Civil Engineering Reference
In-Depth Information
The integral equation is given by
³
ˆ
(14.58)
cu
(,)
Pt
[ (,, ,) (,)
U
PQt
W
t
Q
W
T
(,, ,)
PQt
W
u
(, ]
Q
W
S
S
where
U
and
T
are matrices containing the fundamental solutions.
14.4.3
Numerical implementation
For the solution of the integral equation we discretise the problem in time as well as in
space as for the scalar wave equation. If we discretise the total time into equal (arbitrary
small) steps of size
'
then we have
N
N
¦
¦
u
(,)
Qt
N t
()
u
() ; (,)
Q
t
Qt
N t
()
t
()
Q
(14.59)
n
n
n
n
n
1
n
1
Following the steps for the scalar problem and assuming a constant shape function we
obtain the discretised integral equation for time step
N
as
N
N
¦
¦
(14.60)
³
³
ˆ
cu
()
P
'
U
t
()
Q
dS Q
()
'
T
u
()
Q
dS Q
()
N
Nn
n
Nn
n
n
1
n
1
S
S
where
t
t
n
n
³
³
'
U
U
(,,, ) ;
PQt
W
d
W
'
T
T
(,,, )
PQt
W
d
W
(14.61)
Nn
N
Nn
N
t
t
n
1
n
1
Introducing the space discretisation
J
J
¦
¦
u
(
QN
)
u
;
t
(
QN
)
t
(14.62)
n
j
nj
n
j
nj
j
1
j
1
,
ee
nj
where
ut
are displacements and tractions at
Q
,
,
nn
ut
refer to values of
u
and
t
at
nj
node
j
of element
e
at time step
n
and
N
are shape functions. Substitution of (14.62)
into (14.60) gives
N
J
N
J
E
E
¦¦¦
¦¦¦
e
e
e
e
(14.63)
ˆ
(
P
)
'
'
cu
U
t
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
()
PN J S
(14.64)
Nn
i
j
S
e
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