<s>
Mesh	B-Algorithm
generation	O
is	O
the	O
practice	O
of	O
creating	O
a	O
mesh	B-Algorithm
,	O
a	O
subdivision	O
of	O
a	O
continuous	O
geometric	O
space	O
into	O
discrete	O
geometric	O
and	O
topological	O
cells	O
.	O
</s>
<s>
Mesh	B-Algorithm
cells	O
are	O
used	O
as	O
discrete	O
local	O
approximations	O
of	O
the	O
larger	O
domain	O
.	O
</s>
<s>
Meshes	B-Algorithm
are	O
created	O
by	O
computer	O
algorithms	O
,	O
often	O
with	O
human	O
guidance	O
through	O
a	O
GUI	B-Application
,	O
depending	O
on	O
the	O
complexity	O
of	O
the	O
domain	O
and	O
the	O
type	O
of	O
mesh	B-Algorithm
desired	O
.	O
</s>
<s>
A	O
typical	O
goal	O
is	O
to	O
create	O
a	O
mesh	B-Algorithm
that	O
accurately	O
captures	O
the	O
input	O
domain	O
geometry	O
,	O
with	O
high-quality	O
(	O
well-shaped	O
)	O
cells	O
,	O
and	O
without	O
so	O
many	O
cells	O
as	O
to	O
make	O
subsequent	O
calculations	O
intractable	O
.	O
</s>
<s>
The	O
mesh	B-Algorithm
should	O
also	O
be	O
fine	O
(	O
have	O
small	O
elements	O
)	O
in	O
areas	O
that	O
are	O
important	O
for	O
the	O
subsequent	O
calculations	O
.	O
</s>
<s>
Meshes	B-Algorithm
are	O
used	O
for	O
rendering	O
to	O
a	O
computer	O
screen	O
and	O
for	O
physical	B-Algorithm
simulation	I-Algorithm
such	O
as	O
finite	B-Application
element	I-Application
analysis	I-Application
or	O
computational	B-Architecture
fluid	O
dynamics	O
.	O
</s>
<s>
Meshes	B-Algorithm
are	O
composed	O
of	O
simple	O
cells	O
like	O
triangles	O
because	O
,	O
e.g.	O
,	O
we	O
know	O
how	O
to	O
perform	O
operations	O
such	O
as	O
finite	B-Application
element	I-Application
calculations	O
(	O
engineering	O
)	O
or	O
ray	O
tracing	O
(	O
computer	O
graphics	O
)	O
on	O
triangles	O
,	O
but	O
we	O
do	O
not	O
know	O
how	O
to	O
perform	O
these	O
operations	O
directly	O
on	O
complicated	O
spaces	O
and	O
shapes	O
such	O
as	O
a	O
roadway	O
bridge	O
.	O
</s>
<s>
In	O
structured	O
meshing	O
the	O
mesh	B-Algorithm
is	O
a	O
regular	O
lattice	O
,	O
such	O
as	O
an	O
array	O
,	O
with	O
implied	O
connectivity	O
between	O
elements	O
.	O
</s>
<s>
This	O
page	O
is	O
primarily	O
about	O
unstructured	O
meshes	B-Algorithm
.	O
</s>
<s>
While	O
a	O
mesh	B-Algorithm
may	O
be	O
a	O
triangulation	B-Algorithm
,	O
the	O
process	O
of	O
meshing	O
is	O
distinguished	O
from	O
point	B-Algorithm
set	I-Algorithm
triangulation	I-Algorithm
in	O
that	O
meshing	O
includes	O
the	O
freedom	O
to	O
add	O
vertices	O
not	O
present	O
in	O
the	O
input	O
.	O
</s>
<s>
"	O
Facetting	O
"	O
(	O
triangulating	O
)	O
CAD	B-Application
models	I-Application
for	O
drafting	O
has	O
the	O
same	O
freedom	O
to	O
add	O
vertices	O
,	O
but	O
the	O
goal	O
is	O
to	O
represent	O
the	O
shape	O
accurately	O
using	O
as	O
few	O
triangles	O
as	O
possible	O
and	O
the	O
shape	O
of	O
individual	O
triangles	O
is	O
not	O
important	O
.	O
</s>
<s>
Computer	O
graphics	O
renderings	O
of	O
textures	O
and	O
realistic	O
lighting	O
