Skip to main content
SpringerLink
Log in

Topology synthesis of geometrically nonlinear compliant mechanisms using meshless methods

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

This paper presents a new method for topology optimization of geometrical nonlinear compliant mechanisms using the element-free Galerkin method (EFGM). The EFGM is employed as an alternative scheme to numerically solve the state equations by fully taking advantage of its capability in dealing with large displacement problems. In the meshless method, the imposition of essential boundary conditions is also addressed. The popularly studied solid isotropic material with the penalization (SIMP) scheme is used to represent the nonlinear dependence between material properties and regularized discrete densities. The output displacement is regarded as the objective function and the adjoint method is applied to finding the sensitivity of the design functions. As a result, the optimization of compliant mechanisms is mathematically established as a nonlinear programming problem, to which the method of moving asymptotes (MMA) belonging to the sequential convex programming can be applied. The availability of the present method is finally demonstrated with several widely investigated numerical examples.

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

Similar content being viewed by others

References

  1. Howell, L.L., Compliant Mechanisms. New York: John Wiley & Sons, Inc. 2001.

    Google Scholar 

  2. Howell, L.L. and Midha, A., A method for design of compliant mechanisms with small length flexural pivots. Journal of Mechanical Design, 1994, 116: 280–290.

    Article  Google Scholar 

  3. Bendsøe, M.P. and Sigmund, O., Topology Optimization: Theory, Methods, and Applications. Berlin Heidelberg, New York: Springer), 2003.

    MATH  Google Scholar 

  4. Sigmund, O., On the design of compliant mechanisms using topology optimization. Mechanics of Structures and Machines, 1997, 25(3): 493–524.

    Article  Google Scholar 

  5. Yin, L. and Ananthasuresh, G.K., Design of distributed compliant mechanisms. Mechanics Based Design of Structures and Machines, 2003, 31(2):151–179.

    Article  Google Scholar 

  6. Wang, Y., Chen, S.K. and Wang, X.M. et al., Design of multimaterial compliant mechanisms using level-set methods. Journal of Mechanical Design, 2005, 127(55): 941–956.

    Article  Google Scholar 

  7. Luo, Z., Tong, L.Y., Wang, M.Y. and Wang, S.Y., Shape and topology optimization of compliant mechanisms using a parameterization level set method. Journal of Computational Physics, 2007, 227(1): 680–705.

    Article  MathSciNet  Google Scholar 

  8. Ananthasuresh, G.K., Kota, S. and Gianchandani, Y., A methodical approach to the design of compliant micro-mechanisms//Technical Digest of the Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC June 13–16, 1994, 189–192.

  9. Bendsøe, M.P. and Kikuchi, N., Generating optimal topology in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71: 197–224.

    Article  MathSciNet  Google Scholar 

  10. Bendsøe, M.P. and Sigmund, O., Material interpolation schemes in topology optimization. Archive of Applied Mechanics, 1999, 69: 635–654.

    Article  Google Scholar 

  11. Wang, Y., Wang, X.M. and Guo, D.M., A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192: 227–246.

    Article  MathSciNet  Google Scholar 

  12. Mei, Y.L. and Wang, X.M., A level set method for microstructure design of composite materials. Acta Mechanica Solida Sinica, 2004, 17(3): 239–249.

    Google Scholar 

  13. Buhl, T., Pedersen, C.B.W. and Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization. Structural and Multidisciplinary Optimization, 2000, 19(2): 93–104.

    Article  Google Scholar 

  14. Pedersen, C.B.W., Buhl, T. and Sigmund, O., Topology synthesis of large-displacement compliant mechanisms. International Journal for Numerical Methods in Engineering, 2001, 50: 2683–2705.

    Article  Google Scholar 

  15. Saxena, A. and Ananthasuresh, G.K., Topology synthesis of compliant mechanisms for nonlinear force-deflection and curved path specifications. ASME Journal of Mechanical Design, 2001, 123: 33–42.

    Article  Google Scholar 

  16. Belyeschko, T., Krongauz, Y. and Organ, D. et al., Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 1996, 39: 3–47.

