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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 99
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping
Paper 236

Symmetry of Solutions in Discrete and Continuous Structural Topology Optimization

J.N. Richardson1,2, S. Adriaenssens3, Ph. Bouillard1 and R. Filomeno Coelho1

1BATir, Building, Architecture and Town Planning, Faculty of Applied Sciences, Université Libre de Bruxelles, Brussels, Belgium
2MEMC, Mechanics of Materials and Construction, Faculty of Engineering Sciences, Vrije Universiteit Brussel, Brussels, Belgium
3Department of Civil and Environmental Engineering, Princeton University, Princeton NJ, USA

Full Bibliographic Reference for this paper
J.N. Richardson, S. Adriaenssens, Ph. Bouillard, R. Filomeno Coelho, "Symmetry of Solutions in Discrete and Continuous Structural Topology Optimization", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 236, 2012. doi:10.4203/ccp.99.236
Keywords: structural topology optimization, symmetric topology, asymmetric topology, group theory, symmetry operation, truss topology optimization.

Summary
In this paper symmetry and asymmetry of optimal solutions in symmetric structural topology optimization problems is investigated, based on the choice of variables. Symmetry reduction of structural problems is a well-established technique for structural analysis [1] This has translated into symmetry reduction of structural optimization problems. Asymmetric solutions have often been disregarded, particularly in discrete topology optimization, without investigation of the possible benefits these solutions may offer. A group theory approach is used to formally define the symmetry of the structural problems in the present paper. The symmetry groups are represented by sets of permutation matrices which act on the design variables. This approach allows the set of symmetric structure designs to be described and related to the entire search space. In this paper two of the most widely discussed objective functions are considered: minimization of the mass of the structure and minimization of the (volume constrained) compliance energy for a given loading. It is shown by means of these definitions that, given a symmetric problem with continuous variables, an optimal symmetric solution (if any) necessarily exists. However, it is shown that this does not hold for the case in which the variables are discrete. Two examples of discrete variable problems are investigated to demonstrate these principles. The fist example is a twenty bar, two-dimensional truss with point symmetry group S2. The second is a twenty four bar, three dimensional special truss with dihedral point symmetry group D3. In both cases all possible structures are calculated, including the symmetric subset of the search space. The minimal designs are compared to the minimum symmetric designs. In all cases it can be seen that the asymmetric solutions outperform the symmetric solutions significantly. While it is strongly embedded in the engineering tradition to enforce symmetry in symmetric problems, it has been shown that the discrete nature of the optimization variables calls this practice into question. It is the hope that relaxing this assumption may lead to more efficient and elegant designs in practice.

References
1
A. Zingoni, "Group-theoretic exploitations of symmetry in computational solid and structural mechanics", International Journal for Numerical Methods in Engineering, 79, 253-289, 2009. doi:10.1002/nme.2576

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