Robust functional response-based metamodel optimization considering both location and dispersion effects for aeronautical airfoil designs

Abstract

Airfoils play significant roles in aerodynamic engineering as they are important devices in designing aircraft and engines. For airfoils, geometric design variables often deviate from nominal values that deteriorate airfoil quality. Thus, the design optimization of airfoils under the noises of design variables is important. However, the existing literature does not consider functional responses that can more adequately describe airfoil performances than univariate and multivariate responses. To fill in this gap and further improve airfoil quality, we develop a surrogate-based robust parameter design methodology for the design optimization of airfoils with functional responses. A new optimization criterion that involves both location and dispersion effects associated with the functional feature of quality losses, noises of design variables, and emulator uncertainty is proposed. A Gaussian process (GP) model with functional responses is employed as a surrogate/emulator. An efficient method for computing the proposed criterion is developed. The proposed methodology is applied to a typical airfoil design optimization problem to demonstrate that better and more robust solutions are obtained with the proposed criterion for airfoils, and the computing time is significantly reduced with the proposed estimation method.

Appendices

Appendix 1.

1.1 Proof of Theorem 1

The proof involves three steps. In the first step, we prove that ${Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)$ uniformly converges to ${\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$. We then show that $\underset{n\to \infty, m\to \infty }{\lim}\underset{{\boldsymbol{x}}_c\boldsymbol{\in}{\chi}_c}{\min }{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)=\underset{{\boldsymbol{x}}_c\boldsymbol{\in}{\chi}_c}{\min }{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$ in Step 2 and prove $\underset{n\to \infty, m\to \infty }{\lim }{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)={\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)$ in Step 3.

• Step 1. Proof of uniform convergence of ${Q}_f^{n,m}\left({x}_c\right)$ to ${\mathcal{Q}}_f\left({x}_c\right)$.

We start with the proof of uniform convergence of E_Y[L(Y_{n, m}(x, s))] and Var_Y[L(Y_{n, m}(x, s))] to L(y(x, s)) and 0 for quadratic and non-quadratic loss functions. Then, the uniform convergence of ${Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)$ to ${\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$ can be obtained since E_{x, s}[E_Y[L(Y_{n, m}(x, s))]], E_{x, s}[Var_Y[L(Y_{n, m}(x, s))]] and Var_{x, s}[E_Y[L(Y_{n, m}(x, s))]] uniformly converge to E_{x, s}[L(y(x, s))], zero and Var_{x, s}[L(y(x, s))], respectively.

1.2 Case 1:Typical quadratic loss functions

In this case, the proof is straightforward since ${E}_Y\left[L\left({Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)\right)\right]={\left({\mu}_{Y,n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-T\left(\boldsymbol{s}\right)\right)}^2+{\sigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)$ and ${Var}_Y\left[L\left({Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)\right)\right]=4{\left({\mu}_{Y,n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-T\left(\boldsymbol{s}\right)\right)}^2{\sigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)+2{\sigma}_{Y,n,m}^4\left(\boldsymbol{x},\boldsymbol{s}\right)$, where μ_{Y, n, m}(x, s) and ${\sigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)$ denote the posterior mean and variance at (x, s) of a GP metamodel constructed with an experimental design of size n over a grid in S of size m. Note that Wu and Schaback (Wu and Schaback 1993) prove that μ_{Y, n, m}(x, s) uniformly converges to y(x, s) as n →  ∞ , m → ∞, and μ_{Y, n, m}(x, s) is uniformly bounded by a constant B. Furthermore, Ranjan et al. (Ranjan et al. 2008) prove that ${\sigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)$ uniformly converges to 0 when f(x, s) = 1 as n →  ∞ , m → ∞. Moreover, ${\sigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)$ can be shown to be bounded from above by a constant C_{n, m} depending on n and m (Ranjan et al. 2008; Wang et al. 2020). Then, we obtain that E_Y[L(Y_{n, m}(x, s))] uniformly converges to L(y(x, s)) = (y(x, s) − T(s))² and Var_Y[L(Y_{n, m}(x, s))] uniformly converges to zero since sum and product of uniformly convergent and bounded functions preserve uniform convergence.

1.3 Case 2: Extended bounded and uniformly continuous function

In this case, we prove E_Y[L(Y_{n, m}(x, s))] uniformly converges to L(y(x, s)). Then, Var_Y[L(Y_{n, m}(x, s))] = E_Y[L²(Y_{n, m}(x, s))] − E_Y[L(Y_{n, m}(x, s))]² uniformly converges to zero since L²(∙) is bounded and uniformly continuous and E_Y[L(Y_{n, m}(x, s))] is uniformly bounded.

