1.5 The error of an FE solution

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• the deflection w

• the vertical force T = Hw_

• the load p = −Hw__

i.e., the zero-order, first-order, and second-order derivative of the deflection

w. All three derivatives of w are relevant to the structural analysis, and hence

it is legitimate to ask which of the three errors

Fig. 1.9. The shapes which

The FE method is an approximate method, see Fig. 1.9. As such it must

approximate three functions:

14 1 What are finite elements?

Fig. 1.10. The error in the displacement is zero at the nodes, while the error in

the stresses is zero at the midpoints of the elements. This is a typical pattern in FE

analysis

w −wh error in the deflection

T − Th error in the internal action

p− ph error in the load

is to be minimized? In principle we have already given the answer. The FE

solution aims at minimizing the square of the error of the internal action

T − Th,

_ l

0

(T − Th)2

H

dx =

_ l

0

H(w

_ − w

_

h)2 dx → minimum. (1.45)

Hence an FE solution does not tend to win a beauty contest by imitating

the original shape w as closely as possible nor does it aim to simulate the

loading; rather, the solution tends to minimize the error in the strain energy

(the internal energy).

The load case ph

A closer study of the FE solution reveals that wh is the equilibrium position

of the rope if the distributed load were concentrated at the nodes, fi = p le.

This load case is called the FE load case ph, (see Fig. 1.10).

1.6 A beautiful idea that does not work 15

Of course we would like to know what the consequences are. How far are

the results of the load case ph (= nodal forces) from p (= distributed load)?

Stated otherwise: given the error in the load

r := p − ph (residual forces) (1.46)

how large is the error in the vertical force

Te := T − Th (1.47)

and the difference in the deflection

e := w − wh ? (1.48)

In other words what can be said about the error in the first-order, T − Th =

H(w−w_

h), and zero-order derivative, w−wh, if the error in the second-order

derivative p − ph is known?

The normal procedure is to differentiate the deflection w, yielding the

vertical force T, and to differentiate T to find the load p

w ⇒ T = Hw

_ ⇒ p = −Hw

__

. (1.49)

In a reverse order, the functions must be integrated

w =

__

− p

H

dx dx ⇐ T =

_

−p dx ⇐ p = −Hw

__ (1.50)

and integration smoothes the wrinkles; see Fig. 1.10.

But is there a reliable method to make predictions about the distance

in the first-order derivatives by looking at the distance in the second-order

derivative? The answer is no. Otherwise it would suffice to calculate an approximate

solution on a coarse mesh, and extrapolate from this solution to

the exact solution. In general this seems not to be possible, certainly not in

one step. There exist only different techniques which provide upper or lower

bounds for the error. The development of such error estimators is the subject

matter of adaptive methods.

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