Developing a Bidirectional Evolutionary Topology Algorithm for Continuum Structures with the Objective Functions of Stiffness and Fundamental Frequency with Geometrical Symmetry Constraint

Document Type : Research Article

Authors

Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

Abstract

Topology optimization of structures, seeking the best distribution of mass in the design space to improve the performance and weight of a structure, is one of the most comprehensive issues raised in the field of structural optimization. In addition to the structure stiffness as the most common objective function, frequency optimization is of great importance in automotive and aerospace industries achieved by maximizing the fundamental frequency or the gap between two consecutive eigenfrequencies. The phenomenon of multiple frequencies, mesh dependency of topology responses, checkerboarding, geometric symmetry constraint, and occurrence of artificial localized vibration modes in low-density regions are the most important challenges faced by the designer in stiffness and frequency optimization problems which influence the manufacturability of the design too. In this paper, Bidirectional Evolutionary Structural Optimization (BESO) method which is a successful approach in stiffness problems is applied for a frequency and stiffness problem separately via creating a software package including a Matlab code and Abaqus FE solver linked to each other. Also, in this paper, the effect of geometric symmetry constraint is considered on resulted topologies from stiffness and frequency problems. So the BESO method is applied for modeling a 2D beam and its stiffness and frequency optimization and finally, the optimization results of both objective functions will be compared with the initial structure.

Keywords

Main Subjects

References

[1] MP. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput Methods Appl Mech Eng, Vol. 71, No. 2, pp. 197–224, 1988.
 [2]  M. Zhou, GIN. Rozvany, The COC algorithm. Part II: Topological geometry and generalized shape optimization, Comput Methods Appl Mech Eng, Vol. 89, pp. 197–224, 1991.
[3]   GIN. Rozvany, M. Zhou, T. Birker, Generalized shape optimization without homogenization, Struct Optimiz, Vol. 4, pp. 250–4, 1992.
[4]   O. Sigmund, J. Petersson, Numerical instability in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct Optimiz, Vol. 16, pp. 68–75, 1998.
[5]   A. Ritz, Sufficiency of a finite exponent in SIMP (power law) methods, Struct Multidiscip Optimiz, Vol. 21, pp. 159–63, 2001.
[6]   MP. Bendsoe, O. Sigmund, Topology Optimization: Theory, Methods and Applications, Berlin, Heidelberg: Springer-Verlag, 2003.
[7]   YM. Xie, GP. Steven, A simple evolutionary procedure for structural optimization, Comput Struct, Vol. 49, pp. 885–96, 1993.
[8]   YM. Xie, GP. Steven, Evolutionary Structural Optimization, London: Springer, 1997.
[9]   JA. Sethian, A. Wiegmann, Structural boundary design via level set and immersed interface methods, Journal of Computational Physics, Vol. 163, pp. 489–528, 2000.
[10] MY. Wang, X. Wang, D. Guo, A level set method for structural topology optimization, Comput Methods Appl Mech Eng, Vol. 192, pp. 227–4, 2003.
[11] Yang, X., et al., Bidirectional evolutionary method for stiffness optimization. AIAA journal, 1999. 37(11).
[12] X. Huang, YM. Xie, Convergent and mesh-independent solutions for the bidirectional evolutionary structural optimization method. Finite Elements in Analysis and Design, Vol.43, pp. 1039–49, 2007.
[13] LH. Tenek, I. Hagiwara, Eigenfrequency maximization of plates by optimization of topology using homogenization and mathematical programming,  JSME Int J, Vol.37, pp. 667–77, 1994.
[14] ZD. Ma, HC. Cheng, N. Kikuchi, Topological design for vibrating structures, Comput Methods Appl Mech Eng, Vol. 121, pp. 259–80, 1995.
[15] I. Kosaka, CC. Swan, A symmetry reduction method for continuum structural topology optimization, Computers & Structures, Vol.70, pp.47–61, 1999.
[16] NL. Pedersen, Maximization of eigenvalues using topology optimization, Structural and Multidisciplinary Optimization ,Vol.20, pp. 2–11, 2000.
[17] J. Du, N. Olhoff, Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps, Structural and Multidisciplinary Optimization , Vol.34, pp. 91–110, 2007.
[18] XY. Yang, YM. Xie, GP. Steven, OM. Querin, Topology optimization for frequencies using an evolutionary method, Journal of Structural Engineering, Vol. 125, No. 12, pp. 1432–8, 1999.
[19] Zuo, Z.H., Xie, Y.M. and Huang, X.D. (2010). “An improved bidirectional evolutionary topology optimization method for frequencies”, International Journal of Structural Stability and Dynamics, Vol. 10, No. 1, pp. 55–75.
[20] Xia, L., F. Fritzen, and P. Breitkopf, Evolutionary topology optimization of elastoplastic structures. Structural and Multidisciplinary Optimization, 2017. 55(2): p. 569-581.
[21] Sun, X.F., et al., Topology Optimization of Composite Structure Using Bi-Directional Evolutionary Structural Optimization Method. Procedia Engineering, 2011. 14: p. 2980-2985.
[22] Xia, L., et al., Stress-based topology optimization using bi-directional evolutionary structural optimization method. Computer Methods in Applied Mechanics and Engineering, 2018. 333: p. 356-370.
[23] Da, D., et al., Evolutionary topology optimization of continuum structures with smooth boundary representation. Structural and Multidisciplinary Optimization, 2017: p. 1-17.
 [24] Liu, Q., R. Chan, and X. Huang, Concurrent topology optimization of macrostructures and material microstructures for natural frequency. Materials & Design, 2016. 106: p. 380-390.
[25] Zuo, Z.H., Y.M. Xie, and X. Huang, Evolutionary topology optimization of structures with multiple displacement and frequency constraints. Advances in Structural Engineering, 2012. 15(2): p. 359-372.
 [26] YM. Xie, Xiaodong. Huang, Evolutionary Topology Optimization of Continuum Structures: Methods and Applications, First Edittion, pp. 40-43, New York: Wiley, 2010.
[27] Seyranian, A.P., Lund, E. and Olhoff, N. Multiple eigenvalues in structural optimization problems, Structural Optimization, Vol. 8(4), pp.207–27, 1994.
 
Amirkabir Journal of Mechanical Engineering
Volume 52, Issue 1
April 2020
Pages 249-264

Files

Share

How to cite

Statistics