A simplified shape optimization strategy for blended-wing-body underwater gliders

Abstract

This paper presents a simplified shape optimization (SSO) strategy for blended-wing-body underwater gliders (BWBUGs). In this strategy, the 3-D shape optimization of the BWBUG is approximately simplified into three 2-D airfoil design problems. The three crucial sectional airfoils are firstly selected from the BWBUG and the Class-Shape function Transformation method is used for parameterization of the airfoils. Then, the airfoils are optimized using a kriging-based constrained global optimization (KCGO) method and the optimized BWBUG shape is finally formed with the optimized airfoils. To verify the practicability of the proposed SSO strategy, the shape of the BWBUG is optimized using a direct shape optimization (DSO) strategy as well. Results show that the lift-to-drag ratio of the BWBUG is improved by 4.1728% and 4.4328% with SSO and DSO strategies respectively, which shows that the SSO is comparable with the DSO strategy in the optimization results. However, the CPU time involved in SSO is only about 16% of that in DSO, which illustrates that SSO is an applicable and more efficient strategy for shape design optimization of BWBUGs.

Appendix

1.1 A. Validation of the CFD-based simulation

In order to ensure the accuracy of the simulation, the simulation results are validated with the experiment results provided by Zarruk et al. (2014). They studied the hydrodynamic performance of two 3-D tapered NACA0009 hydrofoils by experiments. Each type of hydrofoil includes four metal models with different materials including stainless steel, aluminum and composite. In this paper, the experiment results of the tapered standard NACA0009 hydrofoil made of stainless steel are compared with the simulation results. The geometry of the hydrofoil owns a trapezoidal planform. The span and base chord are 0.3 m and 0.12 m respectively. Besides, the tip chord is 0.06 m and then the aspect ratio can be calculated as 3.33. The mesh used for numerical simulation is a structured mesh with 440,400 cells (y⁺ = 5). The geometric model and the detailed meshes of boundary layers are shown in Fig. 13.

Fig. 13

Geometric shape of the test model and detailed meshes of the boundary layers

The simulation results are shown in Fig. 14 and they are compared with the experiment results. The comparison illustrates that simulation results are in good agreement with the experimental data. On one hand, the lift coefficients (C_l) obtained by simulations are very close to the experimental results when the angle of attack is less than 4 degrees. As the angle of attack becomes bigger, the simulated lift coefficients are little smaller. On the other hand, the simulated drag coefficients (C_d) have the same trend with the experiment data as the angle of attack varies. To be more specific, the simulated drag coefficients are a little larger than experiment results before 4 degrees. When the angle of attack is bigger than 4 degrees, the simulated results are smaller than experiment results.

Fig. 14

Comparison of the lift and drag coefficients between the experiment and simulation results

To ensure that the simulation results are independent of mesh sizes, the simulation will be performed on different mesh sizes and y⁺ values. The structured mesh with the sizes 1,505,000 (y⁺ = 0.5), 1,043,720 (y⁺ = 1) and 440,400 (y⁺ = 5) are being used for simulating the hydrodynamic performance of the hydrofoil respectively (Reynolds number is 1.0 × 10⁶ and angle of attack is set to 4 degrees). The results shown in Table 6 indicate that the C_l and C_d are almost the same between the finest and coarsest mesh. Specifically, there exist 0.16 and 1.4% differences in C_l and C_d respectively. Considering the efficiency and accuracy, a smaller mesh size is used in this paper.

Table 6 Comparison of lift and drag coefficients with different mesh sizes and y⁺ values

1.2 B. Kriging model

Kriging was initially presented by the Geologist Krige (1951) and Sacks et al. (1989) first applied it to predict the responses of the input based on the simulation results. Kriging predict responses through a constant plus a stochastic process.

$$ f(x)=\mu +Z(x) $$

(14)

where f(x) is the responses of the predictor, μ is the global approximation constant and Z(x) is the stochastic process with the following characteristics.

$$ {\displaystyle \begin{array}{l}\begin{array}{l}\kern7.5em E\left[Z(x)\right]=0\\ {}\kern6em Var\left[Z(x)\right]\kern0.5em ={\sigma}^2\end{array}\\ {} Cov\left[Z\left({x}^i\right),Z\left({x}^j\right)\right]={\sigma}^2R\left({x}^i,{x}^j\right)\end{array}} $$

(15)

where σ² represents the variance of the response and R(xⁱ, x^j) is the correlation function between xⁱ and x^j. In this paper, Gaussian function is selected as the correlation which is described in (16).

$$ R\left({x}^i,{x}^j\right)=\exp \left\{-\sum \limits_{k=1}^{nd}{\theta}_k{\left({x}_k^i-{x}_k^j\right)}^2\right\} $$

(16)

where θ_k are the hyper-parameters which need to be determined by maximum likelihood estimation. Since the correlation function is selected as the Gaussian function, the predicted response can be expressed as (17) and (18).

$$ \widehat{f}(x)=\widehat{\mu}+{r}^T(x){R}^{-1}\left(f-1\widehat{\mu}\right) $$

(17)

$$ \widehat{\mu}=\frac{1^T{R}^{-1}f}{1^T{R}^{-1}1} $$

(18)

where f represents the responses of the sample points, r(x) is the vector of correlation function between the to-be-estimated point x and sample points, and $\widehat{\mu}$ denotes the generalized least square estimator. Besides, the prediction variance of kriging can be estimated as follows:

$$ {s}^2(x)={\sigma}^2\left[1-{r}^T{R}^{-1}r+\frac{{\left(1-{1}^T{R}^{-1}r\right)}^2}{1^T{R}^{-1}1}\right] $$

(19)

$$ {\sigma}^2=\frac{{\left(y-1\widehat{\mu}\right)}^T{R}^{-1}\left(f-1\widehat{\mu}\right)}{n} $$

(20)