{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:28:52Z","timestamp":1760711332198,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T00:00:00Z","timestamp":1709683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Samara National Research University Development Program","award":["PR-NU 2.1-08-2023"],"award-info":[{"award-number":["PR-NU 2.1-08-2023"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This paper describes the development of a methodology for air propeller optimization using Bezier curves to describe blade geometry. The proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span. The goal of optimization is to identify the appropriate geometry of a propeller that reduces the power required to achieve a given thrust. Because the proposed optimization problem is a constrained optimization process, the technique of generating a penalty function was used to convert the process into a nonconstrained optimization. For the optimization process, a variant of the differential evolution algorithm was used, which includes adaptive techniques of the evolutionary operators and a population size reduction method. The aerodynamic characteristics of the propellers were obtained using the similar to blade element momentum theory (BEMT) isolated section method (ISM) and the XFOIL program. Replacing the angle of geometric twist with the angle of attack of the airfoil section as a design variable made it possible to increase the robustness of the optimization algorithm and reduce the calculation time. The optimization technique was implemented in the OpenVINT code and has been used to design helicopter and tractor propellers for unmanned aerial vehicles. The development algorithm was validated experimentally and using CFD numerical method. The experimental tests confirm that the optimized propeller geometry is superior to commercial analogues available on the market.<\/jats:p>","DOI":"10.3390\/computation12030052","type":"journal-article","created":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T04:03:50Z","timestamp":1709697830000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Algorithm for Propeller Optimization Based on Differential Evolution"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2698-1348","authenticated-orcid":false,"given":"Andry","family":"Sedelnikov","sequence":"first","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0893-9878","authenticated-orcid":false,"given":"Evgenii","family":"Kurkin","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"given":"Jose Gabriel","family":"Quijada-Pioquinto","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"given":"Oleg","family":"Lukyanov","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"given":"Dmitrii","family":"Nazarov","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"given":"Vladislava","family":"Chertykovtseva","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"given":"Ekaterina","family":"Kurkina","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-7714-0963","authenticated-orcid":false,"given":"Van Hung","family":"Hoang","sequence":"additional","affiliation":[{"name":"Institute of Aerospace Engineering, Samara National Research University, 34 Moskovskoe Shosse, 443086 Samara, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"137867","DOI":"10.1109\/ACCESS.2020.3012481","article-title":"Comprehensive Optimization of the Unmanned Tilt-Wing Cargo Aircraft with Distributed Propulsors","volume":"8","author":"Chen","year":"2020","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"8","DOI":"10.2514\/1.C036258","article-title":"Considerations in optimal propeller design","volume":"58","author":"Traub","year":"2021","journal-title":"J. Aircr."},{"key":"ref_3","unstructured":"Betz, A. (1966). Introduction to the Theory of Flow Machines, Pergamon Press."},{"key":"ref_4","unstructured":"Betz, A. (1919). Schraubenpropeller Mit Geringstem Energieverlust (Screw Propeller with Least Energy Loss), Akademie der Wissenschaften."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"676","DOI":"10.2514\/3.23779","article-title":"Design of Optimum Propellers","volume":"10","author":"Adkins","year":"1994","journal-title":"J. Propul. Power."},{"key":"ref_6","unstructured":"Aleksandrov, V.L. (1951). Vozdushnye Vinty, Gosudarstvennoe Izdatel\u2019stvo Oboronnoj Promyshlennosti. (In Russian)."