conditions	O
use	O
meshes	B-Algorithm
instead	O
.	O
</s>
<s>
Many	O
mesh	B-Algorithm
generation	O
software	O
is	O
coupled	O
to	O
a	O
CAD	B-Application
system	I-Application
defining	O
its	O
input	O
,	O
and	O
simulation	O
software	O
for	O
taking	O
its	O
output	O
.	O
</s>
<s>
The	O
input	O
can	O
vary	O
greatly	O
but	O
common	O
forms	O
are	O
Solid	O
modeling	O
,	O
Geometric	B-Application
modeling	I-Application
,	O
NURBS	B-Algorithm
,	O
B-rep	B-Algorithm
,	O
STL	O
or	O
a	O
point	B-Algorithm
cloud	I-Algorithm
.	O
</s>
<s>
The	O
terms	O
"	O
mesh	B-Algorithm
generation	O
,	O
"	O
"	O
grid	O
generation	O
,	O
"	O
"	O
meshing	O
,	O
"	O
"	O
and	O
"	O
gridding	O
,	O
"	O
are	O
often	O
used	O
interchangeably	O
,	O
although	O
strictly	O
speaking	O
the	O
latter	O
two	O
are	O
broader	O
and	O
encompass	O
mesh	B-Algorithm
improvement	O
:	O
changing	O
the	O
mesh	B-Algorithm
with	O
the	O
goal	O
of	O
increasing	O
the	O
speed	O
or	O
accuracy	O
of	O
the	O
numerical	B-General_Concept
calculations	I-General_Concept
that	O
will	O
be	O
performed	O
over	O
it	O
.	O
</s>
<s>
In	O
computer	O
graphics	O
rendering	O
,	O
and	O
mathematics	O
,	O
a	O
mesh	B-Algorithm
is	O
sometimes	O
referred	O
to	O
as	O
a	O
tessellation	B-Algorithm
.	O
</s>
<s>
Mesh	B-Algorithm
faces	O
(	O
cells	O
,	O
entities	O
)	O
have	O
different	O
names	O
depending	O
on	O
their	O
dimension	O
and	O
the	O
context	O
in	O
which	O
the	O
mesh	B-Algorithm
will	O
be	O
used	O
.	O
</s>
<s>
In	O
finite	B-Application
elements	I-Application
,	O
the	O
highest-dimensional	O
mesh	B-Algorithm
entities	O
are	O
called	O
"	O
elements	O
,	O
"	O
"	O
edges	O
"	O
are	O
1D	O
and	O
"	O
nodes	O
"	O
are	O
0D	O
.	O
</s>
<s>
In	O
computational	B-Architecture
geometry	O
,	O
the	O
0D	O
points	O
are	O
called	O
vertices	O
.	O
</s>
<s>
Many	O
meshing	O
techniques	O
are	O
built	O
on	O
the	O
principles	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
,	O
together	O
with	O
rules	O
for	O
adding	O
vertices	O
,	O
such	O
as	O
Ruppert	O
's	O
algorithm	O
.	O
</s>
<s>
A	O
distinguishing	O
feature	O
is	O
that	O
an	O
initial	O
coarse	O
mesh	B-Algorithm
of	O
the	O
entire	O
space	O
is	O
formed	O
,	O
then	O
vertices	O
and	O
triangles	O
are	O
added	O
.	O
</s>
<s>
In	O
structured	O
mesh	B-Algorithm
generation	O
the	O
entire	O
mesh	B-Algorithm
is	O
a	O
lattice	O
graph	O
,	O
such	O
as	O
a	O
regular	O
grid	O
of	O
squares	O
.	O
</s>
<s>
In	O
block-structured	O
meshing	O
,	O
the	O
domain	O
is	O
divided	O
into	O
large	O
subregions	O
,	O
each	O
of	O
which	O
is	O
a	O
structured	O
mesh	B-Algorithm
.	O
</s>
<s>
Some	O
direct	O
methods	O
start	O
with	O
a	O
block-structured	O
mesh	B-Algorithm
and	O
then	O
move	O
the	O
mesh	B-Algorithm
to	O
conform	O
to	O
the	O
input	O
;	O
see	O
based	O
on	O
polycube	O
.	O
</s>
<s>
Another	O
direct	O
method	O
is	O
to	O
cut	O
the	O
structured	O
cells	O
by	O
the	O
domain	O
boundary	O
;	O
see	O
