1.4 Projection

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Work is a scalar quantity, as are temperature and pressure. This is nearly the

most important statement that can be made about work. Work is force ×

displacement. Work and energy are the same. The integral

1

2

_ l

0

T2

H

dx , T = Hw

_

, (1.21)

is the internal energy of the rope. It measures the strain energy stored in the

rope.

Energy can also serve as a scale. It is the scale FE methods work with.

Having a scale means having a topology, which in turn defines “far away” and

“nearby”. To measure the length of a vector the Euclidean norm is used:

|x| =

 

x21

+ x22

+ x23

. (1.22)

1.4 Projection 9

Fig. 1.7. All vectors have the same

shadow x

_

In this topology two cities A and B are close neighbors if the difference between

their position vectors a and b (with reference to the origin of a map) is small:

|a − b | “small” =⇒ A and B are neighbors. (1.23)

Projections only make sense if distances can be measured. The shadow x_ of a

3-D vector x is the vector in the plane which has the smallest distance to the

tip of x; see Fig. 1.6. The distance between the original vector and its shadow

is the length of the vector

e = x − x

_

, (1.24)

which points from the tip of the shadow to the tip of the vector x. The shadow

x_ renders this distance a minimum

|e| =

 

(x1 − x_

1)2 + (x2 − x_

2)2 + (x3 − 0)2 = minimum. (1.25)

Any other vector ˜x

_ in the plane has a greater distance from the vector x

|˜e| = |x −˜x

_| > |e| = |x − x

_| . (1.26)

This is the first feature of a projection: the shadow solves a minimum problem.

The second feature is that the residual vector, the error e, is orthogonal to

the x1−x2-plane (assuming that the sun shines from straight above), because

the scalar product between the error and the shadow is zero:

10 1 What are finite elements?

eTx

_ = 0. (1.27)

This is equivalent to saying that the shadow of the error e has no physical extent,

but only if the line of sight coincides with the direction of the projection!

Seen from any other direction the length of e is not zero. Hence a projection

method is blind with respect to errors which lie in the line of sight. All vectors

˜x

that lie “above” the vector x, which differ from x only by an additive term

The third feature is that the result of a projection cannot be improved.

Repeating a projection changes nothing: the shadow of the shadow is the

shadow. Which means that a projection method freezes after the first step,

while other operations, such as squaring a number, can be repeated infinitely

often.

The fourth feature of a projection is that the length of the shadow is

shorter than the length of the original vector; see Fig. 1.7. This is not only

true for vectors, but also for functions: the Fourier series fn(x) of a function

f(x) is the projection of f(x) onto the trigonometric functions in the sense

2

(= 2 n is less than the L2-norm of f:

||fn|| 0 = [

_ l

0

f2

n(x) dx]1/2 ≤ [

_ l

0

f2(x) dx]1/2 = ||f|| 0 . (1.28)

All this applies now to the FEmethod as well: the exact deflection curve w ∈ V

is projected onto a subspace Vh, and the shadow wh is the FE solution.

In the case of the rope the space Vh contains all the deformations which

are expansions in terms of the three unit displacements ϕi(x),

wh(x) = w1 · ϕ1(x) + w2 · ϕ2(x) + w3 · ϕ3(x) , (1.29)

and the FE solution is the solution of the following minimum problem:

Find the deflection

wh(x) = w1 · ϕ1(x) + w2 · ϕ2(x) + w3 · ϕ3(x) (1.30)

in Vh which has the shortest distance (= strain energy) from the exact deflection

w.

