Fig. 1.3 The structural and aerodynamical model of a modern aircraft.

another model is required which is better suited to describing the structural deformation,

for example the one shown in Figure 1.3 on the left. Again, along with the model comes a

partial differential equation, this time from elasticity theory, which can again be solved, for

example by ¬nite elements. But before this can be done, the loads have to be transferred

from one mesh to the other in a physically reasonable way. If this has been done and the

deformation has been computed then we are confronted with another coupling problem.

This time, the deformations have to be transferred from the structural to the aerodynamical

model. If all these problems can be solved we can start to iterate the process until we ¬nd

a steady state, which presumably exists.

Since we have the aerodynamical model, the structural model, and the coupling problem,

one usually speaks in this context of a three-¬eld formulation. As we said earlier, here

we are interested only in the coupling process, which can be described as a scattered data

approximation problem, as follows. Suppose that X denotes the nodes of the structural

mesh and Y the nodes of the aerodynamical mesh (neither actually has to be a mesh).

To transfer the deformations u(x j ) ∈ R3 from X to Y we need to ¬nd a vector-valued

interpolant su,X satisfying su,X (x j ) = u(x j ). Then the deformations of Y are given simply

by su,X (y j ), y j ∈ Y . Conversely, if f (y j ) ∈ R denotes the load at y j ∈ Y then we need

another function s f,Y to interpolate f in Y . The loads on the mesh X are again simply given

by evaluation at X . A few more things have to be said. First of all, if the loads are constant or

if the displacements come from a linear transformation, this situation should be recovered

exactly, which means that our interpolation process has to be exact for linear polynomials.

Furthermore, certain physical entities such as energy and work should be conserved. This

means at least that

f (y) = s f,Y (x)

y∈Y x∈X

and

f (y)su,X (y) = s f,Y (x)u(x),

y∈Y x∈X

6 Applications and motivations

1.00

0.75

0.50

0.25

0.00

25

50

75

00

0 0.25

Fig. 1.4 Steady state of the deformed aircraft.

where the last equation is to be taken component-wise. If the models differ too much then

both equations have to be understood in a more local sense. However, these equations

make it obvious that in certain applications more has to be satis¬ed than just simple point

evaluations. It is important to note that interpolation is crucial in this process since otherwise

each coupling step would result in a loss of energy.

The advantage of this scattered data approach is that it allows us to couple any two

models that have at least some node information. There is no additional information such as

the elements or connectivity of the nodes involved. Moreover, the two models can be quite

different. It often happens that the boundary of the aerodynamical aircraft has no joint node

with the structural model. The latter might even degenerate into a two-dimensional object.

Figure 1.4 shows a typical result for the example from Figure 1.3 based on a speed

M = 0.8, an angle of attack ± = ’0.087—¦ , and an altitude h =10 498 meters. On the left

the deformation of a wing is shown, while the right-hand graph gives the negative pressure

distribution at 77% wing span, for a static and an elastic computation. The difference

between the two pressure distributions indicates that elasticity causes a loss of buoyancy,

which can become critical for highly ¬‚exible structures, as found for example in the case

of a large civil aircraft.

It should, be clear that the coupling process described here is not limited to the ¬eld of

aeroelasticity. It can be applied in any situation where a given problem is decomposed into

several subproblems, provided that these subproblems exchange data over speci¬ed nodes.

1.3 Grid-free semi-Lagrangian advection

In this section we will discuss brie¬‚y how the scattered data approximation can be used to

solve advection equations. For simplicity, we restrict ourselves here to the two-dimensional

case and to the transport equation, which is given by

‚ ‚ ‚

0= u(x, y, t) + a1 (x, y) u(x, y, t) + a2 (x, y) u(x, y, t). (1.1)

‚t ‚x ‚y

1.4 Learning from splines 7

It describes, for example, the advection of a ¬‚uid with velocity ¬eld a = (a1 , a2 ) and it

will serve us as a model problem straight away. Suppose that (x(t), y(t)) describes a curve

for which the function u(t) := u(x(t), y(t), t) is constant, i.e. u(t) = const. Such a curve is

called a characteristic curve for (1.1). Differentiating u yields

‚u ‚u ‚u

0= + x(t) + y (t) ,

™ ™

‚t ‚x ‚y

where x = d x/dt. The similarity to (1.1) allows us to formulate the following approximation

™

scheme for solving the transport equation (1.1) with initial data given by a known function u 0 .

Suppose that we know the distribution u at time tn and at sites X = {(x1 , y1 ), . . . , (x N , y N )}

approximately, meaning that we have a vector u (n) ∈ R N with u (n) ≈ u(x j , y j , tn ). To ¬nd

j

the values of u at site (x j , y j ) and time tn+1 we ¬rst have to ¬nd the upstream point (x ’ , y ’ )

j j

with c := u(x ’ , y ’ , tn ) = u(x j , y j , tn+1 ) and then have to estimate the value c from the

j j

values of u at X and time tn . Hence, in the ¬rst step we have to solve N ordinary differential

equations

(ξ j , · j ) = a(ξ j , · j ), 1 ¤ j ¤ N,

™™

with initial value (ξ (tn+1 ), ·(tn+1 )) = (x j , y j ). The upstream point is the solution at tn , i.e.

(x ’ , y ’ ) = (ξ (tn ), ·(tn )). Since this point will in general not be contained in X , the value

j j

u(x ’ , y ’ , tn ) has to be estimated from u (n) . This can be written as an interpolation problem.

j j

We need to ¬nd a function su that satis¬es su (x j ) = u (n) for 1 ¤ j ¤ N .

j

The method just described is called a semi-Lagrangian method. It is obviously not re-

stricted to a two-dimensional setting. It also applies to advection equations other than the

transport problem (even nonlinear ones), but then an interpolatory step might also be nec-

essary when solving the ordinary differential equations.

Moreover, it is not necessary at all to use the same set of sites X in each time step. It is

much more appropriate to adapt the set X as required.

Finally, if the concept of scattered data approximation is generalized to allow also func-

tionals other than pure point-evaluation functionals, there are plenty of other possibili-

ties for solving partial differential equations. We will discuss some of them later in this

book.

1.4 Learning from splines

The previous sections should have given some insight into the application of scattered data

interpolation and approximation in the multivariate case.

To derive some concepts, we will now have a closer look at the univariate setting. Hence

we will suppose that the data sites are ordered as follows,

X : a < x1 < x2 < . . . < x N < b, (1.2)

and that we have certain data values f 1 , . . . , f N to be interpolated at the data sites. In other

8 Applications and motivations

words, we are interested in ¬nding a continuous function s : [a, b] ’ R with