<s>
Smoothed-particle	B-Application
hydrodynamics	I-Application
(	O
SPH	O
)	O
is	O
a	O
computational	O
method	O
used	O
for	O
simulating	O
the	O
mechanics	O
of	O
continuum	O
media	O
,	O
such	O
as	O
solid	O
mechanics	O
and	O
fluid	O
flows	O
.	O
</s>
<s>
It	O
has	O
been	O
used	O
in	O
many	O
fields	O
of	O
research	O
,	O
including	O
astrophysics	O
,	O
ballistics	B-Application
,	O
volcanology	O
,	O
and	O
oceanography	O
.	O
</s>
<s>
It	O
is	O
a	O
meshfree	B-Algorithm
Lagrangian	O
method	O
(	O
where	O
the	O
co-ordinates	O
move	O
with	O
the	O
fluid	O
)	O
,	O
and	O
the	O
resolution	O
of	O
the	O
method	O
can	O
easily	O
be	O
adjusted	O
with	O
respect	O
to	O
variables	O
such	O
as	O
density	O
.	O
</s>
<s>
By	O
construction	O
,	O
SPH	O
is	O
a	O
meshfree	B-Algorithm
method	I-Algorithm
,	O
which	O
makes	O
it	O
ideally	O
suited	O
to	O
simulate	O
problems	O
dominated	O
by	O
complex	O
boundary	O
dynamics	O
,	O
like	O
free	O
surface	O
flows	O
,	O
or	O
large	O
boundary	O
displacement	O
.	O
</s>
<s>
The	O
lack	O
of	O
a	O
mesh	O
significantly	O
simplifies	O
the	O
model	O
implementation	O
and	O
its	O
parallelization	O
,	O
even	O
for	O
many-core	B-General_Concept
architectures	O
.	O
</s>
<s>
Therefore	O
,	O
overlooking	O
issues	O
of	O
parallel	B-Operating_System
speedup	I-Operating_System
,	O
the	O
simulation	O
of	O
constant-density	O
flows	O
(	O
e.g.	O
,	O
external	O
aerodynamics	O
)	O
is	O
more	O
efficient	O
with	O
grid-based	O
methods	O
than	O
with	O
SPH	O
.	O
</s>
<s>
Smoothed-particle	B-Application
hydrodynamics	I-Application
is	O
being	O
increasingly	O
used	O
to	O
model	O
fluid	O
motion	O
as	O
well	O
.	O
</s>
<s>
However	O
,	O
both	O
grid-based	O
and	O
SPH	O
techniques	O
still	O
require	O
the	O
generation	O
of	O
renderable	O
free	O
surface	O
geometry	O
using	O
a	O
polygonization	O
technique	O
such	O
as	O
metaballs	O
and	O
marching	B-Algorithm
cubes	I-Algorithm
,	O
point	O
splatting	O
,	O
or	O
'	O
carpet	O
 '	O
visualization	O
.	O
</s>
<s>
For	O
gas	O
dynamics	O
it	O
is	O
more	O
appropriate	O
to	O
use	O
the	O
kernel	B-General_Concept
function	I-General_Concept
itself	O
to	O
produce	O
a	O
rendering	O
of	O
gas	O
column	O
density	O
(	O
e.g.	O
,	O
as	O
done	O
in	O
the	O
SPLASH	O
visualisation	O
package	O
)	O
.	O
</s>
<s>
In	O
the	O
typical	O
implementation	O
of	O
both	O
uniform	O
grids	O
and	O
SPH	O
particle	O
techniques	O
,	O
many	O
voxels	B-Algorithm
or	O
particles	O
will	O
be	O
used	O
to	O
fill	O
water	O
volumes	O
that	O
are	O
never	O
rendered	O
.	O
</s>
<s>
Smoothed-particle	B-Application
hydrodynamics	I-Application
's	O
adaptive	O
resolution	O
,	O
numerical	O
conservation	O
of	O
physically	O
conserved	O
quantities	O
,	O
and	O
ability	O
to	O
simulate	O
phenomena	O
covering	O
many	O
orders	O
of	O
magnitude	O
make	O
it	O
ideal	O
for	O
computations	O
in	O
theoretical	O
astrophysics	O
.	O
</s>
<s>
Simulations	O
of	O
galaxy	O
formation	O
,	O
star	O
formation	O
,	O
stellar	O
collisions	O
,	O
supernovae	O
and	O
meteor	B-Language
impacts	O
are	O
some	O
of	O
the	O
wide	O
variety	O
of	O
astrophysical	O
and	O
cosmological	O
uses	O
of	O
this	O
method	O
.	O
</s>
<s>
Another	O
important	O
advantage	O
of	O
meshfree	B-Algorithm
methods	I-Algorithm
in	O
general	O
,	O
and	O
of	O
SPH	O
in	O
particular	O
,	O
is	O
that	O
mesh	O
dependence	O
problems	O
are	O
naturally	O
avoided	O
given	O
the	O
meshfree	B-Algorithm
