Consider a l-D elastic bar problem defined on [0, 4]. The domain is devided into
ID: 1843068 • Letter: C
Question
Explanation / Answer
Using the nodal-based smoothing operation, the strains to be used in Equation (23) is assumedto be the
smoothed
strain for node
k
dened by
e
k
e
(
x
k
)
=
k
e
(
x
)
W
(
x
x
k
)
d
where
W
=
W
W
W
is a diagonal matrix of smoothing function
W
. For simplicity, the smoothingfunction is taken as
W
(
x
x
k
)
=
1
/
A
k
,
x
k
0
,
x
/
k
where
A
k
=
k
d
is the area of smoothing domain for node
k
.Substituting into Equation and integrating by parts, the smoothed strain canbe calculated using
e
k
=
1
A
k
k
e
(
x
)
d
=
1
A
k
k
L
n
u
(
x
)
d
=
e
k
(
u
)
where
k
is the boundary of the smoothing domain for node
k
, and
L
n
is the matrix of the outwardnormal vector on
k
. Equation states the fact that the assumed strain
e
k
is a function of theassumed displacement
u
.Substituting Equation into Equation , the smoothed strain can be expressed in thefollowing matrix form of nodal displacements:
e
k
=
i
N
in
B
i
(
x
k
)
d
i
where
N
in
is the number of nodes in the inuence domain of node
k
(including node
k
). Whenlinear shape functions are used, it is the number of nodes that is directly connected to node
k
inthe triangular mesh (see Figure 1). In Equation , the
B
i
(
x
k
)
is termed as the
smoothed
strainmatrix that is calculated using
B
i
(
x
k
)
=
b
ix
(
x
k
)
00
b
iy
(
x
k
)
b
iy
(
x
k
)
b
ix
(
x
k
)
Using the Gauss integration along each segment of boundary
k
, we have
b
il
=
1
A
k N
s
m
=
1
N
g
n
=
1
w
n
i
(
x
mn
)
n
l
(
x
m
)
(
l
=
x
,
y
)
(30)where
N
s
is the number of segments of the boundary
k
,
N
g
is the number of Gauss points usedin each segment,
w
n
is the corresponding weight number of Gauss integration scheme, and
n
l
isthe unit outward normal corresponding to each segment on the smoothing domain boundary. In theLC-PIM using linear shape functions,
n
g
=
1 is used. The entries in sub-matrices of the stiffnessmatrix
K
in Equation are then expressed as
K
ij
=
N
k
=
1
K
ij
(
k
where the summation means an assembly process as we practice in the FEM, and
K
ij
(
k
)
is thestiffness matrix associated with node
k
that is computed using
K
ij
(
k
)
=
k
B
T
i
D
B
j
d
=
B
T
i
D
B
j
A
k
The entries (in sub-vectors of nodal forces) of the force vector
f
in Equation can be simplyexpressed as
f
i
=
k
N
in
f
i
(
k
)
The above integration is also performed by a summation of integrals over smoothing domains;hence,
f
i
is an assembly of nodal force vectors at the surrounding nodes of node
k
:
f
i
(
k
)
=
t
(
k
)
U
i
ˆ
t
d
+
(
k
)
U
i
b
d
Note again that the force vector obtained in LC-PIM is the same as that in the FEM, if the sameorder of shape functions is used. Therefore, it is shown again that there is no difference between
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.