Civil Engineering Reference
In-Depth Information
For computation of
Dirichlet
problems we use equation (7.3), with a modified right
hand side
> @^ ` > @^ `
1
'
Ut
'
Tu
(11.1)
1
Here
> @ > @
''are the assembled coefficient matrices,
^`
1
,
TU
t
is the first column of the
stiffness matrix
K
M
and
^ `
1
u
is a vector with a unit value in the first row ,i.e
1
0
0
-½
°
°°
®¾
°°
°
¯¿
^`
1
u
(11.2)
#
If we perform the multiplication of
> @^`
1
T
' it can be easily seen that the right hand
side of equation (11.1) is simply the first column of matrix
> @
' . The computation of
the region “stiffness matrix” is therefore basically a solution of
> @^` ^ `
i
'
Ut
F
, with
i
N
right hand sides
^`
i
, where each right hand side corresponds to a column in
> @
F
'
.
Each solution vector
^`
i
t
represents a column in
K
, i.e.,
^` ^`
1
N
ª
º
K
¬
t
t
"
(11.3)
¼
2
For each region
(N)
we have the following relationship between {
t
} and {
u
}:
N
N
^`
^`
N
(11.4)
t
K
u
To compute, for example, the problem of heat flow past an isolator, which is not
impermeable but has conductivity different to the infinite domain, we specify two
regions, an infinite and a finite one, as shown in figure 11.2. Note that the outward
normals of the two regions point in directions opposite to each other (Figure 11.3). First
we compute matrices
K
I
and
K
II
for each region separately and then we assemble the
regions using the conditions for flow balance and uniqueness of temperature in the case
of potential problems and equilibrium and compatibility in the case of elasticity. These
conditions are written as
I
II
I
II
^` ^` ^`
^` ^`
(11.5)
t
t
t
;
u u
The assembled system of equations for the example in Figure 11.2 is simply:
^`
^`
I
II
t
KK
u
(11.6)
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