based	O
on	O
Marching	B-Algorithm
cubes	I-Algorithm
.	O
</s>
<s>
Some	O
types	O
of	O
meshes	B-Algorithm
are	O
much	O
more	O
difficult	O
to	O
create	O
than	O
others	O
.	O
</s>
<s>
Simplicial	O
meshes	B-Algorithm
tend	O
to	O
be	O
easier	O
than	O
cubical	O
meshes	B-Algorithm
.	O
</s>
<s>
An	O
important	O
category	O
is	O
generating	O
a	O
hex	O
mesh	B-Algorithm
conforming	O
to	O
a	O
fixed	O
quad	O
surface	B-Algorithm
mesh	I-Algorithm
;	O
a	O
research	O
subarea	O
is	O
studying	O
the	O
existence	O
and	O
generation	O
of	O
meshes	B-Algorithm
of	O
specific	O
small	O
configurations	O
,	O
such	O
as	O
the	O
tetragonal	O
trapezohedron	O
.	O
</s>
<s>
Because	O
of	O
the	O
difficulty	O
of	O
this	O
problem	O
,	O
the	O
existence	O
of	O
combinatorial	O
hex	O
meshes	B-Algorithm
has	O
been	O
studied	O
apart	O
from	O
the	O
problem	O
of	O
generating	O
good	O
geometric	O
realizations	O
.	O
</s>
<s>
While	O
known	O
algorithms	O
generate	O
simplicial	O
meshes	B-Algorithm
with	O
guaranteed	O
minimum	O
quality	O
,	O
such	O
guarantees	O
are	O
rare	O
for	O
cubical	O
meshes	B-Algorithm
,	O
and	O
many	O
popular	O
implementations	O
generate	O
inverted	O
(	O
inside-out	O
)	O
hexes	O
from	O
some	O
inputs	O
.	O
</s>
<s>
Meshes	B-Algorithm
are	O
often	O
created	O
in	O
serial	O
on	O
workstations	O
,	O
even	O
when	O
subsequent	O
calculations	O
over	O
the	O
mesh	B-Algorithm
will	O
be	O
done	O
in	O
parallel	B-Operating_System
on	O
super-computers	O
.	O
</s>
<s>
This	O
is	O
both	O
because	O
of	O
the	O
limitation	O
that	O
most	O
mesh	B-Algorithm
generators	O
are	O
interactive	O
,	O
and	O
because	O
mesh	B-Algorithm
generation	O
runtime	O
is	O
typically	O
insignificant	O
compared	O
to	O
solver	O
time	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
mesh	B-Algorithm
is	O
too	O
large	O
to	O
fit	O
in	O
the	O
memory	O
of	O
a	O
single	O
serial	O
machine	O
,	O
or	O
the	O
mesh	B-Algorithm
must	O
be	O
changed	O
(	O
adapted	O
)	O
during	O
the	O
simulation	O
,	O
meshing	O
is	O
done	O
in	O
parallel	B-Operating_System
.	O
</s>
<s>
The	O
computational	B-Architecture
domain	O
might	O
not	O
be	O
rectangular	O
,	O
but	O
for	O
the	O
sake	O
of	O
simplicity	O
,	O
the	O
domain	O
is	O
taken	O
to	O
be	O
rectangular	O
.	O
</s>
<s>
The	O
simplest	O
procedure	O
that	O
may	O
be	O
used	O
to	O
produce	O
boundary	O
fitted	O
computational	B-Architecture
mesh	B-Algorithm
is	O
the	O
normalization	O
transformation	O
.	O
</s>
<s>
The	O
advantage	O
of	O
using	O
the	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
is	O
that	O
the	O
solution	O
of	O
grid	O
generating	O
equations	O
can	O
be	O
exploited	O
to	O
generate	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
After	O
extensive	O
work	O
done	O
by	O
Crowley	O
(	O
1962	O
)	O
and	O
Winslow	O
(	O
1966	O
)	O
on	O
PDEs	O
by	O
transforming	O
physical	O
domain	O
into	O
computational	B-Architecture
plane	O
while	O
mapping	O
