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Topology Optimization of Discrete Structures

An Introduction in View of Computational and Nonsmooth Aspects

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Topology Optimization in Structural Mechanics

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 374))

Abstract

We discuss standard problems of topology optimization of discrete structures. This paper is an attempt to provide an (almost) self-contained introduction and stresses the techniques of reformulating problems and mathematical tools needed for a successful numerical treatment.

First, relations between several classical formulations of single load problems are shown in order to illustrate the mathematical techniques used, such as minimax-Theorems, duality etc. Numerical approaches are discussed. The outlined concept is then generalized to the multiple load case where minimization of compliance in the sense of worst case design is considered. We end up with displacement based nonsmooth optimization formulations. In the fourth section the problem of simultaneous optimization of topology and geometry is considered. We illustrate its mathematical treatment as a bilevel problem which again uses nonsmooth optimization. In each section a few numerical examples show the applicability of the proposed approaches.

Since nonsmooth analysis is non-standard in structural optimization, we close with a short introduction to important terms up to the concept of an algorithm.

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References

  1. Dorn, W., Gomory, R. and Greenberg, M.: Automatic Design of Optimal Structures, J. de Mechanique, 3 (1996), 25–52.

    Google Scholar 

  2. Ringertz, U.: A Branch and Bound Algorithm for Topology Optimization of Truss Structures, Eng. Opt., 10 (1986), 111–124.

    Article  Google Scholar 

  3. Lewinski, T., Zhou, M. and Rozvany, G.I.N.: Extended Exact Solutions for Least-Weight Truss Layouts — Part I and Part II, Int. J. Mech. Sci., 36 (1994), 375–419.

    Article  MATH  Google Scholar 

  4. Taylor, J.E.: Truss Topology Design for Elastic/Softening Materials, in: loc. cit. [62], 451–467.

    Google Scholar 

  5. Pedersen, P.: Topology Optimization of Three-dimensional Trusses, in: loc. cit. [62], 19–30.

    Google Scholar 

  6. Topping, B.H.V.: Topology Design of Discrete Structures, in: loc. cit. [62], 517–534.

    Google Scholar 

  7. Zhou, M. and Rozvany, G.I.N.: Iterative COC methods, in: loc. cit. [63], 27–75.

    Google Scholar 

  8. Ben-Tal, A. and Bendsoe, M.P.: A New Method for Optimal Truss Topology Design, SIAM J. Opt., 3 (1993), 322–358.

    MathSciNet  MATH  Google Scholar 

  9. Bendsoe, M.P.: Methods for Optimization of Structural Topology, Shape and Material, Springer, Berlin, New York 1995.

    Google Scholar 

  10. Kirsch, U.: Optimal Topologies of Truss Structures, Comp. Meth. Appl. Mech. Eng., 72 (1989), 15–28.

    Article  MathSciNet  MATH  Google Scholar 

  11. Kirsch, U.: Fundamental Properties of Optimal Topologies, in: loc. cit. [62], 3–18.

    Google Scholar 

  12. Rozvany, G.I.N., Bendsoe, M.P. and Kirsch, U.: Layout Optimization of Structures, Appl. Mech. Rev., Vol. 48, No. 2 (1995).

    Google Scholar 

  13. Haug, E.J. and Arora, J.S.: Applied Optimal Design, Wiley & Sons, New York 1979.

    Google Scholar 

  14. Smith, O. da Silva: Generation of 3D-Ground Structures for Truss Topology Optimization, in: loc. cit. [64], 147–152.

    Google Scholar 

  15. Bazaraa, M.S., Sherali, H.D. and Shetty, C.M.: Nonlinear Programming, 2nd Edition, Wiley & Sons, New York 1993.

    MATH  Google Scholar 

  16. Svanberg, K.: On Local and Global Minima in Structural Optimization, in: New Directions in Optimum Structural Design (Eds. A. Atrek, R.H. Gallagher, K.M. Ragsdell O.C. Zienkiewicz), Wiley & Sons, New York, 1984, 327–341.

    Google Scholar 

  17. Fleury, C.: Sequential Convex Programming for Structural Optimization Problems, in: loc. cit. [63], 531–553.

    Google Scholar 

  18. Nguyen, V.H., Strodiot, J.J. and Fleury, C.: A Mathematical Convergence Analysis of the Convex Linearization Method for Engineering Design Problems, Eng. Opt., 11 (1987), 195–216.

    Article  Google Scholar 

  19. Fleury, C.: CONLIN: An Efficient Dual Optimizer Based on Convex Approximation Concepts, Struct. Opt., 1 (1989), 81–89.

    Article  Google Scholar 

  20. Svanberg, K.: Method of Moving Asymptotes — A New Method for Structural Optimization, Int. J. Num. Meth. Eng., 24 (1987), 359–373.

