<s>
The	O
material	B-Algorithm
point	I-Algorithm
method	I-Algorithm
(	O
MPM	O
)	O
is	O
a	O
numerical	O
technique	O
used	O
to	O
simulate	O
the	O
behavior	O
of	O
solids	O
,	O
liquids	O
,	O
gases	O
,	O
and	O
any	O
other	O
continuum	O
material	O
.	O
</s>
<s>
Unlike	O
other	O
mesh-based	O
methods	O
like	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
or	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
,	O
the	O
MPM	O
is	O
not	O
a	O
mesh	O
based	O
method	O
and	O
is	O
instead	O
categorized	O
as	O
a	O
meshless/meshfree	O
or	O
continuum-based	O
particle	O
method	O
,	O
examples	O
of	O
which	O
are	O
smoothed	B-Application
particle	I-Application
hydrodynamics	I-Application
and	O
peridynamics	O
.	O
</s>
<s>
which	O
makes	O
it	O
a	O
promising	O
and	O
powerful	O
tool	O
in	O
computational	B-Algorithm
mechanics	I-Algorithm
.	O
</s>
<s>
The	O
MPM	O
was	O
originally	O
proposed	O
,	O
as	O
an	O
extension	O
of	O
a	O
similar	O
method	O
known	O
as	O
FLIP	B-Algorithm
(	O
a	O
further	O
extension	O
of	O
a	O
method	O
called	O
PIC	B-Algorithm
)	O
to	O
computational	O
solid	O
dynamics	O
,	O
in	O
the	O
early	O
1990	O
by	O
Professors	O
Deborah	O
L	O
.	O
Sulsky	O
,	O
Zhen	O
Chen	O
and	O
Howard	O
L	O
.	O
Schreyer	O
at	O
University	O
of	O
New	O
Mexico	O
.	O
</s>
<s>
are	O
then	O
updated	O
with	O
these	O
rates	O
depending	O
on	O
integration	B-Algorithm
scheme	I-Algorithm
of	O
choice	O
and	O
a	O
suitable	O
constitutive	O
model	O
.	O
</s>
<s>
The	O
PIC	B-Algorithm
was	O
originally	O
conceived	O
to	O
solve	O
problems	O
in	O
fluid	O
dynamics	O
,	O
and	O
developed	O
by	O
Harlow	O
at	O
Los	O
Alamos	O
National	O
Laboratory	O
in	O
1957	O
.	O
</s>
<s>
One	O
of	O
the	O
first	O
PIC	B-Algorithm
codes	O
was	O
the	O
Fluid-Implicit	O
Particle	O
(	O
FLIP	B-Algorithm
)	O
program	O
,	O
which	O
was	O
created	O
by	O
Brackbill	O
in	O
1986	O
and	O
has	O
been	O
constantly	O
in	O
development	O
ever	O
since	O
.	O
</s>
<s>
Until	O
the	O
1990s	O
,	O
the	O
PIC	B-Algorithm
method	I-Algorithm
was	O
used	O
principally	O
in	O
fluid	O
dynamics	O
.	O
</s>
<s>
Motivated	O
by	O
the	O
need	O
for	O
better	O
simulating	O
penetration	O
problems	O
in	O
solid	O
dynamics	O
,	O
Sulsky	O
,	O
Chen	O
and	O
Schreyer	O
started	O
in	O
1993	O
to	O
reformulate	O
the	O
PIC	B-Algorithm
and	O
develop	O
the	O
MPM	O
,	O
with	O
funding	O
from	O
Sandia	O
National	O
Laboratories	O
.	O
</s>
<s>
The	O
uses	O
of	O
the	O
PIC	B-Algorithm
or	O
MPM	O
method	O
can	O
be	O
divided	O
into	O
two	O
broad	O
categories	O
:	O
firstly	O
,	O
there	O
are	O
many	O
applications	O
involving	O
fluid	O
dynamics	O
,	O
plasma	O
physics	O
,	O
magnetohydrodynamics	O
,	O
and	O
multiphase	O
applications	O
.	O
</s>
<s>
The	O
PIC	B-Algorithm
method	I-Algorithm
has	O
been	O
used	O
to	O
simulate	O
a	O
wide	O
range	O
of	O
fluid-solid	O
interactions	O
,	O
including	O
sea	O
ice	O
dynamics	O
,	O
penetration	O
of	O
biological	O
soft	O
tissues	O
,	O
fragmentation	O
