طراحی بهینه صفحات همسانگرد محدود حاوی گشودگی شش ضلعی با استفاده از الگوریتم‌های فرا ابتکاری
نویسندگان
دانشکده مکانیک، دانشگاه صنعتی شاهرود، شاهرود
چکیده
در این مقاله، طراحی بهینه ورق همسانگرد محدود حاوی گشودگی شش ضلعی تحت بارگذاری درون صفحه‌ای با استفاده از الگوریتم‌های فرا ابتکاری مورد بررسی قرار گرفته است. الگوریتم تکامل تفاضلی و الگوریتم جستجوی هارمونی از دسته الگوریتم‌های تکاملی، الگوریتم بیگ بنگ- بیگ کرانچ از دسته الگوریتم‌های مبتنی بر فیزیک و الگوریتم گرگ خاکستری و الگوریتم اجتماع ذرات از دسته الگوریتم‌های هوش ازدحامی، پنج الگوریتم بهینه‌سازی مورد استفاده در این مقاله هستند. نتایج به‌دست آمده از مقایسه این الگوریتم‌ها، دلالت بر عملکرد بالا در فضای جستجو، سرعت مناسب در همگرایی و رقابتی بودن الگوریتم گرگ خاکستری جهت اجنتاب از نقطه بهینه محلی نسبت به چهار الگوریتم دیگر دارد. در تحلیل ورق همسانگرد محدود حاوی گشودگی شش ضلعی پارامترهایی از جمله انحنای گوشه‌های گشودگی، زاویه چرخش گشودگی، نسبت اضلاع ورق، نسبت اندازه گشودگی به طول مشخصه ورق و نوع بارگذاری به‌عنوان پارامترهای مؤثر بر توزیع تنش محسوب می‌شود. در مطالعه حاضر، روش به‌کار گرفته شده برپایه حل تحلیلی متغیر مختلط موشخیلشویلی است و از نگاشت همنوا برای ساده‌سازی روند حل معادلات استفاده شده است. ورق محدود (نسبت قطر دایره محیط به بزرگ‌ترین ضلع ورق، بزرگ‌تر از 2/0)، همسانگرد و الاستیک خطی درنظر گرفته شده است. از روش اجزای محدود، برای بررسی درستی جواب‌ها استفاده شد. نتایج عددی، تطابق خوبی با نتایج حاصل از حل تحلیلی حاضر دارد. نتایج نشان می‌دهند با انتخاب مناسب پارامترهای بهینه می‌توان مقدار تنش اطراف گشودگی شش ضلعی را به میزان قابل توجهی کاهش داد.
کلیدواژه‌ها
عنوان مقاله [English]
Optimized Design of Finite Isotropic Plates with Hexagonal Cutout by Metaheuristic Algorithms
نویسندگان [English]
- m. h. bayati chaleshtari
- m. jafari
This paper aims at optimizing the finite isotropic plates with the hexagonal cutout subjected to plane loading using metaheuristic optimization algorithms. This research uses Differential Evolution Algorithm (DE) and Harmony Search Algorithm (HSA) from the evolutionary algorithm category, Big Bang- Big Crunch Algorithm (BB-BC) from the physics-based algorithm category, and Grey Wolf Optimizer Algorithm (GWO) and Particle Swarm Optimization (PSO) from the SI algorithm category; then the results of these algorithms are compared with each other. The results indicate that the grey wolf optimizer has the complete performance, short solution time and the ability to avoid local optimums. In the analysis of finite isotropic plate, the effective parameters on stress distribution around the hexagonal cutouts are cutout bluntness, cutout orientation, plate’s aspect ratio, cutout size, and type of loading. In this study, with the assumption of plane stress conditions, the analytical solution of Muskhelishvili’s complex variable method and conformal mapping is utilized. The plate is considered to be finite (the proportion ratio of the diameter of circle circumscribing to the longest plate side should be more than 0.2), isotropic, and linearly elastic. The finite element method has been used to check the accuracy of the results. Numerical results are in a good agreement with those of the present analytical solution. The results show that by selecting the aforementioned parameters properly, less amounts of stress could achieve around the cutout can lead to an increase in the load-bearing capacity of the structure.
کلیدواژه‌ها [English]
- Analytical Solution
- Finite Isotropic Plates
- Hexagonal Cutout
- Metaheuristic Algorithms
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11. Rezaeepazhand, J., and Jafari, M., “Stress Concentration in Metallic Plates with Special Shaped Cutout”, International Journal of Mechanical Sciences, Vol. 52, pp. 96-102, 2010.
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13. Jafari, M., and Ghandi Varnosefaderani, I., “Semi- analytical Solution of Stress Concentration Factor in the Isotropic Plates Containing Two Quasi- rectangular Cutouts”, Modares Mechanical Engineering, Vol. 15, No. 8, pp. 341-350, 2015.
