**Introduction**

NASA rotor 37 is a representative transonic axial-flow compressor rotor that has been used extensively in computational fluid dynamics (CFD) community to test optimisation algorithms and validate CFD codes (Dunham et al. 1998, Duta and Giles 2006, Ameri 2010).

Different approaches have been used to optimise the blade design under different constraints. For example, Samad and Kim (2008) used a three-dimensional Reynolds-Averaged Navier Stokes (RANS) solver to generate the data sets and built global response surfaces using second order polynomials: the Pareto front with respect to pressure ratio and adiabatic efficiency was then found using a multi-objective genetic algorithm coupled with an SQP optimiser. Shahpar et al. (2008) used Multipoint Approximation Method (MAM) to find an optimal blade design. The approach used by Oyama and Obayashi (2002) is also interesting (although for NASA rotor 67 optimisation) as no metamodels were used to reduce the computational cost: the minimum entropy design was found evaluating through direct simulations the points selected by an evolutionary algorithm. Duta and Giles (2006) used adjoint code to study the sensitivity of the mass flow to the twist of the mid-height section of the blade.

The aerodynamic optimisation of NASA rotor 37 compressor blade was considered a good test case for HyGP. First of all because to the best of the author’s knowledge such optimisation has never been attempted using global metamodels generated by genetic programming. Secondly, the problem exhibits a relatively high dimensionality, with 25 input variables, and so in general it is a challenging problem due to the curse of dimensionality. It is important to note that only few examples have been found in literature of the application of genetic programming to design spaces of more than ten dimensions. Nordin et al. (1999) successfully solved a 40 input-variable data-mining problem without any dimensional reduction, whereas Smits et al. (2005) and Vladislavleva (2008) reduced a 23 input-variable problem to a lower dimensionality using a GP-based sensitivity analysis tool before generating the final model using genetic programming.

In the following sections the optimisation process and the resulting optimal blade are described. To assess HyGP metamodels reliability, the optimisation was repeated using global metamodels generated by MLSM.

**Problem description**

The blade parameterisation was done using five engineering parameters available in Rolls-Royce PADRAM code (Shahpar and Lapworth 2003): axial movement of sections along the engine axis (variable XCEN, in mm), circumferential movements of sections (variable DELT, in degrees), solid body rotation of sections (variable SKEW, in degrees), and leading/trailing edge recambering (variables LEMO and TEMO, in degrees). A sketch showing the physical meaning of the input variables is given in Fig. 1A. These design parameters define the shape of a bidimensional airfoil on five stations along the blade span, at 20%, 40%, 60%, 80%, and 100% (tip) of the overall span, as shown in Fig. 1B. The airfoil at the blade root was fixed (station 0%). The total number N of independent design variables is 25. B-spline interpolation was then used through the control stations along the span to generate smooth design perturbations in the radial direction.

Figure 1 : NASA rotor 37 blade parametrisation

The optimisation problem was to find the values of the 25 parameters that maximise the adiabatic efficiency of the blade, defined in Eq. (1):

where is the pressure ratio and is the temperature ratio between outlet and inlet, respectively; is fluid specific heat. Constraints were defined on pressure ratio and mass flow through the rotor: a maximum perturbation of 0.5% with respect to the baseline pressure ratio and to the baseline mass flow was imposed.

**Input data and HyGP settings**

The original range of the design variables ([−5, 5] mm for XCEN and [−0.5, 0.5] degrees for the other variables DELT, SKEW, LEMO and TEMO) was scaled to [1.0, 11.0]. In the scaled design space, a DoE made of 100 points was randomly generated. In order to improve the quality of the random design, a constraint on the minimal distance between points was imposed: the space-filling properties of the scaled DoE are shown in Fig. 2A through a plot of the minimum distance of each point to neighbouring DoE points. The average minimum distance is 14.49, the minimum distance standard deviation is 0.97. For each DoE point, a few CFD simulations were performed with Rolls-Royce SOPHY (Shahpar 2005) to compute the corresponding values of efficiency, pressure ratio and mass flow rate. The data were used as HyGP building data set.

