<s>
In	O
mesh	B-Architecture
generation	I-Architecture
,	O
Delaunay	B-Algorithm
refinement	I-Algorithm
are	O
algorithms	O
for	O
mesh	B-Architecture
generation	I-Architecture
based	O
on	O
the	O
principle	O
of	O
adding	O
Steiner	O
points	O
to	O
the	O
geometry	O
of	O
an	O
input	O
to	O
be	O
meshed	O
,	O
in	O
a	O
way	O
that	O
causes	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
or	O
constrained	B-Algorithm
Delaunay	I-Algorithm
triangulation	I-Algorithm
of	O
the	O
augmented	O
input	O
to	O
meet	O
the	O
quality	O
requirements	O
of	O
the	O
meshing	O
application	O
.	O
</s>
<s>
Delaunay	B-Algorithm
refinement	I-Algorithm
methods	O
include	O
methods	O
by	O
Chew	O
and	O
by	O
Ruppert	O
.	O
</s>
<s>
Chew	B-Algorithm
's	I-Algorithm
second	I-Algorithm
algorithm	I-Algorithm
takes	O
a	O
piecewise	O
linear	O
system	O
(	O
PLS	O
)	O
and	O
returns	O
a	O
constrained	B-Algorithm
Delaunay	I-Algorithm
triangulation	I-Algorithm
of	O
only	O
quality	O
triangles	O
where	O
quality	O
is	O
defined	O
by	O
the	O
minimum	O
angle	O
in	O
a	O
triangle	O
.	O
</s>
<s>
Developed	O
by	O
L	O
.	O
Paul	O
Chew	O
for	O
meshing	O
surfaces	O
embedded	O
in	O
three-dimensional	O
space	O
,	O
Chew	B-Algorithm
's	I-Algorithm
second	I-Algorithm
algorithm	I-Algorithm
has	O
been	O
adopted	O
as	O
a	O
two-dimensional	O
mesh	B-Algorithm
generator	O
due	O
to	O
practical	O
advantages	O
over	O
Ruppert	O
's	O
algorithm	O
in	O
certain	O
cases	O
and	O
is	O
the	O
default	O
quality	O
mesh	B-Algorithm
generator	O
implemented	O
in	O
the	O
freely	O
available	O
Triangle	O
package	O
.	O
</s>
<s>
Chew	B-Algorithm
's	I-Algorithm
second	I-Algorithm
algorithm	I-Algorithm
is	O
guaranteed	O
to	O
terminate	O
and	O
produce	O
a	O
local	O
feature	O
size-graded	O
meshes	B-Algorithm
with	O
minimum	O
angle	O
up	O
to	O
about	O
28.6	O
degrees	O
.	O
</s>
<s>
The	O
algorithm	O
begins	O
with	O
a	O
constrained	B-Algorithm
Delaunay	I-Algorithm
triangulation	I-Algorithm
of	O
the	O
input	O
vertices	O
.	O
</s>
<s>
Ruppert	O
's	O
algorithm	O
takes	O
a	O
planar	O
straight-line	O
graph	O
(	O
or	O
in	O
dimension	O
higher	O
than	O
two	O
a	O
piecewise	O
linear	O
system	O
)	O
and	O
returns	O
a	O
conforming	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
only	O
quality	O
triangles	O
.	O
</s>
<s>
"	O
Ruppert	O
's	O
algorithm	O
for	O
two-dimensional	O
quality	O
mesh	B-Architecture
generation	I-Architecture
is	O
perhaps	O
the	O
first	O
theoretically	O
guaranteed	O
meshing	O
algorithm	O
to	O
be	O
truly	O
satisfactory	O
in	O
practice.	O
"	O
</s>
<s>
The	O
input	O
to	O
a	O
2D	O
finite	B-Application
element	I-Application
method	I-Application
needs	O
to	O
be	O
in	O
the	O
form	O
of	O
triangles	O
that	O
fill	O
all	O
space	O
,	O
and	O
each	O
triangle	O
to	O
be	O
filled	O
with	O
one	O
kind	O
of	O
material	O
–	O
in	O
this	O
example	O
,	O
either	O
"	O
air	O
"	O
or	O
"	O
wing	O
"	O
.	O
</s>
<s>
The	O
simulation	O
time	O
is	O
generally	O
proportional	O
to	O
the	O
number	O
of	O
triangles	O
,	O
and	O
so	O
one	O
wants	O
to	O
minimize	O
the	O
number	O
of	O
triangles	O
,	O
while	O
still	O
using	O
enough	O
triangles	O
to	O
give	O
reasonably	O
accurate	O
results	O
–	O
typically	O
by	O
using	O
an	O
unstructured	B-Algorithm
grid	I-Algorithm
.	O
</s>
<s>
The	O
computer	O
uses	O
Ruppert	O
's	O
algorithm	O
(	O
or	O
some	O
similar	O
meshing	O
algorithm	O
)	O
to	O
convert	O
the	O
polygonal	O
model	O
into	O
triangles	O
suitable	O
for	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
The	O
algorithm	O
begins	O
with	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
the	O
input	O
vertices	O
and	O
then	O
consists	O
of	O
two	O
main	O
operations	O
.	O
</s>
<s>
Without	O
modification	O
Ruppert	O
's	O
algorithm	O
is	O
guaranteed	O
to	O
terminate	O
and	O
generate	O
a	O
quality	O
mesh	B-Algorithm
for	O
non-acute	O
input	O
and	O
any	O
poor-quality	O
threshold	O
less	O
than	O
about	O
20.7	O
degrees	O
.	O
</s>