using	O
Poisson	O
's	O
equation	O
,	O
Thompson	O
et	O
al	O
.	O
</s>
<s>
where	O
,	O
are	O
the	O
co-ordinates	O
in	O
the	O
computational	B-Architecture
domain	O
,	O
while	O
P	O
and	O
Q	O
are	O
responsible	O
for	O
point	O
spacing	O
within	O
D	O
.	O
Transforming	O
above	O
equations	O
in	O
computational	B-Architecture
space	O
yields	O
a	O
set	O
of	O
two	O
elliptic	O
PDEs	O
of	O
the	O
form	O
,	O
</s>
<s>
These	O
systems	O
of	O
equations	O
are	O
solved	O
in	O
the	O
computational	B-Architecture
plane	O
on	O
uniformly	O
spaced	O
grid	O
which	O
provides	O
us	O
with	O
the	O
co-ordinates	O
of	O
each	O
point	O
in	O
physical	O
space	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
grid	O
has	O
to	O
be	O
computed	O
after	O
each	O
time	O
step	O
which	O
adds	O
up	O
to	O
computational	B-Architecture
time	O
.	O
</s>
<s>
Steger	O
and	O
Sorenson	O
(	O
1980	O
)	O
proposed	O
a	O
volume	O
orthogonality	B-Application
method	O
that	O
uses	O
Hyperbolic	O
PDEs	O
for	O
mesh	B-Algorithm
generation	O
.	O
</s>
<s>
For	O
a	O
2-D	O
problem	O
,	O
Considering	O
computational	B-Architecture
space	O
to	O
be	O
given	O
by	O
,	O
the	O
inverse	O
of	O
the	O
Jacobian	O
is	O
given	O
by	O
,	O
</s>
<s>
where	O
represents	O
the	O
area	O
in	O
physical	O
space	O
for	O
a	O
given	O
area	O
in	O
computational	B-Architecture
space	O
.	O
</s>
<s>
Poor	O
selection	O
of	O
may	O
lead	O
to	O
shock	O
and	O
discontinuous	O
propagation	O
of	O
this	O
information	O
throughout	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
While	O
mesh	B-Algorithm
being	O
orthogonal	O
is	O
generated	O
very	O
rapidly	O
which	O
comes	O
out	O
as	O
an	O
advantage	O
with	O
this	O
method	O
.	O
</s>
<s>
This	O
method	O
includes	O
a	O
technique	O
that	O
minimizes	O
grid	O
smoothness	O
,	O
orthogonality	B-Application
and	O
volume	O
variation	O
.	O
</s>
<s>
In	O
this	O
method	O
an	O
alternative	O
grid	O
is	O
generated	O
by	O
a	O
new	O
mesh	B-Algorithm
after	O
each	O
iteration	O
and	O
computing	O
the	O
grid	O
speed	O
using	O
backward	O
difference	O
method	O
.	O
</s>
<s>
Further	O
work	O
needed	O
to	O
be	O
done	O
to	O
minimize	O
the	O
integrals	B-Algorithm
that	O
will	O
reduce	O
the	O
CPU	O
time	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
unstructured	B-Algorithm
grids	I-Algorithm
require	O
a	O
higher	O
cell	O
density	O
in	O
the	O
boundary	O
layer	O
because	O
the	O
cell	O
needs	O
to	O
be	O
as	O
equilateral	O
as	O
possible	O
to	O
avoid	O
errors	O
.	O
</s>
<s>
We	O
must	O
identify	O
what	O
information	O
is	O
required	O
to	O
identify	O
the	O
cell	O
and	O
all	O
the	O
neighbors	O
of	O
the	O
cell	O
in	O
the	O
computational	B-Architecture
mesh	B-Algorithm
.	O
</s>
<s>
We	O
can	O
choose	O
to	O
locate	O
the	O
arbitrary	O
points	O
anywhere	O
we	O
want	O
for	O
the	O
unstructured	B-Algorithm
grid	I-Algorithm
.	O
</s>
<s>
The	O
adaptive	O
method	O
is	O
referred	O
to	O
as	O
‘	O
h’	O