Figure 2 : Minimum distance between points in building data set (A)

and additional data set for model penalisation (B) for efficiency,

pressure ratio and mass flow symbolic regression

The constraint on pressure ratio was recast in normalised form using two inequalities:

where is the baseline pressure ratio. The constraint on the mass mass flow m’ was similarly recast and normalised:

where is the baseline mass flow.

Efficiency was also reformulated, to turn the maximisation problem (max efficiency ) into a minimisation problem (min () ):

The scaled 100-point DoE was fed as a building data set into HyGP to generate the five metamodels of , , , , defined in Eqs. 2, 3, 4, 5, 6: 2 metamodels for pressure ratio, 2 for mass flow and 1 for the reformulated efficiency.

Preliminary tests with standard HyGP settings, not shown here, resulted in really irregular response surfaces that could not be used for optimisation. Metamodels featured low generalisation ability, typical of overfitting. The exclusion of the protected division from the primitives somewhat improved metamodel smoothness, but the introduction of the a penalisation approach based on feasible/unfeasible output proved to be decisive. An additional data set C made of 50 points uniformly distributed in the scaled design space (latin hypercube DoE) was used to bias the search for reliable metamodels of the reformulated efficiency and the four constraints. The minimum distance between points in C is plotted in Fig. 2B: the average minimum distance between points is 17.99, the minimum distance standard deviation is 0.85. On this additional data set metamodels of ηp returning values lower than 1.1 were penalised, as well as metamodels for , , , giving values lower than 0.5.

In the next sections the optimum found using HyGP metamodels is compared with the one returned by MLSM metamodels.

**Result of the optimisation performed with HyGP and GA**

HyGP metamodels were explored using Multiobjective GA with Adaptive Range (ARMOGA) from Rolls-Royce optimisation software SOFT (Shahpar 2002) to find the values of the 25 input parameters that optimise the rotor efficiency within the given constraints. Among the suboptima found, two designs were selected: their accuracy with respect to the responses provided by CFD simulations at the same design point is assessed in Tables 1 and 2.

Table 1: Validation of HyGP optimum 1 found by ARMOGA

HyGP CFD error (%)

ηp 1.125 1.138 -1.14%

c1 0.988 0.981 0.71%

c2 0.994 **1.008 ** -1.39%

c3 0.999 0.999 0.00%

c4 0.997 0.990 0.71%

Table 2: Validation of HyGP optimum 2 found by ARMOGA

HyGP CFD error (%)

ηp 1.091 1.134 -3.79%

c1 0.996 0.973 2.36%

c2 0.999 ** 1.016 ** -1.67%

c3 0.952 **1.002** -4.99%

c4 0.993 0.987 0.61%

In Tables 3 and 4 the actual values of the efficiency , the pressure ratio and the mass flow returned by CFD simulations at the two optima described in Tables 1 and 2 are compared to their corresponding values in the baseline design.

Table 3: Optimum 1: values of , and in baseline design and HyGP optimised design

(values by CFD)

Var. Baseline Optimised rel. variation (%)

η 0.857 0.862 +0.58%

Pr 2.15 2.12 **-1.39%**

m’ 20.1 20.19 +0.5%

Table 4: Optimum 2: values of , and in baseline design and HyGP optimised design

(values by CFD)

Var. Baseline Optimised rel. variation (%)

η 0.857 0.866 +1.05%

Pr 2.15 2.11 **-1.86%**

m’ 20.1 20.24 +0.70%

As shown by Tables 3 and 4, the errors on efficiency, pressure ratio and mass flow is generally under 2%. Optimum 1 stands out for a smaller violation of the pressure ratio constraint (-1.39% variation with respect to baseline pressure ratio against a maximum of 0.5%) with respect to optimum 2, where a higher efficiency is gained (+1.05%) at the cost of more severe violations on pressure ratio and mass flow constraint (-1.86% with respect to baseline pressure ratio, +0.70% with respect to baseline mass flow. In both cases the maximum variation allowed was 0.5%).

