Biomedical Engineering Reference
In-Depth Information
Since the elastic properties are constant across the finite element, Eq. (5.71) be-
comes
1
(
{ }
T
()
e
(5.72)
U
=
VB
[
] [
D B
δ
][
])
2
Using total potential energy theorem for an element (see Hutton 2005) the resulting
matrix relation becomes
{ } { }
T
VB
[][ []
[]
D B
δ
=
f
(5.73)
{ } { }
k
δ
=
f
The element volume V is equal to thickness
t
multiplied by area,
A
. The stiffness
matrix, [
k
] becomes
β
0
0
α
1
1
β
α
2
2
1
v
0
βββ
000
123
β
0
α
Et
3
3
[]
k
=
v
1
0
0 0 0
ααα
(5.74)
123
2
0
αβ
4
Av
(1
−
)
11
00
1
−
v
αααβββ
123123
0
αβ
αβ
22
2
0
33
and {
f
} is the column matrix of the applied forces
f
f
f
f
f
f
1
x
2
x
3
x
{}
=
f
1
y
2
y
3
y
5.5
Numerical Solution of Algebraic Systems
A system of linear or non-linear algebraic equations is produced from the discretisa-
tion, and this is solved by numerical methods. The complexity and size of the set of
equations depends on the dimensionality and geometry of the physical problem. In
this section we present two types of numerical methods:
direct methods
and
itera-
tive methods
.
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