Response surface based optimization; gradient-enhanced kriging of a high-pressure turbine

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optimize a design, this optimization will always be a compromise. The improve-ment of a particular design can go on for a long time, even if the gains become very small. Over time, these increments in performance will no longer be worth the eff ort or computations. At that point, a design is considered to be optimized, although it can remain a sub-optimal de-sign.

A method to deal with these computa-tional simulations is to employ a so-called surrogate model. A surrogate model imitates the behaviour of its original func-tion, and can provide predictions in doing so. In this defi nition, the computational simulations are regarded as the original function (true value) and expensive to obtain, e.g. they take a long time to be solved. The surrogate model is cheap to evaluate and thus allows a quick predic-tion of the funcpredic-tion value.

KRIGING

During the thesis work, a modeling

method known as Kriging was applied. Kriging fi nds its origins in the mining in-dustry (Cressie, 1990). When a particular mineral-fi eld is discovered, it is interest-ing to know what quantity of minerals can be expected in this fi eld. In order to get an estimate, test-drills can be performed. There are only a limited number of test-drills available, because they are expen-sive. Based on these test-drills, the entire lay-out of mineral deposits – and thus the total amount, could be determined using Kriging.

Rolls-Royce Deutschland is a company that uses surrogate modeling in the de-sign optimization of their turbines. The surrogate model does not replace the simulations entirely, but is used in con-junction with it. In the research, the com-putational simulations are equivalent to the expensive test-drills. The amount of minerals is similar to the turbine’s effi -ciency. The goal is now to fi nd the highest effi ciency, using only a few computational simulations.

The effi ciency of a turbine is constrained

by the mass-fl ow through the turbine as well as the reaction. This last constraint is an indication for the amount of en-ergy that is subtracted from the airfl ow through the turbine. For the constraints, a surrogate model can be built. Now, using the one surrogate model for the effi -ciency and two surrogate models for the constraints, the objective function to be optimized is defi ned.

GRADIENT-ENHANCED KRIGING

Kriging uses only the function value (effi -ciency) of the computational simulations. The faculty of Aerospace Engineering, however, developed a promising exten-sion to this method; the use of gradient-information (Dwight, 2012). This is re-ferred to as Gradient-Enhanced Kriging (GEK).

See it this way: you are out skiing and you need to go home, but your view is ob-structed by a snowstorm. It is interesting to know your altitude, but it can be more useful to know the slope of your posi-tion (the gradient) to fi nd your way back

Gradient-Enhanced Kriging of a high-pressure turbine

In the Aerospace industry, before the construction of a new aircraft or engine is even

realized, its performance and characteristics are already known. This knowledge is

obtained through simulations in a wind-tunnel or using computational fl uid dynamics.

But with complex aircraft and engines, the use of these simulations can be limited by

the computational means that are available. For this master thesis, the application

of surrogate modeling, with the design of turbines of aero-engines is studied. This

master thesis was performed with Rolls-Royce Deutschland.

TEXT Ir. Floris Huijbregts, MSc Graduate, Flight Performance and Propulsion

RESPONSE SURFACE BASED OPTIMIZATION

FL IC K R / JO N O ST RO W ER

JUNE 2014 Leonardo Times

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home, located in the valley.

Using the gradient-information in the sur-rogate model, the hope is to get a better prediction and thus fi nd a more optimal turbine design. Of course, this gradient-information has to be evaluated as well, using the computational simulations. It is, therefore, also of interest, if the increase prediction-accuracy weighs out the in-crease in computations to obtain the gra-dients.

TURBINE DESIGN

Its blade shape determines the turbine characteristics –effi ciency, mass fl ow and reaction. All blades in one row are similar, and the shape is defi ned by a set of pa-rameters. The entire three-dimensional shape is given by fourteen parameters; thus, a fourteen-dimensional surrogate model has to be constructed.

Using the experience of Rolls Royce, the fourteen parameters are defi ned as a de-viation from an existing turbine shape. This shape would be further optimized using the surrogate model. The initial sur-rogate model is built based on the obser-vations of fourteen simulations.

OPTIMIZATION

One could stop here and merely use the surrogate model to determine the

opti-mal shape of the turbine. However, be-cause these are predictions, there is a need to improve this prediction as well; and to optimize the surrogate model. In order to do so, the predicted optimum is located. At this optimum, one can evalu-ate the true value; and perform a simula-tion with that particular turbine shape. Adding this piece of information is re-ferred to as adaptive-sampling. The initial fourteen observations are an example of non-adaptive sampling.

When one repeats this exercise, the pre-diction of the optimum can be improved. This method is only applicable when the problem has one local optimum, the glob-al optimum. With multiple locglob-al optima, it would be possible to get trapped at a local optimum, not the global optimum. However, the turbine with which the ex-ercise started was already a near-optimal turbine, based on Rolls Royce’ experience. In addition, the domain over which the variables could vary was relatively small.

OUTCOME

Because of a limitation in calculating the gradients, the gradient-information was added to the center point of the domain only. Subsequently, several iterations adding adaptive samples are performed. These adaptive samples did not contain

any gradient-information.

When Kriging was compared to GEK, it was possible to see that the latter not only converged quicker towards an optimal prediction, the predictions itself were also more accurate. Although the construction of the surrogate model showed to be very sensitive, a clear improvement of GEK was observed over kriging, using the same number of computational simulations.

OUTLOOK

In the end, the addition of a small piece of information showed signifi cant eff ects. Not only Rolls-Royce, but other design or-ganizations can as well benefi t from surro-gate modeling with gradient-information. The interesting thing is, the surrogate model can refl ect any objective function or quantity. With a proper problem defi ni-tion, limited observations of this quantity, GEK can be used in many diff erent areas. Possible improvements in constructing the surrogate model, using the GEK ap-proach, will further boost its possibilities.

PERSONAL EXPERIENCE

Before starting my thesis research, I fi g-ured I wanted to work with an aerospace company. Not only for the experience in the industry, but also to see how re-search in a company compares to that at a university. As an FPP-student, it required some hard work to get used to this more statistical type of assignment. All in all, it is safe to say that I learned many lessons with Rolls-Royce, and not just on engi-neering! x₁ x 2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 x₁ x2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 References

[1] Cressie, N., “The origins of Kriging”, Mathematical Geology, vol. 22, pp/239-252, April 1990

[2] Dwight, R.P., De Baar, J., Azijli, I., “A Tutorial on Adaptive Surrogate Model-ling”, 2012

Figure 1. Visualization of a prediction based on the Kriging and GEK method, for a one-dimensional function

Figure 3. The GEK-prediction of the Rosenbrock function, using the gradient-information at the centre observation.

Figure 2. The Rosenbrock function, used as ex-ample for optimization problems.

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