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On Equivalence Between Optimality Criteria and Projected Gradient Methods with Application to Topology Optimization Problem

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Abstract

The paper demonstrates the equivalence between the optimality criteria (OC) method, initially proposed by Bendsøe and Kikuchi for topology optimization problem, and the projected gradient method. The equivalence is shown using Hestenes definition of Lagrange multipliers. Based on this development, an alternative formulation of the Karush–Kuhn–Tucker (KKT) condition is suggested. Such reformulation has some advantages, which will be also discussed in the paper. For verification purposes the modified algorithm is applied to the design of tension-only structures.

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References

  1. Bendsøe, M.P. and Kikuchi, N., ‘Generating optimal topologies in structural design using a homogenization method,’ Computer Methods in Applied Mechanics and Engineering 71, 1988, 197–224.

    Google Scholar 

  2. Soto, C.A., ‘Applications of structural topology optimization in the automotive industry: Past, present and future,’ in Mang, H.A., Rammerstorfer, F.G. and Eberhardsteiner, J. (eds.), Fifth World Congress on Computational Mechanics, Vienna, Austria, 2002.

  3. Bendsøe, M.P. and Sigmund, O., ‘Topology Optimization,’ Theory, Methods and Applications, Springer, Berlin, 2003.

    Google Scholar 

  4. Maute, K., Topologie- und Formoptimierung von dünnwandigen Tragwerken, Ph.D. Thesis, University of Stuttgart, Stuttgart, Germany, 1998 (in German).

    Google Scholar 

  5. Venkayya, V.B., ‘Generalized optimality criteria method,’ in Kamat, M. (ed.), Structural optimization: status and promise, American Institute of Aeronautics and Astronautics, Washington, DC, 1993.

    Google Scholar 

  6. Haftka, R.T., Elements of Structural Optimization, Kluwer Academic, Dordrecht, The Neatherlands, 1992.

    Google Scholar 

  7. Sigmund, O., Design of materials and structures using topology optimization, Ph.D. Thesis, TU Denmark, 1994.

    Google Scholar 

  8. Luenberger, D., Linear and nonlinear programming, 2nd edn., Addison-Wesley, Reading, MA, 1989.

    Google Scholar 

  9. Gunzburger, M.D., Perspectives in Flow Control and Optimization, SIAM, Philadelphia, 2003.

    Google Scholar 

  10. Hestenes, M.R., Optimization theory. The finite dimensional case, Wiley, New York, 1975.

    Google Scholar 

  11. Rockafellar, R.T., ‘Lagrange multipliers and optimality,’ SIAM Review 35(2), 1993, 183–238.

    Google Scholar 

  12. Bathe, K.-J., Finite element procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996.

    Google Scholar 

  13. Fletcher, R., ‘An exact penalty function for nonlinear programming with inequalities,’ Mathematical Programming 5, 1973, 129–150.

    Google Scholar 

  14. Meiss, K., ‘New developments in engineering application of structural optimization,’ in Schießl P. (ed.), 4th International Ph.D. Symposium, Springer-VDI, Düsseldorf, 2002.

    Google Scholar 

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  1. Institute of Lightweight Structures and Conceptual Design, University of Stuttgart, Pfaffenwaldring 7, 70569, Germany

    Sergey Ananiev

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  1. Sergey Ananiev
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Correspondence to Sergey Ananiev.

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Ananiev, S. On Equivalence Between Optimality Criteria and Projected Gradient Methods with Application to Topology Optimization Problem. Multibody Syst Dyn 13, 25–38 (2005). https://doi.org/10.1007/s11044-005-2530-y

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  • DOI: https://doi.org/10.1007/s11044-005-2530-y

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