隨著全球潛艦技術的不斷進步，潛艦的幾何外形設計對於減阻性能的提升具有重要意義。本研究以伴隨算子方法應用於Joubert通用潛艦幾何結構外形作減阻之優化分析，以細長比7.7的艦體模型在雷諾數1.2×10^7的狀態下，獲取艦體的阻力數據並透過伴隨算子來進行艦艏與艦艉幾何外形減阻優化。為避免幾何外形因優化而造成外形變化過於劇烈，因而造成無法觀察到外形隨優化次數增加的變化。故研究中設定艦體總阻力減少10%為目標進行外形優化並限制艦體外形範圍改變。
經過優化後的外形，其摩擦阻力與壓力阻力都有下降的趨勢，而艦體的摩擦阻力占有總阻力的80%以上。研究發現多次優化的外形雖可減少阻力，但伴隨算子方法建議之潛艦優化外形呈現不平整波浪表面，為此本文利用伴隨算子方法之建議優化的輪廓外形數據，再透過平滑曲線擬合方式重建艦體外形。該方法不僅可以獲得更符合需求的表面平滑外形，總阻力與原始Joubert艦體可減少接近10%。此外本研究也使用伴隨算子方法來同時優化二維(2D)與三維(3D)軸對稱幾何的艦體外形模型，研究結果顯示軸對稱二維與三維模型優化結果有相同的趨勢，它們的總阻力與原始艦體相比都減少接近10%，此研究結果說明軸對稱外形的物體可以用二維模型進行優化以減少計算量與時間成本。三維模型適用複雜且非軸對稱幾何外形的物體，未來可以將模型換成配有後控制面/方向舵/帆罩的艦體外形，能透過本研究所使用的伴隨算子優化方法可根據不同需求進行不同方面的優化設計(例如升力/阻力的優化與降噪的優化等)。
透過設計者與伴隨算子方法建議之優化外形的相互協作，伴隨算子方法建議的優化外形輸出結果不僅可以提供設計者靈感去決定外形要如何設計，也可以縮短研發時間。設計者作為監督伴隨算子優化分析方法的角色，需要判斷伴隨算子優化分析方法的輸出結果是否符合需求，還需要適時修正伴隨算子優化分析方法的輸出結果以符合設計需求。這樣的相互協作方式，不僅能減少設計時間還能減少製造成本。

With the continuous advancement of global submarine technology, the geometric design of submarines plays a significant role in improving drag reduction performance. This study provides a method by using Adjoint Solver to optimize the drag reduction of Joubert generic submarine geometry. The hull model with length and diameter ratio of 7.7 is analyzed under Reynolds number 1.2×10^7 to obtain drag data, and the Adjoint Solver is used to optimize the bow and stern geometry for drag reduction. To avoid excessive geometric changes due to optimization, which may obscure the changes in shape with the number of optimizations, the study sets a goal of reducing total drag of the hull by 10%.
After optimization, total drag including skin frictional drag and pressure drag shows a descending trend and frictional drag accounts for more than 80% of the total drag. However, the shape of a submarine after multiple optimizations results in a wavy surface. The wavy surface shape causes the bow (or stern) unsuitable for submarine design requirements. Therefore, it is proposed to redraw the hull profile using smooth curve fitting to represent the final shape design.
The methods used in this study not only achieves a more suitable shape but also reduces the total drag by nearly 10% compared to the original Joubert hull. In the study, Adjoint Solver was used to optimize both 2D and 3D axisymmetric model. The results indicate that the optimization trends are consistent for both 2D and 3D axisymmetric models. They both can achieve nearly 10% reduction in total drag compared to the original hull. Additionally, the study shows that axisymmetric shapes can be optimized using 2D models to reduce computational and time costs, while 3D models are suitable for complex and non-axisymmetric geometries. In the future, the hull shapes with attached control surfaces, rudders, and sail structures can be optimized in various aspects (such as lift/drag optimization and noise reduction optimization) using the method proposed in this study.
Through the collaboration between designers and Adjoint Solver, the output from the Adjoint Solver can not only inspire designers in determining the optimal shape, but also shorten development time. As supervisors of the Adjoint Solver, need to assess whether the outcomes meet the requirements and make timely adjustments to align the results with design needs. This collaborative approach can reduce both design time and manufacturing costs.

[1] P. N. Joubert, "Some Aspects of Submarine Design Part 1. Hydrodynamics," Australian Department of Defnce, Australia, 2004.
[2] M. Renilson, Submarine Hydrodynamics 2nd ed., New York: Springer, 2018.
[3] K. He, Z. Pan, W. Zhao, J. Wang and D. Wan, "Overview of Research Progress on Numerical Simulation Methods for Turbulent Flows Around Underwater Vehicles.," Journal of Marine Science and Application, pp. 1-22, 2024.
[4] D. F. Myring, "A Theoretical study of body drag in subcritical axisymmeric flow," Aeronautical quarterly, pp. 186-194, 1976.
