results. The
impact velocity is 6150 m/s. Particle interval is set 0.2 mm and
a total of 550,848 particles are used. The linear hardening material
model is used to assure the material parameters are the same in the
two methods. The parameters of Gru
¨neisen equation of state are
taken from Ref. [60]:C¼5300 m/s, S¼1.34, and
g
0
¼2.0. All
material parameters used are listed in Table 4.
Fig. 7 shows the computed results of debris cloud at 8
m
s after
impact and compares with the experimental results. The debris
cloud produced by SPH simulation consists of larger fragments than
by MPM. As discussed in Section 5, larger fragments in SPH results
may be caused by tensile instability which could induce particles
clumping. Furthermore, the configuration of the debris cloud
produced by SPH simulation doesn’t agree well with the experi-
mental results. According to published literature, SPH could
perform better in simulating hypervelocity impact problem with
special treatments to avoid the instability and the inconsistency.
Unfortunately, fewof them have yet been implemented in LS-DYNA
version 970.
In this simulation, GIMP makes no significant difference in
debris cloud configuration than MPM. So MPM is used, which is
more efficient than GIMP.
Table 5 shows that critical time step size in SPH simulation is
shortened very much due to the severe compression of material.
With almost the same initial time step size, SPH has to take much
more time steps than MPM to finish the simulation.
7. Conclusions
A three-dimensional MPM computer code MPM3D is developed
for the simulation of hypervelocity impact problems in this paper,
in which the Johnson–Cook material model and Mie–Gru
¨neisen
equation of state are implemented. The basic formulations and
features of SPH and MPM are compared in detail, and their
performances are investigated numerically by using MPM3D code
and LS-DYNA SPH module. MPM possesses many prominent
features. The formulation of MPM is simple and similar to the
traditional finite element method (FEM). The time consuming
neighbor searching, which is compulsory in most meshfree
methods like SPH and EFG, is not required in MPM. The shape
functions exactly satisfy the constant and linear consistency. MPM
avoid tensile instability that is annoying in SPH. The boundary
conditions can be applied in MPM as easily as in FEM, and contact
algorithm can be efficiently implemented whose cost is linear in
the number of bodies. Because the same regular computational grid
can be used in all time steps, the time step keeps constant in MPM
simulation. The material point method is an efficient and promising
method for simulating the hypervelocity impact problems.
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Table 5
Variation of time step size in debris cloud simulation
D
t
max
(
m
s)
D
t
min
(
m
s) Steps
SPH 0.0282 0.0074 1059
MPM 0.0290 0.0290 276
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