**Comparison with optimum found by MLSM and SQP**

NASA rotor 37 blade optimisation was repeated using MLSM to generate the metamodels for the objective and the four constraints, whereas SQP was used to find the optimum. The same 100-point DoE described in Section “Input data and HyGP settings” was used. All points were used for metamodel building. As a result, the closeness of fit was not optimised: it was instead set to the maximum value (100) in order to encourage local accuracy (Toropov et al. 2005, Loweth et al. 2011). That was possible as the data are not affected by noise, so smoothing was not required. A second order polynomial was chosen as basis for the MLSM.

In Table 5 the values of , , , , – Eqs. (2, 3, 4, 5, 6) – at the optimum found are compared to the responses returned by CFD simulations. In Table 6 the resulting efficiency, pressure ratio and mass flow at the optimum are compared with the same parameters in the baseline design.

Table 5: validation of MLSM optimum found by SQP

MLSM CFD error (%)

ηp 1.132 1.132 -0.05%

c1 0.990 0.987 0.24%

c2 1.000 ** 1.003** -0.29%

c3 0.986 0.995 -0.94%

c4 1.000 0.995 0.54%

Table 6: values of , and in baseline design and in MLSM optimised design

(values by CFD)

Var. Baseline Optimised rel. variation (%)

η 0.857 0.868 1.28%

Pr 2.15 2.13 **-0.78%**

m’ 20.10 20.11 +0.03%

The comparison of the optima found by SQP on MLSM metamodels with optimum 1 found using HyGP metamodels (Table 1) shows that MLSM allowed for a larger increase in the blade efficiency (1.28% against 0.58% produced by HyGP coupled with GA – Table 1) with a less severe violation of the constraint imposed on pressure ratio (-0.78% against -1.39% obtained by GA with HyGP metamodels).

The better performance of MLSM can be explained by the better accuracy of the MLSM technique with respect to HyGP. In particular, the fact that the optimum was not located

on the design space boundary contributed to the superior accuracy of MLSM. Harewood et al. (2007) reported on the possible lack of accuracy in MLSM if the optimum is located on the boundary of the design space. Also, the smoothness of the metamodels resulted in the better performance of SQP search algorithm with respect to the non-deterministic GA search. It is worth noting neither optimisation process was able to improve the blade efficiency without violating at least a constraint.

The optimal set of blade parameters found through HyGP and MLSM metamodels are reported in Tables 7 and 8: the corresponding blade shapes are compared to the

shape of the baseline blade design in Fig. 3.

Table 7: blade parameters in optimum 1 found using HyGP metamodels

Station XCEN DELT SKEW LEMO TEMO

(mm) (°) (°) (°) (°)

0% 0.00000 0.00000 0.00000 0.00000 0.00000

20% 3.83924 -0.44331 -0.42574 0.15586 -0.07605

40% 1.04283 -0.37682 -0.32360 -0.24108 0.47043

60% 1.78793 0.38174 0.05784 0.00936 -0.10840

80% -2.38414 -0.18487 0.32866 -0.39907 0.38807

100% 1.63545 0.27269 -0.39470 0.02017 -0.14352

Table 8: blade parameters in the optimum found using MLSM metamodels

Station XCEN DELT SKEW LEMO TEMO

(mm) (°) (°) (°) (°)

0% 0.00000 0.00000 0.00000 0.00000 0.00000

20% 0.01355 0.01992 -0.04297 -0.02942 -0.10893

40% -1.09720 -0.03056 -0.04254 -0.13860 -0.06543

60% 0.59583 0.01364 0.05910 -0.11524 0.04775

80% 2.29663 -0.02520 0.07393 -0.50000 0.03837

100% 2.08573 -0.11509 0.05538 -0.19514 -0.02887

Figure 3 : Baseline (A) and optimised blade designs

produced by HyGP (B) – Table 7 – and MLSM (C) – Table 8

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