Skip to main content

Advertisement

SpringerLink
Log in

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

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Abbreviations

CSG-TOM:

Constructive solid geometry-based topology optimization methods

SIMP:

Solid isotropic material with penalization

ESO:

Evolutionary structural optimization

MOGA:

Multi-objective genetic algorithm

SBX:

Simulated binary crossover

ToPy:

Topology optimization using python

\(\varvec{n}\) :

Total nodes

\(\varvec{m}\) :

Variable nodes

\(\varvec{k}\) :

Fixed nodes

\(\varvec{n}_{\varvec{void}}\) :

Special nodes

\(\varvec{n}_{\varvec{sym}}\) :

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

References

  • Abaqus Users Manual (2006). “Abaqus”

  • Ahmed F (2012) Topology optimization of compliant systems using constructive solid geometry. Masters Thesis, IIT Kanpur

  • 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 India

  • Bendsoe MP (1989) Optimal shape design as a material distribution problem. Structural optimization 1(4):193–202

    Article  Google Scholar 

  • Bendsoe MP (1995) Topology design of truss structures. Springer, Berlin

    Book  Google Scholar 

  • Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  MathSciNet  MATH  Google Scholar 

  • Braid IC, Lang CA (1974) Computer-aided design of mechanical components with volume building bricks. Automatica 10(6):635–642

    Article  Google Scholar 

  • Cuillière JC, Francois V, Drouet JM (2013) Automatic mesh generation and transformation for topology optimization methods. Comput Aided Des 45(12):1489–1506

  • 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 1843–1850

  • Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38

  • Deb K, Agrawal RB (1995) Simulated binary crossover for continuous search space. Complex Syst 9(2):115–148

  • Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6:82–197

    Article  Google Scholar 

  • Deb K, Datta R (2012) Hybrid evolutionary multi-objective optimization and analysis of machining operations. Eng Optim 44(6):685–706

    Article  MathSciNet  Google Scholar 

  • Guide MUS (1998) The mathworks. Inc, Natick, MA, 5

  • 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 981–992

  • Hazewinkel M (ed) (2001) Bezier curve. Encyclopedia of mathematics. Springer, Berlin. ISBN: 978-1-55608-010-4

  • Howell LL (2001) Compliant mechanisms. Wiley-Interscience, New York

    Google Scholar 

  • Huang X, Xie M (2010) Evolutionary topology optimization of continuum structures: methods and applications. Wiley, New York

    Book  MATH  Google Scholar 

  • Hunter W (2009) Topology optimization using python based open source software. https://code.google.com/p/topy/

  • 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: 0045-7825

  • 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):091007

    Article  Google Scholar 

  • Lee DT, Schachter BJ (1980) Two algorithms for constructing a Delaunay triangulation. Int J Comput Inf Sci 9(3):219–242

    Article  MathSciNet  MATH  Google Scholar 

  • MATLAB users guide, “Mathworks, 2013”. http://www.mathworks.in/help/pde/ug/decsg.html

  • Michell AGM (1904) LVIII. The limits of economy of material in frame-structures. Lond Edinb Dublin Philos Mag J Sci 8(47):589–597

  • 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):535–559

  • 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):559–573

    Article  MATH  Google Scholar 

  • Rozvany GIN, Pomezanski V, Gaspar Z, Querin OM (2005) Some pivotal issues in structural topology optimization. In: Proceedings of WCSMO, 6

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424

    Article  Google Scholar 

  • Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596

    Article  MathSciNet  MATH  Google Scholar 

  • Sokolowski J, Zolesio JP (1992) Introduction to shape optimization. Springer, Berlin

  • 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 :0045-7825

  • Van Rossum G, Drake FL (2010) The python language reference. Python Software Foundation, Delaware

  • 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: 0045-7825

  • Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784

  • Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896

  • Xie YM, Steven GP (1994) Optimal design of multiple load case structures using an evolutionary procedure. Eng Comput 11(4):295–302

    Article  MATH  Google Scholar 

  • Xie YM, Steven GP (1996) Evolutionary structural optimization for dynamic problems. Comput Struct 58(6):1067–1073

    Article  MATH  Google Scholar 

  • 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):183–192

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank Prof. Kalyanmoy Deb, Michigan State University and the computational facilities provided by KanGAL, IIT Kanpur. Part of the work has been jointly supported by the Department of Biotechnology, India and the Swedish Governmental Agency for Innovation Systems.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India

    Anmol Pandey, Rituparna Datta & Bishakh Bhattacharya

Authors
  1. Anmol Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rituparna Datta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bishakh Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rituparna Datta.

Ethics declarations

Conflict of interest

None.

Additional information

Communicated by V. Loia.

Appendix 1

Appendix 1

In ‘Topology Repair Operation’, the geometries are viewed as graph of connected nodes. A bunch of inter-connected nodes is termed as a ‘Set’. Following steps are taken to obtain the final geometry:

  1. 1.

    Find the ‘Sets’ of inter-connected nodes.

  2. 2.

    Find the ‘Special Sets’ containing k fixed nodes and non-design constrained sets. If they fall in the same set, then go to Step 7.

  3. 3.

    Using the initial triangulation data, find the connectivity between each pair of sets which were removed by the optimization variables.

  4. 4.

    Obtain the graph of connectivity between sets. Weight of connection between any two sets is through the nodes joined by the minimum edge length.

  5. 5.

    Using breadth first search (BFS) from each set in the graph, find the minimum connections required to join all the ‘Special Sets’. The absolute values of widths are taken for this new connection.

  6. 6.

    Join the pair of nodes for each new connection made to form a single connected ‘Set’.

  7. 7.

    Ignore all other sets and form the final geometry using the singly connected ‘Set’.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pandey, A., Datta, R. & Bhattacharya, B. Topology optimization of compliant structures and mechanisms using constructive solid geometry for 2-d and 3-d applications. Soft Comput 21, 1157–1179 (2017). https://doi.org/10.1007/s00500-015-1845-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-015-1845-8

Keywords

Search

Navigation