BMCS based Robust Design Method

The tools and methods discussed in earlier sections are employed next to build an efficient method for robust design. We call this method Bayesian Monte Carlo based Robust Design and the steps involved are shown in figure 5.10. In this method we start with an initial

Chapter 5 Robust Design against Manufacturing Variations 81

Figure 5.10: Flowchart for Robust Design of Compressor Blades against Manufacturing

Uncertainty

combined array dataset D_n ∈ x˜ ∈ _Rp+q_{, generated using the LHS technique. The choice} of a combined array is intuitive for the problem under consideration since the noise factors (manufacturing variations) are a subset of the design factors (design space), i.e. E ∈χ. The grid generator and CFD solver are used to generate the initial dataset [Xn,Yn]. The dataset is employed for tuning the covariance function as discussed in section 3.4. The BMCS method is employed, using the emulator obtained from the previous step, to calculate the mean and standard deviation of the pressure loss. This is followed by multiobjective optimization to seek the Pareto optimal set. The CFD analysis is performed on points on the Pareto front and these new points are appended to the existing dataset, which is subsequently used to update the baseline Gaussian process emulator. This process is repeated till we meet some predefined convergence criteria.

The above mentioned approach alleviates most of the concerns expressed regarding exist- ing robust design strategies discussed earlier. Specifically, the present approach uses a com- putationally cheap emulator to improve the efficiency of robust design search. The method

Chapter 5 Robust Design against Manufacturing Variations 82 also provides the option of updates which ensures that the designer spends effort in the area of interest (near the Pareto front) and the emulator prediction improves with each iteration. We will illustrate this improvement in the numerical studies section. In the next section, we discuss the results obtained by applying the proposed method to a compressor blade design problem.

5.4

Numerical Studies and Results

We next conduct a robust design optimization study for a compressor blade. We employ the Bayesian Monte Carlo based method proposed in the flowchart shown in figure 5.10. To start with, we use the Gaussian process emulator which was trained to initial dataset of 100 points. NSGA-II is then employed in conjunction with this emulator to search the entire design space. BMCS is carried out for all points in the population at each generation to evaluate the performance statistics (design objectives). The objectives are then ranked to obtain Pareto-optimal solutions. We use a population size of 100 points with 100 generations for this study.

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0 0.01 0.02 0.03 0.04 0.05 0.06

Normalized Mean Pressure Loss

Normalized Standard Deviation of Pressure Loss

All Explored Points

Points on Final Pareto Front Final Pareto Front

Chapter 5 Robust Design against Manufacturing Variations 83 A low-crowding algorithm, which maximizes the Euclidean distance between the Pareto points, is used to select points which are then verified by running full scale CFD simulations Kumar et al. (2006). These points are updated to the existing dataset and the Gaussian process emulator is re-trained on the new dataset. The Gaussian emulator based NSGA-II searches are repeated for updates until a convergence criterion is met. Here the convergence criterion is a function of the improvement made in predicting the Pareto front after each update. When the observed improvement is less than a user defined value the search is terminated. Figure 5.11 contains the initial dataset and subsequent updates points which are collectively shown as the explored points. The figure also shows the final Pareto Front after ten updates. Note that exact aerodynamic analysis using the high fidelity CFD code was conducted only at the 336 points shown in the figure.

Figure 5.12: Predicted posterior mean versus original values for final emulator for modeling manufacturing variations

We had mentioned earlier that at each iteration the surrogate model is updated with CFD solutions at the points selected from the Pareto front. Hence, at the final iteration we had 336 points for training the Gaussian emulator. We conducted model validation on the final surrogate model to illustrate improvements from the initial surrogate model and build confidence in our predictions. Figure 5.12 shows the plot of predicted values versus actual CFD values using the leave-one-out validation test. The regression coefficient for a linear

Chapter 5 Robust Design against Manufacturing Variations 84 model fit to the leave-one-out results is R2 = 0.978, which has considerably improved as compared to R2 _{= 0.916 for the initial dataset. An SCVR test was also conducted for the}

final surrogate model. Figure 5.13 shows the SCVR plot. It can be observed that most of the points lie between +1σ and −1σ. Though there are a few outliers this suggests a good model fit. Figure 5.14 compares the initial Pareto front with the final converged front. It

Figure 5.13: SCV Rivalues using Leave-One-Out validation for final emulator for modeling

manufacturing variations

can be noted that significant improvement in the Pareto front is obtained.