<s>
A	O
mesh	B-Algorithm
is	O
a	O
representation	O
of	O
a	O
larger	O
geometric	O
domain	O
by	O
smaller	O
discrete	O
cells	O
.	O
</s>
<s>
Meshes	B-Algorithm
are	O
commonly	O
used	O
to	O
compute	O
solutions	O
of	O
partial	O
differential	O
equations	O
and	O
render	O
computer	O
graphics	O
,	O
and	O
to	O
analyze	O
geographical	O
and	O
cartographic	O
data	O
.	O
</s>
<s>
A	O
mesh	B-Algorithm
partitions	O
space	O
into	O
elements	O
(	O
or	O
cells	O
or	O
zones	O
)	O
over	O
which	O
the	O
equations	O
can	O
be	O
solved	O
,	O
which	O
then	O
approximates	O
the	O
solution	O
over	O
the	O
larger	O
domain	O
.	O
</s>
<s>
This	O
cell	O
shape	O
consists	O
of	O
3	O
sides	O
and	O
is	O
one	O
of	O
the	O
simplest	O
types	B-Algorithm
of	I-Algorithm
mesh	I-Algorithm
.	O
</s>
<s>
A	O
triangular	O
surface	B-Algorithm
mesh	I-Algorithm
is	O
always	O
quick	O
and	O
easy	O
to	O
create	O
.	O
</s>
<s>
It	O
is	O
most	O
common	O
in	O
unstructured	B-Algorithm
grids	I-Algorithm
.	O
</s>
<s>
In	O
most	O
cases	O
a	O
tetrahedral	O
volume	O
mesh	B-Algorithm
can	O
be	O
generated	O
automatically	O
.	O
</s>
<s>
These	O
are	O
effectively	O
used	O
as	O
transition	O
elements	O
between	O
square	O
and	O
triangular	O
faced	O
elements	O
and	O
other	O
in	O
hybrid	O
meshes	B-Algorithm
and	O
grids	O
.	O
</s>
<s>
A	O
hexahedron	O
,	O
a	O
topological	O
cube	B-Application
,	O
has	O
8	O
vertices	O
,	O
12	O
edges	O
,	O
bounded	O
by	O
6	O
quadrilateral	O
faces	O
.	O
</s>
<s>
For	O
the	O
same	O
cell	O
amount	O
,	O
the	O
accuracy	O
of	O
solutions	O
in	O
hexahedral	O
meshes	B-Algorithm
is	O
the	O
highest	O
.	O
</s>
<s>
Some	O
other	O
advantages	O
of	O
structured	O
grid	O
over	O
unstructured	O
are	O
better	O
convergence	B-Algorithm
and	O
higher	O
resolution	O
.	O
</s>
<s>
An	O
unstructured	B-Algorithm
grid	I-Algorithm
is	O
identified	O
by	O
irregular	O
connectivity	O
.	O
</s>
<s>
Compared	O
to	O
structured	O
meshes	B-Algorithm
,	O
for	O
which	O
the	O
neighborhood	O
relationships	O
are	O
implicit	O
,	O
this	O
model	O
can	O
be	O
highly	O
space	O
inefficient	O
since	O
it	O
calls	O
for	O
explicit	O
storage	O
of	O
neighborhood	O
relationships	O
.	O
</s>
<s>
It	O
should	O
be	O
noted	O
,	O
though	O
,	O
that	O
the	O
storage	O
requirements	O
of	O
a	O
structured	O
grid	O
and	O
of	O
an	O
unstructured	B-Algorithm
grid	I-Algorithm
are	O
within	O
a	O
constant	O
factor	O
.	O
</s>
<s>
It	O
integrates	O
the	O
structured	O
meshes	B-Algorithm
and	O
the	O
unstructured	O
meshes	B-Algorithm
in	O
an	O
efficient	O
manner	O
.	O
</s>
<s>
Those	O
parts	O
of	O
the	O
geometry	O
that	O
are	O
regular	O
can	O
have	O
structured	O
grids	O
and	O
those	O
that	O
are	O
complex	O
can	O
have	O
unstructured	B-Algorithm
grids	I-Algorithm
.	O
</s>
<s>
A	O
mesh	B-Algorithm
is	O
considered	O
to	O
have	O
higher	O
quality	O
if	O
a	O
more	O
accurate	O
solution	O
is	O
calculated	O
more	O
quickly	O
.	O
</s>
<s>
Decreasing	O
the	O
mesh	B-Algorithm
size	O
always	O
increases	O
the	O
accuracy	O
but	O
also	O
increases	O
computational	O
cost	O
.	O
</s>
<s>
For	O
discretization	O
error	O
,	O
a	O
given	O
mesh	B-Algorithm
is	O
a	O
discrete	O
approximation	O
of	O
the	O
space	O
,	O
and	O
so	O
can	O
only	O
provide	O
an	O
approximate	O
solution	O
,	O
even	O
when	O
equations	O
are	O
solved	O
exactly	O
.	O
</s>
<s>
(	O
In	O
computer	O
graphics	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
the	O
number	O
of	O
rays	O
fired	O
is	O
another	O
source	O
of	O
discretization	O
error	O
.	O
)	O
</s>
<s>
For	O
solution	O
error	O
,	O
for	O
PDEs	O
many	O
iterations	O
over	O
the	O
entire	O
