Skip to main content
SpringerLink
Log in

A simple alternative formulation for structural optimisation with dynamic and buckling objectives

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

Structural topology optimisation has mainly been applied to strength and stiffness objectives, due to the ease of calculating the sensitivities for such problems. In contrast, dynamic and buckling objectives require time consuming central difference schemes, or inefficient non-gradient algorithms, for calculation of the sensitivities. Further, soft-kill algorithms suffer from numerous numerical issues, such as localised artificial modes and mode switching. This has resulted in little focus on structural topology optimisation for dynamic and buckling objectives. In this work it is found that nominal stress contours can be derived from applying the vibration and buckling mode shapes as displacement fields, defined as the dynamic and buckling von Mises stress, respectively. This paper shows that there is an equivalence between the dynamic von Mises stress and the frequency sensitivity numbers for element removal and addition in bidirectional evolutionary structural optimisation. Likewise, it was found that the contours of buckling von Mises stress and buckling sensitivity numbers are analogous; therefore, an equivalence is shown for element removal and addition. The examples demonstrate consistent resulting topologies from the two different formulations for both dynamic and buckling criteria. This article aims to develop a simple alternative, based on visual correlation with a mathematical verification, for topology optimisation with dynamic and buckling criteria.

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
Fig. 20
Fig. 21

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  • Bendsoe MP, Sigmund O (2004) Topology optimization: theory methods and applications. Springer, Germany

    Book  MATH  Google Scholar 

  • Browne PA, Budd C, Gould NIM, Kim HA, Scott JA (2012) A fast method for binary programming using first-order derivatives, with application to topology optimization with buckling constraints. Int J Numer Methods Eng 92:1026–1043

    Article  MathSciNet  MATH  Google Scholar 

  • Cui C, Ohmori H, Sasaki M (2003) Computational morphogenesis of 3D structures by extended ESO method. J Inter Assoc Shell Spatial Struct 44:51–61

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Du J, Olhoff N (2007) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struc Multidisc Optim 34:91–110

    Article  MathSciNet  MATH  Google Scholar 

  • Gong S, Cheng M, Zhang J, He R (2012) Study on modal topology optimization method of continuum structure based on EFG method. Journal Comput Meth 9:1–12

    MathSciNet  Google Scholar 

  • Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43:1039–1049

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Huang X, Zuo ZH, Xie YM (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88:357–364

    Article  Google Scholar 

  • Li Q, Steven GP, Xie YM (1999) On equivalence between stress criterion and stiffness criterion in evolutionary structural optimization. Struct Optim 18:67–73

    Article  Google Scholar 

  • Lindgaard E, Dahl J (2013) On compliance and buckling objective functions in topology optimization of snap-through problems. Struct Multidisc Optim 47:409–421

    Article  MathSciNet  MATH  Google Scholar 

  • Lindgaard E, Lund E (2010) Nonlinear buckling optimization of composite structures. Computer Methods Appl Mech Eng 199:2319–2330

    Article  MathSciNet  MATH  Google Scholar 

  • Lindgaard E, Lund E (2011) A unified approach to nonlinear buckling optimisation of composite structures. Comput Struct 89: 357–370

    Article  MATH  Google Scholar 

  • Ma ZD, Cheng HC, Kikuchi N (1994) Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization. Comput Systems Eng 5:77–89

    Article  Google Scholar 

  • Manickarajah D, Xie Y M, Steven GP (1995) A simple method for the optimisation of columns, frames and plates against buckling. In: Kitipornchai S, Hancock GJ, Bradford MA (eds) Structural stability and design. Balkema, Rotterdam, pp 175–180

  • Manickarajah D, Xie YM, Steven GP (1998) An evolutionary method for optimization of plate buckling resistance. Finite Elem Anal Des 29:205–230

    Article  MATH  Google Scholar 

  • Min S, Kikuchi N (1997) Optimal reinforcement design of structures under the buckling load using homogenization design method. Struct Eng Mech 5:565–576

    Article  Google Scholar 

  • Munk DJ, Vio GA, Steven GP (2016a) A new formulation for the BESO method to solve the Zhou-Rozvany problem, (under review). Struct Multidiscip Optim

  • Munk DJ, Vio GA, Steven GP (2016b) A novel moving iso-surface threshold technique for the vibration optimisation of structures subjected to dynamic loading. AIAA J. (accepted)

  • Munk DJ, Vio GA, Steven GP (2015b) Topology and shape optimization methods using evolutionary algorithms: a review. Struc Multidisc Optim 52:613–631

  • Neves MM, Rodrigues H, Guedes JM (1995) Generalized topology design of structures with a buckling load criterion. Struct Optim 10:71–78

    Article  Google Scholar 

  • Niu F, Xu S, Cheng G (2011) A general formulation of structural topology optimization for maximizing structural stiffness. Struct Multidiscip Optim 43:561–572

    Article  Google Scholar 

  • Ohmori H, Futai H, Iijima T, Muto A, Hasegawa H (2005) Application of computational morphogenesis to structural design. In: Proceedings of frontiers of computational sciences symposium, pp 45–52

  • Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidisc Optim 20:2–11

  • Querin OM, Steven GP, Xie YM (1998) Evolutionary structural optimization (ESO) using a bidirectional algorithm. Eng Comput 15:1031–1048

    Article  MATH  Google Scholar 

  • Rozvany G (2009) A critical review of established methods of structural topology optimization. Struct Multidisc Optim 37:217–237

    Article  MathSciNet  MATH  Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struc Multidisc Optim 48:1031–1055

    Article  Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75

    Article  Google Scholar 

  • Simitses GJ, Kamat MP, Smith CV (1973) Strongest column by the finite element displacement method. AIAA J 116:1231–1232

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, Germany

    Book  MATH  Google Scholar 

  • Zhao C, Steven GP, Xie XM (1996) Evolutionary natural frequency optimization of thin plate bending vibration problems. Struct Optim 11:244–251

    Article  Google Scholar 

  • Zhou M (2004) Topology optimization for shell structures with linear buckling responses. Paper presented at the WCCM VI, Beijing, China, 5–10 September 2004, pp 795–800

Download references

Author information

Authors and Affiliations

  1. AMME, The University of Sydney, Sydney, NSW 2006, Australia

    David J. Munk, Gareth A. Vio & Grant P. Steven

Authors
  1. David J. Munk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gareth A. Vio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Grant P. Steven
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David J. Munk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Munk, D.J., Vio, G.A. & Steven, G.P. A simple alternative formulation for structural optimisation with dynamic and buckling objectives. Struct Multidisc Optim 55, 969–986 (2017). https://doi.org/10.1007/s00158-016-1544-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-016-1544-9

Keywords

Search

Navigation