Abstract    The main purpose of the present work is multi-objective shape optimization of a projectile tip for impacting and normal penetrating. Velocity drop, weight and inner volume of projectile have been considered as three conflicting objective functions. For this purpose, at the first step, finite element modeling was done using ABAQUS/Explicit and projectile penetration was examined in different geometric dimensions. Hammersley sequence sampling was employed for designing computer experiments. In the next step, results of the FEM were employed as raw data for MLF-type neural network training to achieve a mathematical model which is able to describe velocity drop behavior. Projectile weight and Inner volume were also expressed in explicit mathematical form using geometric relations. Obtained mathematical models were used as conflicting objective functions for multi-objective optimization of projectile tip using modified NSGA-II. Finally, it is shown that some interesting and important relationships as useful optimal design principles involved in the performance of projectile impact have been discovered by Pareto based multi-objective optimization.

Keywords    Projectile, Impact, Finite element method, Neural Networks, Multi-objective optimization , NSGA-II.

چکیده    هدف اصلی در این مقاله بهینه سازی شکل نوک یک پرتابه در برخورد و نفوذ قائم با در نظر گرفتن افت سرعت، وزن و حجم‌ داخلی پرتابه به عنوان سه تابع هدف متضاد است. برای این منظور، ابتدا با استفاده از نرم فزار ABAQUS/Explicit مدل‌سازی اجزای محدود انجام و با نظر گرفتن ابعاد هندسی مختلف، نفوذ پرتابه بررسی شد. آزمایش‌های کامپیوتری با استفاده از دنباله‌ی همرسلی طرحی شد. در قدم بعدی از مجموعه‌ی نتایج مدل‌سازی اجزای محدود، به عنوان داده‌های خام برای آموزش و آزمایش شبکه‌های عصبی فیدفوروارد به منظور دست‌یابی به یک مدل ریاضی برای بیان افت سرعت استفاده شد. برای وزن و حجم پرتابه نیز روایط صریح ریاضی توسعه داده شد. از روابط به دست آمده برای بهینه‌سازی چندهدفی با استفاده از الگوریتم ژنتیک استفاده شد. نتایج به دست آمده روابط سودمندی را در طراحی بهینه نمایش می‌دهد که تنها با به کارگیری بهینه‌سازی چندهدفی مدل ریاضی استخراج شده از نتایج حل اجزای محدود قابل تحصیل است.

