<s>
In	O
the	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
meshfree	B-Algorithm
methods	I-Algorithm
are	O
those	O
that	O
do	O
not	O
require	O
connection	O
between	O
nodes	O
of	O
the	O
simulation	O
domain	O
,	O
i.e.	O
</s>
<s>
a	O
mesh	B-Algorithm
,	O
but	O
are	O
rather	O
based	O
on	O
interaction	O
of	O
each	O
node	O
with	O
all	O
its	O
neighbors	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
original	O
extensive	O
properties	O
such	O
as	O
mass	O
or	O
kinetic	O
energy	O
are	O
no	O
longer	O
assigned	O
to	O
mesh	B-Algorithm
elements	O
but	O
rather	O
to	O
the	O
single	O
nodes	O
.	O
</s>
<s>
Meshfree	B-Algorithm
methods	I-Algorithm
enable	O
the	O
simulation	O
of	O
some	O
otherwise	O
difficult	O
types	O
of	O
problems	O
,	O
at	O
the	O
cost	O
of	O
extra	O
computing	O
time	O
and	O
programming	O
effort	O
.	O
</s>
<s>
The	O
absence	O
of	O
a	O
mesh	B-Algorithm
allows	O
Lagrangian	O
simulations	O
,	O
in	O
which	O
the	O
nodes	O
can	O
move	O
according	O
to	O
the	O
velocity	O
field	O
.	O
</s>
<s>
Numerical	B-Algorithm
methods	I-Algorithm
such	O
as	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
,	O
finite-volume	B-Algorithm
method	I-Algorithm
,	O
and	O
finite	B-Application
element	I-Application
method	I-Application
were	O
originally	O
defined	O
on	O
meshes	O
of	O
data	O
points	O
.	O
</s>
<s>
In	O
such	O
a	O
mesh	B-Algorithm
,	O
each	O
point	O
has	O
a	O
fixed	O
number	O
of	O
predefined	O
neighbors	O
,	O
and	O
this	O
connectivity	O
between	O
neighbors	O
can	O
be	O
used	O
to	O
define	O
mathematical	O
operators	O
like	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
But	O
in	O
simulations	O
where	O
the	O
material	O
being	O
simulated	O
can	O
move	O
around	O
(	O
as	O
in	O
computational	O
fluid	O
dynamics	O
)	O
or	O
where	O
large	O
deformations	O
of	O
the	O
material	O
can	O
occur	O
(	O
as	O
in	O
simulations	O
of	O
plastic	O
materials	O
)	O
,	O
the	O
connectivity	O
of	O
the	O
mesh	B-Algorithm
can	O
be	O
difficult	O
to	O
maintain	O
without	O
introducing	O
error	O
into	O
the	O
simulation	O
.	O
</s>
<s>
If	O
the	O
mesh	B-Algorithm
becomes	O
tangled	O
or	O
degenerate	O
during	O
simulation	O
,	O
the	O
operators	O
defined	O
on	O
it	O
may	O
no	O
longer	O
give	O
correct	O
values	O
.	O
</s>
<s>
The	O
mesh	B-Algorithm
may	O
be	O
recreated	O
during	O
simulation	O
(	O
a	O
process	O
called	O
remeshing	O
)	O
,	O
but	O
this	O
can	O
also	O
introduce	O
error	O
,	O
since	O
all	O
the	O
existing	O
data	O
points	O
must	O
be	O
mapped	O
onto	O
a	O
new	O
and	O
different	O
set	O
of	O
data	O
points	O
.	O
</s>
<s>
Meshfree	B-Algorithm
methods	I-Algorithm
are	O
intended	O
to	O
remedy	O
these	O
problems	O
.	O
</s>
<s>
Meshfree	B-Algorithm
methods	I-Algorithm
are	O
also	O
useful	O
for	O
:	O
</s>
<s>
Then	O
we	O
can	O
use	O
these	O
definitions	O
of	O
and	O
its	O
spatial	O
and	O
temporal	O