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"012060","DOI":"10.1088\/1757-899X\/1024\/1\/012060","article-title":"Multi-Disciplinary Framework for Propeller Blade Design","volume":"1024","author":"Breitsamter","year":"2021","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4723","DOI":"10.2514\/1.J059317","article-title":"Airfoil Design Parameterization and Optimization Using B\u00e9zier Generative Adversarial Networks","volume":"58","author":"Chen","year":"2020","journal-title":"AIAA J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"95","DOI":"10.2514\/1.36055","article-title":"Optimization of Propeller Bases Propulsion Systems","volume":"46","author":"Gur","year":"2009","journal-title":"J. Aircr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.2514\/1.C032859","article-title":"Constrained and Unconstrained Propeller Blade Optimization","volume":"52","author":"Dorfling","year":"2015","journal-title":"J. Aircr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1016\/j.cja.2019.11.005","article-title":"Aeroacoustic and Aerodynamic Optimization of Propeller Blades","volume":"33","author":"Yu","year":"2020","journal-title":"Chin. J. Aeronaut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"159986","DOI":"10.1109\/ACCESS.2019.2950453","article-title":"An Energy Efficiency Optimization Method for Fixed Pitch Propeller Electric Aircraft Propulsion Systems","volume":"7","author":"Wang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.jhydrol.2007.05.014","article-title":"Multi-objective automatic calibration of SWAT using NSGA-II","volume":"341","author":"Bekele","year":"2007","journal-title":"J. Hydrol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2792","DOI":"10.1007\/s11431-010-4087-0","article-title":"Optimization design study of low-Reynolds-number high-lift airfoils for the high-efficiency propeller of low-dynamic vehicles in stratosphere","volume":"53","author":"Ma","year":"2010","journal-title":"Sci. China Technol. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Slavik, S., Klesa, J., and Brabec, J. (2020). Propeller Selection by Means of Pareto-Optimal Sets Applied to Flight Performance. Aerospace, 7.","DOI":"10.3390\/aerospace7030021"},{"key":"ref_16","unstructured":"Fang, B.R. (1997). Design of Aircraft Aerodynamic Configuration, Chinese Aviation Industry Press."},{"key":"ref_17","unstructured":"Colozza, A. (1998). High Altitude Propeller Design and Analysis Overview, Federal Data Systems. NASA\/CR 98-208520."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1016\/j.ast.2015.06.011","article-title":"High Altitude Propeller Design and Analysis","volume":"45","author":"Morgado","year":"2015","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Burdett, T.A., and Van Treuren, K.W. (2012, January 11\u201315). A Theoretical and Experimental Comparison of Optimizing Angle of Twist Using BET and BEMT. Proceedings of the ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. Volume 6: Oil and Gas Applications; Concentrating Solar Power Plants; Steam Turbines; Wind Energy, Copenhagen, Denmark.","DOI":"10.1115\/GT2012-68350"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Oliveira, H.A., de Matos, J.G., Ribeiro, L.A.d.S., Saavedra, O.R., and Vaz, J.R.P. (2023). Assessment of Correction Methods Applied to BEMT for Predicting Performance of Horizontal-Axis Wind Turbines. Sustainability, 15.","DOI":"10.3390\/su15087021"},{"key":"ref_21","first-page":"440","article-title":"On The Vortex Theory of Screw Propellers","volume":"123","author":"Goldstein","year":"1929","journal-title":"Proc. R. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zhong, W., Wang, T.G., Zhu, W.J., and Shen, W.Z. (2019). Evaluation of Tip Loss Corrections to AD\/NS Simulations of Wind Turbine Aerodynamic Performance. App. Sci., 9.","DOI":"10.3390\/app9224919"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1049\/iet-rpg.2012.0283","article-title":"Vortex Methods to Answer The Need For Improved Understanding And Modelling of Tip-Loss Factors","volume":"7","author":"Branlard","year":"2013","journal-title":"IET Renew. Power Gener."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.egypro.2017.07.028","article-title":"Tip Loss Factor Effects on Aerodynamic Performances of Horizontal Axis Wind Turbine","volume":"118","author":"Elkhchine","year":"2017","journal-title":"Energy Procedia"},{"key":"ref_25","unstructured":"Shaidakov, V.I. (1996). Aerodinamika Vinta v kol\u2019ce: Uch Posobie, MAI. (In Russian)."