Civil Engineering Reference
In-Depth Information
where
>
N
is the assembled left hand side and
^`
o
B
F
contains the right hand side
due to given
boundary conditions for region
N
. Vector
^`
co
t
contains the heat flow
at the coupled nodes and vector
^`
f
x
either temperatures or heat flow at the free
nodes of region
N
, depending on the boundary conditions prescribed (
Dirichlet
or
Neumann
).
2.
Solution of system with unit values applied at the interface nodes
The second problem to be solved for each region is to obtain the solution due to
Dirichlet
boundary condition of unit value applied at each of the interface nodes in
turn and zero prescribed values at the free nodes. The equations to be solved are
-
N
½
>@
^`
^`
t
°
°
N
cn
^`
(11.8)
B
F
n
1, 2
!
N
®
¾
c
N
fn
n
x
°
°
¯
¿
where
^`
n
F
is the right hand side computed for a unit value of
u
at node
n.
The
vector
^`
cn
t
contains the heat flow at the coupled nodes and
^`
f
x
the temperature
or heat flow at the free nodes, for the case of unit
Dirichlet
boundary conditions at
node
n
.
N
c
equations are obtained where
N
c
is the number of interface nodes in the
case of the potential problem (in the case of elasticity problems it refers to the
number of interface degrees of freedom). Note that the left hand side of the system
of equations, [
B
]
N
, is the same for the first and second problem and that
^`
n
simply corresponds to the
n
th
column of
> @
F
' .
After the solution of the first two problems
^`
c
and
^`
f
t
x
can be expressed in terms of
^`
c
u
by:
-
^`
^`
N
½ -
^`
^`
N
½
t
t
ª
N
º
K
A
°
° °
°
c
c
0
N
^`
«
u
»
®
¾ ®
¾
c
N
N
N
(11.9)
«
»
°
x
° °
x
°
¬
¼
¯
¿ ¯
¿
f
f
0
where
^`
c
u
contains the temperatures at the interface nodes of region
N
and the
N
N
matrices
K
and
A
are defined by:
N
N
ª
º
ª
º
^`
^`
^`
^`
(11.10)
N
N
K
t
"
t
;
A
x
"
x
«
»
«
»
¬
c
1
cN
¼
¬
c
1
cN
¼
c
c
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