nature	O
of	O
the	O
method	O
.	O
</s>
<s>
Over	O
the	O
past	O
years	O
,	O
different	O
corrections	O
have	O
been	O
introduced	O
to	O
improve	O
the	O
accuracy	O
of	O
the	O
SPH	O
solution	O
,	O
leading	O
to	O
the	O
RKPM	B-Algorithm
by	O
Liu	O
et	O
al	O
.	O
</s>
<s>
The	O
Smoothed-Particle	B-Application
Hydrodynamics	I-Application
(	O
SPH	O
)	O
method	O
works	O
by	O
dividing	O
the	O
fluid	O
into	O
a	O
set	O
of	O
discrete	O
moving	O
elements	O
,	O
referred	O
to	O
as	O
particles	O
.	O
</s>
<s>
These	O
particles	O
interact	O
through	O
a	O
kernel	B-General_Concept
function	I-General_Concept
with	O
characteristic	O
radius	O
known	O
as	O
the	O
"	O
smoothing	O
length	O
"	O
,	O
typically	O
represented	O
in	O
equations	O
by	O
.	O
</s>
<s>
From	O
the	O
work	O
done	O
in	O
the	O
80	O
's	O
and	O
90	O
's	O
on	O
numerical	O
integration	O
of	O
point-like	O
particles	O
in	O
large	O
accelerators	O
,	O
appropriate	O
time	O
integrators	O
have	O
been	O
developed	O
with	O
accurate	O
conservation	O
properties	O
on	O
the	O
long	O
term	O
;	O
they	O
are	O
called	O
symplectic	B-Algorithm
integrators	I-Algorithm
.	O
</s>
<s>
The	O
most	O
popular	O
in	O
the	O
SPH	O
literature	O
is	O
the	O
leapfrog	B-Algorithm
scheme	O
,	O
which	O
reads	O
for	O
each	O
particle	O
:	O
</s>
<s>
Other	O
symplectic	B-Algorithm
integrators	I-Algorithm
exist	O
(	O
see	O
the	O
reference	O
textbook	O
)	O
.	O
</s>
<s>
density	O
regularization	O
and	O
numerical	O
viscosity	B-Application
,	O
respectively	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
description	O
of	O
hydrodynamic	O
flows	O
require	O
a	O
convenient	O
treatment	O
of	O
diffusive	O
processes	O
to	O
model	O
the	O
viscosity	B-Application
in	O
the	O
Navier	O
–	O
Stokes	O
equations	O
.	O
</s>
<s>
the	O
artificial	O
viscosity	B-Application
was	O
used	O
to	O
deal	O
with	O
high	O
Mach	O
number	O
fluid	O
flows	O
.	O
</s>
<s>
Here	O
,	O
is	O
controlling	O
a	O
volume	O
viscosity	B-Application
while	O
acts	O
similar	O
to	O
the	O
Neumann	O
Richtmeyr	O
artificial	O
viscosity	B-Application
.	O
</s>
<s>
The	O
artificial	O
viscosity	B-Application
also	O
has	O
shown	O
to	O
improve	O
the	O
overall	O
stability	O
of	O
general	O
flow	O
simulations	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
not	O
only	O
stabilize	O
inviscid	O
simulations	O
but	O
also	O
to	O
model	O
the	O
physical	O
viscosity	B-Application
by	O
this	O
approach	O
.	O
</s>
<s>
This	O
approach	O
introduces	O
the	O
bulk	O
viscosity	B-Application
.	O
</s>
<s>
The	O
particle-based	O
nature	O
of	O
SPH	O
makes	O
it	O
ideal	O
to	O
combine	O
with	O
a	O
particle-based	O
gravity	O
solver	O
,	O
for	O
instance	O
tree	O
gravity	O
code	O
,	O
particle	B-Algorithm
mesh	I-Algorithm
,	O
or	O
particle-particle	B-Algorithm
particle-mesh	I-Algorithm
.	O
</s>
<s>
stands	O
for	O
the	O
gradient	O
of	O
the	O
kernel	B-General_Concept
function	I-General_Concept
evaluated	O
at	O
the	O
initial	O
reference	O
configuration	O
.	O
</s>
<s>
The	O
anti-symmetric	O
property	O
of	O
the	O
derivative	O
of	O
the	O
kernel	B-General_Concept
function	I-General_Concept
will	O
ensure	O
the	O
momentum	O
conservation	O
for	O
each	O
pair	O
of	O
interacting	O
particles	O
and	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
element	I-Algorithm
method	I-Algorithm
,	O
used	O
for	O
simulating	O
granular	O
materials	O
,	O
is	O
related	O
to	O
SPH	O
.	O
</s>