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22. Bazehhour, B. G., and Rezaeepazhand, J., “Torsion of Tubes with Quasi-polygonal Holes using Complex Variable Method”, Mathematics and Mechanics of Solids, Vol. 19, pp. 260-276, 2014.
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26. Liu, Y., Jin, F., and Li, Q. A., “Strength-based Multiple Cutout Optimization in Composite Plates using Fixed Grid Finite Element Method”, Composite Structures, Vol. 73, pp. 403-412, 2006.
27. Kradinov, V., Madenci, E., and Ambur, D. R., “Application of Genetic Algorithm for Optimum Design of Bolted Composite Lap Joints”, Composite Structures, Vol. 77, pp. 148-159, 2007.
28. Jafari, M., and Rohani, A., “Optimization of Perforated Composite Plates under Tensile Stress using Genetic Algorithm”, Journal of Composite Materials, Vol. 52, pp. 96-102, 2015.
29. Suresh, S., Sujit, P. B., and Rao, A. K., “Particle Swarm Optimization Approach for Multi-objective Composite Box-beam Design”, Composite Structures, Vol. 81, pp. 598-605, 2007.
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33. Jianqiao, C., Yuanfu, T., Rui, G., Qunli, A., and Xiwei, G., “Reliability Design Optimization of Composite Structures Based on PSO Together with FEA”, Chinese Journal of Aeronautics, Vol. 26, pp. 343-349, 2013.
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35. Storn, R., and Price, K., “Minimizing the Real Functions of the ICEC’96 Contest by Differential Evolution”, International Conference. on Evolutionary Computation, Nagoya, Japan, 1996.
36. Storn, R., and Price, K., “Differential Evolution-a Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces”, Journal of Global Optimization, Vol. 11, No. 4, pp. 341-359, 1997.
37. Erol, O. K., and Eksin, I., “A New Optimization Method: Big Bang-Big Crunch”, Advances in Engineering Software, Vol. 37, pp. 106-111, 2006.
38. Camp, C. V., “Design of Space Trusses using Big Bang-Big Crunch Optimization”, Journal of Structural Engineering, Vol. 133, pp. 999-1008, 2007.
39. Mirjalili, S., Mirjalili, S. M., and Lewis, A., “GreyWolf Optimizer”, Advances in Engineering Software, Vol. 69, pp. 46-61, 2014.
40. Song, H. M., Sulaiman, M. H., and Mohamed, M. R., “Anapplication of Grey Wolf Optimizer for Solving Combined Economic Emission Dispatch Problems”, International Review on Modelling and Simulations, Vol. 7, No.5, pp. 838-844, 2014.
41. Saremi, S., Mirjalili, S., and Mirjalili, S. M., “Evolutionary Population Dynamics and Grey Wolf Optimizer”, Neural Computing and Applications, Vol. 26, No. 5, pp. 1257-1263, 2015.
42. Geem, Z. W., Kim, J. H., and Loganathan, G. V., “A New Heuristic Optimization Algorithm, Harmony Search”, Journal of Simulation, Vol. 76, pp. 60-68, 2001.
43. Geem, Z. W., Fesanghary, M., Choi, J., Saka, M. P., Williams, J. C., Ayvaz, M. T., Li, L., Ryu, S., and Vasebi, A., Recent Advances in Harmony Search, pp. 127-142, In: Kosinski, V., (Eds.), Advance in Evolutionary Algorithms, I-Teach Education and Publishing, Vienna , Austria, 2008.
44. Lee, K. S., and Geem, Z. W., “A New Meta-heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice”, Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 3902-3933, 2005.
45. Yang, X., Yuan, J., and Mao, H., “A Modified Particle Swarm Optimizer with Dynamic Adaptation”, Applied Mathematics and Computation, Vol. 189, pp. 1205-1213, 2007.
46. Chan, F. T. S., and Tiwari, M. K., Swarm Intelligence: Focus on Ant and Particle Swarm Optimization, I-teach Education and Publishing, Vienna, Austria, 2007.
47. Kennedy, J., and Eberhart, R., “Particle Swarm Optimization”, Proceeding of IEEE International Conference on Neural Networks, NewYork, USA, July, 1995.
2. Heywood, R. B., Designing by Photoelasticity, Chapman and Hall, London, 1952.
3. Muskhelishvili, N. I., Some Basic Problems of Mathematical Theory of Elasticity, Netherlands: Noordhooff, Groningen, Holland, pp. 56-104, 1953.
