1.40 Interpolation

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In some sense the FE method is structural analysis with polynomials, i.e.,

functions such as

u(x) = x + 3x2 u(x, y) = 1+xy + x5 y7 . (1.562)

Polynomials are very versatile functions, and easy to handle, but if the displacement

u is assumed to be zero in the first span and to increase linearly

in the second span, two distinct polynomials are needed. Interpolation with

piecewise polynomials, as in Fig. 1.143, is therefore the basic procedure of FE

analysis.

For a mathematician these hat functions, or more generally these nodal

unit displacements ϕi, are the real finite elements.16 The structural elements

are only considered a convenient tool to generate the nodal unit displacements,

the “real” finite elements.

Indeed the term finite element is not unique. When we speak of linear

elements we mean the shape functions. But when we speak of plate or shell

elements we mean the structural element.

The characteristic feature of the FE method is that the shape functions

have a finite support, because they are nonzero only over a small region of the

structure while the basis functions of a Fourier series such as

16 “The use of the concept finite element may seem deceptive. In principle we subdivide

the domain into elements, that is geometric objects, while by finite elements

we mean functions.” [51]

Fig. 1.142. A polygon

is a more stable

198 1 What are finite elements?

Fig. 1.143. Construction of a polygon from hat functions ϕi

wn(x) =

_n

k=1

_

ak · sin kπ x

l

+ bk · cos kπ x

l

_

, (1.563)

are everywhere oscillatory.

Hence, in this sense the FE method is actually a method of finite functions.

This is similar to the three-moment equation, where by a smart choice of

redundants the bandwidth of the flexibility matrix F = [δij] can be kept

small (Fig. 1.144 a). If instead all interior supports were removed (Fig. 1.144

b), the structure would be statically determinate as well but the flexibility

matrix

δ ij =

_ l

0

MiMj

EI

dx (1.564)

would be fully populated and certainly ill-conditioned, and therefore susceptible

to rounding errors, because—given that the number of spans is large—the

moments Mi and Mj , and thus the numbers δij of adjacent nodes, would be

nearly identical.

1.41 Polynomials 199

Shape functions ϕi that extend over the whole structure—like the deflection

curves of the redundants Xi = 1 in Fig. 1.144 b—are not a good choice for

the FE method. The overlap between the shape functions must be kept small.

In this sense the nodal unit displacements are a better choice, because they

are almost orthogonal, and hence lead to much better-conditioned systems of

equations.

The FE method could rightly be called an interpolation method if there

were not the problem that the function to be interpolated is not known. It

would not be claiming too much to say that

The whole theory of finite elements is only concerned with the question of

what the best choice for the unknown nodal deflections wi might be?

Here best does not necessarily mean that the interpolating function passes

through the nodes of the original curve, just that the difference between the

FE stresses and the true stresses is minimized. This is the difference between

a “normal” interpolation and an FE interpolation.

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