    Article  MathSciNet  MATH  Google Scholar 

  21. Svanberg, K.: A Globally Convergent Version of MMA Without Linesearch, in: loc. cit. [64], 9–16.

    Google Scholar 

  22. Achtziger, W., Bendsoe, M., Ben-Tal, A. and Zowe, J.: Equivalent Displacement Based Formulations for Maximum Strength Truss Topology Design, Impact of Computing in Science and Engineering, 4 (1992), 315–345.

    Article  MathSciNet  MATH  Google Scholar 

  23. Svanberg, K.: Optimal Truss Sizing Based on Explicit Taylor Series Expansion, Struct. Opt., 2 (1990), 153–162.

    Article  Google Scholar 

  24. Svanberg, K.: Global Convergence of the Stress Ratio Method for Truss Sizing, Struct. Opt., 8 (1994), 60–68.

    Article  Google Scholar 

  25. Taylor, J.E.: Maximum Strength Elastic Structural Design, Proc. ASCE, 95 (1969), 653–663.

    Google Scholar 

  26. Taylor, J.E. and Rossow, M.P.: Optimal Truss Design Based on an Algorithm Using Optimality Criteria, Int. J. Solids Struct., 13 (1977), 913–923.

    Article  Google Scholar 

  27. Cox, H.L.: The Design of Structures for Least Weight, Pergamon Press, Oxford 1965.

    Google Scholar 

  28. Hemp, W.S.: Optimum Structures, Clarendon Press, Oxford, U.K 1973.

    Google Scholar 

  29. Achtziger, W.: Optimierung von einfach and mehrfach belasteten Stabwerken, Bayreuther Mathematische Schriften, 46 (1993), in German.

    Google Scholar 

  30. Oberndorfer, J.M., Achtziger, W. and Hörnlein, H.R.E.M.: Two Approaches for Truss Topology Optimization: A Comparison for Practical Use, Struct. Opt., 11 (1996), 137144.

    Google Scholar 

  31. Sankaranaryanan, S., Haftka, R. and Kapania, R.K.: Truss Topology Optimization with Stress and Displacement Constraints, in: loc. cit. [62], 71–78.

    Google Scholar 

  32. Cheng, G. and Jiang, Z.: Study on Topology Optimization with Stress Constraints, Eng. Opt., 20 (1992), 129–148.

    Article  Google Scholar 

  33. Ringertz, U.: Newton Methods for Structural Optimization, Technical Report, No. 8819 (1988), Department of Leightweight Structures, The Royal Institute of Technology, Stockholm, Sweden.

    Google Scholar 

  34. Hörnlein, H.R.E.M.: Ein Algorithmus zur Strukturoptimierung von Fachwerkkonstruktionen, Diplomarbeit, Ludwigs-Maximilian-Universität, München, Germany 1979, in German.

    Google Scholar 

  35. Pedersen, P.: On the Minimum Mass Layout of Trusses, Symposium on Structural Optimization, AGARD-CP-36–70 (1970), AGARD Conf. Proc.

    Google Scholar 

  36. Smith, O. da Silva: Topology Optimization of Trusses with Local Stability Constraints and Multiple Loading Conditions, Technical Report, No. 517 (1996), Danish Center for Applied Mathematics and Mechanics (DCAMM), Technical University of Denmark, Lyngby, Denmark.

    Google Scholar 

  37. Achtziger, W.: Truss Topology Optimization Including Bar Properties Different for Tension and Compression, Struct. Opt., 12 (1996), 63–74.

    Article  Google Scholar 

  38. Petersson, J. and Klarbring, A.: Saddle Point Approach to Stiffness Optimization of Discrete Structures Including Unilateral Contact, Control and Cybernetics, 3 (1994), 461–479.

    MathSciNet  Google Scholar 

  39. Klarbring, A., Petersson, J. and Rönnquist, M.: Truss Topology Optimization Involving Unilateral Contact, J. Opt. Theory Appl., 87 (1) (1995).

    Google Scholar 

  40. Koevara, M., Zibulevsky, M. and Zowe, J.: Mechanical Design Problems with Unilateral Contact, Technical Report, No. 190 (1996), Institute of Applied Mathematics, University of Erlangen-Nuremberg, Germany, submitted.

    Google Scholar 

  41. Wilkinson, J.H.: The Algebraic Eigenvalue Problem, Clarendon Press, Oxford 1965.

    MATH  Google Scholar 

  42. Rockafellar, R.T.: Convex Analysis, Princeton University Press, Princeton, N.J. 1970.

    Google Scholar 

  43. Achtziger, W.: Multiple Load Truss Topology and Sizing Optimization: Some Properties of Minimax Compliance, Technical Report 1994–41 (1994), Institute of Mathematics, The Technical University of Denmark, submitted.

    Google Scholar 

  44. Jarre, F., Koevara, M. and Zowe, J.: Interior Point Methods for Mechanical Design Problems, Technical Report, No. 173 (1996), Institute of Applied Mathematics, University of Erlangen-Nuremberg, Germany, submitted.

