78 HIGHER-ORDER ELEMENTS
Interpolation functions for these and other three-dimensional
elements can be seen in Connor and Brebbia.
1
3.6 ORDER OF INTERPOLATION FUNCTIONS
We must always ensure that the expression for u in terms of the
curvilinear coordinates contains a complete polynomial in x, y,
z.
That
is by suitably specialising the nodal potentials,
" =
ΣΜ
(
3
·
54
)
we can reproduce
u = c
1
+ c
2
x + c
3
y + c
4
z (3.55)
If the nodal potentials are taken according to (3.55) and substituted
into (3.54) we find,
" = Σ Φ&ι +
c
2
x
i + Wi + <^ΐ) (3.56)
Note that in order to be able to reproduce (3.55) everywhere in the
element we have to have,
Σ
Φι
= 1 (3.57)
and
Σ *ιΦι
=
χ>
Σ
yi<t>i = y>
Σ
2
A·
= z
(3.58)
Relationship (3.57) is satisfied by any interpolation function, but
equations (3.58) are only satisfied if the order of
u
is at least the same
as the order of the x, y, z functions.
The same restriction applies to finite elements as well as boundary
ones.
REFERENCES
1.
Connor, J. J. and Brebbia, C. A., Finite Element Techniques for Fluid Flow,
Newnes-Butterworths, London, (1976)
2.
Brebbia, C. A. and Wrobel, L., The Boundary Element Method'. In Computer
Methods in Fluids, Ed. C. Taylor, Pentech Press, (1979)
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