1.50 Warning

Back

We want to close this introductory chapter with a warning note. A reviewing

engineer who has followed our analysis up to this point is perhaps now tempted

to request that in the future the design engineer document the FE load case

ph so that he, the reviewing engineer, can quickly check how close the FE

solution is to the true solution. Theoretically such a proposition makes sense,

but on the other hand one must be careful not to overinterpret the FE load

case ph.

In real structures the load case ph is seemingly miles away from the original

load case p. If we really take the trouble to plot the load case ph, we

are surprised how little the FE load case and the original load case have in

common. This is why commercial FE programs do not show the load case ph.

Any user not well acquainted with FE theory would doubt the FE results.

The real secret of the FE method is that nonetheless the results are accurate,

and if we as structural engineers want to have more faith in FE methods,

we must deal with this question more intensively. Foremost this is a problem

of structural mechanics and not of mathematics.

What do we mean by near and far in structural mechanics? What level of

uncertainty can we allow, and when would we lose focus?19

What we see, if we concentrate on plate bending problems, and what we

can compare are the loads, the load case p, and the load case ph, i.e., (we

simplify somewhat) the fourth-order derivatives of the two deflection surfaces

wh and w. But the bending moments are the second-order derivatives

m =

_ _

pdΩ dΩ , mh =

_ _

ph dΩ dΩ , (1.665)

and because integration smoothes out the wrinkles, the bending moments

of the FE solution are in relatively good agreement with the exact bending

moments. No reviewing engineer has the tools to make guesses about the

deviations in the bending moments by looking at the discrepancies in the

load, the fourth-order derivatives.

If on the uppermost floor, the fourth floor, the walls deviate by 20 cm

from their position, how large then is the deviation on the second floor? The

19 Read the paper by B¨urg and Schneider [55] on the design of a simple flat garage

roof by 32 different professional engineers!

236 1 What are finite elements?

Fig. 1.173. Beam: a) load case p, b) load case ph and nodal unit displacements

deviation on the fourth floor we can measure. The deviation on the second

floor we cannot. About the latter we can only speculate. This is the problem.

There are examples in mathematics which should warn us. A linear system

of equations Ku = f can only be solved approximately on a computer. It

seems then that the residual r = Kuc −f is an appropriate measure for the

error of the computed solution uc. But there are examples where a solution uc

with a larger residual is closer to the exact solution than a computer solution

with a smaller residual.

Fortunately such ill-conditioned problems are rare in structural mechanics.

Most of our problems are well behaved. We solve elliptic boundary value

problems and we may assume that a small residual (probably) indicates a

small error. Instinctively we also rely on St. Venant’s principle.

In boundary element analysis the residual forces p − ph are zero because

the BE solution satisfies the differential equation. Hence it seems that BE

solutions should be more accurate than FE solutions. They are in general but

only by a rather small margin. If we compare the stress resultants between

a BE solution and an FE solution, then we are surprised to see how good

for example the agreement between the bending moments of an FE solution

and a BE solution are. This is the point where one begins to speculate: how

significant are the residual forces p − ph really? Is the whole interpretation

not too naive?

Yes and no. The FE solution is the solution of a load case ph. This interpretation

is correct, and as demonstrated by adaptive methods it makes sense

1.50 Warning 237

to refine the mesh in regions where the residual forces p−ph are large. Could

it then be that we are looking at the wrong data because an FE program does

not focus on the distance p−ph but on the distance in the strain energy? No,

the data we inspect are correct, only an FE program looks at the data from

a slightly different perspective.

As engineers, we tend to say the distance is large if the distributed loads

p and ph deviate by a large margin. But not so the FE program. It does not

compare the shape of p and of ph, it does not measure the gap |p − ph|, but

it measures the action.

The FE program lets the residual forces p−ph act through the nodal unit

displacements ϕi, and only if the residual forces are orthogonal

_ l

0

(p − ph) ϕi dx = 0, i= 1, 2, . . . , n (1.666)

to all functions ϕi is the distance p − ph well tuned.

Imagine that a triangular area (p) is to be covered with a stack of needles

(point forces fi). Calculus provides no tool to measure the distance between

the triangular area and the area covered by the needles, because (mathematical)

needles are infinitely thin. But FE analysis offers an ingenious way to

give a meaning to it by applying the principle of virtual displacements.

The beam problem in Fig. 1.173 may exemplify this. In the naive sense,

ph will never converge to the triangular load p, because ph = 0. But the nodal

forces fi and moments which are the real ph do converge to p in the finite

element sense, i.e., in the weak sense

_ l

0

pϕi dx =

∞_

j=1

fj ϕi(xj) i = 1, 2, . . . ,∞. (1.667)

Hence we may not take the residual forces prima facie to insist on a good match

between p and ph. In FE analysis, action is what counts—we approximate

actions not functions (!)—and the action we do not see on the screen.

2. What are boundary elements?

The boundary element method (BE method) is an integral equation method,

or as we could say as well an influence function method. It is based on the

fact that in linear problems the boundary values uniquely determine the displacements

and stresses inside a structure such as the frame in Fig. 2.1, so

that it suffices to discretize the edge with boundary elements only.

In one form or the other the idea is applied each day. If for example two

moments, M(0) and M(l), act at the ends of a beam the bending moment

M(x) can be generated from these boundary values with the help of a simple

ruler; see Figure 2.2. If the beam carries also a distributed load p, then a

curved ruler is used instead, see Fig. 2.2.

The key to this technique is that the linear bending moment M(x) =

ax+b is a homogeneous solution of the equation −M__ = 0, and the quadratic

bending moment is a particular solution of the differential equation −M__ = p.

Technically the BE method applies influence functions (= integrals) to

relate the displacements and tractions on the boundary with the displacements

and stresses at internal points.

Fig. 2.1. The boundary values w,w

_

,M, V and the distributed load p suffice to

calculate any value in the interior

240 2 What are boundary elements?

Fig. 2.2. Influence functions for bending moments

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