Biomedical Engineering Reference
In-Depth Information
12.2.5
Finite Element Discretization
Hyperelastic Warping is based on an FE discretization of the template image.
The FE method uses “shape functions” to describe the element shape and the ar-
bitrary variations in configuration over the element domain [34]. In Hyperelastic
Warping, an FE mesh is constructed to correspond to all or part of the template
image (either a rectilinear mesh, or a mesh that conforms to a particular structure
of interest in the template image). The template intensity field
T
is interpolated
to the nodes of the FE mesh. The template intensity field is convected with the
FE mesh and thus the nodal values do not change. As the FE mesh deforms,
the values of the target intensity field
S
are queried at the current location of the
nodes of the template FE mesh. To apply an FE discretization to Eq. (12.15), an
isoparametric conforming FE approximation is introduced for the variations
η
and
∆
u
:
N
nodes
N
nodes
η
e
≡
η
|
e
=
N
j
(
ξ
)
η
j
,
u
e
≡
u
|
e
=
N
j
(
ξ
)
u
j
,
(12.19)
j
=
1
j
=
1
where the subscript
e
specifies that the variations are restricted to a particular
element with domain
e
, and
N
nodes
is the number of nodes composing each
element. Here,
ξ
∈
, where
:
={
(
−
1
,
1)
×
(
−
1
,
1)
×
(
−
1
,
1)
}
is the bi-unit
cube and
N
j
are the isoparametric shape functions (having a value of “1” at
their specific node and varying to “0” at every other node). The gradients of the
variation
η
are discretized as
N
nodes
N
nodes
B
j
η
j
,
∇η
=
B
NL
j
(12.20)
∇
s
η
=
η
j
.
j
=
1
j
=
1
Where
B
L
and
B
NL
are the linear and nonlinear strain-displacement matrices,
respectively, in Voigt notation [1]. With the use of appropriate Voigt notation,
the linearized Eq. (12.15) can be written, for an assembled FE mesh, as:
N
nodes
N
nodes
N
nodes
(
K
R
(
ϕ
∗
)
+
K
I
(
ϕ
∗
))
ij
·
u
j
=
(
F
ext
(
ϕ
∗
)
+
F
int
(
ϕ
∗
))
i
(12.21)
i
=
1
j
=
1
i
=
1
Equation (12.21) is a system of linear algebraic equations. The term in paren-
theses on the left-hand side is the (symmetric) tangent stiffness matrix.
u
is
the vector of unknown incremental nodal displacements - for an FE mesh of
8-noded hexahedral elements in three dimensions,
u
has length [8
×
3
×
N
el
],
Where
N
el
is the number of elements in the mesh.
F
ext
is the vector of external
