Biomedical Engineering Reference
In-Depth Information
To approximately describe the variation of the field variable, for example the
nodal displacement field u, within the finite element, interpolation functions,
mostly polynomials, are introduced which are based on the displacement values of
the element nodes. Requirements of such interpolation functions are that they are
continuous over the finite element, differentiable as required by the weak form, and
complete, i.e. include all lower-order terms. Furthermore, they should be differ-
entiable as required by the weak form, and complete, i.e. include all lower-order
terms, and finally, they should meet the values at the element nodes.
Within the isoparametric concept, the same shape functions for interpolating
the geometry and the field variables are employed:
X
e
¼
X
x
e
¼
X
n
n
N
i
ð
X
Þ
X
i
N
i
ð
X
Þ
x
i
:
ð
3
:
354
Þ
and
i
¼
1
i
¼
1
From that, for a finite element, the displacement field u within the element is
approximated by
u
ð
X
Þ
u
h
ð
X
Þ¼
x
e
X
e
¼
X
n
N
i
ð
X
Þ
u
i
ð
3
:
355
Þ
i
¼
1
and
u
i
¼
x
i
X
i
ð
3
:
356
Þ
in accordance to (
3.48
) where N
i
ð
X
Þ
is the shape or interpolation function at the i-
th node, and u
i
is the unknown corresponding displacement vector of the i-th node,
and n denotes the number of nodes of the element.
Virtual variation of the displacement field approximation of (
3.355
) leads to
du
ð
X
Þ
du
h
ð
X
Þ¼
X
n
N
i
ð
X
Þ
du
i
:
ð
3
:
357
Þ
i
¼
1
Using the approximated displacement field solution (
3.357
), and substituting
in (
3.352
), the discrete approximation for the equilibrium for a finite element is
given by
0
¼
Z
dV du
i
Z
oX
e
S
X
t
X
n
n
o
N
i
ð
X
Þ
ox
N
i
ð
X
Þ
dA du
i
i
¼
1
i
¼
1
X
e
ð
3
:
358
Þ
Z
N
i
ð
X
Þ
k dV du
i
þ
Z
X
e
X
n
X
n
N
i
ð
X
Þ
v qdV du
i
i
¼
1
i
¼
1
X
e