method	O
if	O
mesh	B-Algorithm
refinement	I-Algorithm
is	O
used	O
,	O
‘	O
r’	O
method	O
if	O
the	O
number	O
of	O
grid	O
point	O
is	O
fixed	O
and	O
not	O
redistributed	O
and	O
‘	O
p’	O
if	O
the	O
order	O
of	O
solution	O
scheme	O
is	O
increased	O
in	O
finite-element	O
theory	O
.	O
</s>
<s>
The	O
problem	O
is	O
the	O
connectivity	O
hampers	O
if	O
mesh	B-Algorithm
point	O
movement	O
is	O
very	O
large	O
.	O
</s>
<s>
Usually	O
the	O
cells	O
are	O
polygonal	O
or	O
polyhedral	O
and	O
form	O
a	O
mesh	B-Algorithm
that	O
partitions	O
the	O
domain	O
.	O
</s>
<s>
Simplicial	O
meshes	B-Algorithm
may	O
be	O
of	O
any	O
dimension	O
and	O
include	O
triangles	O
(	O
2D	O
)	O
and	O
tetrahedra	O
(	O
3D	O
)	O
as	O
important	O
instances	O
.	O
</s>
<s>
Cubical	O
meshes	B-Algorithm
is	O
the	O
pan-dimensional	O
category	O
that	O
includes	O
quads	O
(	O
2D	O
)	O
and	O
hexes	O
(	O
3D	O
)	O
.	O
</s>
<s>
In	O
3D	O
,	O
4-sided	O
pyramids	O
and	O
3-sided	O
prisms	O
appear	O
in	O
conformal	O
meshes	B-Algorithm
of	O
mixed	O
cell	O
type	O
.	O
</s>
<s>
The	O
mesh	B-Algorithm
is	O
embedded	O
in	O
a	O
geometric	O
space	O
that	O
is	O
typically	O
two	O
or	O
three	O
dimensional	O
,	O
although	O
sometimes	O
the	O
dimension	O
is	O
increased	O
by	O
one	O
by	O
adding	O
the	O
time-dimension	O
.	O
</s>
<s>
Higher	O
dimensional	O
meshes	B-Algorithm
are	O
used	O
in	O
niche	O
contexts	O
.	O
</s>
<s>
One-dimensional	O
meshes	B-Algorithm
are	O
useful	O
as	O
well	O
.	O
</s>
<s>
A	O
significant	O
category	O
is	O
surface	B-Algorithm
meshes	I-Algorithm
,	O
which	O
are	O
2D	O
meshes	B-Algorithm
embedded	O
in	O
3D	O
to	O
represent	O
a	O
curved	O
surface	O
.	O
</s>
<s>
One	O
can	O
make	O
a	O
polyhedral	O
Voronoi	B-Architecture
diagram	I-Architecture
mesh	B-Algorithm
by	O
dualizing	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
simplicial	O
mesh	B-Algorithm
.	O
</s>
<s>
One	O
can	O
create	O
a	O
cubical	O
mesh	B-Algorithm
by	O
generating	O
an	O
arrangement	O
of	O
surfaces	O
and	O
dualizing	O
the	O
intersection	O
graph	O
;	O
see	O
spatial	O
twist	O
continuum	O
.	O
</s>
<s>
Sometimes	O
both	O
the	O
primal	O
mesh	B-Algorithm
and	O
its	O
dual	O
mesh	B-Algorithm
are	O
used	O
in	O
the	O
same	O
simulation	O
;	O
see	O
Hodge	O
star	O
operator	O
.	O
</s>
<s>
This	O
arises	O
from	O
physics	O
involving	O
divergence	B-Application
and	O
curl	O
(	O
mathematics	O
)	O
operators	O
,	O
such	O
as	O
flux	B-General_Concept
&	O
vorticity	O
or	O
electricity	O
&	O
magnetism	O
,	O
where	O
one	O
variable	O
naturally	O
lives	O
on	O
the	O
primal	O
faces	O
and	O
its	O
counterpart	O
on	O
the	O
dual	O
faces	O
.	O
</s>
<s>
Three-dimensional	O
meshes	B-Algorithm
created	O
for	O
finite	B-Application
element	I-Application
analysis	I-Application
need	O
to	O
consist	O
of	O
tetrahedra	O
,	O
pyramids	O
,	O
prisms	O
or	O
hexahedra	O
.	O
</s>
<s>
Those	O
used	O
for	O
the	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
