Soft Computing

, Volume 21, Issue 5, pp 1157–1179 | Cite as

Topology optimization of compliant structures and mechanisms using constructive solid geometry for 2-d and 3-d applications

  • Anmol Pandey
  • Rituparna Datta
  • Bishakh Bhattacharya
Methodologies and Application


This research focuses on the establishment of a constructive solid geometry-based topology optimization (CSG-TOM) technique for the design of compliant structure and mechanism. The novelty of the method lies in handling voids, non-design constraints, and irregular boundary shapes of the design domain, which are critical for any structural optimization. One of the most popular models of multi-objective genetic algorithm, non-dominated sorting genetic algorithm is used as the optimization tool due to its ample applicability in a wide variety of problems and flexibility in providing non-dominated solutions. The CSG-TOM technique has been successfully applied for 2-D topology optimization of compliant mechanisms and subsequently extended to 3-D cases. For handling these cases, a new software framework involving optimization routine for geometry and mesh generation with FEA solver has been developed. The efficacy of the approach has been demonstrated for 2-D and 3-D geometries and also compared with state of the art techniques.


Structural and topology optimization Finite element analysis (FEA) Multi-objective genetic algorithms Compliant structures 

Abbreviations and symbols


Constructive solid geometry-based topology optimization methods


Solid isotropic material with penalization


Evolutionary structural optimization


Multi-objective genetic algorithm


Simulated binary crossover


Topology optimization using python


Total nodes


Variable nodes


Fixed nodes


Special nodes


Symmetry nodes

\(\eta _i\)

Volume fraction of topology in ith generation

\(\varvec{\alpha }\)

Volume correction factor

\(\varvec{\varepsilon }\)

Ratio of required volume fraction to current volume fraction

\(W_{-}, W_{+}\)

Width selection operator

\(\lambda \)

