Civil Engineering Reference
In-Depth Information
For elasticity problems there is more than one unknown per node, so columns are
numbered according to the degree of freedom, rather than node number. For two-
dimensional elasticity problems, each node has two degrees of freedom and the
incidences of element 3 are expanded to
destinations
shown in table 7.2.
Table 7.2
Destinations
Element
Node 1
Node 2
x
y
x
y
1
1
2
3
4
2
3
4
5
6
3
5
6
7
8
4
7
8
9
10
5
9
10
11
12
6
11
12
13
14
7
13
14
1
2
For element 3 the destination vector is (/5,6,7,8/) and the assembly is
o
Destinatio
n
Numbers
1
2
3
4
5
6
7
8
!
ª
º
3
3
3
3
«
'
T
'
T
'
T
'
T
»
xx
11
yx
11
xx
21
yx
21
«
»
3
3
3
3
'
T
'
T
'
T
'
T
«
»
xy
11
yy
11
xy
21
yy
21
(7.6)
>@
«
»
3
3
3
3
'
T
'
T
'
T
'
T
'
T
«
xx
12
yx
12
xx
22
yx
22
»
3
3
3
3
«
»
'
T
'
T
'
T
'
T
xy
12
yy
12
xy
22
yy
22
«
»
#
#
#
#
#
#
#
#
#
#
#
#
#
#
«
»
¬
¼
Note that destination numbers are now used for numbering the columns.
Coming back to potential problems and assuming that, as in the introductory example
solved with the Trefftz method, the flux
t
is known on all boundary nodes and solution
u
is required, we assemble the left hand side, perform the matrix multiplication on the
right and solve the system of equations. Alternatively, multiplication ['
U
]{
t
} can be
made element by element at the assembly level, without explicitly creating the matrix
['
U
], therefore saving on storage space. This would also allow us to consider
discontinuous distribution of normal gradients or tractions. For the simple example in
Figure 7.1, equation (7.1) can be replaced by
>
^`^`
(7.7)
'
T
u
F
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