بهینه‌سازی توپولوژی خرپا با استفاده از برنامه‌ریزی صفر و یک (دودویی)
نوع مقاله : مقاله پژوهشی
نویسندگان
دانشگاه تربیت مدرس
چکیده
با اعمال محدودیت جابجایی بهینه‌سازی توپولوژی به یک مسئله بهینه‌سازی غیرخطی تبدیل می‌شود. برای تبدیل مسئله برنامه‌ریزی غیرخطی به یک مسئله برنامه‌ریزی خطی، از متغیرهای طراحی گسسته استفاده می‌شود و مسئله بهینه‌سازی توپولوژی به مسئله برنامه‌ریزی اعداد صحیح دودویی (صفر و یک) تبدیل می‌گردد. مثال‌های متعددی حل شده‌اند و با نمونه‌های منتشر شده در مقالات پیشین مقایسه گشته‌اند. با حل مسائل مطرح شده مشاهده می‌شود که روش ارائه شده در این تحقیق برای بهینه‌سازی توپولوژی خرپا منجر به نتایجی بهتر و در بعضی از موارد، مشابه نتایج تحقیقات قبلی اما با محاسبات کمتر می‌شود. با این وجود، ثابت شده است که نتایج بدست آمده در این مقاله بهینه سراسری هستند در حالی که روش‌های بکار رفته در تحقیقات پیشین نمی‌توانند بهینگی سراسری خود را اثبات کنند.
کلیدواژه‌ها
عنوان مقاله [English]
Truss Topology Optimization via Zero-One (Binary) Programming
نویسندگان [English]
- kaveh mirzapour
- hamid moharrami
In this paper, a new method has been proposed for the topology optimization of trusses. The method takes advantage of the equation of equilibrium between internal and external forces in the flexibility method of structural analysis. The internal forces are written as the multiplication of cross-sections of members into their stresses. The stress constraints (i.e. limits on the stress values) are then imposed on the problem and eventually, the topology optimization of trusses ends up in a Linear Programming (LP) problem. The solution to the LP problem is straightforward and results in a global optimum. Accordingly, the outcome of our formulation is a global optimum.
When the displacement constraints are included among the constraints, the truss topology optimization turns into a nonlinear optimization problem. To convert the problem to a linear programming problem, we used discrete design variables and converted the problem to a binary (zero-one) integer programming. Several examples were solved and compared to the published examples in the literature. It was observed that our method of truss topology optimization ends up with the same results as the previous research works, but with much less calculations. Nevertheless, our results are proved to be the global optimum, whereas the methods used in the literature cannot prove their global optimality
کلیدواژه‌ها [English]
- Truss Topology Optimization
- Linear Programming
- Stress Constraints
- Displacement Constraints
- Binary (Zero-One) Integer Programming
- Klein, B., “Direct Use of Extremal Principles in Solving Certain Optimizing Problems Involving Inequalities”,Journal of the Operations Research Society of America 3, No. 2 (1955): 168-175.
- Dorn, W. S., “Automatic Design of Optimal Structures”, de Mecanique, 3:25–52, 1964.
- Farrell, J. J., “On The Analysis of Structural Design”, Journal of Mathematical Analysis and Applications, 25(2):285–295, 1969.
- Saka, M. P., “Shape Optimization of Trusses”, Journal of the Structural Division, 106(5):1155–1174, 1980.
- Sakamoto, J., and Oda, J., “A Technique of Optimal Layout Design for Truss Structures Using Genetic Algorithm”, In 34th Structures, Structural Dynamics and Materials Conference, page 1582, 1993.
- Ohsaki, M., and Katoh, N., “Topology Optimization of Trusses with Stress and Local Constraints on Nodal Stability and Member Intersection”, Structural and Multidisciplinary Optimization, 29(3):190–197, 2005.
- Faustino, A. M., Júdice, J.J., Ribeiro, I. M., and Neves, A. S., “An Integer Programming Model for Truss Topology Optimization”, Investigação Operacional, 26(1):111–127, 2006.
- Stolpe, M., “On The Reformulation of Topology Optimization Problems as Linear or Convex Quadratic Mixed 0-1 Programs”, Optimization and Engineering, 8(2):163–192, 2007.
- Hajela, P., Lee, E., and Lin, C. Y., “Genetic Algorithms in Structural Topology Optimization”, Springer Netherlands, Dordrecht, pages 117–133,
- Kaveh, A., and Kalatjari, V., “Topology Optimization of Trusses Using Genetic Algorithm, Force Method and Graph Theory”, International Journal for Numerical Methods in Engineering, 58(5):771–791, 2003.
- Richardson, J. N., Adriaenssens, S., Bouillard, P., and Coelho, R. F., “Multi Objective Topology Optimization of Truss Structures with Kinematic Stability Repair”, Structural and Multidisciplinary Optimization, 46(4):513–532, 2012.
- Wu, Y., Li, Q., Hu, Q., and Borgart, A., “Size and Topology Optimization for Trusses with Discrete Design Variables by Improved Firefly Algorithm”, Mathematical Problems in Engineering, 2017, 2017.
- De Klerk, E., Roos, C., and Terlaky, T., “Semi-Definite Problems in Truss Topology Optimization”, Delft University of Technology, Faculty of Technical Mathematics and Informatics, 1995.
- Przemieniecki, J. S., “Theory of Matrix Structural Analysis”, Courier Corporation, 1985.
- Petersen, C. C., “A Note on Transforming The Product of Variables to Linear Form in Linear Programs”, Diskussionspapier, Purdue University, 1971.
- Glover, F., “Improved Linear Integer Programming Formulations of Nonlinear Integer Problems”, Management Science, 22(4):455–460, 1975.
- Glover, F., “An Improved MIP Formulation for Products of Discrete and Continuous Variables”, Journal of Information and Optimization Sciences, 5(1):69–71, 1984.
- Aslam, K., Structural Analysis. Cengage Learning, 2018.
- Rozvany, G. I., “Difficulties in Truss Topology Optimization with Stress, Local Buckling and System Stability Constraints”,Structural Optimization, 11(3), pp.213-217. 1996.
- Sved, G., and Ginos, Z., “Structural Optimization Under Multiple Loading”, International Journal of Mechanical Sciences, 10(10):803–805, 1968.
- Haftka, R. T., and Gürdal, Z., “Elements of Structural Optimization”, Springer Science & Business Media, 2012.
- Rao, S. S., “Engineering Optimization: theory and practice”, John Wiley & Sons, 2019.
)