Biomedical Engineering Reference
In-Depth Information
17
Shape functions and numerical
integration
17.1
Introduction
In the previous chapter the shape functions
N
i
have hardly been discussed in any
detail. The key purpose of this chapter is first to introduce isoparametric shape
functions, and second to outline numerical integration of the integrals appearing
in the element coefficient matrices and element column. Before this can be done
it is useful to understand the minimum requirements to be imposed on the shape
functions. The key question involved is, what conditions should at least be satisfied
such that the approximate solution of the boundary value problems, dealt with in
the previous chapter, generated by a finite element analysis, converges to the exact
solution at mesh refinement. The answer is:
(i) The shape functions should be smooth within each element
e
, i.e. shape functions
are not allowed to be discontinuous within an element.
(ii) The shape functions should be continuous across each element boundary.
This con-
dition does not always have to be satisfied, but this is beyond the scope of the present
topic.
(iii) The shape functions should be complete, i.e. at element level the shape functions
should enable the representation of uniform gradients of the field variable(s) to be
approximated.
Conditions
(i)
and
(ii)
allow that the gradients of the shape functions show finite
jumps across the element interface. However, smoothness in the element inte-
rior assures that all integrals in which gradients of the unknown function, say
u
,
occur can be evaluated. In Fig.
17.1
(a) an example is given of an admissible shape
function. In this case the derivative of the shape function is discontinuous over
the element boundary, however the jump is finite. In Fig.
17.1
(b) the discontinu-
ous shape function at the element boundary leads to an infinite derivative and the
integrals in the weighted residual equations can no longer be evaluated.
Completeness
When the finite element mesh is refined further and further, at
the element level the exact solution becomes more and more linear in the coordi-
nates and its derivatives approach constant values in each element. To ensure that