can	O
consist	O
of	O
arbitrary	O
polyhedra	O
.	O
</s>
<s>
Those	O
used	O
for	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
consist	O
of	O
piecewise	O
structured	O
arrays	O
of	O
hexahedra	O
known	O
as	O
multi-block	O
structured	O
meshes	B-Algorithm
.	O
</s>
<s>
3-sided	O
prisms	O
are	O
used	O
for	O
boundary	O
layers	O
conforming	O
to	O
a	O
tet	O
mesh	B-Algorithm
of	O
the	O
far-interior	O
of	O
the	O
object	O
.	O
</s>
<s>
Surface	B-Algorithm
meshes	I-Algorithm
are	O
useful	O
in	O
computer	O
graphics	O
where	O
the	O
surfaces	O
of	O
objects	O
reflect	O
light	O
(	O
also	O
subsurface	B-Algorithm
scattering	I-Algorithm
)	O
and	O
a	O
full	O
3D	B-Algorithm
mesh	I-Algorithm
is	O
not	O
needed	O
.	O
</s>
<s>
Surface	B-Algorithm
meshes	I-Algorithm
are	O
also	O
used	O
to	O
model	O
thin	O
objects	O
such	O
as	O
sheet	O
metal	O
in	O
auto	O
manufacturing	O
and	O
building	O
exteriors	O
in	O
architecture	O
.	O
</s>
<s>
High	O
(	O
e.g.	O
,	O
17	O
)	O
dimensional	O
cubical	O
meshes	B-Algorithm
are	O
common	O
in	O
astrophysics	O
and	O
string	O
theory	O
.	O
</s>
<s>
What	O
is	O
the	O
precise	O
definition	O
of	O
a	O
mesh	B-Algorithm
?	O
</s>
<s>
However	O
,	O
some	O
mathematical	O
objects	O
are	O
clearly	O
meshes	B-Algorithm
:	O
a	O
simplicial	O
complex	O
is	O
a	O
mesh	B-Algorithm
composed	O
of	O
simplices	O
.	O
</s>
<s>
cubical	O
)	O
meshes	B-Algorithm
are	O
conformal	O
,	O
meaning	O
they	O
have	O
the	O
cell	O
structure	O
of	O
a	O
CW	O
complex	O
,	O
a	O
generalization	O
of	O
a	O
simplicial	O
complex	O
.	O
</s>
<s>
A	O
mesh	B-Algorithm
need	O
not	O
be	O
simplicial	O
because	O
an	O
arbitrary	O
subset	O
of	O
nodes	O
of	O
a	O
cell	O
is	O
not	O
necessarily	O
a	O
cell	O
:	O
e.g.	O
,	O
three	O
nodes	O
of	O
a	O
quad	O
does	O
not	O
define	O
a	O
cell	O
.	O
</s>
<s>
An	O
intersection	O
being	O
more	O
than	O
one	O
cell	O
is	O
sometimes	O
forbidden	O
and	O
rarely	O
desired	O
;	O
the	O
goal	O
of	O
some	O
mesh	B-Algorithm
improvement	O
techniques	O
(	O
e.g.	O
</s>
<s>
In	O
some	O
contexts	O
,	O
a	O
distinction	O
is	O
made	O
between	O
a	O
topological	O
mesh	B-Algorithm
and	O
a	O
geometric	O
mesh	B-Algorithm
whose	O
embedding	O
satisfies	O
certain	O
quality	O
criteria	O
.	O
</s>
<s>
Important	O
mesh	B-Algorithm
variants	O
that	O
are	O
not	O
CW	O
complexes	O
include	O
non-conformal	O
meshes	B-Algorithm
where	O
cells	O
do	O
not	O
meet	O
strictly	O
face-to-face	O
,	O
but	O
the	O
cells	O
nonetheless	O
partition	O
the	O
domain	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
is	O
an	O
octree	B-Data_Structure
,	O
where	O
an	O
element	O
face	O
may	O
be	O
partitioned	O
by	O
the	O
faces	O
of	O
adjacent	O
elements	O
.	O
</s>
<s>
Such	O
meshes	B-Algorithm
are	O
useful	O
for	O
flux-based	O
simulations	O
.	O
</s>
<s>
In	O
overset	O
grids	O
,	O
there	O
are	O
multiple	O
conformal	O
meshes	B-Algorithm
that	O
overlap	O
