Skip to main content
SpringerLink
Log in

Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume. The design variables describe the cross sectional areas of the bars. This class of problems is well-studied for continuous bar areas. We consider here the difficult situation that the truss must be built from pre-produced bars with given areas. This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem.

In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem. The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained. We focus on the implementation details but also establish finite convergence of the branch-and-bound method. The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I. The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function. This is one reason for making the resulting branch-and-bound method very efficient. The paper closes with several large-scale numerical examples. These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization.

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. Achtziger, W.: Topology optimization of discrete structures: An introduction in view of computational and nonsmooth aspects. In: Rozvany, G.I.N. (ed.) Topology Optimization in Structural Mechanics, pp. 57–100. Springer, Vienna (1997)

    Google Scholar 

  2. Achtziger, W., Stolpe, M.: Global optimization of truss topology with discrete bar areas—Part I: Theory of relaxed problems. Comput. Optim. Appl. (2007). doi: 10.1007/s10589-007-9138-5

    MATH  Google Scholar 

  3. Achtziger, W., Stolpe, M.: Truss topology optimization with discrete design variables—guaranteed global optimality and benchmark examples. Struct. Multidiscip. Optim. 34(1), 1–20 (2007)

    Article  MathSciNet  Google Scholar 

  4. Ben-Tal, A., Nemirovski, A.: Potential reduction polynomial time method for truss topology design. SIAM J. Optim. 4(3), 596–612 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Tal, A., Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim. 7(4), 991–1016 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben-Tal, A., Nemirovski, A.: Structural design. In: Handbook of Semidefinite Programming, pp. 443–467. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

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

    Google Scholar 

  8. Bollapragada, S., Ghattas, O., Hooker, J.N.: Optimal design of truss structures by logical-based branch and cut. Oper. Res. 49(1), 42–51 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12(4), 979–1006 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Groenwold, A.A., Stander, N., Snyman, J.A.: A pseudo-discrete rounding method for structural optimization. Struct. Optim. 11, 218–227 (1996)

    Article  Google Scholar 

  11. Hajela, P., Lee, E.: Genetic algorithms in truss topology optimization. Int. J. Solids Struct. 32(22), 3341–3357 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1993)

    Google Scholar 

  13. ILOG Inc.: ILOG Cplex 9.0 user’s manual and reference manual. (2004) http://www.ilog.com

  14. Kane, C., Schoenauer, M.: Topological optimum design using genetic algorithms. Control Cybern. 25(5), 1059–1087 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Land, A.H., Doig, A.G.: An automatic method for solving discrete programming problems. Econometrica 28, 497–520 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  16. The MathWorks: Matlab user guides. http://www.mathworks.com (2005)

  17. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, New York (1999)

    MATH  Google Scholar 

  18. Ringertz, U.: A branch and bound algorithm for topology optimization of trusses. Eng. Optim. 10, 111–124 (1986)

    Article  Google Scholar 

  19. Ringertz, U.: On methods for discrete structural optimization. Eng. Optim. 13, 44–64 (1988)

    Google Scholar 

  20. Stolpe, M.: On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs. Optim. Eng. 8, 163–192 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stolpe, M., Kawamoto, A.: Design of planar articulated mechanisms using branch and bound. Math. Program. 103(2), 357–398 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stolpe, M., Svanberg, K.: Modeling topology optimization problems as linear mixed 0–1 programs. Int. J. Numer. Methods Eng. 57(5), 723–739 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, University of Dortmund, FB Mathematics, LS X, 44221, Dortmund, Germany

    Wolfgang Achtziger

  2. Department of Mathematics, Technical University of Denmark, Matematiktorvet Building 303 S, 2800, Kgs. Lyngby, Denmark

    Mathias Stolpe

Authors
  1. Wolfgang Achtziger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mathias Stolpe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wolfgang Achtziger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Achtziger, W., Stolpe, M. Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results. Comput Optim Appl 44, 315–341 (2009). https://doi.org/10.1007/s10589-007-9152-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9152-7

Keywords

Search

Navigation