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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