Abstract
In this paper the concept of extended optimality, or hyperoptimality, is adopted. By following this idea, a new compliance–volume product is suggested as objective. The volume appearing in the product is also raised to the power of a new design parameter which can be set to different values. In such manner design concepts with different volume fractions can be generated by using the approach of extended optimality. Both manufacturing constraints and unilateral contact constraints are included in the proposed method. The manufacturing constraints are implemented by adjusting the move limits such that the draw directions are satisfied. Both one draw direction as well as split draw constraints are considered. The contact conditions are modeled by the augmented Lagrangian approach such that the Jacobian in the Newton algorithm as well as in the adjoint equation becomes symmetric. The design parametrization is done by the SIMP model and Sigmund’s filter is utilized when the sensitivities are calculated. The proposed method is very robust and efficient. This is demonstrated by solving problems in both 2D and 3D. The numerical results are also compared to solutions obtained by performing compliance optimization with a constraint on the volume fraction.
Similar content being viewed by others
Notes
A free student version of Topo4abq is available at www.fema.se.
References
Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to newton like solution methods. Comput Methods Appl Mech Eng 92:353–375
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Heidelberg
Edwards CS, Kim HA, Budd CJ (2007) An evaluative study on ESO and SIMP for optimising a cantilever tie-beam. Struct Multidisc Optim 34:403–414
Facchinei F, Jiang H, Qi L (1999) A smoothing method for mathematical programs with equilibrium constraints. Math Program 85:107–134
Ireman P, Klarbring A, Strömberg N (2009) Gradient theory of damage coupled to frictional contact and wear, and its numerical treatment. Comput Model Eng Sci 52(2):125–158
Lewiński T, Rozvany GIN (2008) Analytical benchmarks for topological optimization iv: square-shaped line support. Struct Multidisc Optim 36:143–158
Lewiński T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts - part II: unsymmetric cantilivers. Int J Mech Sci 36:399–419
Michell AGM (1904) The limits of economy of material in frame structures. Philos Mag 8:589–597
Rozvany GIN (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Struct Multidisc Optim 15:42–48
Rozvany GIN (2009a) A critical review of established methods of structural topology optimizaton. Struct Multidisc Optim 37:217–237
Rozvany GIN (2009b) Traditional vs. extended optimality in topology optimization. Struct Multidisc Optim 37:319–323
Rozvany GIN, Querin, OM, Gaspar, Z, Pomezanski V (2002) Extended optimality in topology design. Struct Multidisc Optim 24:257–261
Rozvany GIN, Querin OM, Gaspar Z, Pomezanski V (2005) Erratum for the brief note “extended optimality in topology design”. Struct Multidisc Optim 30:504
Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21:120–127
Strömberg N (1997) An augmented Lagrangian method for fretting problems. Eur J Mech A Solids 16:573–593
Strömberg N, Klarbring A (2010) Topology optimization of structures in unilateral contact. Struct Multidisc Optim 41:57–64
Tanskanen P (2002) The evolutionary optimization method: theoretical aspects. Comput Methods Appl Mech Eng 191:5485–5498
Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336
Acknowledgements
I am most grateful to the reviewers’ comments, which really helped me to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Strömberg, N. Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance–volume product. Struct Multidisc Optim 42, 341–350 (2010). https://doi.org/10.1007/s00158-010-0502-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-010-0502-1