of	O
gas-filled	O
canisters	O
,	O
dispersion	O
of	O
atmospheric	O
pollutants	O
,	O
multiscale	O
simulations	O
coupling	O
molecular	O
dynamics	O
with	O
MPM	O
,	O
and	O
fluid-membrane	O
interactions	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
PIC-based	O
FLIP	B-Algorithm
code	O
has	O
been	O
applied	O
in	O
magnetohydrodynamics	O
and	O
plasma	O
processing	O
tools	O
,	O
and	O
simulations	O
in	O
astrophysics	O
and	O
free-surface	O
flow	O
.	O
</s>
<s>
One	O
subset	O
of	O
numerical	O
methods	O
are	O
Meshfree	B-Algorithm
methods	I-Algorithm
,	O
which	O
are	O
defined	O
as	O
methods	O
for	O
which	O
"	O
a	O
predefined	O
mesh	O
is	O
not	O
necessary	O
,	O
at	O
least	O
in	O
field	O
variable	O
interpolation	O
"	O
.	O
</s>
<s>
Ideally	O
,	O
a	O
meshfree	B-Algorithm
method	I-Algorithm
does	O
not	O
make	O
use	O
of	O
a	O
mesh	O
"	O
throughout	O
the	O
process	O
of	O
solving	O
the	O
problem	O
governed	O
by	O
partial	O
differential	O
equations	O
,	O
on	O
a	O
given	O
arbitrary	O
domain	O
,	O
subject	O
to	O
all	O
kinds	O
of	O
boundary	O
conditions	O
,	O
"	O
although	O
existing	O
methods	O
are	O
not	O
ideal	O
and	O
fail	O
in	O
at	O
least	O
one	O
of	O
these	O
respects	O
.	O
</s>
<s>
Meshless	B-Algorithm
methods	I-Algorithm
,	O
which	O
are	O
also	O
sometimes	O
called	O
particle	O
methods	O
,	O
share	O
a	O
"	O
common	O
feature	O
that	O
the	O
history	O
of	O
state	O
variables	O
is	O
traced	O
at	O
points	O
(	O
particles	O
)	O
which	O
are	O
not	O
connected	O
with	O
any	O
element	O
mesh	O
,	O
the	O
distortion	O
of	O
which	O
is	O
a	O
source	O
of	O
numerical	O
difficulties.	O
"	O
</s>
<s>
As	O
can	O
be	O
seen	O
by	O
these	O
varying	O
interpretations	O
,	O
some	O
scientists	O
consider	O
MPM	O
to	O
be	O
a	O
meshless	B-Algorithm
method	I-Algorithm
,	O
while	O
others	O
do	O
not	O
.	O
</s>
<s>
PIC	B-Algorithm
methods	I-Algorithm
may	O
be	O
based	O
on	O
either	O
the	O
strong	O
form	O
collocation	O
or	O
a	O
weak	O
form	O
discretisation	O
of	O
the	O
underlying	O
partial	O
differential	O
equation	O
(	O
PDE	O
)	O
.	O
</s>
<s>
Those	O
based	O
on	O
the	O
strong	O
form	O
are	O
properly	O
referred	O
to	O
as	O
finite-volume	O
PIC	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Those	O
based	O
on	O
the	O
weak	O
form	O
discretisation	O
of	O
PDEs	O
may	O
be	O
called	O
either	O
PIC	B-Algorithm
or	O
MPM	O
.	O
</s>
<s>
Several	O
versions	O
of	O
MPM	O
include	O
Generalized	O
Interpolation	O
Material	B-Algorithm
Point	I-Algorithm
Method	I-Algorithm
;	O
Convected	O
Particle	O
Domain	O
Interpolation	O
Method	O
;	O
Convected	O
Particle	O
Least	O
Squares	O
Interpolation	O
Method	O
.	O
</s>
<s>
One	O
stability	O
problem	O
in	O
MPM	O
that	O
does	O
not	O
occur	O
in	O
FEM	O
is	O
the	O
cell-crossing	O
errors	O
and	O
null-space	O
errors	O
because	O
the	O
number	O
of	O
integration	B-Algorithm
points	I-Algorithm
(	O
material	O
points	O
)	O
does	O
not	O
remain	O
constant	O
in	O
a	O
cell	O
.	O
</s>
