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Shape optimization of underwater wings with a new multi-fidelity bi-level strategy

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Abstract

This paper proposes a new multi-fidelity bi-level optimization (MFBLO) strategy for shape designs of underwater wings. Firstly, hydrodynamic analyses of the wing planform and sections are decoupled for constructing a bi-level shape optimization frame, which includes an upper-level task merely concerning the wing planform design and several lower-level tasks only related to the section designs. By doing this, the shape design optimization gets remarkable benefits from the reduction of dimension and computational costs. Secondly, the bridge function method combined with three multi-fidelity data fusion approaches CC1, CC2, and CC3 are proposed to conduct the bi-level optimization, respectively. After comparison analyses, CC2 shows higher computational efficiency and accuracy, which is more appropriate for the bi-level shape optimization frame. Finally, compared with the single-level optimization with the fixed planform or sections and the conventional high-dimensional optimization, the proposed MFBLO needs less computation budget and gets higher lift-drag ratio, showing its outstanding advantages in handling the shape optimization of underwater wings.

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Abbreviations

u, l :

Upper and lower

i, j :

Indexes

_:

Separator

*:

Optimal values

b :

Wing span,

c r, c t :

Root and tip chords

λ :

Leading edge angle

S w :

Planform area

ns :

Number of sections

n :

Dimension of problems

V :

Velocity

α :

Angle of attack (AOA)

F, f, G, g, H, h, x :

Objective functions, constraints and design variables of bi-level optimizations

ind, eff:

Induced and effective,

prof,∞:

Profile and far field

HF, LF:

High and low fidelities

^:

Predicted values

c, t/c :

Chord, relative thickness of sections

Sp :

Spanwise position of sections

w li_j :

jth Parameter of ith section shape curve function

indV :

Induced velocities in VLM

w :

Area weight coefficients

N :

Looping index of optimizations

L, D, C L, C D :

Lift, drag, and their coefficients of UWs

l, d, C l, C d :

Lift, drag, and their coefficients of sections

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Acknowledgments

The authors are also grateful to members of the research group for the implementation of some existing multi-fidelity optimization algorithms.

Funding

Supports are from the National Natural Science Foundation of China (Grant No. 51875466 and Grant No. 51805436) and China Postdoctoral Science Foundation (Grant No. 2018M643726).

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Authors and Affiliations

  1. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China

    Siqing Sun, Baowei Song, Peng Wang, Huachao Dong & Xiao Chen

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  1. Siqing Sun
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  4. Huachao Dong
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Correspondence to Peng Wang.

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Appendices

Appendix 1. Four test cases

Two different wing planform shapes, including straight and swept wings, are selected as basic geometries, and Table 4 lists the parameters. Besides, two different hydrofoils, including symmetric and cambered sections, are chosen as fundamental profiles, as Fig. 19 shows. By combining them arbitrarily, four different test wings are generated, and Table 4 also gives the nomenclatures of the cases. To simulate actual working condition for the UW, the AOAs are set from 0 to 8 degree, and meanwhile, the far-field velocity is set to 1 m/s.

Table 4 Geometry parameters of wings
Full size table
Fig. 19
figure 19

Fundamental profile. a Symmetric. b Cambered

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Appendix 2. Five mathematical examples

Table 5 Five MF mathematical examples and their expressions
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Appendix 3. Validation of Q3DL, Q3D, and RANS simulations

The simulation results need to be validated with the experiment results to ensure the accuracy. Hence, we adopt experiment results of “Type I-SS” UW, which are tested by Zarruk et al. (2014) in a water tunnel at University of Tasmania. Type I-SS UW is made by stainless steel, whose sections are NACA0009 hydrofoils. Besides, the mesh used for RANS simulation is a structured mesh with 438,870 cells (y+ = 5). The shape and the computational mesh of Type I-SS UW are shown in Fig. 20. The simulation results are shown in Fig. 21. The comparison illustrates that RANS simulation results are in good agreement with the experimental data. Besides, Q3DL and Q3D solvers have the same trend with the experiment data as the angle of attack varies. Thus, RANS is regarded as the highest-fidelity solver, and Q3D is a high-fidelity solver, whereas Q3DL has the lowest fidelity.

Fig. 20
figure 20

Shape and computational mesh of “Type I-SS” UW. a Geometric model of “Type I-SS” UW. b Detailed meshes of boundary layers

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Fig. 21
figure 21

Simulation results of Q3DL, Q3D, RANS, and experiment

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To make sure the simulation results are independent to mesh size, simulated results with different mesh sizes and y + values are compared. The mesh with the sizes 2,205,450 (y+ = 0.5), 1,141,380 (y+ = 1) and 438,870 cells (y+ = 5) are used for simulating the hydrodynamic performance of Type I-SS UW, respectively. Besides, Reynolds number and AOA is set to 1.0 × 106 and 4 degrees, respectively. The results shown in Table 6 indicate that CL and CD are almost the same between the finest and coarsest mesh. There are 0.18% and 1.7% differences in CL and CD. Hence, compromising between the efficiency and accuracy, the mesh with the sizes 438,870 cells (y+ = 5) are used in this paper.

Table 6 Validation of mesh independence
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Sun, S., Song, B., Wang, P. et al. Shape optimization of underwater wings with a new multi-fidelity bi-level strategy. Struct Multidisc Optim 61, 319–341 (2020). https://doi.org/10.1007/s00158-019-02362-z

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