Biomedical Engineering Reference
In-Depth Information
Therefore Eq. (
2.58
) becomes,
ffi
¼ 2de
T
r
d e
T
r
ð
2
:
61
Þ
simplifying the second term in Eq. (
2.52
),
2
3
5
dt ¼
Z
2
3
Z
Z
Z
t
2
t
2
dX
1
2
4
4
5
dt
d e
T
r
de
T
rdX
ð
2
:
62
Þ
t
1
t
1
X
X
Equation (
2.52
) now becomes,
2
3
Z
q
Z
ffi
dX
Z
de
T
r dX
þ
Z
du
T
b dX
þ
Z
C
t
t
2
4
5dt ¼ 0
ð
2
:
63
Þ
du
T
u
du
T
t dC
X
t
1
X
X
To satisfy Eq. (
2.63
) for all possible choices of the integrand of the time
integration has to be null, leading to the following expression,
q
Z
ffi
dX
Z
de
T
r dX
þ
Z
du
T
b dX
þ
Z
du
T
u
du
T
t dC ¼ 0
ð
2
:
64
Þ
X
X
X
C
t
This last equation is known as the 'Galerkin weak form', which can also be
viewed as the principle of virtual work. The principle of virtual work states that if
a solid body is in equilibrium, the virtual work produced by the body inner stresses
and the body applied external forces should vanish when the body experiments a
virtual displacement. Considering the stress-strain relation, r ¼ c e, and the strain-
displacement relation, e ¼ Lu, Eq. (
2.64
) can be rearranged in the following
expression,
Z
d L
ð
T
cL
ð
dX
Z
du
T
b dX
Z
du
T
t dC
þ
Z
ffi
dX ¼ 0
ð
2
:
65
Þ
qdu
T
u
X
X
C
t
X
which is the generic Galerkin weak form written in terms of displacement, very
useful in solid mechanical problems. In static problems the fourth term of
Eq. (
2.65
) disappears.
2.3 Discrete System of Equations
The discrete equations for meshless methods are obtained from the principle of
virtual work by using the meshless shape functions as trial and test functions. The
domain X is discretized in a nodal distribution, and each node possesses an