In FE analysis the strain energy is usually expressed

a(w,w) :=

_ l

0

H (w

_)2 dx =

_ l

0

T2

H

dx . (1.31)

If

e(x) = w(x) − wh(x) (1.32)

Fig. 1.7.

parallel to the line of sight (i.e., projection), have the same shadow; see

L -norm) of the Fourier series f

of the L -scalar product, and according to Bessel’s inequality the length

1.4 Projection 11

is the error of the FE solution, then the FE solution is that function in Vh for

which the strain energy of the error e(x) becomes a minimum:

a(e, e) =

1

2

_ l

0

(T − Th)2

H

dx = minimum. (1.33)

Any other function wh in Vh has a larger distance—in terms of energy—than

the FE solution. This property of the FE solution wh can also be expressed

as follows, see (7.413) p. 572,

a(e, e) ≤ a(w − vh, w − vh) for all vh ∈ Vh . (1.34)

We also know that the strain energy of the FE solution is always less than

the strain energy of the exact solution:

a(wh, wh) =

_ l

0

T2

h

H

dx <

_ l

0

T2

H

dx = a(w,w) , (1.35)

h

0 < a(w,w) = a(wh + e,wh + e)

= a(wh, wh) + 2 a(e,wh)

_ _ _

=0

+a(e, e)

_ _ _

>0

, (1.36)

where

a(e,wh) =

_ l

0

(T − Th) Th

H

dx = 0 (1.37)

is a consequence of the Galerkin orthogonality

a(e, ϕi) = 0 i = 1, 2, 3 (1.38)

i.e., the fact that the error e is orthogonal in terms of the strain energy to all

unit displacements ϕi, and therefore also to wh = w1 · ϕ1 + w2 · ϕ2 + w3 · ϕ3.

Hence the strain energy or internal energy is the metric FE methods work

with. Distance is measured in this metric and therefore also convergence.

The internal energy induces a topology on the space V which is even a

norm on this space, because it separates the elements of V . Two functions w1

and w2 are identical if and only if their distance in terms of the strain energy

is zero:

1

2

_ l

0

(T1 − T2)2

H

dx =

1

2

_ l

0

H (w

_

1

− w

_

2)2 dx = 0 ⇔ w1 = w2 (1.39)

that is if w1 − w2 has zero energy.

inequality follows directly from

i.e., the shadow w has a shorter length (= strain energy) than w. This

12 1 What are finite elements?

Fig. 1.8. A small deflection

curve can hide a large strain

energy

A function w is small in this metric if its energy (essentially the square of

the first derivative) is small, and the exact deflection w and the FE solution

wh are close in this metric if the strain energy of the error

e(x) = w(x) − wh(x) (e = error) (1.40)

is small

1

2

_ l

0

T2

e

H

dx =

1

2

_ l

0

H(w

_ − w

_

h)2 dx = small =⇒ e(x) = small . (1.41)

This energy metric makes more sense than a naive metric that considers a

function such as w(x) = sin(8π x) a “small” function (see Fig. 1.8), while for

the FE method it is a “large” function, because the strain energy due to the

rapid oscillations is large

_ 1

0

w(x)2 dx = 0.5 ,

1

2

_ 1

0

Hw

_(x)2 dx = 316 · H . (1.42)

Hence from an engineering standpoint it makes more sense to classify functions

with regard to the strain energy than their amplitude or their L2-norm.

A better strategy would it be to base the metric on both components, the

zero-order and the first-order derivative. This leads to the so-called Sobolev

norms, which, depending on the index n, measure the derivatives up to order

||w||n =

__ l

0

_

w(x)2 + w

_(x)2 + . . . + w(n)(x)2

_

dx

_1/2

(1.43)

and classify functions according to this metric. By increasing the index n

different topologies can be generated on V . In the same way the distance

between two vectors does not depend on the difference of the first two components

alone, |a−b| =

_

(a1 − b1)2 (which would be a very crude topology)

but on the difference of all components

n

1.5 The error of an FE solution 13

the FE program can represent

constitute a subset Vh of V

|a − b| =

_

(a1 − b1)2 + (a2 − b2)2 + . . . + (an − bn)2 . (1.44)

This metric generates the finest possible topology, just as in a lottery the prize

money increases, the more figures on a ticket agree with the number drawn.

Remark 1.1. Later it will be seen that in so-called load cases δ when displacements

are prescribed the projection is no longer orthogonal but “skew” this

implies that the length of the shadow (the strain energy) will be greater than

the strain energy of the exact solution; see Sect. 1.38, p. 187. This is to be

expected: the stiffer a structure the greater the strain energy developed by

displacing a support.

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