geometrically	O
and	O
do	O
not	O
partition	O
the	O
domain	O
;	O
see	O
e.g.	O
,	O
Overflow	O
,	O
the	O
OVERset	O
grid	O
FLOW	O
solver	O
.	O
</s>
<s>
So-called	O
meshless	O
or	O
meshfree	B-Algorithm
methods	I-Algorithm
often	O
make	O
use	O
of	O
some	O
mesh-like	O
discretization	O
of	O
the	O
domain	O
,	O
and	O
have	O
basis	O
functions	O
with	O
overlapping	O
support	O
.	O
</s>
<s>
Sometimes	O
a	O
local	O
mesh	B-Algorithm
is	O
created	O
near	O
each	O
simulation	O
degree-of-freedom	O
point	O
,	O
and	O
these	O
meshes	B-Algorithm
may	O
overlap	O
and	O
be	O
non-conformal	O
to	O
one	O
another	O
.	O
</s>
<s>
Implicit	O
triangulations	B-Algorithm
are	O
based	O
on	O
a	O
delta	O
complex	O
:	O
for	O
each	O
triangle	O
the	O
lengths	O
of	O
its	O
edges	O
,	O
and	O
a	O
gluing	O
map	O
between	O
face	O
edges	O
.	O
</s>
<s>
Many	O
meshes	B-Algorithm
use	O
linear	O
elements	O
,	O
where	O
the	O
mapping	O
from	O
the	O
abstract	O
to	O
realized	O
element	O
is	O
linear	O
,	O
and	O
mesh	B-Algorithm
edges	O
are	O
straight	O
segments	O
.	O
</s>
<s>
A	O
primary	O
goal	O
for	O
higher-order	O
elements	O
is	O
to	O
more	O
accurately	O
represent	O
the	O
domain	O
boundary	O
,	O
although	O
they	O
have	O
accuracy	O
benefits	O
in	O
the	O
interior	O
of	O
the	O
mesh	B-Algorithm
as	O
well	O
.	O
</s>
<s>
One	O
of	O
the	O
motivations	O
for	O
cubical	O
meshes	B-Algorithm
is	O
that	O
linear	O
cubical	O
elements	O
have	O
some	O
of	O
the	O
same	O
numerical	O
advantages	O
as	O
quadratic	O
simplicial	O
elements	O
.	O
</s>
<s>
In	O
the	O
isogeometric	B-Algorithm
analysis	I-Algorithm
simulation	O
technique	O
,	O
the	O
mesh	B-Algorithm
cells	O
containing	O
the	O
domain	O
boundary	O
use	O
the	O
CAD	B-Application
representation	O
directly	O
instead	O
of	O
a	O
linear	O
or	O
polynomial	O
approximation	O
.	O
</s>
<s>
Improving	O
a	O
mesh	B-Algorithm
involves	O
changing	O
its	O
discrete	O
connectivity	O
,	O
the	O
continuous	O
geometric	O
position	O
of	O
its	O
cells	O
,	O
or	O
both	O
.	O
</s>
<s>
The	O
same	O
kinds	O
of	O
operations	O
are	O
done	O
for	O
cubical	O
(	O
quad/hex	O
)	O
meshes	B-Algorithm
,	O
although	O
there	O
are	O
fewer	O
possible	O
operations	O
and	O
local	O
changes	O
have	O
global	O
consequences	O
.	O
</s>
<s>
E.g.	O
,	O
for	O
a	O
hexahedral	O
mesh	B-Algorithm
,	O
merging	O
two	O
nodes	O
creates	O
cells	O
that	O
are	O
not	O
hexes	O
,	O
but	O
if	O
diagonally-opposite	O
nodes	O
on	O
a	O
quadrilateral	O
are	O
merged	O
and	O
this	O
is	O
propagated	O
into	O
collapsing	O
an	O
entire	O
face-connected	O
column	O
of	O
hexes	O
,	O
then	O
all	O
remaining	O
cells	O
will	O
still	O
be	O
hexes	O
.	O
</s>
<s>
In	O
adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
,	O
elements	O
are	O
split	O
(	O
h-refinement	B-Algorithm
)	O
in	O
areas	O
where	O
the	O
function	O
being	O
calculated	O
has	O
a	O
high	O
gradient	O
.	O
</s>
<s>
Meshes	B-Algorithm
are	O
also	O
