NEWTON ITERATION 1 Formulation

In Section 5.2 our approximation of Theodorsen’s function corresponded to a zeroth-order Taylor approximation about the previous iterate. We might logically try to use a first-order Taylor approximation instead. In this case it would be sensible to use the - form of our section model (Eq. 5.2.1), because we will then only need to expand Theodorsen’s function in terms of (and not ). In the - form we would need to consider the effect of the function in both variables. So, we have

(( ) ( ( )) ( ) ) (5.3.1)

Expanding ( ) around we obtain:

Chapter 5 – page 116

Denoting ( ) and | , we substitute this Eq. 5.3.2 into Eq. 5.3.1, with all free variables evaluated at :

(( ) ( ( ))

( ) )

(5.3.3)

This can be rearranged to

(( ) ( )

( )

)

(5.3.4)

which is a polynomial multiparameter eigenvalue problem, cubic in and quadratic in . We can linearise it with the same methods we used in Chapter 4. Defining a new eigenvector [ ], we obtain the linear problem

( ) (5.3.5) with coefficients [ ] [ ] [ ] (5.3.6)

Chapter 5 – page 117

However, to evaluate the matrices , and we required one value that we have not yet obtained: | . We can obtain this derivative in two ways: either by differentiating Theodorsen’s function analytically, or by using finite-difference aproximations. In the general case (when we have a system with a truly unstructured component) we will have to use finite-differences, and so we use this method here. For convenience we use a first-order accurate forward difference scheme

( ) ( )

(5.3.7)

for some small increment . Eq. 5.3.5 can be solved using the operator determinant method. This completes our formulation: we can now iterate on Eq. 5.3.5, updating , and as the iteration progresses. Algorithm 5.6 is an implementation of the Newton iteration.

Algorithm 5.6

1 initialise , initial guesses , , increment , tolerances and , matrices , , , , , , matrix size , Theodorsen’s function ( )

2 3

evaluate: ( ) and ⁄ ( ( ) )

construct the matrices

[ ] [ ] [ ]

4 compute the set of solutions { ( )} , { ( )} , { ( )} to

( )

( ̅ ̅ ̅ ) ̅

5 select eigenvalue-eigenvector pair { ( )}, { ( )} , { ( )} such

Chapter 5 – page 118 6 7 evaluate ( ) if( ) ( ) and ‖(( ) ( ( )) ( ) ) ‖ 8 9

then return , and

else , goto (2)

5.3.2 Numerical experiments

As in Section 5.2, we now simulate the section model with Theodorsen aerodynamics, with parameter values given in Chapter 2. However, this time we are using the - form, Eq. 5.2.1, because this is the form the method is derived for. Figure 5.8 shows a contour plot with nine different iteration paths (from nine different initial guesses) superimposed. All of the iterations converge to the first flutter point at , . Figure 5.9 shows a logarithmic convergence plot for an iteration converging to the first flutter point, with an initial guess of , . As can be seen, the convergence rate of this iteration is relatively uniform, and very fast – only four iterations2 are required to solve the problem to a residual and increment tolerance of . The convergence is second-order: the gradient of the plot can be measured as . This is expected, as Newton-type methods are well-known to have second-order convergence [1]. The fact that we are using finite- differences to compute the derivatives has not noticeably affected the convergence rate: the finite-difference increments are sufficiently small that the computed derivative is effectively exact.

Figures 5.10 and 5.11 show a numerical convergence analysis of the semi-structured Newton’s method for the section model with Theodorsen aerodynamics. The observed convergence behaviour is extremely good, and is equivalent to that of the Picard iteration. There are no points that do not convergence, and the basin of attraction around the first flutter point is unbounded upwards and to the right. Overall, we obtain the excellent convergence basins of the Picard iteration, but with second-order convergence.

2

We can only plot three points on the plot as we must take the final iterate as the converged result for the convergence analysis ( in Eq. 5.2.17) leaving us with only four points to work with (three iterates and the initial guess), and as we are taking differences between these iterates, we only have three differences to plot.

Chapter 5 – page 119

Figure 5.8: Contour plot with nine different iteration paths of the Newton’s method

Figure 5.9: Logarithmic convergence plot for an Newton iteration converging to the first flutter point ( , ).

Chapter 5 – page 120

Figure 5.10: Numerical converge analysis of Newton’s method ( - section model).

Figure 5.11: Numerical converge analysis of Newton’s method ( - section model, wide field of view).

Chapter 5 – page 121 5.4 HIGHER-ORDER METHODS

The previous two methods have formed a logical progression: that of increasing order in the Taylor series approximation of Theodorsen’s function. We might then ask whether it is worth increasing the order of approximation any further, to second-order or beyond. For the system we have been dealing with – the section model with Theodorsen aerodynamics – there is no real motivation to do so, as the existing methods have such good convergence properties (see Section 5.2.4 and Section 5.3.2). However, for more complex systems, where there may be multiple physical flutter points, there is more reason to look at more advanced methods. We would be particularly interested in approximations that are sufficiently good that they may be able to model multiple nearby flutter points simultaneously. We have, in fact, already met such approximations, in the form of our approximations to Theodorsen’s function in Chapter 2. These approximations are so good that they are able to model all physical flutter points (those inhabiting the upper right quarter plane) with a high level of accuracy – though the price is a significantly enlarged problem to solve. However, there is a subtle difference between these approximations and the Taylor approximations we have used earlier in this chapter. This is that these high-accuracy approximations are entirely global: they cannot be updated with data from a previous iteration. Hence they cannot be used in an iterative method like the previous two that we have presented.

We would ideally look for methods somewhere in between the two: that contained some globally-fixed terms, and also some local terms that could be iteratively refined. This would lead to an iterative solver that, while costly in terms of time per iteration, would converge very fast, and (more importantly) would be able to accurately identify specific flutter points to avoid or to home in on. This is the major disadvantage of the previous two solvers we have presented, in that no amount of fiddling with the selection parameters will cause their iterates to reliably converge to a given flutter point (e.g. the first flutter point). The iterates cannot reliably be made to escape the basin of attraction that they initially find themselves in. This is because the approximation of Theodorsen’s function that is being used is not accurate enough to give any information about flutter points in any location outside the immediate vicinity of the current iterate. Another possible solution to this problem is to use a multi-resolution approach, with an initial global approximation to estimate the flutter point locations, and then iterative local approximations to refine these estimates. There is a

Chapter 5 – page 122

significant amount of potential here for the devising of methods with global or near-global convergence – something that would be extremely valuable in industrial applications.

However, as of yet there has been no research into developing these kinds of approximations for Theodorsen’s function – the emphasis has been on devising functions that approximate Theodorsen’s function globally, with no local dependence. There has previously been no motivation for considering any other forms of approximation. However, with the development of these semistructured methods for flutter problems, there is now significant impetus for the development of a much broader class of approximations.