Structural and Multidisciplinary Optimization

, Volume 32, Issue 4, pp 313–326 | Cite as

Evolutionary truss topology optimization using a graph-based parameterization concept

  • M. GigerEmail author
  • P. Ermanni
Research Paper


A novel parameterization concept for the optimization of truss structures by means of evolutionary algorithms is presented. The main idea is to represent truss structures as mathematical graphs and directly apply genetic operators, i.e., mutation and crossover, on them. For this purpose, new genetic graph operators are introduced, which are combined with graph algorithms, e.g., Cuthill–McKee reordering, to raise their efficiency. This parameterization concept allows for the concurrent optimization of topology, geometry, and sizing of the truss structures. Furthermore, it is absolutely independent from any kind of ground structure normally reducing the number of possible topologies and sometimes preventing innovative design solutions. A further advantage of this parameterization concept compared to traditional encoding of evolutionary algorithms is the possibility of handling individuals of variable size. Finally, the effectiveness of the concept is demonstrated by examining three numerical examples.


Truss topology optimization Mathematical graph Structural optimization Parameterization Evolutionary algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Azid IA, Kwan ASK, Seetharamu KN (2002) An evolutionary approach for layout optimization of three-dimensional truss. Struct Multidisc Optim 24(4):333–337CrossRefGoogle Scholar
  2. Balakrishnan R, Ranganathan K (2000) A textbook of graph theory. Springer, Berlin Heidelberg New York, pp 1–227zbMATHGoogle Scholar
  3. Bentley Peter J (ed) (1999) Evolutionary design by computers. Morgan Kaufmann, San Francisco CaliforniazbMATHGoogle Scholar
  4. Chen W-K (1997) Graph theory and its engineering applications. World Scientific, River Edge, New JerseyGoogle Scholar
  5. Cheng G, Guo X (1997) \(\epsilon\)-relaxed approach in structural topology optimization. Struct Multidisc Optim 13(4):258–266Google Scholar
  6. Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: ACM Proceedings of the 24th National Conference. Association of Computing Machinery, New YorkGoogle Scholar
  7. Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J Mec 3:25–52Google Scholar
  8. Foulds LR (1994) Graph theory applications. Springer, Berlin Heidelberg New YorkGoogle Scholar
  9. Fowler PW, Guest SD (2000) A symmetry extension of maxwell’s rule for rigidity of frames. Int J Solids Struct 37:1793–1804zbMATHCrossRefMathSciNetGoogle Scholar
  10. Giger M, Ermanni P (2005) Development of cfrp racing motorcycle rims using a heuristic evolutionary algorithm approach. Struct Multidisc Optim 30:54–65CrossRefGoogle Scholar
  11. Gill PE, Murray W, Sanders MA (2002) SNOPT: an sqp algorithm for large-scale constrained optimization. SIAM J Optim 12(4):979–1006 (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  12. Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading, MassachusettszbMATHGoogle Scholar
  13. Gou X, Cheng G, Yamazaki K (2001) A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints. Struct Multidisc Optim 22:364–372CrossRefGoogle Scholar
  14. Gou X, Liu W, Li H (2003) Simultaneous shape and topology optimization of truss under local and global stability constraints. Acta Mech Solida SinicaGoogle Scholar
  15. Hajela P (1992) Stochastic search in structural optimization: genetic algorithms and simulated annealing. In: Structural optimization: status and promise. Prog Astronaut Aeronaut 150:611–637Google Scholar
  16. Hajela P, Lee J (1995) Genetic algorithms in truss topological optimization. Int J Solids Struct 32(22):3341–3357zbMATHCrossRefMathSciNetGoogle Scholar
  17. Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan, Ann Arbor, MIGoogle Scholar
  18. Kawamoto A, Bendsøe MP, Sigmund O (2004) Planar articulated mechanism design by graph theoretical enumeration. Struct Multidisc OptimGoogle Scholar
  19. Kawamura H, Ohmori H, Kito N (2002) Truss topology optimization by a modified genetic algorithm. Struct Multidisc Optim 23:467–472CrossRefGoogle Scholar
  20. König O (2004) Evolutionary design optimization: tools and applications. PhD thesis, Swiss Federal Institute of Technology, Zürich, 2004. Diss. ETH No. 15486Google Scholar
  21. Lemonge ACC (1999) Application of genetic algorithms in structural optimization problems. Ph.D. thesis, Federal University of Rio de Janeiro, Brazil, 1999. Program of Civil Engineering-COOPEGoogle Scholar
  22. Lemonge ACC, Barbosa HJC (2004) An adaptive penalty scheme for genetic algorithms in structural optimization. Int J Numer Meth EngngGoogle Scholar
  23. Lingyun W, Mei Z, Guangming W, Guang M (2004) Truss optimization on shape and sizing with frequency constraints based on genetic algorithm. Comput Mech 35(5):361–368CrossRefGoogle Scholar
  24. Pedersen NL, Nielsen AK (2003) Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and buckling. Struct Multidisc Optim 25(5–6):436–445CrossRefGoogle Scholar
  25. Pyrz M (2004) Evolutionary algorithm integrating stress heuristics for truss optimization. Opt Eng 5(1):45–57zbMATHCrossRefMathSciNetGoogle Scholar
  26. Rozvany GIN (1997) Aims, scope basic concepts and methods of topology optimization. In: Rozvany GIN (ed) Topology optimization in structural mechanics, CISM courses and lectures, No. 374. Springer, Berlin Heidelberg New York, pp 1–55Google Scholar
  27. Rozvany GIN (2001) Stress ratio and compliance based methods in topology optimization—a critical review. Struct Multidisc Optim 21:109–119CrossRefGoogle Scholar
  28. Rozvany GIN (2003) On design-dependent constraints and singular topologies. Struct Multidisc Optim 21:164–172CrossRefGoogle Scholar
  29. Ryoo J, Hajela P (2004) Handling variable string lengths in ga-based structural topology optimization. Struct Multidisc Optim 26:318–325CrossRefGoogle Scholar
  30. Siek JG, Lee LQ, Lumsdaine A (2002) The boost graph library. C++ in-depth series. Addison WesleyGoogle Scholar
  31. Stolpe M, Svanberg K (2003) A note on stress-constrained truss topology optimization. Struct Multidisc Optim 25:62–64CrossRefGoogle Scholar
  32. Wang D, Zhang WH, Jiang JS (2002) Combined shape and sizing optimization of truss structures. Comput Mech 29:307–312zbMATHCrossRefGoogle Scholar
  33. West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  34. Wu S-J, Chow P-T (1995) Steady-state genetic algorithms for discrete optimization of trusses. Comput Struct 56(6):979–991zbMATHCrossRefGoogle Scholar
  35. Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Centre of Structure TechnologiesZurichSwitzerland

Personalised recommendations