Biomedical Engineering Reference
In-Depth Information
Z
r
r
ð
S
w
Þ
dV
¼
Z
X
S
ð
w
r
r
Þ
dV
þ
Z
X
ðr
r
S
Þ
w dV
X
Z
S
ð
w
r
r
Þ
dV
þ
Z
w
ðr
r
S
Þ
dV
:
ð
3
:
350
Þ
X
X
In equation (
3.350
), employing integration by parts reduces the continuity of
the variable u, i.e. the derivatives of u are first order (cf. from (
3.347
) and (
3.351
)
that the expression
r
r
S has been reduced to S).
Substitution of (
3.350
)in(
3.349
) and using G
AUSS
's theorem to express
r
r
ð
w
S
Þ
and using C
AUCHY
's stress theorem to introduce the von N
EUMANN
boundary
condition (
3.93
), leads to the weak form of the boundary value problem in the
current configuration
Z
S
ð
w
r
r
Þ
dV
Z
t
w dA
Z
w
k dV
þ
Z
w
v q dV
¼
0
ð
3
:
351
Þ
X
oX
X
X
where the surface integral is integrated only over the boundary surface oX
:
Whereas (
3.347
) requires strict fulfillment at every material point of the body,
the weaker form (
3.351
), needs to be satisfied for the entire body in a global sense.
In structural analysis, replacing w with the virtual displacement field du (
3.351
)
leads to the principle of virtual work
Z
S
ð
du
r
r
Þ
dV
Z
t
du dA
Z
du
k dV
þ
Z
du
v q dV
¼
0
: ð
3
:
352
Þ
X
oX
X
X
Equation (
3.352
) is formulated for the whole body.
3.3.4 Approximation of the Solution
The considered continuum body domain X is discretized in a finite number of finite
elements
X
X
h
¼
[
n
X
e
ð
3
:
353
Þ
e
¼
1
where n is the total number of elements and X
h
denotes the approximated body
domain; the summation symbol is replaced to indicate the process of assembly of
the single element contributions.
