Abstract
Our paper considers a classic problem in the field of Truss Topology Design, the goal of which is to determine the stiffest truss, under a given load, with a bound on the total volume and discrete requirements in the cross-sectional areas of the bars. To solve this problem we propose a new two-stage Branch and Bound algorithm.
In the first stage we perform a Branch and Bound algorithm on the nodes of the structure. This is based on the following dichotomy study: either a node is in the final structure or not. In the second stage, a Branch and Bound on the bar areas is conducted. The existence or otherwise of a node in this structure is ensured by adding constraints on the cross-sectional areas of its incident bars. In practice, for reasons of stability, free bars linked at free nodes should be avoided. Therefore, if a node exists in the structure, then there must be at least two incident bars on it, unless it is a supported node. Thus, a new constraint is added, which lower bounds the sum of the cross-sectional areas of bars incident to the node. Otherwise, if a free node does not belong to the final structure, then all the bar area variables corresponding to bars incident to this node may be set to zero. These constraints are added during the first stage and lead to a tight model. We report the computational experiments conducted to test the effectiveness of this two-stage approach, enhanced by the rule to prevent free bars, as compared to a classical Branch and Bound algorithm, where branching is only performed on the bar areas.
Similar content being viewed by others
Notes
In the objective function, to simplify, we consider twice the compliance.
\(|\mathcal{A}|\) denotes the cardinal of set \(\mathcal{A}\).
References
Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2005)
Achtziger, W.: Optimization with variable sets of constraints and an application to truss design. Comput. Optim. Appl. 15(1), 69–96 (2000)
Achtziger, W., Stolpe, M.: Truss topology optimization with discrete design variables—guaranteed global optimality and benchmark examples. Struct. Multidiscip. Optim. 34, 1–20 (2007)
Achtziger, W., Stolpe, M.: Global optimization of truss topology with discrete bar areas. Part I. Theory of relaxed problems. Comput. Optim. Appl. 40(2), 247–280 (2008)
Achtziger, W., Stolpe, M.: Global optimization of truss topology with discrete bar areas. Part II. Implementation and numerical results. Comput. Optim. Appl. 44(2), 315–341 (2009)
Alvarez, F., López, J., Ramírez, H.: Interior proximal algorithm with variable metric for second-order cone programming: applications to structural optimization and support vector machines, Optimization Online, June (2009)
Bastos, F., Cerveira, A., Gromicho, J.: Using optimization to solve truss topology design problems. Investig. Oper. 22, 123–156 (2005)
Ben-Tal, A., Bendsøe, M.: A new method for optimal truss topology design. SIAM J. Optim. 3, 322–358 (1993)
Ben-Tal, A., Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim. 7, 991–1016 (1997)
Ben-Tal, A., Nemirovski, A.: Potential reduction polynomial time method for truss topology design. SIAM J. Optim. 3, 596–612 (1994)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics, Philadelphia (2001)
Bollapragada, S., Ghattas, O., Hoocker, J.: Optimal design of truss structures by logic-based branch and cut. Oper. Res. 49, 42–51 (2001)
Borchers, B.: CSDP 3.2 User’s Guide. Optim. Methods Softw. 11, 597–611 (1999)
Calvel, S., Mongeau, M.: Topology optimization of a mechanical component subject to dynamical constraints. Optimization Online, August (2005)
Cerveira, A., Bastos, F.: Semidefinite relaxations and Lagrangian duality in truss topology design problem. Int. J. Math. Stat. 9, 12–25 (2011)
de Klerk, E., Roos, C., Terlaky, T.: Semi-definite problems in truss topology optimization. Tech. report Nr. 95–128, Faculty of Technical Mathematics and Informatics, Delft University of Technology, (1995)
Faustino, A., Júdice, J., Ribeiro, I., Neves, A.: An integer programming model for truss topology optimization. Investig. Oper. 26, 111–127 (2006)
Haslinger, J., Kočvara, M., Leugering, G., Stingl, M.: Multidisciplinary free material optimization. SIAM J. Appl. Math. 70, 2709–2728 (2010)
Kočara, M.: Truss topology design with integer variables made easy? Optimization Online (2010)
Rasmussen, M., Stolpe, M.: A note on stress-constrained truss topology optimization. J. Struct. Multidisciplin. Optim. 27(1–2), 136–137 (2004)
Rasmussen, M., Stolpe, M.: Global optimization of discrete truss topology design problems using a parallel cut-and-branch method. Comput. Struct. 86, 13–14 (2008). 1527–1538
Rozvany, G.: Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct. Multidiscip. Optim. 11(3), 213–217 (1996)
Stingl, M., Kočvara, M., Leugering, G.: Free material optimization with fundamental eigenfrequency constraints. SIAM J. Optim. 20, 524–547 (2009)
Stolpe, M.: Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound. Int. J. Numer. Methods Biomed. Eng. 61(8), 1270–1309 (2004)
Zhou, M.: Difficulties in truss topology optimization with stress and local buckling constraints. Struct. Multidiscip. Optim. 11(1), 134–136 (1996)
Acknowledgements
This work is supported by National Funding from FCT—Fundação para a Ciência e a Tecnologia, under the project: PEst-OE/MAT/UI0152.
Author information
Authors and Affiliations
Corresponding author
Appendix: Convex quadratically constrained problem
Appendix: Convex quadratically constrained problem
Theorem 1
Consider the following non-convex problem
and consider the following quadratically constrained minimization problem
with \(\underline{V}_{i}=L_{i} s_{i}\) and \(\overline{V}_{i}=U_{i}s_{i}\), i=1,…,m. Similarly to Theorem 4.1 in [4], the following holds:
-
(a)
In all cases, whether \((\overline{R})\) or (Q2P) possess a solution or not, we have the identity \(\mathit{inf}(\overline{R})=-\mathit{inf}(\mathit{Q2P})\), where \(\mathit{inf}(\overline{R})=-\infty\) (inf(Q2P)=−∞) if \(\overline{R}\) (resp. Q2P) is infeasible and \(\mathit{inf}(\overline{R})=\min(\overline{R})\) (resp. \(\mathit{inf} (\mathit{Q2P})=\min(\overline{R})\)) otherwise.
-
(b)
Let (u,r,γ,λ) be optimal for (Q2P) with corresponding Lagrange-multipliers \(\tau^{-}, \tau^{+} \in \mathbb{R}^{m}_{+}\). Then (a,u) is optimal for \((\overline{R})\) where
$$a_i = \tau^-\underline{V}_i+ \tau^+ \overline{V}_i \qquad \forall i= 1,\ldots,m. $$ -
(c)
The following statements are equivalent:
-
(c1)
Problem \((\overline{R})\) possesses a solution.
-
(c2)
Problem \((\overline{R})\) is feasible.
-
(c3)
Problem (Q2P) is bounded.
-
(c4)
Problem (Q2P) possesses a solution.
-
(c1)
Rights and permissions
About this article
Cite this article
Cerveira, A., Agra, A., Bastos, F. et al. A new Branch and Bound method for a discrete truss topology design problem. Comput Optim Appl 54, 163–187 (2013). https://doi.org/10.1007/s10589-012-9487-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-012-9487-6