4. Savin, G. N., Stress Concentration Around holes, Pergamon Press, NewYork, 1961.
5. Lekhnitskii, S. G., Anisotropic plates, Second Edittion, NewYork, 1968.
6. Pilkey, W. D., Peterson’s Stress Concentration Factors, John Wiley & Sons Inc, Second Edition, NewYork, 1997.
7. Theocaris, P. S., and Petrou, L., “Stress Distributions and Intensities at Corners of Equilateral Triangular Holes”, International Journal of Fracture, Vol. 31, pp. 271-289, 1986.
8. Batista, M., “On the Stress Concentration Around Hole in an Infinite Plate Subject to Uniform Load at Infinity”, International Journal of Mechanical Sciences, Vol. 53, pp. 254-261, 2011.
9. Banerjee, M., Jain, N. K., and Sanyal, S., “Stress Concentration in Isotropic and Orthotropic Composite Plates with Center Circular Hole Subjected to Transverse Static Loading”, International Journal Mechanical India Engineering, Vol. 3, pp. 109-113, 2013.
10. Darwish, F., Gharaibeh, M., and Tashtoush, G., “A Modified Equation for the Stress Concentration Factor in Countersunk Holes”, Journal of Mechanics A/Solids, Vol. 36, pp. 94-103, 2012.
11. Rezaeepazhand, J., and Jafari, M., “Stress Concentration in Metallic Plates with Special Shaped Cutout”, International Journal of Mechanical Sciences, Vol. 52, pp. 96-102, 2010.
12. Daoust, J., and Hoa, S. V., “An Analytical Solution for Anisotropic Plates Containing Triangular Holes”, Composite Structures, Vol. 19, pp. 107-130, 1991.
13. Jafari, M., and Ghandi Varnosefaderani, I., “Semi- analytical Solution of Stress Concentration Factor in the Isotropic Plates Containing Two Quasi- rectangular Cutouts”, Modares Mechanical Engineering, Vol. 15, No. 8, pp. 341-350, 2015.
14. Asmar, G. H., and Jabbour, T. G., “Stress Analysis of Anisotropic Plates Containing Rectangular Holes”, International Journal of Mechanics and Solids, Vol. 2, pp. 59-84, 2007.
15. Rao, D. K. N., Babu, M. R., Reddy, K. R. N., and Sunil, D., “Stress Around Square and Rectangular Cutouts in Symmetric Laminates”, Composite Structures, Vol. 92, No. 12, pp. 2845- 2859, 2010.
16. Jain, N. K., and Mittal, N. D., “Finite Element Analysis for Stress Concentration and Deflection in Isotropic, Orthotropic and Laminated Composite Plates with Central Circular Hole Under Transverse Static Loading”, Materials Science and Engineering, Vol. 498, No. 1, pp. 115-124, 2008.
17. Ukadgaonker, V. G., and Rao, D. K. N., “Stress Distribution Around Triangular Holes in Anisotropic Plates”, Composite Structures, Vol. 45, pp. 171-183, 1999.
18. Sharma, D. S., “Stress Distribution Around Polygonal Holes”, International Journal of Mechanical Sciences, Vol. 65, pp. 115-124, 2012.
19. Linc, C., and Koc, C., “Stress and Strength Analysis of Finite Composite Laminates with Elliptical Holes”, Journal of Composite Materials, Vol. 22, pp. 373-85, 1988.
20. Woo, C. W., and Chan, L. W. S., “Boundary Collocation Method for Analyzing Perforated Plate Problems” Engineering Fracture Mechanics, Vol. 43, pp. 757-768, 1992.
21. Xu, X., Sun, L., and Fan, X., “Stress Concentration of Finite Composite Laminates with Elliptical Hole”, Composite Structure, Vol. 57, No. 2, pp. 29-34, 1995.
22. Bazehhour, B. G., and Rezaeepazhand, J., “Torsion of Tubes with Quasi-polygonal Holes using Complex Variable Method”, Mathematics and Mechanics of Solids, Vol. 19, pp. 260-276, 2014.
23. Pan, Z., Cheng, Y., and Liu, J., “Stress Analysis of a Finite Plate with a Rectangular Hole Subjected to Uniaxial Tension using Modified Stress Functions”, International Journal of Mechanical Sciences, Vol. 75, pp. 265-277, 2013.
24. Jafari, M., and Ardalani, E., “Analytical Solution to Calculate the Stress Distribution Around Triangular Hole in Finite Isotropic Plates under in- Plane Loading”, Modares Mechanical Engineering, Vol. 15, No. 5, pp. 165-175, 2015.
25. Lemanh, T., and Lee, J., “Stacking Sequence Optimization for Maximum Strengths of Laminated Composite Plates Using Genetic Algorithm and Isogeometric Analysis”, Composite Structures, Vol. 116, pp. 357-363, 2014.
26. Liu, Y., Jin, F., and Li, Q. A., “Strength-based Multiple Cutout Optimization in Composite Plates using Fixed Grid Finite Element Method”, Composite Structures, Vol. 73, pp. 403-412, 2006.