derivatives	O
to	O
write	O
the	O
equation	O
being	O
simulated	O
in	O
finite	B-Algorithm
difference	I-Algorithm
form	O
,	O
then	O
simulate	O
the	O
equation	O
with	O
one	O
of	O
many	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
In	O
this	O
simple	O
example	O
,	O
the	O
steps	O
(	O
here	O
the	O
spatial	O
step	O
and	O
timestep	O
)	O
are	O
constant	O
along	O
all	O
the	O
mesh	B-Algorithm
,	O
and	O
the	O
left	O
and	O
right	O
mesh	B-Algorithm
neighbors	O
of	O
the	O
data	O
value	O
at	O
are	O
the	O
values	O
at	O
and	O
,	O
respectively	O
.	O
</s>
<s>
Generally	O
in	O
finite	O
differences	O
one	O
can	O
allow	O
very	O
simply	O
for	O
steps	O
variable	O
along	O
the	O
mesh	B-Algorithm
,	O
but	O
all	O
the	O
original	O
nodes	O
should	O
be	O
preserved	O
and	O
they	O
can	O
move	O
independently	O
only	O
by	O
deforming	O
the	O
original	O
elements	O
.	O
</s>
<s>
If	O
even	O
only	O
two	O
of	O
all	O
the	O
nodes	O
change	O
their	O
order	O
,	O
or	O
even	O
only	O
one	O
node	O
is	O
added	O
to	O
or	O
removed	O
from	O
the	O
simulation	O
,	O
that	O
creates	O
a	O
defect	O
in	O
the	O
original	O
mesh	B-Algorithm
and	O
the	O
simple	O
finite	B-Algorithm
difference	I-Algorithm
approximation	O
can	O
no	O
longer	O
hold	O
.	O
</s>
<s>
Smoothed-particle	B-Application
hydrodynamics	I-Application
(	O
SPH	O
)	O
,	O
one	O
of	O
the	O
oldest	O
meshfree	B-Algorithm
methods	I-Algorithm
,	O
solves	O
this	O
problem	O
by	O
treating	O
data	O
points	O
as	O
physical	O
particles	O
with	O
mass	O
and	O
density	O
that	O
can	O
move	O
around	O
over	O
time	O
,	O
and	O
carry	O
some	O
value	O
with	O
them	O
.	O
</s>
<s>
Then	O
we	O
can	O
use	O
these	O
definitions	O
of	O
and	O
its	O
spatial	O
derivatives	O
to	O
write	O
the	O
equation	O
being	O
simulated	O
as	O
an	O
ordinary	O
differential	O
equation	O
,	O
and	O
simulate	O
the	O
equation	O
with	O
one	O
of	O
many	O
numerical	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
earliest	O
meshfree	B-Algorithm
methods	I-Algorithm
is	O
smoothed	B-Application
particle	I-Application
hydrodynamics	I-Application
,	O
presented	O
in	O
1977	O
.	O
</s>
<s>
In	O
the	O
1990s	O
a	O
new	O
class	O
of	O
meshfree	B-Algorithm
methods	I-Algorithm
emerged	O
based	O
on	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
first	O
method	O
called	O
the	O
diffuse	B-Algorithm
element	I-Algorithm
method	I-Algorithm
(	O
DEM	O
)	O
,	O
pioneered	O
by	O
Nayroles	O
et	O
al.	O
,	O
utilized	O
the	O
MLS	B-Algorithm
approximation	O
in	O
the	O
Galerkin	O
solution	O
of	O
partial	O
differential	O
equations	O
,	O
with	O
approximate	O
derivatives	O
of	O
the	O
MLS	B-Algorithm
function	O
.	O
</s>
<s>
Thereafter	O
Belytschko	O
pioneered	O
the	O
Element	O
Free	O
Galerkin	O
(	O
EFG	O
)	O
method	O
,	O
which	O
employed	O
MLS	B-Algorithm
with	O
Lagrange	O
multipliers	O
to	O
enforce	O
boundary	O
conditions	O
,	O
higher	O
