Skip to main content
SpringerLink
Log in

Design and verification of auxetic microstructures using topology optimization and homogenization

  • Special
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The design of mechanical microstructures having auxetic behaviour is proposed in this paper using techniques of topology optimization for compliant mechanisms. A robust hybrid algorithm based on evolutionary algorithms and local search steps is used. The result may need verification in order to accommodate needs not taken into account in the topology optimization. Therefore, a numerical homogenization scheme is used in order to show that the final design still has the wished negative Poisson’s property.

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. Allaire G., Jouve F., Maillot H.: Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim. 28(2–3), 87–98 (2004)

    MATH  MathSciNet  Google Scholar 

  2. Andreassen E., Lazarov B.S., Sigmund O.: Design of manufacturable 3D extremal elastic microstructure. Mech. Mater. 69(1), 1–10 (2014)

    Article  Google Scholar 

  3. ASME: Guide for Verification and Validation in Computational Solid Mechanics. American Society of Mechanical Engineers (2006)

  4. Bendsøe M.P.: Optimal shape design as a material distribution problem. Struct. Multidiscip. Optim. 1(4), 193–202 (1989)

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Bendsøe M.P., Sigmund O.: Topology Optimization—Theory, Methods and Applications, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  7. Bendsøe M.P., Sigmund O.: Topology Optimization. Theory, Methods and Applications, 2nd edn, pp. 94–98. Springer, Berlin (2003)

    Google Scholar 

  8. Canfield S., Frecker M.: Topology optimization of compliant mechanical amplifiers for piezoelectric actuators. Struct. Multidiscip. Optim. 20(4), 269–279 (2000)

    Article  Google Scholar 

  9. Dascalu C., Bilbie G., Agiasofitou E.K.: Damage and size effects in elastic solids: a homogenization approach. Int. J. Solids Struct. 45(2), 409–430 (2008)

    Article  MATH  Google Scholar 

  10. Drosopoulos, G.A., Wriggers, P., Stavroulakis, G.E.: Contact analysis in multi-scale computational homogenization. In: Jirásek, M., Allix, O., Moës, N., Oliver, J. (eds.) Proceedings of 3rd International Conference on Computational Modeling of Fracture and Failure of Materials and Structures (CFRAC), p. 226. Prague, Czech Republic (2013)

  11. Drosopoulos, G.A., Wriggers, P., Stavroulakis, G.E.: Incorporation of contact mechanics in multi-level computational homogenization for the study of composite materials. In: Proceedings of 3rd International Conference on Computational Contact Mechanics (ICCCM). Lecce, Italy (2013)

  12. Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: Micro Machine and Human Science, 1995. MHS ’95, Proceedings of the Sixth International Symposium on, pp. 39–43 (1995)

  13. Frecker M., Kikuchi N., Kota S.: Topology optimization of compliant mechanisms with multiple outputs. Struct. Multidiscip. Optim. 17(4), 269–278 (1999)

    Article  Google Scholar 

  14. Geers M.G.D., Kouznetsova V., Brekelmans W.A.M.: Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. 234(7), 2175–2182 (2010)

    Article  MATH  Google Scholar 

  15. Herakovich C.T.: Composite laminates with negative through-the-thickness Poisson’s ratios. J. Compos. Mater. 18(5), 447–455 (1984)

    Article  Google Scholar 

  16. Hill R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963)

    Article  MATH  Google Scholar 

  17. Holland J.H.: Outline for a logical theory of adaptive systems. J. ACM 9(3), 297–314 (1962)

    Article  MATH  Google Scholar 

  18. Holland J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and artificial Intelligence. MIT Press, Cambridge, MA (1992)

    Google Scholar 

  19. Kaminakis N.T., Stavroulakis G.E.: Topology optimization for compliant mechanisms, using evolutionary-hybrid algorithms and application to the design of auxetic materials. Compos. Part B Eng. 43(6), 2655–2668 (2012)

    Article  Google Scholar 

  20. Kanouté P., Boso D., Chaboche J., Schrefler B.: Multiscale methods for composites: a review. Arch. Comput. Methods Eng. 16(1), 31–75 (2009)

    Article  MATH  Google Scholar 

  21. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Neural Networks, 1995. Proceedings, IEEE International Conference on, vol. 4, pp. 1942–1948 (1995)

  22. Kouznetsova, V.: Computational Homogenization for the Multi-Scale Analysis of Multi-Phase materials. PhD thesis. Technical University Eindhoven, The Netherlands (2002)

  23. Krishnan, G., Ananthasuresh, G.K.: Evaluation and design of displacement-amplifying compliant mechanisms for sensor applications. J. Mech. Des. 130(10), 102304 (2008)

  24. Larsen U., Sigmund O., Bouwstra S.: Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. J. Micro Electro. Mech. Syst. 6(2), 99–106 (1997)