27. Kradinov, V., Madenci, E., and Ambur, D. R., “Application of Genetic Algorithm for Optimum Design of Bolted Composite Lap Joints”, Composite Structures, Vol. 77, pp. 148-159, 2007.
28. Jafari, M., and Rohani, A., “Optimization of Perforated Composite Plates under Tensile Stress using Genetic Algorithm”, Journal of Composite Materials, Vol. 52, pp. 96-102, 2015.
29. Suresh, S., Sujit, P. B., and Rao, A. K., “Particle Swarm Optimization Approach for Multi-objective Composite Box-beam Design”, Composite Structures, Vol. 81, pp. 598-605, 2007.
30. Alonso, M. G., and Duysinx, p., “Particle Swarm Optimization (PSO): an Alternative Method for Composite Optimization”, 10th World Congress on Structural and Multidisciplinary Optimization, Orlando, Florida, USA, May 19 -24, 2013.
31. Chang, N., Wang, W., Yang, W., and Wang, J., “Ply Stacking Sequence Optimization of Composite Laminate by Permutation Discrete Particle Swarm Optimization”, Structural and Multidisciplinary Optimization, Vol. 41, No. 2, pp. 179-187, 2010.
32. Hudson, C., Carruthers, J., and Robinson, M., “Multiple Objective Optimization of Composite Sandwich Structures for Rail Vehicle Floor Panels”, Composite Structures, Vol. 92, No. 9, pp. 2077‐2082, 2010.
33. Jianqiao, C., Yuanfu, T., Rui, G., Qunli, A., and Xiwei, G., “Reliability Design Optimization of Composite Structures Based on PSO Together with FEA”, Chinese Journal of Aeronautics, Vol. 26, pp. 343-349, 2013.
34. Kathiravan, R., and Ganguli, R., “Strength Design of Composite Beam using Gradient and Particle Swarm Optimization”, Composite Structures, Vol. 81, pp. 471-479, 2007.
35. Storn, R., and Price, K., “Minimizing the Real Functions of the ICEC’96 Contest by Differential Evolution”, International Conference. on Evolutionary Computation, Nagoya, Japan, 1996.
36. Storn, R., and Price, K., “Differential Evolution-a Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces”, Journal of Global Optimization, Vol. 11, No. 4, pp. 341-359, 1997.
37. Erol, O. K., and Eksin, I., “A New Optimization Method: Big Bang-Big Crunch”, Advances in Engineering Software, Vol. 37, pp. 106-111, 2006.
38. Camp, C. V., “Design of Space Trusses using Big Bang-Big Crunch Optimization”, Journal of Structural Engineering, Vol. 133, pp. 999-1008, 2007.
39. Mirjalili, S., Mirjalili, S. M., and Lewis, A., “GreyWolf Optimizer”, Advances in Engineering Software, Vol. 69, pp. 46-61, 2014.
40. Song, H. M., Sulaiman, M. H., and Mohamed, M. R., “Anapplication of Grey Wolf Optimizer for Solving Combined Economic Emission Dispatch Problems”, International Review on Modelling and Simulations, Vol. 7, No.5, pp. 838-844, 2014.
41. Saremi, S., Mirjalili, S., and Mirjalili, S. M., “Evolutionary Population Dynamics and Grey Wolf Optimizer”, Neural Computing and Applications, Vol. 26, No. 5, pp. 1257-1263, 2015.
42. Geem, Z. W., Kim, J. H., and Loganathan, G. V., “A New Heuristic Optimization Algorithm, Harmony Search”, Journal of Simulation, Vol. 76, pp. 60-68, 2001.
43. Geem, Z. W., Fesanghary, M., Choi, J., Saka, M. P., Williams, J. C., Ayvaz, M. T., Li, L., Ryu, S., and Vasebi, A., Recent Advances in Harmony Search, pp. 127-142, In: Kosinski, V., (Eds.), Advance in Evolutionary Algorithms, I-Teach Education and Publishing, Vienna , Austria, 2008.
44. Lee, K. S., and Geem, Z. W., “A New Meta-heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice”, Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 3902-3933, 2005.
45. Yang, X., Yuan, J., and Mao, H., “A Modified Particle Swarm Optimizer with Dynamic Adaptation”, Applied Mathematics and Computation, Vol. 189, pp. 1205-1213, 2007.
46. Chan, F. T. S., and Tiwari, M. K., Swarm Intelligence: Focus on Ant and Particle Swarm Optimization, I-teach Education and Publishing, Vienna, Austria, 2007.
47. Kennedy, J., and Eberhart, R., “Particle Swarm Optimization”, Proceeding of IEEE International Conference on Neural Networks, NewYork, USA, July, 1995.
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