Symmetry condition operator


  1. Abaqus Users Manual (2006). “Abaqus”Google Scholar
  2. Ahmed F (2012) Topology optimization of compliant systems using constructive solid geometry. Masters Thesis, IIT KanpurGoogle Scholar
  3. Ahmed, F., Bhattacharya, B., & Deb, K. (2013, January). Constructive Solid Geometry Based Topology Optimization Using Evolutionary Algorithm. In Proceedings of Seventh International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA 2012) (pp. 227–238). Springer IndiaGoogle Scholar
  4. Bendsoe MP (1989) Optimal shape design as a material distribution problem. Structural optimization 1(4):193–202CrossRefGoogle Scholar
  5. Bendsoe MP (1995) Topology design of truss structures. Springer, BerlinCrossRefGoogle Scholar
  6. Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetCrossRefMATHGoogle Scholar
  7. Braid IC, Lang CA (1974) Computer-aided design of mechanical components with volume building bricks. Automatica 10(6):635–642CrossRefGoogle Scholar
  8. Cuillière JC, Francois V, Drouet JM (2013) Automatic mesh generation and transformation for topology optimization methods. Comput Aided Des 45(12):1489–1506Google Scholar
  9. Datta R, Deb K (2011) Multi-objective design and analysis of robot gripper configurations using an evolutionary-classical approach. In: Proceedings of the 13th annual conference on genetic and evolutionary computation. ACM, pp 1843–1850Google Scholar
  10. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38Google Scholar
  11. Deb K, Agrawal RB (1995) Simulated binary crossover for continuous search space. Complex Syst 9(2):115–148Google Scholar
  12. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6:82–197CrossRefGoogle Scholar
  13. Deb K, Datta R (2012) Hybrid evolutionary multi-objective optimization and analysis of machining operations. Eng Optim 44(6):685–706MathSciNetCrossRefGoogle Scholar
  14. Guide MUS (1998) The mathworks. Inc, Natick, MA, 5Google Scholar
  15. Hamza K, Saitou K (2004) Optimization of constructive solid geometry via a tree-based multi-objective genetic algorithm. In: Genetic and evolutionary computation—GECCO 2004. Springer, Berlin, pp 981–992Google Scholar
  16. Hazewinkel M (ed) (2001) Bezier curve. Encyclopedia of mathematics. Springer, Berlin. ISBN: 978-1-55608-010-4Google Scholar
  17. Howell LL (2001) Compliant mechanisms. Wiley-Interscience, New YorkGoogle Scholar
  18. Huang X, Xie M (2010) Evolutionary topology optimization of continuum structures: methods and applications. Wiley, New YorkCrossRefMATHGoogle Scholar
  19. Hunter W (2009) Topology optimization using python based open source software.
  20. Jakiela MJ, Chapman C, Duda J, Adewuya A, Saitou K (2000) Continuum structural topology design with genetic algorithms. Comput Methods Appl Mech Eng 186(2–4):339–356. ISSN: 0045-7825Google Scholar
  21. Kudikala R, Kalyanmoy D, Bhattacharya B (2009) Multi-objective optimization of piezoelectric actuator placement for shape control of plates using genetic algorithms. J Mech Des 131(9):091007CrossRefGoogle Scholar
  22. Lee DT, Schachter BJ (1980) Two algorithms for constructing a Delaunay triangulation. Int J Comput Inf Sci 9(3):219–242MathSciNetCrossRefMATHGoogle Scholar
  23. MATLAB users guide, “Mathworks, 2013”.
  24. Michell AGM (1904) LVIII. The limits of economy of material in frame-structures. Lond Edinb Dublin Philos Mag J Sci 8(47):589–597Google Scholar
  25. Nishiwaki S, Frecker M, Seungjae M, Kikuchi N (1998) Topology optimization of compliant mechanisms using the homogenization method. Int J Numer Methods Eng 42(3):535–559Google Scholar
  26. Querin OM, Young V, Steven GP, Xie YM (2000) Computational efficiency and validation of bi-directional evolutionary structural optimisation. Comput Methods Appl Mech Eng 189(2):559–573CrossRefMATHGoogle Scholar
  27. Rozvany GIN, Pomezanski V, Gaspar Z, Querin OM (2005) Some pivotal issues in structural topology optimization. In: Proceedings of WCSMO, 6Google Scholar
  28. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424CrossRefGoogle Scholar
  29. Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596MathSciNetCrossRefMATHGoogle Scholar
  30. Sokolowski J, Zolesio JP (1992) Introduction to shape optimization. Springer, BerlinGoogle Scholar
  31. Tavakoli R (2014) Multimaterial topology optimization by volume constrained Allen–Cahn system and regularized projected steepest descent method. Comput Methods Appl Mech Eng 276:534–565. ISSN :0045-7825Google Scholar
  32. Van Rossum G, Drake FL (2010) The python language reference. Python Software Foundation, DelawareGoogle Scholar
  33. Wang SY, Tai K (2005) Structural topology design optimization using genetic algorithms with a bit-array representation. Comput Methods Appl Mech Eng 194(36–38):3749–3770. ISSN: 0045-7825Google Scholar
  34. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784Google Scholar
  35. Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896Google Scholar
  36. Xie YM, Steven GP (1994) Optimal design of multiple load case structures using an evolutionary procedure. Eng Comput 11(4):295–302CrossRefMATHGoogle Scholar
  37. Xie YM, Steven GP (1996) Evolutionary structural optimization for dynamic problems. Comput Struct 58(6):1067–1073CrossRefMATHGoogle Scholar
  38. Young V, Querin OM, Steven GP, Xie YM (1999) 3D and multiple load case bi-directional evolutionary structural optimization (BESO). Struct Optim 18(2–3):183–192CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Anmol Pandey
    • 1
  • Rituparna Datta
    • 1
  • Bishakh Bhattacharya
    • 1
  1. 1.Department of Mechanical EngineeringIndian Institute of Technology KanpurKanpurIndia

Personalised recommendations