References      1.        Marom, I. and S. Bodner, "Projectile perforation of multi-layered beams", DTIC Document, (1978) 2.        Corran, R., P. Shadbolt, and C. Ruiz, "Impact loading of plates—an experimental investigation", International Journal of Impact Engineering,. Vol. 1, No. 1, (1983), 3-22. 3.        Nurick, G. and C. Walters. "The ballistic penetration of multiple thin plates separated by an air gap", SEM Spring Conference on Experimental Mechanics, (1990). 4.        Gupta, N. and V. Madhu, "Normal and oblique impact of a kinetic energy projectile on mild steel plates", International Journal of Impact Engineering, Vol. 12, No. 3, (1992), 333-343. 5.        Gupta, N. and V. Madhu, "An experimental study of normal and oblique impact of hard-core projectile on single and layered plates", International Journal of Impact Engineering, Vol. 19, No. 5, (1997), 395-414. 6.        Børvik, T., "Perforation of 12mm thick steel plates by 20mm diameter projectiles with flat, hemispherical and conical noses: part I: experimental study", International Journal of Impact Engineering, Vol. 27, No. 1, (2002), 37-64. 7.        Børvik, T., et al., "Perforation of 12mm thick steel plates by 20mm diameter projectiles with flat, hemispherical and conical noses: part II: numerical simulations", International Journal of Impact Engineering, Vol. 27, No. 1, (2002), 37-64. 8.        Nilakantan, G., "Finite element analysis of projectile size and shape effects on the probabilistic penetration response of high strength fabrics", Composite Structures, Vol. 94, No. 5, (2012). 1846-1854. 9.        Wei, Z, "Experimental investigation on the ballistic performance of monolithic and layered metal plates subjected to impact by blunt rigid projectiles", International Journal of Impact Engineering, Vol. 49, (2012), 115-129. 10.     Iqbal, M.A., "Experimental and numerical studies of double-nosed projectile impact on aluminum plates", International Journal of Impact Engineering, Vol. 54, (2013), 232-245. 11.     Deb, K., "A fast and elitist multiobjective genetic algorithm: NSGA-II", Evolutionary Computation, IEEE Transactions on, Vol. 6, No. 2, (2002), 182-197. 12.     Nariman-Zadeh, N., "Inverse modelling of multi-objective thermodynamically optimized turbojet engines using GMDH-type neural networks and evolutionary algorithms", Engineering Optimization, Vol. 37, No. 5, (2005), 437-462. 13.     Osyczka, A., "Multicriteria optimization for engineering design", Design Optimization, Vol. 1, (1985), 193-227. 14.     Fonseca, C.M. and P.J. Fleming. "Genetic algorithms for multiobjective optimization: Formulation, discussion and generalization", in Proceedings of the fifth international conference on genetic algorithms, San Mateo, California, (1993) 15.     Coello, C. A. C., G. B. Lamont, and D. A. Van Veldhuizen, "Evolutionary algorithms for solving multi-objective problems", Springer,  Vol. 5. (2007):. 16.     Khakhali, A., "Reliability-based robust multi-objective crashworthiness optimisation of S-shaped box beams with parametric uncertainties", International Journal of Crashworthiness, Vol. 15, No. 4, (2010), 443-456. 17.     Khalkhali, A. and H. Safikhani, "Pareto based multi-objective optimization of a cyclone vortex finder using CFD, GMDH type neural networks and genetic algorithms", Engineering Optimization, Vol. 44, No. 1, (2012), 105-118. 18.     Åström, K. and P. Eykhoff, "System identification—a survey", Automatica, Vol. 7, No. 2, (1971), 123-162. 19.     Sanchez, E., T. Shibata, and L.A. Zadeh, "Genetic algorithms and fuzzy logic systems: Soft computing perspectives", World Scientific Publishing Company Incorporated, Vol. 7, (1997) 20.     Sacks, J., "Design and analysis of computer experiments", Statistical Science, Vol. 4, No. 4, (1989), 409-423. 21.     Simpson, T. W., "Metamodels for computer-based engineering design: survey and recommendations", Engineering With Computers, Vol. 17, No. 2, (2001), 129-150. 22.     Tang, B., "Orthogonal array-based Latin hypercubes", Journal of the American Statistical Association, Vol. 88, No. 424, (1993), 1392-1397. 23.     McKay, M. D., R. J. Beckman, and W. J. Conover, "Comparison of three methods for selecting values of input variables in the analysis of output from a computer code", Technometrics, Vol. 21, No. 2, (1979), 239-245. 24.     Kalagnanam, J. R. and U. M. Diwekar, "An efficient sampling technique for off-line quality control", Technometrics, Vol. 39, No. 3, (1997), 308-319. 25.     Johnson, G.R. and W.H. Cook. "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures" in Proceedings of the 7th International Symposium on Ballistics, The Hague, Netherlands: International Ballistics Committee, (1983). 26.     Jaspers, S. and J. Dautzenberg, "Material behaviour in conditions similar to metal cutting: flow stress in the primary shear zone", Journal of Materials Processing Technology, Vol. 122, No. 2, (2002), 322-330. 27.     Wang, G.G. and S. Shan, "Review of metamodeling techniques in support of engineering design optimization", Journal of Mechanical Design, Vol. 129, No. 4, (2007),  370. 28.     Cybenko, G., "Approximation by superpositions of a sigmoidal function", Mathematics of Control, Signals, and Systems (MCSS), Vol. 2, No. 4, (1989), 303-314. 29.     Hornik, K., M. Stinchcombe, and H. White, "Multilayer feedforward networks are universal approximators", Neural Networks, Vol. 2, No. 5, (1989), 359-366. Coello, C. and A.D. Christiansen, "Multiobjective optimization of trusses using genetic algorithms", Computers & Structures, Vol. 75, No. 6, (2000), 647-660.