order	O
numerical	B-Algorithm
quadrature	I-Algorithm
in	O
the	O
weak	B-Algorithm
form	I-Algorithm
,	O
and	O
full	O
derivatives	O
of	O
the	O
MLS	B-Algorithm
approximation	O
which	O
gave	O
better	O
accuracy	O
.	O
</s>
<s>
Around	O
the	O
same	O
time	O
,	O
the	O
reproducing	B-Algorithm
kernel	I-Algorithm
particle	I-Algorithm
method	I-Algorithm
(	O
RKPM	O
)	O
emerged	O
,	O
the	O
approximation	O
motivated	O
in	O
part	O
to	O
correct	O
the	O
kernel	O
estimate	O
in	O
SPH	O
:	O
to	O
give	O
accuracy	O
near	O
boundaries	O
,	O
in	O
non-uniform	O
discretizations	O
,	O
and	O
higher-order	O
accuracy	O
in	O
general	O
.	O
</s>
<s>
Notably	O
,	O
in	O
a	O
parallel	O
development	O
,	O
the	O
Material	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
were	O
developed	O
around	O
the	O
same	O
time	O
which	O
offer	O
similar	O
capabilities	O
.	O
</s>
<s>
Material	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
are	O
widely	O
used	O
in	O
the	O
movie	O
industry	O
to	O
simulate	O
large	O
deformation	O
solid	O
mechanics	O
,	O
such	O
as	O
snow	O
in	O
the	O
movie	O
Frozen	O
.	O
</s>
<s>
RKPM	O
and	O
other	O
meshfree	B-Algorithm
methods	I-Algorithm
were	O
extensively	O
developed	O
by	O
Chen	O
,	O
Liu	O
,	O
and	O
Li	O
in	O
the	O
late	O
1990s	O
for	O
a	O
variety	O
of	O
applications	O
and	O
various	O
classes	O
of	O
problems	O
.	O
</s>
<s>
The	O
following	O
numerical	B-Algorithm
methods	I-Algorithm
are	O
generally	O
considered	O
to	O
fall	O
within	O
the	O
general	O
class	O
of	O
"	O
meshfree	O
"	O
methods	O
.	O
</s>
<s>
The	O
primary	O
areas	O
of	O
advancement	O
in	O
meshfree	B-Algorithm
methods	I-Algorithm
are	O
to	O
address	O
issues	O
with	O
essential	O
boundary	O
enforcement	O
,	O
numerical	B-Algorithm
quadrature	I-Algorithm
,	O
and	O
contact	O
and	O
large	O
deformations	O
.	O
</s>
<s>
The	O
common	O
weak	B-Algorithm
form	I-Algorithm
requires	O
strong	O
enforcement	O
of	O
the	O
essential	O
boundary	O
conditions	O
,	O
yet	O
meshfree	B-Algorithm
methods	I-Algorithm
in	O
general	O
lack	O
the	O
Kronecker	O
delta	O
property	O
.	O
</s>
<s>
This	O
make	O
essential	O
boundary	O
condition	O
enforcement	O
non-trivial	O
,	O
at	O
least	O
more	O
difficult	O
than	O
the	O
Finite	B-Application
element	I-Application
method	I-Application
,	O
where	O
they	O
can	O
be	O
imposed	O
directly	O
.	O
</s>
<s>
Several	O
methods	O
have	O
been	O
developed	O
to	O
impose	O
the	O
essential	O
boundary	O
conditions	O
weakly	B-Algorithm
,	O
including	O
Lagrange	O
multipliers	O
,	O
Nitche	O
's	O
method	O
,	O
and	O
the	O
penalty	O
method	O
.	O
</s>
<s>
As	O
for	O
quadrature	B-Algorithm
,	O
nodal	O
integration	O
is	O
generally	O
preferred	O
which	O
offers	O
simplicity	O
,	O
efficiency	O
,	O
and	O
keeps	O
the	O
meshfree	B-Algorithm
method	I-Algorithm
free	O
of	O
any	O
mesh	B-Algorithm
(	O
as	O
opposed	O
to	O
using	O