[5] P. N. Joubert, "Some Aspects of Submarine Design Part 2. Shape of a Submarine 2026," Defence Science and Technology Organisation, Australia, 2006.
[6] M. B. Jones, L. P. Erm, A. Valiyff and S. M. Henbest, "Skin-Friction Measurement on a Model Submarine.," Australian Department of Defnce, Australia, 2013.
[7] D. B. Clarke, D. Butler, C. L. Ellis and P. A. Brandner, "Htdrodynamic Measurements on the Joubert Hull in AMC Cavitation Tunnel with CFD Determined Blockage Corrections," in 20th Australia Fluid Mechanics Conference, Perth, Australia, 2016.
[8] H. Tober, Evaluation of drag estimation methods for ship hulls., 2020.
[9] M. Moonesun, M. Javadi, P. Charmdooz and K. U. Mikhailoich, "Evaluation of submarine model test in towing tank and comparison with CFD and experimental formulas for fully submerged resistance," Indian Journal of Geo-Marine Sciences Vol.42, pp. 1049-1056, 2013.
[10] M. Moonesun, Y. Korol and H. Dalayeli, "CFD Analysis on the Bare Hull Form of Submarines for Minimizing the," International Journal of Maritime Technology 3, pp. 1-16, 2015.
[11] M. Moonesun, A. Mahdian, Y. M. Korol, M. Dadkhah and M. M. Javadi, "Concepts in submarine shape design.," Indian Journal of Geo-Marine Sciences Vol.45, pp. 100-104, 2016.
[12] M. M. Karim, M. M. Rahman and M. A. Alim, "Numerical computation of viscous drag for axisymmetric underwater vehicles.," Jurnal Mekanikal, No.26, pp. 9-21, 2008.
[13] M. Tabatabaei Malazi, S. Tumse, M. Ozgoren and B. Sahin, "A Computational Investigation of the Influence of Seafloor Conditions on the Turbulent Flow Characteristics of an Autonomous Underwater Vehicle.," Arabian Journal for Science and Engineering, pp. 1-17, 2024.
[14] 李均培, 陳建宏 且 辛敬業, “軸對稱旋轉體在不同紊流模型下的流場計算研究,” Journal of Taiwan Society of Naval Architects and Marine Engineers, Vol.36, No.4, pp. 169-178, 2017.
[15] M. Moonesun, Y. M. Korol and A. Brazhko, "CFD analysis on the equations of submarine stern shape.," Journal of Taiwan Society of Naval Architects and Marine Engineers 34.1, pp. 21-32, 2015.
[16] A. Jameson, "Aerodynamic design via control theory.," Journal of scientific computing Vol. 3, pp. 233-260, 1988.
[17] D. Hill, "Adjoint systems and their role in the receptivity problem for boundary layers," Journal of Fluid Mechanics Vol. 292, pp. 183-204, 1995.
[18] J. Munoz-Paniagua, J. Garcia, A. Crespo and F. Laspougeas, "Aerodynamic optimization of the nose shape of a train using the adjoint method.," Journal of Applied Fluid Mechanics Vol.8, pp. 601-612, 2015.
[19] D. G. Cacuci, R. Fang, M. Ilic and M. C. Badea, "A heat conduction and convection analytical benchmark for adjoint solution verification of computational fluid dynamics codes used in reactor design," Nuclear Science and Engineering Vol. 182, pp. 452-480, 2016.
[20] C. Othmer, "A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows," International journal for numerical methods in fluids Vol. 58, pp. 861-877, 2008.
[21] R. Corral and F. Gisbert, "Profiled End Wall Design Using an Adjoint Navier–Stokes Solver," Jounal of Turbomachinery Vol. 130, p. 21011, 2008.
[22] C. Z. Wang, K. R. Nagisetty, F. Montanari and D. C. Hill, "Application of adjoint solver to optimization of fin heat exchanger.," Turbo Expo: Power for Land, Sea, and Air Vol. 56734, 2015.
[23] H. Day, D. Ingham, L. Ma and M. Pourkashanian, "Adjoint based optimisation for efficient VAWT blade aerodynamics using CFD," Journal of Wind Engineering and Industrial Aerodynamics Vol. 208, 2021.
[24] S. Zhang, T. Tezdogan, B. Zhang, L. Xu and Y. Lai, "Hull form optimisation in waves based on CFD technique. ," Ships and Offshore Structures Vol. 13, pp. 149-164, 2018.
[25] A. Nazemian and P. Ghadimi, "Shape optimisation of trimaran ship hull using CFD-based simulation and adjoint solver," Ships and Offshore Structures Vol. 17, pp. 359-373, 2022.
[26] F. J. Kelecy, "Adjoint Shape Optimization for Aerospace Application," NASA Ames Research Center, 2021.
[27] "ANSYS FLUENT Adjoint Solver R14.5," ANSYS, Inc., 10 2012. [Online]. Available: https://www.scribd.com/document/135734516/Fluent-Adjoint-Solver-14-5.