Civil Engineering Reference
In-Depth Information
or taking the sum outside the integral
N
N
¦
¦
³
³
ˆ
(14.41)
cu
()
P
'
U
q
()
Q
dS Q
()
'
T
u
()
Q
dS Q
()
N
Nn
n
Nn
n
n
1
n
1
S
S
For each time step
N
we get an integral equation. In a well posed boundary value
problem either
u
or
q
is specified on the boundary and the values of
u
and
q
are known at
the beginning of the analysis (
t=
0). Furthermore the integral equation (11.41) must be
satisfied for any source point
P.
If we ensure the satisfaction at a discrete number of
points
P
i
then we can get for each time step
N
as many equations that are necessary to
compute the unknowns. Similar to static problems we specify the points
P
i
to be the
node points of the boundary element mesh (point collocation). To solve the integral
equation we introduce the discretisation in space of Chapter 3:
J
J
¦
¦
e
e
uQ
(
)
N u
;
qQ
(
)
N q
(14.42)
n
j
nj
n
j
nj
j
1
j
1
e
e
where
uq
refer to values of
u
and
q
at node
j
of element
e
at time step
n
and
N
j
are shape functions. Substitution of (14.42)
into (14.41) gives
uq
are pressure and pressure gradients at
Q
;
,
nn
,
nj
nj
N
E
J
N
E
J
¦¦¦
¦¦¦
(14.43)
e
e
e
e
cu
ˆ
(
P
)
'
U
q
'
T
u
N
i
ijNn
nj
ijNn
nj
n
1
e
1
j
1
n
1
e
1
j
1
where
³
e
ijNn
' '
U
U
()
P
N
J
dS
(14.44)
Nn
i
j
S
e
and
e
ijNn
³
' '
T
T
()
P
N
J
dS
(14.45)
Nn
i
j
S
e
J
is the Jacobian and
E
is the number of Elements.
If we define vectors
^`
n
u
and
^`
q
to contain all nodal values of pressure and
pressure gradient at the nodes at time increment
N
we can rewrite Equation (14.43) in
matrix form
N
N
¦¦
>@
^`
>@
^`
Tu
Uq
(14.46)
n
n
n
n
n
1
n
1
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