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.paerosci.2006.04.001","article-title":"The Aerodynamics of Propellers","volume":"42","author":"Wald","year":"2006","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_27","unstructured":"(2024, January 10). XFOIL Subsonic Airfoil Development System. Available online: https:\/\/web.mit.edu\/drela\/Public\/web\/xfoil\/."},{"key":"ref_28","unstructured":"Munguia, J., and Van Treuren, W. (2019). Designing Small Propellers for Optimum Efficiency, Baylor University."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1115\/1.2841318","article-title":"Turbomachinery Blade Design Using a Navier-Stokes Solver and Artificial Neural Net-work","volume":"121","author":"Pierret","year":"1999","journal-title":"J. Turbomach."},{"key":"ref_30","unstructured":"Segui, M.M., Castelar, Y., and Botez, R.M. (2019, January 2\u20139). Wing Airfoils Generation Based on a New Parametric Curve for Aerodynamic Optimization Application. Proceedings of the CASI AERO-2019 Conference, Big Sky, MT, USA."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bajpai, R.P., and Chandrasekhar, U. (2017). Innovative Design and Development Practices in Aerospace and Automotive Engineering, Springer. Lecture Notes in Mechanical Engineering.","DOI":"10.1007\/978-981-10-1771-1"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"012254","DOI":"10.1088\/1742-6596\/891\/1\/012254","article-title":"Turbine Blade Profile Design Method Based on Bezier Curves","volume":"891","author":"Alexeev","year":"2017","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Hao, X., Zhang, W., Liu, X., and Liu, J. (2008, January 7\u201310). Aerodynamic and Aeroacoustic Optimization of Wind Turbine Blade by a Genetic Algorithm. Proceedings of the 46th AIAA Aerospace Sciences Meeting and Exhibit,AIAA 2008-1331, Reno, NV, USA.","DOI":"10.2514\/6.2008-1331"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"173","DOI":"10.3103\/S106879982102001X","article-title":"Determination of the Optimal Aerodynamic Shape for a Propeller Blade Based on Parametric Optimization","volume":"64","author":"Borovkov","year":"2021","journal-title":"Russ. Aeronaut. (Iz VUZ)"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"32","DOI":"10.3846\/aviation.2022.16471","article-title":"Airfoil Shape Optimization using B\u00e9zier Curve and Genetic Algorithm","volume":"26","author":"Kiral","year":"2022","journal-title":"Aviation"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Xin, P., Dawei, L., Jixiang, S., and Yonghong, L. (2014, January 15\u201316). Airfoil Aerodynamic Optimization Based on an Improved Genetic Algorithm. Proceedings of the Fifth International Conference on Intelligent Systems Design and Engineering Applications, Hunan, China.","DOI":"10.1109\/ISDEA.2014.37"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1177\/0954410017704217","article-title":"Optimal Design and Experiment of Propellers for High Altitude Airship","volume":"232","author":"Jiao","year":"2017","journal-title":"Proc. Inst. Mech. Eng. Part. G J. Aerosp. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bocii, L.S., Di Noia, L.P., and Rizzo, R. (2019). Optimization of the Energy Storage of Series-Hybrid Propelled Aircraft by means of Integer Differential Evolution. Aerospace, 6.","DOI":"10.3390\/aerospace6050059"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"108389","DOI":"10.1016\/j.oceaneng.2020.108389","article-title":"Hydrodynamic Optimization of High-Performance Blade Sections for Stall Regulated Hydrokinetic Turbines Using Differential Evolution Algorithm","volume":"220","author":"Muratoglu","year":"2021","journal-title":"Ocean Eng."},{"key":"ref_40","unstructured":"Spera, D.A. (2008). Models of Lift and Drag Coefficients of Stalled and Unstalled Airfoils in Wind Turbines and Wind Tunnels, GRC. Technical Report NASA\/CR-2008-215434."},{"key":"ref_41","unstructured":"Quijada Pioquinto, J.G., Kurkin, E.I., Nazarov, D.V., Lukyanov, O.E., and Chertykovtseva, V.O. (2024). Programma OpenVINT Optimzacii Formy Lopasti Vozdushnogo Vinta [OpenVINT Code for Optimizing the Shape of a Propeller Blade], RU 2024610972, FIPS. Available online: https:\/\/new.fips.ru\/registers-doc-view\/fips_servlet?DB=EVM&DocNumber=2024610972&TypeFile=html."