    Article  Google Scholar 

  25. Liu, Q.: Literature Review: Materials with Negative Poisson’s Ratios and Potential Applications to Aerospace and Defence. Technical Report DSTO-GD-0472, Air Vehicles Division, Defence Science and Technology Organisation, Department of Defence, Australian Goverment (2006)

  26. Mankame N.D., Ananthasuresh G.K.: Synthesis of contact-aided compliant mechanisms for non-smooth path generation. Int. J. Numer. Methods Eng. 69(12), 2564–2605 (2007)

    Article  MATH  Google Scholar 

  27. Massart T.J., Peerlings R.H.J., Geers M.G.D.: Structural damage analysis of masonry walls using computational homogenization. Int. J. Damage Mech. 16(2), 199–226 (2007)

    Article  Google Scholar 

  28. Mehta V., Frecker M., Lesieutre G.A.: Stress relief in contact-aided compliant cellular mechanisms. J. Mech. Des. 131(9), 091009 (2009)

    Article  Google Scholar 

  29. Nakshatrala P.B., Tortorelli D.A., Nakshatrala K.B.: Nonlinear structural design using multiscale topology optimization. Part i: static formulation. Comput. Methods Appl. Mech. Eng. 261–262, 167–176 (2013)

    Article  MathSciNet  Google Scholar 

  30. Osher S., Sethian J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pettermann H.E., Suresh S.: A comprehensive unit cell model: a study of coupled effects in piezoelectric 1–3 composites. Int. J. Solids Struct. 37(39), 5447–5464 (2000)

    Article  MATH  Google Scholar 

  32. Pindera M.J., Khatam H., Drago A.S., Bansal Y.: Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Compos. Part B Eng. 40(5), 349–378 (2009)

    Article  Google Scholar 

  33. Price K.V., Storn R.M., Lampinen J.A.: Differential Evolution—A practical Approach to Global Optimization. Natural Computing Series. Springer, Berlin (2005)

    Google Scholar 

  34. Rozvany G.I.N.: Structural Design Via Optimality Criteria. Kluwer, Dordrecht (1989)

    Book  MATH  Google Scholar 

  35. Rozvany G.I.N.: A critical review of established methods of structural topology optimization. Struct. Multidiscip. Optim. 37(3), 217–237 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sanchez-Palencia E.: Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics. Springer, Berlin (1980)

    Google Scholar 

  37. Sethian J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press, Cambridge, MA (1999)

    MATH  Google Scholar 

  38. Sigmund O.: Some Inverse Problems in Topology Design of Materials and Mechanisms, pp. 277–284. Kluwer, Dordrecht (1996)

    Google Scholar 

  39. Sigmund O.: A 99 line topology optimization code written in Matlab. Struct. Multidiscip. Optim. 21(2), 120–127 (2001)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  41. Sigmund O., Torquato S.: Composites with extremal thermal expansion coefficients. Appl. Phys. Lett. 69(21), 3203–3205 (1996)

    Article  Google Scholar 

  42. Sigmund O., Torquato S.: Design of smart composite materials using topology optimization. Smart Mater. Struct. 8, 365–379 (1999)

    Article  Google Scholar 

  43. Sokolowski J., Zochowski A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  44. Storn R., Price K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  45. Suquet P., Sawczuk A., Bianchi G.: Local and Global Aspects in the Mathematical Theory of Plasticity, Plasticity Today: Modelling, Methods and Applications. Elsevier, London (1985)

    Google Scholar 

  46. Svanberg K.: The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Eng. 24(2), 359–373 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  47. Theocaris P.S., Stavroulakis G.E., Panagiotopoulos P.D.: Negative Poisson’s ratios in composites with star-shaped inclusions: a numerical homogenization approach. Arch. Appl. Mech. 67(4), 274–286 (1997)

    Article  MATH  Google Scholar 

  48. Yoon G., Kim Y., Bendsøe M., Sigmund O.: Hinge-free topology optimization with embedded translation-invariant differentiable wavelet shrinkage. Struct. Multidiscip. Optim. 27(3), 139–150 (2004)

    Article  Google Scholar 

  49. Zohdi T.I., Wriggers P.: An Introduction to Computational Micromechanics. Springer, Berlin (2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mechanics and Optimization, Technical University of Crete, Chania, Greece

    Nikos T. Kaminakis & Georgios E. Stavroulakis

  2. Institute of Continuum Mechanics, Leibniz University of Hannover, Hannover, Germany

    Georgios A. Drosopoulos

Authors
  1. Nikos T. Kaminakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georgios A. Drosopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Georgios E. Stavroulakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikos T. Kaminakis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaminakis, N.T., Drosopoulos, G.A. & Stavroulakis, G.E. Design and verification of auxetic microstructures using topology optimization and homogenization. Arch Appl Mech 85, 1289–1306 (2015). https://doi.org/10.1007/s00419-014-0970-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-014-0970-7

Keywords

Search

Navigation