# A new Branch and Bound method for a discrete truss topology design problem

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## 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.

## Keywords

Truss topology design Stiffness Semi-definite programming Quadratic programming Global optimization Branch and Bound## Notes

### Acknowledgements

This work is supported by National Funding from FCT—Fundação para a Ciência e a Tecnologia, under the project: PEst-OE/MAT/UI0152.

## References

- 1.Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett.
**33**, 42–54 (2005) MathSciNetCrossRefzbMATHGoogle Scholar - 2.Achtziger, W.: Optimization with variable sets of constraints and an application to truss design. Comput. Optim. Appl.
**15**(1), 69–96 (2000) MathSciNetCrossRefzbMATHGoogle Scholar - 3.Achtziger, W., Stolpe, M.: Truss topology optimization with discrete design variables—guaranteed global optimality and benchmark examples. Struct. Multidiscip. Optim.
**34**, 1–20 (2007) MathSciNetCrossRefGoogle Scholar - 4.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) MathSciNetCrossRefzbMATHGoogle Scholar - 5.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) MathSciNetCrossRefzbMATHGoogle Scholar - 6.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) Google Scholar
- 7.Bastos, F., Cerveira, A., Gromicho, J.: Using optimization to solve truss topology design problems. Investig. Oper.
**22**, 123–156 (2005) Google Scholar - 8.Ben-Tal, A., Bendsøe, M.: A new method for optimal truss topology design. SIAM J. Optim.
**3**, 322–358 (1993) MathSciNetCrossRefzbMATHGoogle Scholar - 9.Ben-Tal, A., Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim.
**7**, 991–1016 (1997) MathSciNetCrossRefzbMATHGoogle Scholar - 10.Ben-Tal, A., Nemirovski, A.: Potential reduction polynomial time method for truss topology design. SIAM J. Optim.
**3**, 596–612 (1994) CrossRefGoogle Scholar - 11.Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics, Philadelphia (2001) CrossRefzbMATHGoogle Scholar
- 12.Bollapragada, S., Ghattas, O., Hoocker, J.: Optimal design of truss structures by logic-based branch and cut. Oper. Res.
**49**, 42–51 (2001) MathSciNetCrossRefzbMATHGoogle Scholar - 13.Borchers, B.: CSDP 3.2 User’s Guide. Optim. Methods Softw.
**11**, 597–611 (1999) MathSciNetCrossRefGoogle Scholar - 14.Calvel, S., Mongeau, M.: Topology optimization of a mechanical component subject to dynamical constraints. Optimization Online, August (2005) Google Scholar
- 15.Cerveira, A., Bastos, F.: Semidefinite relaxations and Lagrangian duality in truss topology design problem. Int. J. Math. Stat.
**9**, 12–25 (2011) MathSciNetGoogle Scholar - 16.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) Google Scholar
- 17.Faustino, A., Júdice, J., Ribeiro, I., Neves, A.: An integer programming model for truss topology optimization. Investig. Oper.
**26**, 111–127 (2006) Google Scholar - 18.Haslinger, J., Kočvara, M., Leugering, G., Stingl, M.: Multidisciplinary free material optimization. SIAM J. Appl. Math.
**70**, 2709–2728 (2010) MathSciNetCrossRefzbMATHGoogle Scholar - 19.Kočara, M.: Truss topology design with integer variables made easy? Optimization Online (2010) Google Scholar
- 20.Rasmussen, M., Stolpe, M.: A note on stress-constrained truss topology optimization. J. Struct. Multidisciplin. Optim.
**27**(1–2), 136–137 (2004) Google Scholar - 21.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 CrossRefGoogle Scholar - 22.Rozvany, G.: Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct. Multidiscip. Optim.
**11**(3), 213–217 (1996) Google Scholar - 23.Stingl, M., Kočvara, M., Leugering, G.: Free material optimization with fundamental eigenfrequency constraints. SIAM J. Optim.
**20**, 524–547 (2009) MathSciNetCrossRefzbMATHGoogle Scholar - 24.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) MathSciNetzbMATHGoogle Scholar - 25.Zhou, M.: Difficulties in truss topology optimization with stress and local buckling constraints. Struct. Multidiscip. Optim.
**11**(1), 134–136 (1996) Google Scholar