coarsened	O
,	O
removing	O
elements	O
for	O
efficiency	O
.	O
</s>
<s>
The	O
multigrid	O
method	O
does	O
something	O
similar	O
to	O
refinement	O
and	O
coarsening	O
to	O
speed	O
up	O
the	O
numerical	O
solve	O
,	O
but	O
without	O
actually	O
changing	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
Moving	O
nodes	O
to	O
improve	O
quality	O
is	O
called	O
"	O
smoothing	O
"	O
or	O
"	O
r-refinement	B-Algorithm
"	O
and	O
increasing	O
the	O
order	O
of	O
elements	O
is	O
called	O
"	O
p-refinement.	O
"	O
</s>
<s>
If	O
the	O
object	O
deforms	O
enough	O
,	O
the	O
entire	O
object	O
is	O
remeshed	O
and	O
the	O
current	O
solution	O
mapped	O
from	O
the	O
old	O
mesh	B-Algorithm
to	O
the	O
new	O
mesh	B-Algorithm
.	O
</s>
<s>
The	O
field	O
is	O
highly	O
interdisciplinary	O
,	O
with	O
contributions	O
found	O
in	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
and	O
engineering	O
.	O
</s>
<s>
Meshing	O
R&D	O
is	O
distinguished	O
by	O
an	O
equal	O
focus	O
on	O
discrete	O
and	O
continuous	O
math	O
and	O
computation	O
,	O
as	O
with	O
computational	B-Architecture
geometry	O
,	O
but	O
in	O
contrast	O
to	O
graph	O
theory	O
(	O
discrete	O
)	O
and	O
numerical	B-General_Concept
analysis	I-General_Concept
(	O
continuous	O
)	O
.	O
</s>
<s>
Mesh	B-Algorithm
generation	O
is	O
deceptively	O
difficult	O
:	O
it	O
is	O
easy	O
for	O
humans	O
to	O
see	O
how	O
to	O
create	O
a	O
mesh	B-Algorithm
of	O
a	O
given	O
object	O
,	O
but	O
difficult	O
to	O
program	O
a	O
computer	O
to	O
make	O
good	O
decisions	O
for	O
arbitrary	O
input	O
a	O
priori	O
.	O
</s>
<s>
Many	O
mesh	B-Algorithm
generation	O
researchers	O
were	O
first	O
users	O
of	O
meshes	B-Algorithm
.	O
</s>
<s>
Mesh	B-Algorithm
generation	O
continues	O
to	O
receive	O
widespread	O
attention	O
,	O
support	O
and	O
funding	O
because	O
the	O
human-time	O
to	O
create	O
a	O
mesh	B-Algorithm
dwarfs	O
the	O
time	O
to	O
set	O
up	O
and	O
solve	O
the	O
calculation	O
once	O
the	O
mesh	B-Algorithm
is	O
finished	O
.	O
</s>
<s>
This	O
has	O
always	O
been	O
the	O
situation	O
since	O
numerical	O
simulation	O
and	O
computer	O
graphics	O
were	O
invented	O
,	O
because	O
as	O
computer	O
hardware	O
and	O
simple	O
equation-solving	O
software	O
have	O
improved	O
,	O
people	O
have	O
been	O
drawn	O
to	O
larger	O
and	O
more	O
complex	O
geometric	B-Application
models	I-Application
in	O
a	O
drive	O
for	O
greater	O
fidelity	O
,	O
scientific	O
insight	O
,	O
and	O
artistic	O
expression	O
.	O
</s>
<s>
This	O
is	O
in	O
keeping	O
with	O
the	O
interdisciplinary	O
nature	O
of	O
the	O
research	O
required	O
to	O
make	O
progress	O
,	O
and	O
also	O
the	O
wide	O
variety	O
of	O
applications	O
that	O
make	O
use	O
of	O
meshes	B-Algorithm
.	O
</s>
<s>
An	O
independent	O
annual	O
conference	O
from	O
1992-2021	O
,	O
and	O
a	O
SIAM	O
workshop	O
concurrent	B-Operating_System
with	O
SIAM	O
PP	O
or	O
SIAM	O
CS&E	O
since	O
2022	O
.	O
</s>
