Biomedical Engineering Reference
In-Depth Information
''influence-domain'', which imposes the nodal connectivity between the neigh-
bouring nodes. The meshless trial function u
ð
x
I
Þ
is given by,
u
ð
x
I
Þ
¼
X
n
u
i
ð
x
I
Þ
u
i
ð
2
:
66
Þ
i¼1
being u
i
ð
x
I
Þ
the meshless approximation or interpolation function and u
i
are the
nodal displacements of the n nodes belonging to the influence-domain of interest
node x
I
. Considering the NNRPIM, it is known that the NNRPIM interpolation
function satisfies the condition,
u
i
ð
x
j
Þ
¼d
ij
ð
2
:
67
Þ
where d
ij
is the Kronecker delta, d
ij
¼ 1ifi ¼ j and d
ij
¼ 0ifi
6
¼ j. Following
Eq. (
2.66
), the test functions (or virtual displacements) are defined as,
du
ð
x
I
Þ
¼
X
n
u
i
ð
x
I
Þ
du
i
ð
2
:
68
Þ
i¼1
where du
i
are the nodal values for the test function.
2.3.1 Weak Form of Galerkin
The meshless formulation can be established in terms of a weak form of the
differential equation under consideration, Eq. (
2.64
). In the solid mechanics con-
text this implies the use of the virtual work equation.
L ¼
Z
r de dX
Z
b
du dX
Z
t
du dC
þ
q
Z
ffi
dX ¼ 0
du
T
u
ð
2
:
69
Þ
X
X
C
X
The virtual deformation de is defined by,
de ¼ Bdu
ð
2
:
70
Þ
where B is the deformation matrix. Thus, the virtual work of the first term in
Eq. (
2.69
), using Eq. (
2.70
), can be expressed as,
L
1
¼
Z
X
du
T
B
T
r dX
ð
2
:
71
Þ
The strain vector can be divided in two parts, the linear part and the nonlinear
part,