    Article  Google Scholar 

  17. Belytschko, T., Lu, Y.Y. and Gu, L., Element free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37: 229–256.

    Article  MathSciNet  Google Scholar 

  18. Liu, G.R., Mesh Free Method: Moving Beyond the Finite Element Method. USA: CRC Press, 2002.

    Book  Google Scholar 

  19. Cho, S. and Kwak, J., Continuum-based topology design optimization using meshfree method. 5th World Congress of Structural and Multidisciplinary Optimization, Lido di Jesolo, Italy, May 19–23, 2003.

  20. Cho, S. and Kwak, J., Topology design optimization of geometrically non-linear structures using meshfree method. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 5909–5925.

    Article  MathSciNet  Google Scholar 

  21. Gu, Y.T. and Liu, G.R., Hybrid boundary point interpolation methods and their coupling with the element free Galerkin method. Engineering Analysis with Boundary Elements, 2003, 27(11): 905–917.

    Article  Google Scholar 

  22. Zhang, X., Liu, X.H. and Song, K.Z. et al., Least-square collocation meshless method. International Journal for Numerical Methods in Engineering, 2001, 51(11): 1089–1100.

    Article  MathSciNet  Google Scholar 

  23. Belytschko, T., Organ, D. and Krongauz, Y., Coupled finite element-element-free Galerkin method. Computational Mechanics, 1995, 17(3): 186–195.

    Article  MathSciNet  Google Scholar 

  24. Zhang, X., Liu, X. and Lu, M.W. et al., Imposition of essential boundary conditions by displacement constraint equations in meshless methods. Communications in Numerical Methods in Engineering, 2001, 17(3): 165–178.

    Article  Google Scholar 

  25. Rozvany, G.I.N., Topology Optimization in Structural Mechanics. Berlin Heidelberg: Springer, 1997.

    Book  Google Scholar 

  26. Kota, S., Lu, K.J. and Kreiner, Z. et al., Design and application of compliant mechanisms for surgical tools. Journal of Biomechanical Engineering, 2005, 127(8): 981–989.

    Google Scholar 

  27. Luo, Z., Du, Y.X. and Chen, L.P. et al., Continuum topology optimization for monolithic compliant mechanisms of micro-actuators. Acta Mechanica Solida Sinica, 2006, 19(1): 58–68.

    Article  Google Scholar 

  28. Zhou, M. and Rozvany, G.I.N., The COC algorithm, part II: topological, geometry and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering, 1991, 89: 197–122.

    Article  Google Scholar 

  29. Svanberg, K., The method of moving asymptotes: a new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24: 359–73.

    Article  MathSciNet  Google Scholar 

  30. Svanberg, K., A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization, 2002, 12: 555–573.

    Article  MathSciNet  Google Scholar 

  31. Sigmund, O. and Petersson, J., Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structure and Multidisciplinary Optimization, 1998, 16: 68–75.

    Article  Google Scholar 

  32. Luo, Z., Chen, L.P. and Yang, J.Z. et al., Compliant mechanism design using multi-objective topology optimization scheme of continuum structures. Structure and Multidisciplinary Optimization, 2005, 30(2): 142–154.

    Article  Google Scholar 

  33. Diaz, A.R. and Sigmund, O., Checkerboard patterns in layout optimization. Structural and Multidisciplinary Optimization, 1995, 10: 40–45.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Science & Engineering, Huazhong University of Science & Technology, Wuhan, 430074, China

    Yixian Du, Liping Chen & Zhen Luo

  2. School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

    Zhen Luo

Authors
  1. Yixian Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhen Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yixian Du.

Additional information

Project supported in part by the National ‘973’ Key Fundamental Research Projects of China (No. 2003CB716207), the National ‘863’ High-Tech Development Projects of China (No. 2006AA04Z162), and also the Australian Research Council (No. ARC-DP0666683).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Du, Y., Chen, L. & Luo, Z. Topology synthesis of geometrically nonlinear compliant mechanisms using meshless methods. Acta Mech. Solida Sin. 21, 51–61 (2008). https://doi.org/10.1007/s10338-008-0808-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10338-008-0808-3

Key words

Search

Navigation