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A holistic topology and shape optimization approach with an application to steel profiles

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Abstract

This paper addresses the problem of combining topology and shape optimization approaches by exploiting suitable methods from both discrete as well as nonlinear optimization. The topology decisions are made iteratively within the general optimization process by a branch-and-bound algorithm. In every node of the branch-and-bound tree a sequence of nonlinear subproblems which consist of a shape and topology optimization component are solved by using sequential quadratic programming (SQP). The topology component follows the solid isotropic material with penalization (SIMP) idea. One important application of the here presented approach in engineering consists of assisting the product development process in early stages. Here, we consider the design of multi-chambered sheet metal profiles. This work deals with the construction of the branch-and-bound tree and how effective node selection rules can be obtained, as well as with the resulting nonlinear subproblems and the applied SQP method. Finally, numerical results for different load scenarios and topology constraints are presented.

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References

  • Achterberg T, Koch T, Martin A (2005) Branching rules revisited. Oper Res Lett 33

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224

    Article  MathSciNet  MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods, and applications. Springer

  • Byrd RH, Nocedal J, Waltz RA (2008) Steering exact penalty methods for nonlinear programming. Optim Methods Softw 23(2):197-213

    Article  MathSciNet  MATH  Google Scholar 

  • Casas E (1986) Control of an elliptic problem with pointwise state constraints. SIAM J Control Optim 24(6):1309-1318

    Article  MathSciNet  MATH  Google Scholar 

  • Conn AR, Gould NIM, Toint PL (1991) Convergence of quasi-newton matrices generated by the symmetric rank one update. Math Program 50(2):177-195

    Article  MathSciNet  MATH  Google Scholar 

  • Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1-38

    Article  MathSciNet  Google Scholar 

  • Dorn WC, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J Mecanique 3:25-52

    Google Scholar 

  • Fügenschuh A, Fügenschuh M (2008) Integer linear programming models for topology optimization in sheet metal design. Math Methods Oper Res 68(2):313-331

    Article  MathSciNet  MATH  Google Scholar 

  • Göllner T, Günther U, Hess W, Martin A, Ulbrich S (2011) Topology and geometry optimization of branched sheet metal products. In: Proceedings in applied mathematics and mechanics

  • Guillaume P, Masmoudi M (1994) Computation of high order derivatives in optimal shape design. Numer Math 67(2):231-250

    Article  MathSciNet  MATH  Google Scholar 

  • Haslinger J, Mäkinen RAE (2003) Introduction to shape optimization: theory, approximation, and computation. SIAM

  • Horn BM, Lüthen H, Pfetsch ME, Ulbrich S (2017) Geometry and topology optimization of sheet metal profiles by using a branch-and-bound framework. Mater Sci Eng Technol 48:27-40

    Google Scholar 

  • Land AH, Doig AG (1960) An automatic method of solving discrete programming problems. Econometrica 28

  • Locatelli M, Schoen F (2013) Global optimization: theory, algorithms, and applications. MOS-SIAM series on optimization

  • Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization, vol 18. Wiley, New York

    MATH  Google Scholar 

  • Powell MJD (1978) The convergence of variable metric methods for nonlinearly constrained optimization calculations. In: Mangasarian OL, Meyer RR, Robinson SM (eds) Nonlinear programming 3. Academic Press, pp 27-63

  • Schrijver A (1986) Theory of linear and integer programming. Wiley, Chichester

    MATH  Google Scholar 

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. J Struct Mech 25(4):493-524

    Google Scholar 

  • Wolsey LA (1998) Integer programming. Wiley-Interscience series in discrete mathematics and optimization

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Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) Collaborative Research Centre 666 “Integral Sheet Metal Design with Higher Order Bifurcations – Development, Production, Evaluation”.

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Authors and Affiliations

  1. Darmstadt, Germany

    Ute Günther

  2. Munich, Germany

    Wolfgang Hess

  3. Research Group Optimization, Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Benjamin M. Horn & Hendrik Lüthen

Authors
  1. Ute Günther
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  2. Wolfgang Hess
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  3. Benjamin M. Horn
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  4. Hendrik Lüthen
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Correspondence to Benjamin M. Horn.

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Günther, U., Hess, W., Horn, B.M. et al. A holistic topology and shape optimization approach with an application to steel profiles. Struct Multidisc Optim 57, 1267–1281 (2018). https://doi.org/10.1007/s00158-017-1809-y

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  • DOI: https://doi.org/10.1007/s00158-017-1809-y

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