Over each element the functions for u and p can be approximated
using the following interpolation functions:
U = 0
Note that the functions for u and p are generally different. For
consistence it may be better to take the functions for p of one order
less than those for u. The interpolation functions are the ones
discussed in Chapter 3. u
are the nodal displacements and
We can substitute the above functions into (5.37) and obtain the
following equation for the point i:
c,u,+ Σ (| P*4>
= Σ (f
where the summation from j = 1 to N indicates summation over the
N elements on the surface and Γ, is the surface of the) element. The
summation from; = 1 to M is carried out over the internal cells and
is the volume of each of them.
The integrals in equation (5.40) are usually solved numerically and
the functions Ψ and φ are expressed in some of the homogeneous
system of coordinates described in Chapter
The integrals need then
to be written in terms of the homogeneous system, to be called ξ, η.
dr = (absolute value of |G|) άξάη
where G has been defined previously.
Due to the difficulties in integrating equation (5.40) it is usual to