mesh	B-Algorithm
are	O
required	O
.	O
</s>
<s>
The	O
choice	O
of	O
mesh	B-Algorithm
element	O
type	O
affects	O
both	O
discretization	O
and	O
solution	O
error	O
.	O
</s>
<s>
A	O
coarse	O
mesh	B-Algorithm
may	O
provide	O
an	O
accurate	O
solution	O
if	O
the	O
solution	O
is	O
a	O
constant	O
,	O
so	O
the	O
precision	O
depends	O
on	O
the	O
particular	O
problem	O
instance	O
.	O
</s>
<s>
One	O
can	O
selectively	O
refine	O
the	O
mesh	B-Algorithm
in	O
areas	O
where	O
the	O
solution	O
gradients	O
are	O
high	O
,	O
thus	O
increasing	O
fidelity	O
there	O
.	O
</s>
<s>
A	O
faster	O
rate	O
of	O
convergence	B-Algorithm
means	O
smaller	O
error	O
with	O
fewer	O
iterations	O
.	O
</s>
<s>
A	O
mesh	B-Algorithm
of	O
inferior	O
quality	O
may	O
leave	O
out	O
important	O
features	O
such	O
as	O
the	O
boundary	O
layer	O
for	O
fluid	O
flow	O
.	O
</s>
<s>
The	O
discretization	O
error	O
will	O
be	O
large	O
and	O
the	O
rate	O
of	O
convergence	B-Algorithm
will	O
be	O
impaired	O
;	O
the	O
solution	O
may	O
not	O
converge	O
at	O
all	O
.	O
</s>
<s>
A	O
mesh	B-Algorithm
convergence	B-Algorithm
study	O
consists	O
of	O
refining	O
elements	O
and	O
comparing	O
the	O
refined	O
solutions	O
to	O
the	O
coarse	O
solutions	O
.	O
</s>
<s>
If	O
further	O
refinement	O
(	O
or	O
other	O
changes	O
)	O
does	O
not	O
significantly	O
change	O
the	O
solution	O
,	O
the	O
mesh	B-Algorithm
is	O
an	O
"	O
Independent	O
Grid.	O
"	O
</s>
<s>
If	O
the	O
accuracy	O
is	O
of	O
the	O
highest	O
concern	O
then	O
hexahedral	O
mesh	B-Algorithm
is	O
the	O
most	O
preferable	O
one	O
.	O
</s>
<s>
The	O
density	O
of	O
the	O
mesh	B-Algorithm
is	O
required	O
to	O
be	O
sufficiently	O
high	O
in	O
order	O
to	O
capture	O
all	O
the	O
flow	O
features	O
but	O
on	O
the	O
same	O
note	O
,	O
it	O
should	O
not	O
be	O
so	O
high	O
that	O
it	O
captures	O
unnecessary	O
details	O
of	O
the	O
flow	O
,	O
thus	O
burdening	O
the	O
CPU	O
and	O
wasting	O
more	O
time	O
.	O
</s>
<s>
Whenever	O
a	O
wall	O
is	O
present	O
,	O
the	O
mesh	B-Algorithm
adjacent	O
to	O
the	O
wall	O
is	O
fine	O
enough	O
to	O
resolve	O
the	O
boundary	O
layer	O
flow	O
and	O
generally	O
quad	O
,	O
hex	O
and	O
prism	O
cells	O
are	O
preferred	O
over	O
triangles	O
,	O
tetrahedrons	O
and	O
pyramids	O
.	O
</s>
<s>
Based	O
on	O
the	O
skewness	O
,	O
smoothness	O
,	O
and	O
aspect	B-Device
ratio	I-Device
,	O
the	O
suitability	O
of	O
the	O
mesh	B-Algorithm
can	O
be	O
decided	O
.	O
</s>
<s>
The	O
skewness	O
of	O
a	O
grid	O
is	O
an	O
apt	O
indicator	O
of	O
the	O
mesh	B-Algorithm
quality	O
and	O
suitability	O
.	O
</s>
<s>
Having	O
a	O
large	O
aspect	B-Device
ratio	I-Device
can	O
result	O
in	O
an	O
interpolation	O
error	O
of	O
unacceptable	O
magnitude	O
.	O
</s>
<s>
See	O
also	O
mesh	B-Architecture
generation	I-Architecture
and	O
principles	B-Algorithm
of	I-Algorithm
grid	I-Algorithm
generation	I-Algorithm
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
flipping	O
and	O
smoothing	O
are	O
powerful	O
tools	O
for	O
adapting	O
a	O
poor	O
mesh	B-Algorithm
into	O
a	O
good	O
mesh	B-Algorithm
.	O
</s>
<s>
Mesh	B-Algorithm
smoothing	O
enhances	O
element	O
shapes	O
and	O
overall	O
mesh	B-Algorithm
quality	O
by	O
adjusting	O
the	O
location	O
of	O
mesh	B-Algorithm
vertices	O
.	O
</s>
<s>
In	O
mesh	B-Algorithm
smoothing	O
,	O
core	O
features	O
such	O
as	O
non-zero	O
pattern	O
of	O
the	O
linear	O
system	O
are	O
preserved	O
as	O
the	O
topology	O
of	O
the	O
mesh	B-Algorithm
remains	O
invariant	O
.	O
</s>
<s>
Laplacian	B-Algorithm
smoothing	I-Algorithm
is	O
the	O
most	O
commonly	O
used	O
smoothing	O
technique	O
.	O
</s>
