Skip to main content

Topology optimization for finding shell structures manufactured by deep drawing


This paper presents a new approach for optimizing shell structures considering their mid surface design including cut-outs. Therefore we introduced a manufacturing constraint to the 3D topology optimization based on the density method in order to receive an optimized structure without undercuts and with a constant wall thickness, so that these structures can be manufactured by deep drawing in one step. It is shown that introducing cut-outs while increasing the shell thickness can improve the performance of shell structures considering their stiffness at a constant mass.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25


  • Allaire G, Jouve F, Michailidis G (2013) Casting constraints in structural optimization via level-set method. Proc. of 10th World Congress on Structural and Multidisciplinary Optimization, Orlando, USA

  • Ansola R, Canales J, Tarrago JA, Rasmussen J (2002) On simultaneous shape and material layout optimization of shell structures. Struct Multidiscip Optim 24:175–184

    Article  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202

    Article  Google Scholar 

  • Bertsch C, Cislino AP, Langer S, Reese S (2008) Topology optimization of 3D elastic structures using boundary elements. Proc Appl Math Mech 8:10771–10772

    Article  Google Scholar 

  • Bletzinger KU, Ramm E (2014) A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Struct Multidiscip Optim 49:873–895

    MathSciNet  Article  Google Scholar 

  • Boljanovic V (2014) Sheet metal forming processes and die design. Industrial Press, South Norwalk

    Google Scholar 

  • Dienemann R, Schumacher A, Fiebig S (2016) Topology and shape optimization of sheet metals with integrated deep-drawing-simulation. Proc. of 12th World Conference on Computational Mechanics, Seoul, Korea

  • Falkenberg P, Franke T, Fiebig S, Vietor T (2015) Consideration of adhesive joints for a multi-material topology optimization approach, 20th International Conference on Composite Materials, Copenhagen, Denmark

  • Fiebig S, Sellschopp J, Manz H, Vietor T, Axmann K, Schumacher A (2015) Future challenges for topology optimization for the usage in automotive lightweight design technologies. Proc. of 11th World Congress on Structural and Multidisciplinary Optimization, Sydney, Australia

  • Fleury C (1989) CONLIN: an efficient dual optimizer based on convex approximation concepts. Struct Optim 1:81–89

    Article  Google Scholar 

  • Franke T, Vietor T, Fiebig S, Horstmann GM (2015) Robust and production-oriented topology optimization of cast parts including manufacturing restrictions and process simulation, NAFEMS seminar Optimierung und Robust Design, Wiesbaden, Germany

  • Guest J, Zhu M (2012) Casting and milling restrictions in topology optimization via projection-based algorithms. Proceedings of the ASME 2012 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, Chicago, IL, USA

  • Harzheim L, Graf G (2006) A review of optimization of cast parts using topology optimization II—topology optimization with manufacturing constraints. Struct Multidiscip Optim 31:388–399

    Article  Google Scholar 

  • Hassani B, Tavakkoli SM, Ghasemnejad H (2013) Simultaneous shape and topology optimization of shell structures. Struct Multidiscip Optim 48:221–233

    MathSciNet  Article  MATH  Google Scholar 

  • Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41:605–620

    Article  Google Scholar 

  • Lochner-Aldinger I, Schumacher A (2014) Homogenization method. In: Adriaenssens S, Block P, Veenendaal D, Williams C (eds) Shell structures for architecture—form finding and optimization. Routledge, New York

    Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Multidiscip Optim 16:68–75

    Article  Google Scholar 

  • Sigmund O, Aage N, Andreassen, E (2016) On the (non-) optimality of Michell structures. Struct Multidiscip Optim 54(2):361–373

  • Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    MathSciNet  Article  MATH  Google Scholar 

  • Xia Q, Shi T, Wang MY, Liu S (2009) A level set based method for the optimization of cast part. Struct Multidiscip Optim 41:735–747

    Article  Google Scholar 

  • Zhou M, Fleury F, Patten S, Stannard N, Mylett D, Gardner S (2011) Topology optimization—practical aspects for industrial applications. Proc. of 9th World Congress on Structural and Multidisciplinary Optimization, Shizuoka, Japan

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to R. Dienemann.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dienemann, R., Schumacher, A. & Fiebig, S. Topology optimization for finding shell structures manufactured by deep drawing. Struct Multidisc Optim 56, 473–485 (2017).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Topology optimization
  • Sheet metals
  • Deep drawing
  • Manufacturing constraint
  • Thin walled structures