Structural and Multidisciplinary Optimization

, Volume 50, Issue 4, pp 537–560 | Cite as

Element deformation scaling for robust geometrically nonlinear analyses in topology optimization

RESEARCH PAPER

Abstract

Geometrically nonlinear structural analyses in conventional density-based Topology Optimization (TO) may fail due to excessive deformation, concerning in particular compression in low stiffness parts (void) of the domain. This limits the application of TO in the field of realistic large deflection mechanisms, actuators and multi-stable structures.

This paper investigates the source of the instabilities that may be encountered using the conventional strategy to scale the stiffness of finite elements using the density (e.g. SIMP). Based on the findings, we propose a new design interpolation, called Element Deformation Scaling (EDS), to obtain more robust structural analyses for geometrically nonlinear TO. Instead of scaling the stiffness, EDS scales the local internal displacements, and therefore, the deformation, in a low-density finite element. This ensures that, even for extremely deformed finite elements, the internal displacements remain in the range of applicability of the material model and finite element description.

The effectiveness of the proposed method is compared with the conventional approach (e.g. SIMP) and the Element Connectivity Parameterization (ECP) method using several numerical experiments using path-following techniques. The proposed method, EDS, is demonstrated to lead to more robust structural analyses than the other approaches. However, EDS still has limitations. These limitations are discussed in detail.

Keywords

Topology optimization Robust analyses Geometrical nonlinearities Material interpolation Design interpolation Element deformation scaling (EDS) 

References

  1. Allaire G, Bonnetier E, Francfort G, Jouve F (1997) Shape optimization by the homogenization method. Numer Math 76(1):27–68MathSciNetCrossRefMATHGoogle Scholar
  2. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetCrossRefMATHGoogle Scholar
  3. Bathe KJ (1996) Finite element procedures vol 2. Prentice hall Englewood Cliffs, NJGoogle Scholar
  4. Bellini PX, Chulya A (1987) An improved automatic incremental algorithm for the efficient solution of nonlinear finite element equations. Comput Struct 26(1):99–110CrossRefMATHGoogle Scholar
  5. Belytschko T, Xiao SP, Parimi C (2003) Topology optimization with implicit functions and regularization. Int J Numer Methods Eng 57(8):1177–1196CrossRefMATHGoogle Scholar
  6. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202CrossRefGoogle Scholar
  7. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224CrossRefGoogle Scholar
  8. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer Verlag, Berl HeidelbergGoogle Scholar
  9. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26-27):3443–3459CrossRefMATHGoogle Scholar
  10. Bruns TE, Tortorelli DA (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57(10):1413–1430CrossRefMATHGoogle Scholar
  11. Buhl T, Pedersen CBW, Sigmund O (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscip Optim 19(2):93–104CrossRefGoogle Scholar
  12. Crisfield MA (1991) Nonlinear finite element analysis of solids and structures. Volume 1:. Essentials. Wiley, New York NY (United States)Google Scholar
  13. Du Y, Chen L, Luo Z (2008) Topology synthesis of geometrically nonlinear compliant mechanisms using meshless methods. Acta Mech Solida Sin 21(1):51–61CrossRefGoogle Scholar
  14. Geers MGD (1999) Enhanced solution control for physically and geometrically non-linear problems. Part I–the subplane control approach. Int J Numer Methods Eng 46(2):177–204MathSciNetCrossRefMATHGoogle Scholar
  15. Gurtin ME (1981) An introduction to continuum mechanics, vol 158. Academic PrGoogle Scholar
  16. Horn BKP (1987) Closed-form solution of absolute orientation using unit quaternions. J Opt Soc Am A 4(4):629–642MathSciNetCrossRefGoogle Scholar
  17. Langelaar M (2006) Design optimization of shape memory alloy structures. Delft University of Technology, PhD thesisGoogle Scholar
  18. Langelaar M, Maute K, Van Dijk NP, Van Keulen F (2010) Investigation of instabilities arising in internal element connectivity parameterization , In: 2 nd International Conference on Engineering Optimization. Lisboa, PortugalGoogle Scholar
  19. Langelaar M, Yoon GH, Kim YY, Van Keulen F (2011) Topology optimization of planar shape memory alloy thermal actuators using element connectivity parameterization. Int J Numer Methods Eng 88(9):817–840CrossRefMATHGoogle Scholar
  20. Maute K, Ramm E (1995) Adaptive topology optimization. Struct Multidiscip Optim 10(2):100–112CrossRefGoogle Scholar
  21. Maute K, Schwarz S, Ramm E (1998) Adaptive topology optimization of elastoplastic structures. Struct Multidiscip Optim 15(2): 81–91CrossRefGoogle Scholar
  22. Nelson RB (1976) Simplified calculation of eigenvector derivatives. AIAA J 14(9):1201–1205MathSciNetCrossRefMATHGoogle Scholar
  23. Pajot JM, Maute K (2006) Topology optimization of geometrically nonlinear structures including thermo-mechanical coupling. University of Colorado, PhD thesisGoogle Scholar
  24. Ragon SA, Gürdal Z, Watson LT (2002) A comparison of three algorithms for tracing nonlinear equilibrium paths of structural systems. IntJSolids Struct 39(3):689–698CrossRefMATHGoogle Scholar
  25. Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Multidiscip Optim 4(3):250–252CrossRefGoogle Scholar
  26. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124CrossRefGoogle Scholar
  27. Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318CrossRefMATHGoogle Scholar
  28. Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetCrossRefMATHGoogle Scholar
  29. Van Miegroet L, Duysinx P (2007) Stress concentration minimization of 2D filets using X-FEM and level set description. Struct Multidiscip Optim 33(4):425–438CrossRefGoogle Scholar
  30. Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1-2):227–246CrossRefMATHGoogle Scholar
  31. Yoon GH, Kim YY (2005a) Element connectivity parameterization for topology optimization of geometrically nonlinear structures. Int J Solids Struct 42(7):1983–2009CrossRefMATHGoogle Scholar
  32. Yoon GH, Kim YY (2005b) The element connectivity parameterization formulation for the topology design optimization of multiphysics systems. Int J Numer Methods Eng 64(12):1649–1677CrossRefMATHGoogle Scholar
  33. Yoon GH, Kim YY (2007) Topology optimization of material-nonlinear continuum structures by the element connectivity parameterization. Int J Numer Methods Eng 69(10):2196–2218MathSciNetCrossRefMATHGoogle Scholar
  34. Yoon GH, Kim YY, Langelaar M, Van Keulen F (2008) Theoretical aspects of the intemal element connectivity parameterization approach for topology optimization. Int J Numer Methods Eng 76(6):775–797MathSciNetCrossRefMATHGoogle Scholar
  35. Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: Topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1-3):309–336CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • N. P. van Dijk
    • 1
  • M. Langelaar
    • 2
  • F. van Keulen
    • 2
  1. 1.Uppsala UniversityUppsalaSweden
  2. 2.Delft University of TechnologyCD DelftThe Netherlands

Personalised recommendations