# Variable decomposition approach applied to multi-objective optimization for minimum powering of commercial ships

## Abstract

As computational fluid dynamics has matured to the point where it is widely accepted as a key tool for ship hull form design, development of simulation-based design (SBD) has been strongly motivated in the past decades. Although many successful demonstrations of SBD were presented, most cases just deal with minimization of total resistance with a formulation of single-objective optimization problem. Once the interest is in minimization of ship-scale delivered power or effective power, issue related to accuracy of the simulation appears critical in many cases, which yield unconvincing results to hull form designers. The method we propose in this paper aims at overcoming the issues. Instead of just counting on predicted power from the simulation and solve a single-objective optimization problem, we first introduce variable decomposition approach to decompose a target ship performance function into terms including embedded parameters, then formulate and solve a multi-objective optimization problem (MOOP). Any scheme to solve MOOP can be applied. In the following, an overview of the present approach is given and results are presented and discussed through comparison with available experimental fluid dynamics data and detailed analysis of flow and integral parameters. The effectiveness of the present approach is also discussed.

## Keywords

Variable decomposition approach Multiobjective optimization RANS-CFD Energy saving device Minimum powering Commercial ship## List of symbols

*L*_{PP}Length between perpendiculars (m)

*L*_{WL}Length at the waterline (m)

*B*Breadth moulded (m)

*d*Draft moulded (m)

- \(\nabla\)
Displacement (m

^{3})- \({C_{\text{B}}}=\frac{\nabla }{{{L_{{\text{PP}}}}Bd}}\)
Block coefficient

*V*_{S}Ship speed (m/s)

- \(\rho\)
Density of water (kg/m

^{3})- \(g\)
Gravitational acceleration (m/s

^{2})- \(\nu\)
Kinematic viscosity

- \(Fr=\frac{{{V_{\text{S}}}}}{{\sqrt {g{L_{{\text{PP}}}}} }}\)
Froude number

- \(Re=\frac{{{V_{\text{S}}}{L_{{\text{PP}}}}}}{\nu }\)
Reynolds number

*R*Total resistance (N)

*1*+*k*Form factor

- \(D\)
Diameter of propeller (m)

*n*Propeller rate of revolution (rps)

*V*_{a}Advance speed of propeller (m/s)

- \(J=\frac{{{V_{\text{a}}}}}{{nD}}\)
Advance ratio

- \(T\)
Propeller thrust (N)

- \({K_t}=\frac{T}{{\rho {n^2}{D^4}}}\)
Thrust coefficient

- Q
Propeller torque (N m)

- \({K_{\text{q}}}=\frac{Q}{{\rho {n^2}{D^5}}}\)
Torque coefficient

- \({\text{SFC}}\)
Skin friction correction

*t*Thrust deduction factor, e.g., \(t=\frac{{T - (R - {\text{SFC}})}}{T}\)

*w*_{n}Nominal wake, \({w_{\text{n}}}=\frac{{\int_{0}^{{2\pi }} {\int_{{\frac{{{d_h}}}{2}}}^{{\frac{D}{2}}} {ur{\text{d}}r{\text{d}}\theta } } }}{{\frac{\pi }{4}({D^2} - d_{h}^{2})}}\), where the origin is the center of propeller

*w*_{e}Effective wake fraction

- \({w_T}=\frac{{{V_{\text{S}}} - {V_{\text{a}}}}}{{{V_{\text{S}}}}}\)
Taylor wake fraction in model scale

*w*_{s}Estimated wake fraction of ship

*Q*(*O*)Propeller torque in open water (N m)

- \({\eta _0}=\frac{{T{V_{\text{a}}}}}{{2\pi nQ(O)}}\)
Propeller open water efficiency

- \({\eta _{\text{R}}}=\frac{{Q(O)}}{Q}\)
Relative rotative efficiency

- \({\eta _{\text{P}}}=\frac{{1 - t}}{{1 - {w_T}}}{\eta _0}{\eta _{\text{R}}}\)
Propulsive efficiency

- \({C_{{\text{th}}}}=T\Bigg/\left( {\frac{{\rho {V_{\text{a}}}^{2}}}{2}\frac{{\pi {D^2}}}{4}} \right)\)
Thrust coefficient

## Notes

### Acknowledgements

This work has been supported by Grant-in-Aid for Scientific Research, Japan (Project nos. 24360363 and 15H04217). The authors would like to express their appreciation to those who concern for the support and encouragement. The authors’ appreciation is extended to Mr. Kenichi Kume (NMRI) for extensive CFD data (duct series self-propulsion simulation by using unstructured meshing tool and CFD); and Mr. Yoshihisa Okada and Mr. Kenta Katayama (Nakashima Propeller Co., LTD) for valuable advice on duct optimization problem.

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