Civil Engineering Reference
In-Depth Information
The transformation from {Y
1
, Y
2
, Y
3
} to {Z
1
, Z
2
, Z
3
} is similar to that from {X
1
, X
2
, X
3
}
to {Y
1
, Y
2
, Y
3
}, except that there is a difference in node labels so that no rotation of the
triangle for the transformation from {X
1
, X
2
, X
3
} to {Z
1
, Z
2
, Z
3
} will be induced. Hence, in
the GETMe node smoothing of triangular meshes, the two steps will always be combined
into one single operation. The triangles have to be transformed without a change in area;
however, in the transformation, the nodes are displaced normal to the edges, and the area of
the triangle will be increased. As the centroid of the triangle will not move in the transfor-
mation, the area can be restored by scaling the transformed vertices with a factor ρ given by
Area T
Area T
()
()
scalingfactor
,
ρ=
and the scaled triangle T* whose vertices after scaling are adjusted to
*
XC ZCk
=+ −
ρ
(
),
=
123
, ,;
CentroidC
=++
(
XXX/
)
3
(6.4)
k
k
1
2
3
Let's take a look at the transformation of triangle T = {(0,0), (10,0), (1,2)} with area =
10 units and α-quality = 0.3646, as shown in Figure 6.13. After the first step, T is trans-
formed to T′ with α increased to 0.7915, and the second step further brings α to 0.9434
for T″ whose area has been increased to 29.1 times that of the original triangle T. The area
of triangle T″ is brought back to 10 units by adjusting its vertices with the scaling factor ρ
without altering the α-quality, as shown in Figure 6.13.
Given a triangle T, the GETMe transformation is formally defined as
Transformation GETMe: T ↦ T*,
with T* being computed using formula 6.4
Similar to QL smoothing and LO scheme, GETMe transformation works on each interior
node in turn. Given a triangular mesh of N triangles,
T
= {Δ
i
, i = 1,N}, for a given node
x
,
the patch of triangles surrounding node
x
is given by
P
(
x
) = {Δ
k
∈
T
;
x
∈ Δ
k
},
x
∈ Δ
k
means that
x
is one of the vertices of triangle Δ
k
T´´
T*
T
T´
Figure 6.13
GETMe transformation.