By uniform convergence of μ_{Y, n, m}(x, s) and ${\upsigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)$, we have $\underset{\boldsymbol{x},\boldsymbol{s}}{\sup}\mid {\mu}_{Y,n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-y\left(\boldsymbol{x},\boldsymbol{s}\right)\mid <{\upvarepsilon}_0$ and $\underset{\boldsymbol{x},\boldsymbol{s}}{\sup }{\upsigma}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)<{\upvarepsilon}_1$ for some ε₀, ε₁ > 0. Then, $\underset{n\to \infty, m\to \infty }{\lim}\underset{\boldsymbol{x},\boldsymbol{s}}{\sup }P\left(\left|{Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-y\left(\boldsymbol{x},\boldsymbol{s}\right)\right|\ge c\right)=0$ for any c > 0 since $P\left(|{Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-y\left(\boldsymbol{x},\boldsymbol{s}\right)|\ge c\right)\le \frac{1}{c^2}E\left[{\left({Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-y\left(\boldsymbol{x},\boldsymbol{s}\right)\right)}^2\right]\le \frac{1}{c^2}\left[{\hat{\sigma}}_{Y,n,m}^2\left(\boldsymbol{x},\boldsymbol{s}\right)+4B\left|{\mu}_{Y,n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)-y\left(\boldsymbol{x},\boldsymbol{s}\right)\right|\right]$, where ∣μ_{Y, n, m}(x, s) ∣  ≤ B and |y(x, s)| ≤ B. Similarly, $\underset{\boldsymbol{x},\boldsymbol{s}}{\sup }P\left(|L\left({Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)\right)-L\left(y\left(\boldsymbol{x},\boldsymbol{s}\right)\right)|>\varepsilon \right)\le \frac{1}{{c_{\varepsilon}}^2}\left({\upvarepsilon}_1+4B{\upvarepsilon}_0\right)$ since P(| L(Y_{n, m}(x, s)) − L(y(x, s))| >ε) ≤ P(| Y_{n, m}(x, s) − y(x, s)| >c_ε) due to the uniform continuity of L(y) ( i.e., if ∣L(Y_{n, m}(x, s)) − L(y(x, s)) ∣  > ε, there exists c_ε > 0 that ∣Y_{n, m}(x, s) − y(x, s) ∣  > c_ε). Finally, the proof of $\underset{n\to \infty, m\to \infty }{\lim}\underset{\boldsymbol{x},\boldsymbol{s}}{\sup}\left|{E}_Y\left[L\left({Y}_{n,m}\left(\boldsymbol{x},\boldsymbol{s}\right)\right)\right]-L\left(y\left(\boldsymbol{x},\boldsymbol{s}\right)\right)\right|=0$ can be easily obtained by modifying the bounded convergence theorem in Durret (Durrett 2019). This gives that E_Y[L(Y_{n, m}(x, s))] uniformly converges to L(y(x, s)).

• Step 2: Proof of $\underset{n\to \infty, m\to \infty }{\lim}\underset{{\boldsymbol{x}}_c\boldsymbol{\in}{\chi}_c}{\min }{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)=\underset{{\boldsymbol{x}}_c\boldsymbol{\in}{\chi}_c}{\min }{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$.

Proof:

We replace minimum with infimum for proof. Since ${Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)$ uniformly converges to ${\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$, there exists ε > 0, N ∈ ℕ₊ such that $\forall n>N,\left|{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)-{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)\right|<\upvarepsilon$. This gives ${\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)-\upvarepsilon <{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)<{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)+\upvarepsilon$ . Choose δ = 1/n, pick up x_{c, 0} such that ${\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,0}\right)\le \operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)+1/n$. Then, we have ${Q}_f^{n,m}\left({\boldsymbol{x}}_{c,0}\right)<{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,0}\right)+\upvarepsilon \le \operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)+1/n+\upvarepsilon$. Since $\operatorname{inf}{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)\le {Q}_f^{n,m}\left({\boldsymbol{x}}_{c,0}\right)$, we obtain $\operatorname{inf}{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)<{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,0}\right)+\upvarepsilon \le \operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)+1/n+\upvarepsilon$. By taking limits, we obtain $\underset{n\to \infty, m\to \infty }{\lim}\operatorname{inf}{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)\le \operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$ if both of them are finite.

On the other hand, pick up x_{c, 1} such that ${Q}_f^{n,m}\left({\boldsymbol{x}}_{c,1}\right)\mathbf{\le}\operatorname{inf}{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)+\frac{1}{n}$. Since ${\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,1}\right)-\upvarepsilon <{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,1}\right)$ and $\operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)\le {\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,1}\right)$, we have $\operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)-\upvarepsilon <{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,1}\right)\mathbf{\le}\operatorname{inf}{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)+\frac{1}{n}$. By taking limits, we obtain $\underset{n\to \infty, m\to \infty }{\lim}\operatorname{inf}{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)\ge \operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$. Thus, we get $\underset{n\to \infty, m\to \infty }{\lim }{Q}_f^{n,m}\left({\boldsymbol{x}}_c\right)=\operatorname{inf}{\mathcal{Q}}_f\left({\boldsymbol{x}}_c\right)$. If both infimums are attainable, we complete the proof.

