# Analytical solutions for the minimum weight design of trusses by cylindrical algebraic decomposition

Special

First Online:

Received:

Accepted:

- 117 Downloads
- 1 Citations

## Abstract

In this study, a method for the analytical evaluation of globally optimal solutions for the minimum weight design of trusses is presented. The basis of the methodology is the cylindrical algebraic decomposition algorithm, in tandem with powerful symbolic computation for the discovery of stationary points. Certain final answers to well-known benchmark problems are produced, while future improvements in both the algorithm implementation and the computer capabilities may allow the solution of even more difficult problems. To the best of our knowledge, no similar attempt can be found in the literature.

## Keywords

Truss weight minimization Analytical methods Cylindrical algebraic decomposition## References

- 1.Feury, C., Geradin, M.: Optimality criteria and mathematical programming in structural weight optimization. Comput. Struct.
**8**(1), 7–17 (1978)CrossRefMATHGoogle Scholar - 2.Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing. Springer, New York (2003)CrossRefMATHGoogle Scholar
- 3.Collins, G.E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Lect. Notes Comput. Sci.
**33**, 134–183 (1975)CrossRefGoogle Scholar - 4.Strzeboński, A.: Solving systems of strict polynomial inequalities. J. Symb. Comput.
**29**, 471–480 (2000)MathSciNetCrossRefMATHGoogle Scholar - 5.Brown, C.W.: QEPCAD B—a program for computing with semi-algebraic sets. Sigsam Bull.
**37**(4), 97–108 (2003)CrossRefMATHGoogle Scholar - 6.Dolzmann, A., Sturm, T.: Redlog: computer algebra meets computer logic. Sigsam Bull.
**31**(2), 2–9 (1997)CrossRefGoogle Scholar - 7.Collins, G.E.: Quantifier Elimination by cylindrical algebraic decomposition—twenty years of progress. In: Caviness, B.F., Johnson, J.R. (eds.) Texts and Monographs in Symbolic Computation, pp. 8–23. Springer, New York (1998)Google Scholar
- 8.Bradford, R., Davenport, J.H., England, M., McCallum, S., Wilson, D.: Cylindrical algebraic decompositions for boolean combinations. Proc. ISSAC
**13**, 125–132 (2013)MathSciNetMATHGoogle Scholar - 9.Fotiou, I.A., Parrilo, P.A., Morari, M.: Nonlinear parametric optimization using cylindrical algebraic decomposition. In: Proceedings of 44th IEEE Conference on Decision and Control, pp. 3735–3740 (2005)Google Scholar
- 10.Constrained Optimization. Wolfram Mathematica \(\textregistered \) Tutorial Collection. http://www.wolfram.com/learningcenter/tutorialcollection/ConstrainedOptimization/ConstrainedOptimization.pdf. Extracted on 20/11/2016
- 11.Tong, W.H., Jiang, J.S., Liu, G.R.: Solution existence of the optimization problem of truss structures with frequency constraints. Int. J. Solids Struct.
**37**(30), 4043–4060 (2000)CrossRefMATHGoogle Scholar - 12.Zuo, W., Bai, J., Li, B.: A hybrid OC-GA approach for fast and global truss optimization with frequency constraints. Appl. Soft Comput.
**14**, 528–535 (2014)CrossRefGoogle Scholar - 13.Ray, T., Saini, P.: Engineering design optimization using a swarm with an intelligent information sharing among individuals. Eng. Optim.
**33**(6), 735–748 (2001)CrossRefGoogle Scholar - 14.Ray, T., Liew, K.M.: Society and civilization: an optimization algorithm based on the simulation of social behavior. IEEE Trans. Evol. Comput.
**7**(4), 386–396 (2003)CrossRefGoogle Scholar - 15.Hernendez, S.: Multiobjective structural optimization. In: Kodiyalam, S., Saxena, M. (eds.) Geometry and Optimization Techniques for Structural Design, pp. 341–362. Elsevier, Amsterdam (1994)Google Scholar
- 16.Liu, H., Zixing, C., Wang, Y.: Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl. Soft Comput.
**10**, 629–640 (2010)CrossRefGoogle Scholar - 17.Brown, C.W., Davenport, J.H.: The complexity of quantifier elimination and cylindrical algebraic decomposition. In: Proceedings of ISSAC ’07, pp. 54–60. ACM (2007)Google Scholar
- 18.Lee, K.S., Geem, Z.W.: A new structural optimization method based on the harmony search algorithm. Comput. Struct.
**82**, 781–798 (2004)CrossRefGoogle Scholar - 19.Sonmez, M.: Articial bee colony algorithm for optimization of truss structures. Appl. Soft Comput.
**11**, 2406–2418 (2011)CrossRefGoogle Scholar - 20.Lamberti, L.: An efficient simulated annealing algorithm for design optimization of truss structures. Comput. Struct.
**86**, 1936–1953 (2008)CrossRefGoogle Scholar

## Copyright information

© Springer-Verlag GmbH Germany 2017