Civil Engineering Reference
In-Depth Information
However (3.58) now involves numerical evaluation of integrals involving 1
/r
(held in
the [
B
] matrix, see 2.76) which do not have simple polynomial representations. Therefore,
in contrast to plane problems, it will be impossible to evaluate (3.60) exactly by numerical
means, and accuracy may deteriorate as
r
approaches zero. Provided integration points
do not lie on the
r
=
0 axis, however, reasonable results are usually achieved using a
similar order of quadrature to that used in plane analysis. Customised numerical integration
schemes for axisymmetric elements are available (Griffiths, 1991), but are not used in
this text.
3.7.6 Plane steady laminar fluid flow
It was shown in (2.126) that a fluid element has a “stiffness” or conductivity matrix defined
in 2D by,
[
T
]
T
[
K
][
T
] d
x
d
y
[
k
c
]
=
(3.61)
and the similarity to (3.40) is obvious. The matrix
deriv
simply contains the derivatives
of the element shape functions with respect to
(x, y)
which were previously needed in
the analysis of solids and formed by the sequence (3.47) to (3.48), while the constitutive
matrix [
K
] (called
kay
in program terminology) contains the permeability (or conductivity)
properties of the element in the form
k
x
0
0
k
y
kay
=
(3.62)
Numerical integration of the conductivity matrix in planar problems is completed by
the sequence,
dtkd=MATMUL(MATMUL(TRANSPOSE(deriv),kay),deriv)
nip
(3.63)
kc =
det
i
*weights(i)*dtkd
i
i
=
1
By comparison with (3.51) it will be seen that these physically very different problems
are likely to require similar solution algorithms.
3.7.7 Mass matrix formation
The mass matrix was shown in Chapter 2, for example (2.71), to take the general form
=
ρ
[
N
]
T
[
N
] d
x
d
y
[
m
m
]
(3.64)
where [
N
] holds the shape functions.
