4. Template-Based Optimal Profile Fitting
4.3 A Fast Converging Minimisation Algorithm
As mentioned in the last section, the optimal profile fitting of a 2D sectional airfoil profiles needs to do unconstrained minimisation in a 3D search space with X-axis shift, 7-axis shift and rotation as the three independent variables. In this section, we present a generic fast converging multi-dimensional minimisation algorithm.
There are a number of multi-dimensional minimisation algorithms, such as downhill simplex search method, Hooke-Jeeves pattern search method, Powell conjugate direction method, Cauchy method, Newton method, conjugate gradient methods, and variable metric methods [10, 11].
However, for our particular minimisation problem the selection of the methods is limited due to the complication of the index function.
The index function in the optimal profile fitting, as given by Equation (4), is the weighted sum distance from the measurement points to the template. The template is represented by a series of points. The three position parameters of the template, i.e., X-axis shift, F-axis shift and
rotation, are independent variables of the index function. As the airfoil profiles are odd shaped, there is no analytical expression available for the index function. As a matter of fact, the value of the index function can only be obtained through some numerical methods. Therefore, any existing minimisation method which uses analytical expression or any derivatives of the index function will not be applicable to our problem.
To solve airfoil optimal profile fitting problem, we developed an intuitive direct search method. It turns out that the new method is an improved version of Hooke-Jeeves pattern search method [12], with a faster convergence rate. The basic idea is that we take a point in the search space as a base point and explore around it to find a right direction and a right step to move the base point towards the minimum point. In order to obtain a fast convergence rate, the direction is adjusted and the step is increased repeatedly before each move of the base point. The following is the outline of the multi-dimensional minimisation algorithm. An illustrative flow chart of the algorithm is shown in Figure 9.
Step 1 Set initial base point and initial search span. The search span may be different for each coordinate direction in the search space.
Step 2 Test the index function values of the surrounding points of base point with search span. Find the best point (i.e. with the lowest index) among the surrounding points. In this step, all the coordinate directions in the search space have to be exhausted.
Step 3 Compare base point with best point. If base point is better (i.e.
with a smaller index) than best point (which means the minimum point is within the search span), jump to Step 9. Otherwise, continue.
Step 4 Take best point as direction point.
Step 5 Along the direction from base point to direction point, compute temporary base point by extrapolation with a ratio, say 2.5. That means temporary base point will be away from base point 2.5 times of the distance between base point and direction point.
Step 6 Test the index function values of the surrounding points of temporary base point with search span. Find best point among the surrounding points. Again, all the coordinate directions in the search space have to be considered in this step.
Step 7 Compare direction point with best point. If best point is better than direction point (which means that the direction of last extrapolation
is probably right and it is worth to further increase the move step of base point), jump to Step 4. Otherwise, continue.
Step 8 Take direction point as base point and jump to Step 2.
Step 9 Reduce search span, say, 10 times.
Step 10 If search span is not sufficiently small, jump to Step 2. Otherwise, continue.
Step 11 Take base point as the minimum point. End of the program.
Stepl
1 r
Step2
4 -^u '
Step 4,5,6
^ C § t e p J J >
Step8
' '
Step9
<^epV)>-
Stepll
Figure 9 Flow chart of direct search minimisation algorithm.
In the above minimisation algorithm, the loop involving Steps 2-8 is for repeated move of the base point towards the minimum point. The inner loop involving Steps 4-7 is for repeated increase of the moving step for the base point. The loop involving Steps 2, 3, 9 and 10 is for repeated reduction of the search span around the base point.
Notice that with the inner loop involving Steps 4-7, the moving step of the base point towards the minimum point increases very fast. If the moving direction remains the same in the iterations, the step increases faster than an exponential function and is given by:
n
l = /fZal (5)
ô=o
where / is the moving step of the base point, X is the distance from base point to direction point at the first iteration (the search span at the moving direction), a is the extrapolation ratio selected in Steps 5 and n is the iteration number of the loop involving Steps 4-7.
With this fast increase of moving step of the base point, a large reduction rate for the search span, say 10 times, can be used in Step 10.
With the features of fast increase of the moving step for base point and large reduction rate for search span, the above algorithm gives a faster convergence rate than that proposed by Hooke-Jeeves [12].
Minimum Point
PO: Initial base point P5, P9: Base points P2, P4, P6, P8, P10:
Temporary base points P1.P3, P5, P7, P9:Best points around base points
Figure 10 Direct search minimisation in 2-D space.
Figure 10 shows the minimisation process of the algorithm for a two- dimensional case with the independent variables Xj and x2. Notice the fast movement of the base point toward the minimum point starting from P0 to P5 and then to P9. As P9 is "better" than all its surrounding points, the search span is reduced 10 times. Then, taking P9 as a new base point, the minimisation process will continue with the reduced search span. The minimisation process ends when the search span is sufficiently small.