# 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.

## Keywords

High-dimensional expensive problem Bi-level optimization Multi-fidelity surrogate models Underwater wing design## Nomenclature of important variables

## Subscripts

*u*,*l*Upper and lower

*i*,*j*Indexes

- _
Separator

## Superscripts

- *
Optimal values

## Symbols

*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}*j*th Parameter of*i*th section shape curve function- ind
*V* 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

## Notes

### Acknowledgments

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

### Funding information

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).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

## References

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