• Step 3: Proof of $\underset{n\to \infty, m\to \infty }{\lim }{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)={\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)$

Note that Step 2 implies that $\underset{n\to \infty, m\to \infty }{\lim }{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)={\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)$ and Step 1 implies that $\underset{n\to \infty, m\to \infty }{\lim }{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)={\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)$. Then, there exists ε₀ > 0, ε₁ > 0, N₀ ∈ ℕ₊, and M₀ ∈ ℕ₊ such that ∀n > N₀, ∀ m > M₀, $\left|{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)-{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)\right|<{\upvarepsilon}_1$ and $\left|{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)-{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)\right|<{\upvarepsilon}_0$. Thus, $\left|{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)-{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)\right|<\left|{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)-{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)|+|{Q}_f^{n,m}\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)-{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)\right|<{\upvarepsilon}_0+{\upvarepsilon}_1$, indicating that $\underset{n\to \infty, m\to \infty }{\lim }{\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}^{n,m}\right)={\mathcal{Q}}_f\left({\boldsymbol{x}}_{c,\mathrm{opt}}\right)$.

Appendix 2.

1.1 Comparisons between the proposed and existing methodologies for normally distributed noises in design variables

In this appendix, noises in design variables are assumed to follow independently normal distributions N(x_{c, i}, 0.01), i = 1, 2, 3 to demonstrate advantages of the proposed methodology in designing airfoils for other types of probability distributions.

We utilize a random sample of x of size 1000 in the proposed computation method and employ the GP model constructed in Sect. 4.1 to perform the proposed method. To compare optimal solutions obtained with the proposed Q_f (using w₁ = 0,0.5,0.625, 0.75) and the existing μ_Q (i.e., Q_f with w₁ = 1), we present associated location and dispersion effects in Table 9, and maximum expected quality loss Q_max and lengths of associated 95%PIs at Q_max in Table 10. From Table 9, we see that all dispersion effects (${\sigma}_Q^2$, ${\sigma}_s^2$,${\sigma}_y^2$, and ${\sigma}_x^2$) obtained with the proposed Q_f using w₁ = 0, 0.5, 0.625, 0.75 are all reduced while μ_Q ’ s are slightly increased. This indicates the proposed Q_f can give more robust solutions to the spatial variability, emulator uncertainty, and noises in design variables than that obtained with the existing criterion μ_Q. Table 10 indicates that the proposed Q_f using w₁ = 0, 0.5, 0.625,0.75 gives solutions with substantially smaller Q_max and narrower 95% PIs at Q_max as RI_Q ≥ 22% and RI_Q ≥ 26%. This again indicates more robust and better solutions of airfoil design are obtained with the proposed Q_f.

Table 9 Location effects μ_Q and dispersion effects obtained with the proposed Q_f (using w₁ = 0,0.5,0.625, 0.75) and the existing μ_Q (normally distributed noises in design variables)

Table 10 The maximum expected quality loss Q_max and corresponding 95% PI obtained with the proposed and the existing criteria (normally distributed noises in design variables)

Table 11 presents optimal solutions obtained with the proposed Q_f (using w₁ = 0,0.5,0.625, 0.75) and the existing μ_Q (i.e., Q_f with w₁ = 1), which are close to that obtained with uniform noises in Table 7. This indicates optimal solutions obtained in the proposed method are robust to the type of probability distributions of manufacturing errors in the airfoil system. It also indicates that the angle of attack is the most important design variable in influencing dispersion effects and smaller values of w₁ give an optimal design with a larger angle of attack. Analysis of why the angle of attack decreases with w₁ can be found in Sect. 4.2 from an aerodynamic point of view. Since plots of Q(s) and 95% PIs over S for the optimal solutions are similar to that in Fig. 5, we do not present them to save space.

Table 11 Optimal solutions obtained with the proposed Q_f (using w₁ = 0,0.5,0.625, 0.75) and the existing μ_Q (normally distributed noises in design variables)

Appendix 3.

1.1 Training data used in constructing the GP model for the airfoil system

In this appendix, we present all training data of 30 experimental runs in constructing the GP model for the airfoil system in Section 4.

Table 12 Training data of 30 experimental runs (Run 1–Run 10)

Table 13 (continued) Training data of 30 experimental runs (Run 11–Run 20)

Table 14 (continued) Training data of 30 experimental runs (Run 21–Run 30)