Structural and Multidisciplinary Optimization

, Volume 25, Issue 3, pp 199–214 | Cite as

Topology optimization of frame structures with flexible joints

  • H. Fredricson
  • T. Johansen
  • A. Klarbring
  • J. Petersson
Research paper

Abstract

A method for structural topology optimization of frame structures with flexible joints is presented. A typical frame structure is a set of beams and joints assembled to carry an applied load. The problem considered in this paper is to find the stiffest frame for a given mass. By introducing design variables for beams and joints, a mass distribution for optimal structural stiffness can be found. Each beam can have several design variables connected to its cross section. One of these is an area-type design variable which is used to represent the global size of the beam. The other design variables are of length ratio type, controlling the cross section of the beam. Joints are flexible elements connecting the beams in the structure. Each joint has stiffness properties and a mass. A framework for modelling these stiffnesses is presented and design variables for joints are introduced. We prove a theorem which can be interpreted as the fact that the removal of structural elements, e.g. joints or beams, can be modelled by a small strictly positive material amount assigned to the element. This is needed for the computations of sensitivities used in the applied gradient based iterative method. Both two and three dimensional problems, as well as multiple load cases and multiple mass constraints, are treated.

Keywords

topology optimization frame structures joint modelling flexible joints 

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References

  1. 1.
    Achtziger, W. 1998: Multiple-load truss topology and sizing optimization: Some properties of minimax compliance. J. Optim. Theory Appl. 98, 255–280 Google Scholar
  2. 2.
    Achtziger, W. 1999: Local stability of trusses in the context of topology optimization Part I: Exact modelling. Struct. Optim. 17, 235–246 Google Scholar
  3. 3.
    Argyris, J.H.; Balmer, H.; Doltsinis, J.St.; Dunne, P.C.; Haase, M.; Kleiber, M.; Malejannakis, G.A.; Mlejnek, H.-P.; Müller, M.; Scharpf, D.W. 1979: Finite element method – The natural approach. Comput. Methods Appl. Mech. Eng. 17/18, 1–106 Google Scholar
  4. 4.
    Aubin, J.-P.; Frankowska, H. 1990: Set-Valued Analysis. Boston: Birkhäuser Google Scholar
  5. 5.
    Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M. 1993: Nonlinear Programming: Theory and Algorithms. New York: John Wiley and Sons Google Scholar
  6. 6.
    Bendsøe, M.P. 1995: Optimization of Structural Topology, Shape, and Material. Berlin, Heidelberg, New York: Springer-Verlag Google Scholar
  7. 7.
    Bendsøe, M.P.; Ben-Tal, A.; Zowe, J. 1994: Optimization methods for truss geometry and topology design. Struct. Optim. 7, 141–159 Google Scholar
  8. 8.
    Cameron, T.M.; Thirunavukarasu, A.C.; El-Sayed, M.E.M. 2000: Optimization of frame structures with flexible joints. Struct. Multidisc. Optim. 19, 204–213 Google Scholar
  9. 9.
    Cheng, G.D.; Guo, X. 1997: ε-relaxed approach in structural topology optimization. Struct. Optim. 13, 258–266 Google Scholar
  10. 10.
    Daniel, J.W. 1973: Stability of the solution of definite quadratic programs. Math. Program. 5, 41–53 Google Scholar
  11. 11.
    Eschenauer, H.A.; Olhoff, N. 2001: Topology optimization of continuum structures: A review. Appl. Mech. Rev. 54, 331–390 Google Scholar
  12. 12.
    Fredricson, H. 2002: Optimization methods for vehicle body structures. Int. Rep. LIU-TEK-LIC, 2002: 24 Dept. Mech. Eng., Linköping University Google Scholar
  13. 13.
    Jivotovski, G. 2000: A gradient based heuristic algorithm and its application to discrete optimization of bar structures. Struct. Multidisc. Optim. 19, 237–248 Google Scholar
  14. 14.
    Jonsson, D.; Johansen, T. 1997: Conceptual design of a gearbox housing using topology optimization. Thesis work carried out at Volvo Technological Development, Int. Rep. LiTH–IKP–EX–1447, Dept. Mech. Eng., Linköping University Google Scholar
  15. 15.
    Memari, A.M.; Madhkan, M. 1999: Optimal design of steel frames subject to gravity and seismic codes’ prescribed lateral forces. Struct. Optim. 18, 56–66 Google Scholar
  16. 16.
    Nishigaki, H.; Nishiwaki, S.; Amago, T.; Kojima, Y.; Kikuchi, N. 2001: First order analysis – new CAE tools for automotive body designers. SAE Paper, No. 2001-01-0768 Google Scholar
  17. 17.
    Sergeyev, O.; Pedersen, P. 1996: On design of joint positions for minimum mass 3D frames. Struct. Optim. 11, 95–101 Google Scholar
  18. 18.
    Sergeyev, O.; Mroz, Z. 1998: Optimal joint positions and stiffness distribution for minimum mass frames with damping constraints. Struct. Multidisc. Optim. 16, 231–245 Google Scholar
  19. 19.
    Svanberg, K. 1987: The method of moving asymptotes – a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 359–373Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  • H. Fredricson
    • 1
  • T. Johansen
    • 2
  • A. Klarbring
    • 3
  • J. Petersson
    • 3
  1. 1.Volvo Car Corp.GöteborgSweden
  2. 2.Volvo Technology Corp.GöteborgSweden
  3. 3.Department of Mechanical EngineeringLinköping UniversityLinköpingSweden

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