    Google Scholar 

  45. Zibulevsky, M. and Ben-Tal, A.: On a New Class of Augmented Lagrangian Methods for Large Scale Convex Programming Problems, Technical Report, 2/93 (1993), Opt. Lab., Technion (Israel Inst. of Technology ), Haifa, Israel.

    Google Scholar 

  46. Ben-Tal, A. and Nemirovskii, A.: Potential Reduction Polynomial Time Method for Truss Topology Design, SIAM J. Opt., 4 (1994), 596–612.

    MathSciNet  MATH  Google Scholar 

  47. Ben-Tal, A. and Roth, G.: A Truncated log Barrier Algorithm for Large-Scale Convex Programming and Minmax Problems: Implementation and Computational Results, Optim. Meth. Software, 6 (1996), 283–312.

    Article  Google Scholar 

  48. Murty, K.G.: Linear Programming, Wiley & Sons, New York 1983.

    Google Scholar 

  49. Levy, R.: Fixed Point Theory and Structural Optimization, Eng. Opt., 17 (1991), 251–261.

    Article  Google Scholar 

  50. Maxwell, J.C.: On Reciprocal Figures, Frames and Diagrams of Forces, Scientific Papers, 2 (1890), Cambridge Univ. Press, Cambridge, U.K., 175–177.

    Google Scholar 

  51. Michell, A.G.M.: The Limits of Economy of Material in Frame Structures, Philosophical Magazine, Series 6, Vol. 8 (1904), 589–597.

    Google Scholar 

  52. Achtziger, W.: Minimax Compliance Truss Topology Subject to Multiple Loadings, in: loc. cit. [62], 43–54.

    Google Scholar 

  53. Achtziger, W.: Multiple Load Truss Optimization: Properties of Minimax Compliance and Two Nonsmooth Approaches, in: loc. cit. [64], 123–128.

    Google Scholar 

  54. Gauvin, J. and Dubeau, F.: Differential Properties of the Marginal Function in Mathematical Programming, Math. Prog. Study, 19 (1982), 101–119.

    Article  MathSciNet  Google Scholar 

  55. Schramm, H. and Zowe, J.: A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, SIAM J. Opt., Vol. 2 (1992), 121–152.

    MathSciNet  MATH  Google Scholar 

  56. Ben-Tal, A., Kocvara, M. and Zowe, J.: Two Nonsmooth Methods for Simultaneous Geometry and Topology Design of Trusses, in: loc. cit. [62], 31–42.

    Google Scholar 

  57. Kocvara, M. and Zowe, J.: How To Optimize Mechanical Structures Simultaneously with Respect to Topology and Geometry, in: loc. cit. [64], 135–140.

    Google Scholar 

  58. Kocvara, M. and Zowe, J.: How Mathematics Can Help in Design of Mechanical Structures, in: Numerical Analysis (Eds. D. Griffiths and G. Watson), Longman Scientific and Technical, 1996, 76–93.

    Google Scholar 

  59. Clarke, F.H.: Optimization and Nonsmooth Analysis, Wiley & Sons, New York 1983.

    Google Scholar 

  60. Zowe, J.: Nondifferentiable Optimization, in: Computational Mathematical Programming (Ed. K. Schittkowski ), Springer, Berlin, 1985, 323–359.

    Chapter  Google Scholar 

  61. Lemaréchal, C.: Nondifferentiable Optimization, in: Handbooks in Operations Research and Management Science (Eds. G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd), Vol. 1, Elsevier Science Publishers, North-Holland, 1989, 529–572.

    Google Scholar 

  62. Bendsoe, M.P. and Mota Soares, C.A. (Eds.): Topology Optimization of Structures, Kluwer Academic Publishers, Dordrecht 1993.

    Google Scholar 

  63. Rozvany, G.I.N. (Ed.): Optimization of Large Structural Systems, NATO ASI Series, Kluwer Academic Publishers, Dordrecht, The Netherlands 1993.

    Google Scholar 

  64. Olhoff, N. and Rozvany, G.I.N. (Eds.): WCSMO-1, First World Congress of Structural and Multidisciplinary Optimization, Pergamon, Elsevier, Oxford, U.K. 1995.

    Google Scholar 

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Authors and Affiliations

  1. University of Erlangen-Nuremberg, Erlangen, Germany

    W. Achtziger

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  1. W. Achtziger
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Editors and Affiliations

  1. Essen University, Germany

    G. I. N. Rozvany

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© 1997 Springer-Verlag Wien

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Achtziger, W. (1997). Topology Optimization of Discrete Structures. In: Rozvany, G.I.N. (eds) Topology Optimization in Structural Mechanics. International Centre for Mechanical Sciences, vol 374. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2566-3_2

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  • DOI: https://doi.org/10.1007/978-3-7091-2566-3_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82907-3

  • Online ISBN: 978-3-7091-2566-3

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