Gauss	B-Algorithm
quadrature	I-Algorithm
,	O
which	O
necessitates	O
a	O
mesh	B-Algorithm
to	O
generate	O
quadrature	B-Algorithm
points	O
and	O
weights	O
)	O
.	O
</s>
<s>
Nodal	O
integration	O
however	O
,	O
suffers	O
from	O
numerical	O
instability	O
due	O
to	O
underestimation	O
of	O
strain	O
energy	O
associated	O
with	O
short-wavelength	O
modes	O
,	O
and	O
also	O
yields	O
inaccurate	O
and	O
non-convergent	O
results	O
due	O
to	O
under-integration	O
of	O
the	O
weak	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
One	O
major	O
advance	O
in	O
numerical	B-Algorithm
integration	I-Algorithm
has	O
been	O
the	O
development	O
of	O
a	O
stabilized	O
conforming	O
nodal	O
integration	O
(	O
SCNI	O
)	O
which	O
provides	O
a	O
nodal	O
integration	O
method	O
which	O
does	O
not	O
suffer	O
from	O
either	O
of	O
these	O
problems	O
.	O
</s>
<s>
The	O
method	O
is	O
based	O
on	O
strain-smoothing	O
which	O
satisfies	O
the	O
first	O
order	O
patch	B-Algorithm
test	I-Algorithm
.	O
</s>
<s>
More	O
recently	O
,	O
a	O
framework	O
has	O
been	O
developed	O
to	O
pass	O
arbitrary-order	O
patch	B-Algorithm
tests	I-Algorithm
,	O
based	O
on	O
a	O
Petrov	O
–	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
One	O
recent	O
advance	O
in	O
meshfree	B-Algorithm
methods	I-Algorithm
aims	O
at	O
the	O
development	O
of	O
computational	O
tools	O
for	O
automation	O
in	O
modeling	O
and	O
simulations	O
.	O
</s>
<s>
Because	O
a	O
triangular	O
mesh	B-Algorithm
can	O
be	O
generated	O
automatically	O
,	O
it	O
becomes	O
much	O
easier	O
in	O
re-meshing	O
and	O
hence	O
enables	O
automation	O
in	O
modeling	O
and	O
simulation	O
.	O
</s>
<s>
This	O
allows	O
easy	O
error	O
estimation	O
for	O
generally	O
complicated	O
problems	O
,	O
as	O
long	O
as	O
a	O
triangular	O
mesh	B-Algorithm
can	O
be	O
generated	O
.	O
</s>
<s>
The	O
W2	O
formulation	O
has	O
also	O
led	O
to	O
the	O
development	O
of	O
combination	O
of	O
meshfree	O
techniques	O
with	O
the	O
well-developed	O
FEM	O
techniques	O
,	O
and	O
one	O
can	O
now	O
use	O
triangular	O
mesh	B-Algorithm
with	O
excellent	O
accuracy	O
and	O
desired	O
softness	O
.	O
</s>
<s>
A	O
typical	O
such	O
a	O
formulation	O
is	O
the	O
so-called	O
smoothed	B-Algorithm
finite	I-Algorithm
element	I-Algorithm
method	I-Algorithm
(	O
or	O
S-FEM	O
)	O
.	O
</s>
<s>
It	O
is	O
a	O
general	O
perception	O
that	O
meshfree	B-Algorithm
methods	I-Algorithm
are	O
much	O
more	O
expensive	O
than	O
the	O
FEM	O
counterparts	O
.	O
</s>
<s>
The	O
recent	O
study	O
has	O
found	O
however	O
,	O
some	O
meshfree	B-Algorithm
methods	I-Algorithm
such	O
as	O
the	O
S-PIM	O
and	O
S-FEM	O
can	O
be	O
much	O
faster	O
than	O
the	O
FEM	O
counterparts	O
.	O
</s>
<s>
Nodal	O
integration	O
has	O
been	O
proposed	O
as	O
a	O
technique	O
to	O
use	O
finite	B-Application
elements	I-Application
to	O
emulate	O
a	O
meshfree	O
behaviour	O
.	O
</s>