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s10589-012-9498-3","article-title":"A penalty function-based differential evolution algorithm for constrained global optimization","volume":"54","author":"Ali","year":"2013","journal-title":"Comput. Optim. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0166-3615(92)90132-7","article-title":"Development of Bezier-based curves","volume":"20","year":"1992","journal-title":"Comput. Ind."},{"key":"ref_44","unstructured":"Zherejov, V.V., and Kusumov, A.N. (1997). Aehrodinamicheskij Raschet Nesushchego Vinta Vertoleta. Uch Posobie Po Kursovomu I Diplomnomu Proektirovaniyu, KGTU. (In Russian)."},{"key":"ref_45","unstructured":"Kravec, A.S. (1941). Kharakteristiki Vozdushnykh Vintov, Uchebnoe Posobie, Gos. Izd. Oboronnoj Promyshlennosti. (In Russian)."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/j.advengsoft.2010.05.002","article-title":"Bezier-PARSEC: An Optimized Aerofoil Parameterization for Design","volume":"41","author":"Derksen","year":"2010","journal-title":"Adv. Eng. Softw."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Espinosa Barcenas, O.U., Quijada Pioquinto, J.G., Kurkina, E., and Lukyanov, O. (2023). Surrogate aerodynamic wing modeling based on a multilayer perceptron. Aerospace, 10.","DOI":"10.3390\/aerospace10020149"},{"key":"ref_48","first-page":"2183902","article-title":"Hydrodynamic performance of a rim-driven thruster improved with gap geometry adjustment","volume":"17","author":"Lin","year":"2023","journal-title":"Eng. Appl. Comput. Fluid. Mech."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Eltaeib, T., and Mahmood, A. (2018). Differential Evolution: A Survey and Analysis. App. Sci., 8.","DOI":"10.3390\/app8101945"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Tanabe, R., and Fukunaga, A.S. (2014, January 6\u201311). Improving the search performance of SHADE using linear population size reduction. Proceedings of the IEEE Congress on Evolutionary Computation (CEC), Beijing, China.","DOI":"10.1109\/CEC.2014.6900380"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Espinosa Barcenas, O.U., Quijada Pioquinto, J.G., Kurkina, E., and Lukyanov, O. (2022). Multidisciplinary analysis and optimization method for conceptually designing of electric flying-wing unmanned aerial vehicles. Drones, 6.","DOI":"10.3390\/drones6100307"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1177\/2472630317690318","article-title":"Continuous adaptive population reduction (CAPR) for differential evolution optimization","volume":"22","author":"Wong","year":"2017","journal-title":"SLAS Technol."},{"key":"ref_53","unstructured":"Iman, R.L. (2008). Encyclopedia of Quantitative Risk Analysis and Assessment, John Wiley Sons."},{"key":"ref_54","unstructured":"Zielinksi, K., Weitkemper, P., Laur, R., and Kammeyer, K.D. (2023, November 02). Examination of Stopping Criteria for Differential Evolution Based on a Power Allocation Problem. Available online: https:\/\/api.semanticscholar.org\/CorpusID:13358798."},{"key":"ref_55","unstructured":"Sharma, H. (2023, November 02). Lightweight Pipelining in Python. Using Joblib for Storing the Machine Learning Pipeline to a File. Available online: https:\/\/towardsdatascience.com\/lightweight-pipelining-in-python-1c7a874794f4."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.swevo.2016.05.003","article-title":"Review of differential evolution population size","volume":"32","author":"Piotrowski","year":"2017","journal-title":"Swarm Evol. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Lukyanov, O.E., Espinosa Barcenas, O.U., and Zolotov, D.V. (2021, January 23\u201325). Experimental model of an electric power plant for small UAV\u2019s automatic control systems. Proceedings of the 2021 International Scientific and Technical Engine Conference, Samara, Russia.","DOI":"10.1109\/EC52789.2021.10016802"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:09:45Z","timestamp":1760105385000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,6]]},"references-count":57,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["computation12030052"],"URL":"https:\/\/doi.org\/10.3390\/computation12030052","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,3,6]]}}}