Biomedical Engineering Reference
In-Depth Information
in the undeformed mesh, the element in the deformed mesh that contained
the node is located using a direct search. The local coordinates of the eight
nodes of the element containing node
N
are assembled into an 8
×
3 matrix
φ
(
ξ
i
,η
i
,ζ
i
), where
ξ
i
,η
i
,
and
ζ
i
are the local element coordinates of the nodes
composing the element; for instance, node 1 has local coordinates (
−
1,1,1).
The local coordinates are related to the global coordinates via the interpolating
polynomial coefficients arising from the shape functions as follows [40]:
[
φ
]
=
[
G
][
α
]
⇔
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
111
· · ·
· · ·
111
1
x
1
y
1
z
1
xy
1
yz
1
xz
1
xyz
1
α
1
β
1
γ
· · ·
· · ·
α
8
· · · · ·
·
·
·
=
· · · · ·
·
·
·
1
x
8
y
8
z
8
xy
8
yz
8
xz
8
xyx
8
β
8
γ
8
(12.29)
Here,
α
is an 8
×
3 matrix containing the polynomial coefficients and (
x
i
,
y
i
,
z
i
)
are the coordinates of node
i
in the global coordinate system. The matrix
α
is
then determined for each node
N
in the reset mesh:
[
α
]
=
[
G
]
−
1
[
φ
]
.
(12.30)
The local element coordinates (
ξ
N
,η
N
,ζ
N
) of node
N
follow from
α
and the
global coordinates (
x
N
,
y
N
,
z
N
):
⎡
⎤
α
1
β
1
γ
α
2
β
2
γ
2
· · ·
· · ·
· · ·
α
8
⎣
⎦
[
ζ
N
η
N
ξ
N
]
=
[1
x
N
y
N
z
N
xy
N
yz
N
xz
N
xyz
N
]
(12.31)
β
8
γ
8
The interpolated value then follows from the local coordinates, the nodal val-
ues and the trilinear shape functions. For example, the interpolated template
intensity is computed using
8
T
N
(
ξ
N
,η
N
,ζ
N
)
=
T
i
h
i
(
ξ
N
,η
N
,ζ
N
)
,
(12.32)
i
=
1
where the
T
i
are nodal intensity values and
h
i
are the shape functions corre-
sponding to each node evaluated at (
ξ
N
,η
N
,ζ
N
). The displacements
u
(
X
) are
interpolated using the same procedure. Note that this interpolation strategy is
consistent with the shape functions used in the FE solution process.
