Abstract
In this paper symmetry and asymmetry of optimal solutions in symmetric structural optimization problems is investigated, based on the choice of variables. A group theory approach is used to define the symmetry of the structural problems in a general way. This approach allows the set of symmetric structures to be described and related to the entire search space of the problem. A relationship between the design variables and the likelihood of finding symmetric or asymmetric solutions to problems is established. It is shown that an optimal symmetric solution (if any) does not necessarily exist in the case of discrete variable problems, regardless of the size of the discrete, countable set from which variables can be chosen. Finally a number of examples illustrate these principles on simple truss structures with discrete topology and sizing variables.
Similar content being viewed by others
Notes
The notation \(\mathbf {x}_{\iota \kappa }\) refers to the characteristics of the vector under the symmetry transformations \(\iota \) and \(\kappa \).
The vertical gaps in the graph are a result of groups of binary numbers whose equivalent structures share common mechanical instabilities. These instabilities are considered non-feasible and therefore not plotted.
References
Achtziger W, Stolpe M (2007) Truss topology optimization with discrete design variables—guaranteed global optimality and benchmark examples. Struct Multidisc Optim 34(1):1–20
Bai Y, Klerk E, Pasechnik D, Sotirov R (2008) Exploiting group symmetry in truss topology optimization. Optim Eng 10(3):331–349
Balasubramanian K (1995) Graph theoretical perception of molecular symmetry. Chem Phys Lett 232(56):415–423
Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin
Cheng G, Liu X (2011) Discussion on symmetry of optimum topology design. Struct Multidisc Optim 44:713–717
Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37(5):447–465
Evgrafov A (2005) On globally stable singular truss topologies. Struct Multidisc Optim 29:170–177
Guo X, Ni C, Cheng G, Du Z (2012) Some symmetry results for optimal solutions in structural optimization. Struct Multidisc Optim 46:631–645. doi:10.1007/s00158-012-0802-8
Hamermesh M (1989) Group theory and its application to physical problems. In: Dover books on physics and chemistry. Dover, New York
Healey TJ (1988) A group-theoretic approach to computational bifurcation problems with symmetry. Comput Methods Appl Mech Eng 67(3):257–295
Ikeda K, Murota K (1991) Bifurcation analysis of symmetric structures using block-diagonalization. Comput Methods Appl Mech Eng 86:215–243
Kangwai R, Guest S, Pellegrino S (1999) An introduction to the analysis of symmetric structures. Comput Struct 71(6):671–688
Kaveh A, Nikbakht M (2011) Analysis of space truss towers using combined symmetry groups and product graphs. Acta Mech 218:133–160
Kaveh A, Nikbakht M, Rahami H (2010) Improved group theoretic method using graph products for the analysis of symmetric-regular structures. Acta Mech 210:265–289
Kirsch U (1990) On singular topologies in optimum structural design. Struct Multidisc Optim 2(3):133–142
Kosaka I, Swan CC (1999) A symmetry reduction method for continuum structural topology optimization. Comput Struct 70(1): 47–61
Leech J (1970) How to use groups. Am J Phys 38:273–273
Øystein J, Sellers J (2006) Partitions with parts in a finite set. Int J Number Theory 2(3):455–468
Renton JD (1964) On the stability analysis of symmetrical frameworks. Q J Mech Appl Math 17:175–197
Richardson J, Adriaenssens S, Bouillard P, Filomeno Coelho R (2012) Multiobjective topology optimization of truss structures with kinematic stability repair. Struct Multidisc Optim 46(4):513–532
Rozvany G (2010) On symmetry and non-uniqueness in exact topology optimization. Struct Multidisc Optim 43:297–317
Rozvany GIN, Birker T (1994) On singular topologies in exact layout optimization. Struct Multidisc Optim 8:228–235
Stolpe M (2010) On some fundamental properties of structural topology optimization problems. Struct Multidisc Optim 41(5):661–670
Svanberg K (1984) On local and global minima in structural optimization. In: Atrek E, Gallagher RH, Ragsdell KM, Zienkiewicz OC (eds) New directions in optimal structural design. Wiley, New York
Zingoni A (2009) Group-theoretic exploitations of symmetry in computational solid and structural mechanics. Int J Numer Methods Eng 79(February):253–289
Zingoni A, Pavlovic M, Zlokovic G (1993) Application of group theory to the analysis of space frames. In: Parke G , Howard C (eds) Space structures. Thomas Telford, London, pp 1334–1347
Zingoni A, Pavlovic MN, Zlokovic GM (1995) A symmetry-adapted flexibility approach for multi-storey space frames: general outline and symmetry-adapted redundants. Struct Eng Rev 7(2):107–119
Zloković Ð (1989) Group theory and G-vector spaces in structures: vibrations, stability, and status. In: Ellis Horwood series in civil engineering. E. Horwood, New York
Acknowledgement
The first author would like to thank the Fonds National de la Recherche Scientifique (FNRS, Belgium) for financial support of this research.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Richardson, J.N., Adriaenssens, S., Bouillard, P. et al. Symmetry and asymmetry of solutions in discrete variable structural optimization. Struct Multidisc Optim 47, 631–643 (2013). https://doi.org/10.1007/s00158-012-0871-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-012-0871-8