Mathematical Methods of Operations Research

, Volume 68, Issue 2, pp 313–331 | Cite as

Integer linear programming models for topology optimization in sheet metal design

  • Armin FügenschuhEmail author
  • Marzena Fügenschuh
Original Article


The process of designing new industrial products is in many cases solely based on the intuition and experience of the responsible design engineer. The aid of computers is restricted to visualization and manual manipulation tools. We demonstrate that the design process for conduits, which are made out of sheet metal plates, can be supported by mathematical optimization models and solution techniques, leading to challenging optimization problems. The design goal is to find a topology that consists of several channels with a given cross section area using a minimum amount of sheet metal and, at the same time, maximizing its stiffness. We consider a mixed integer linear programming model to describe the topology of two dimensional slices of a three dimensional sheet metal product. We give different model formulations, based on cuts and on multicommodity flows. Numerical results for various test instances are presented.


Integer programming Topology optimization Discretization Multicommodity flow Separation Sheet metal products 


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  1. Achterberg T (2004) SCIP—a framework to constraint and mixed integer programming. ZIB Report 04-19, BerlinGoogle Scholar
  2. Applegate DL, Bixby RE, Chvatal V, Cook WJ (2006) The traveling salesman problem: a computational study. Princeton University Press, 606 ppGoogle Scholar
  3. 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
  4. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin, p 370Google Scholar
  5. Birkhofer H, Fügenschuh A, Günther U, Junglas D, Martin A, Sauer T, Ulbrich S, Wäldele M, Walter S (2006) Topology- and shape-optimization of branched sheet metal products. In: Haasis H, Kopfer H, Schönberger J(eds) Operations research proceedings 2005. Springer, Berlin, pp 327–336CrossRefGoogle Scholar
  6. Dobbs W, Felton LP (1969) Optimization of truss geometry. Proc ASCE 95: 2105–2119Google Scholar
  7. Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. Journal de Mechanique 3: 25–52Google Scholar
  8. Ehrgott M (2000) Multicriteria optimization. Springer, BerlinzbMATHGoogle Scholar
  9. Groche P, Breitenbach Gv, Jöckel M, Zettler A (2003) New tooling concepts for the future roll forming applications. ICIT Conference. Bled, YugoslaviaGoogle Scholar
  10. Gross D, Hauger W, Schnell W (2005) Technische Mechanik, Bd. 2: Elastostatik. Springer, Berlin (In German)Google Scholar
  11. Hochbaum D (2008) The pseudoflow algorithm: a new algorithm for the maximum flow problem. Oper Res (to appear)Google Scholar
  12. Kang Z, Zhang C, Cheng G (2005) Structural topology optimization considering mass moment of inertia. In: Proc. of the 6th world congresses of structural and multidisciplinary optimization, Rio de Janeiro, Paper No. 4611Google Scholar
  13. Kessel S, Fröng D (1998) Technische Mechanik—technical mechanics. Teubner Verlag, StuttgartGoogle Scholar
  14. Kohn RV, Strang G (1986a) Optimal design in elasticity and plasticity. Numer Methods Eng 22: 183–188zbMATHCrossRefMathSciNetGoogle Scholar
  15. Kohn RV, Strang G (1986b) Optimal design and relaxation of variational problems. In: Communications in pure and applied mathematics, vol 39, pp 1–25 (Part I), pp 139–182 (Part II), pp 353–357 (Part III)Google Scholar
  16. Kühhorn A, Silber G (2000) Technische Mechanik für Ingenieure. Hüthig Verlag, Heidelberg (In German)Google Scholar
  17. Meyberg K, Vachenauer P (1997) Höhere Mathematik 1. Springer, Berlin, Heidelberg (In German)zbMATHGoogle Scholar
  18. Michell AGM (1904) The limits of economy of material in frame structures. Philos Mag 8(6): 589–597Google Scholar
  19. Nemhauser G, Wolsey L (1998) Integer programming and combinatorial optimization. Wiley, PhiladelphiaGoogle Scholar
  20. Rozvany GIN (2001) Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Struct Multidiscip Optim 21(2): 90–108CrossRefMathSciNetGoogle Scholar
  21. Stolpe M (2007) On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs. Optim Eng 8(2): 163–192CrossRefMathSciNetGoogle Scholar
  22. Stolpe M, Svanberg K (2003) Modeling topology optimization problems as linear mixed 0-1 programs. Int J Numer Methods Eng 57(5): 723–739zbMATHCrossRefMathSciNetGoogle Scholar
  23. Wunderling R (1996) Paralleler und Objektorientierter Simplex-Algorithmus. ZIB Report TR 96-09, BerlinzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Darmstadt University of TechnologyDarmstadtGermany

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