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Structural and Multidisciplinary Optimization

, Volume 58, Issue 6, pp 2411–2429 | Cite as

A novel approach to discrete truss design problems using mixed integer neighborhood search

  • Mohammad ShahabsafaEmail author
  • Ali Mohammad-Nezhad
  • Tamás Terlaky
  • Luis Zuluaga
  • Sicheng He
  • John T. Hwang
  • Joaquim R. R. A. Martins
Research Paper
  • 389 Downloads

Abstract

Discrete truss sizing problems are very challenging to solve due to their combinatorial, nonlinear, non-convex nature. Consequently, truss sizing problems become unsolvable as the size of the truss grows. To address this issue, we consider various mathematical formulations for the truss design problem with the objective of minimizing weight, while the cross-sectional areas of the bars take only discrete values. Euler buckling constraints, Hooke’s law, and bounds for stress and displacements are also considered. We propose mixed integer linear optimization (MILO) reformulations of the non-convex mixed integer models. The resulting MILO models are not solvable with existing MILO solvers as the size of the problem grows. Our novel methodology provides high-quality solutions for large-scale real truss sizing problems, as demonstrated through extensive numerical experiments.

Keywords

Mixed integer linear optimization Truss design problem Euler buckling constraint Neighborhood search mixed integer linear optimization 

Notes

Funding information

This research was supported by Air Force Office of Scientific Research Grant No. FA9550-15-1-0222.

References

  1. Achtziger W (1999a) Local stability of trusses in the context of topology optimization part I: exact modelling. Structural Optimization 17(4):235–246Google Scholar
  2. Achtziger W (1999b) Local stability of trusses in the context of topology optimization part II: a numerical approach. Structural Optimization 17(4):247–258Google Scholar
  3. Achtziger W, Bendsøe MP, Ben-Tal A, Zowe J (1992) Equivalent displacement based formulations for maximum strength truss topology design. IMPACT of Computing in Science and Engineering 4(4):315–345MathSciNetCrossRefGoogle Scholar
  4. Achtziger W, Stolpe M (2006) Truss topology optimization with discrete design variables—guaranteed global optimality and benchmark examples. Struct Multidiscip Optim 34(1):1–20MathSciNetCrossRefGoogle Scholar
  5. Achtziger W, Stolpe M (2007a) Global optimization of truss topology with discrete bar areas—part I: theory of relaxed problems. Comput Optim Appl 40(2):247–280CrossRefGoogle Scholar
  6. Achtziger W, Stolpe M (2007b) Global optimization of truss topology with discrete bar areas—part II: implementation and numerical results. Comput Optim Appl 44(2):315–341CrossRefGoogle Scholar
  7. Achtziger W, Stolpe M (2007c) Truss topology optimization with discrete design variables—guaranteed global optimality and benchmark examples. Struct Multidiscip Optim 34(1):1–20MathSciNetCrossRefGoogle Scholar
  8. Barbosa HJ, Lemonge AC, Borges CC (2008) A genetic algorithm encoding for cardinality constraints and automatic variable linking in structural optimization. Eng Struct 30(12):3708–3723CrossRefGoogle Scholar
  9. Bendsøe MP, Ben-Tal A (1993) Truss topology optimization by a displacements based optimality criterion approach. In: Rozvany G (ed) Optimization of Large Structural Systems volume 231 of NATO ASI Series. Springer, Netherlands, pp 139–155CrossRefGoogle Scholar
  10. Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Structural Optimization 7(3):141–159CrossRefGoogle Scholar
  11. Bendsøe MP, Sigmund O (2013) Topology optimization: theory, methods, and applications. Springer Science & Business Media, BerlinzbMATHGoogle Scholar
  12. Bland JA (2001) Optimal structural design by ant colony optimization. Eng Optim 33(4):425–443CrossRefGoogle Scholar
  13. Brooks TR, Kenway GKW, Martins JRRA (2018) UCRM: an aerostructural model for the study of flexible transonic aircraft wings. AIAA Journal (In press)Google Scholar
  14. Cai J, Thierauf G (1993) Discrete optimization of structures using an improved penalty function method. Eng Optim 21(4):293– 306CrossRefGoogle Scholar
  15. Camp C, Farshchin M (2014) Design of space trusses using modified teaching–learning based optimization. Eng Struct 62-63:87–97CrossRefGoogle Scholar
  16. Camp CV (2007) Design of space trusses using big bang–big crunch optimization. J Struct Eng 133 (7):999–1008CrossRefGoogle Scholar
  17. Camp CV, Bichon BJ (2004a) Design of space trusses using ant colony optimization. J Struct Eng 130 (5):741–751CrossRefGoogle Scholar
  18. Camp CV, Bichon BJ (2004b) Design of space trusses using ant colony optimization. J Struct Eng 130 (5):741–751CrossRefGoogle Scholar
  19. Cerveira A, Agra A, Bastos F, Gromicho J (2009) New branch and bound approaches for truss topology design with discrete areas. In: Proceedings of the american conference on applied mathematics. Recent advances in applied mathematics, pp 228–233Google Scholar
  20. De Klerk E, Roos C, Terlaky T (1995) Semi-definite problems in truss topology optimization. Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report, pp 95–128Google Scholar
  21. Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. Journal de Mecanique 3:25–52Google Scholar
  22. Gill PE, Murray W, Saunders MA (2005) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM review 47(1):99–131MathSciNetCrossRefGoogle Scholar
  23. Glover F (1975) Improved linear integer programming formulations of nonlinear integer problems. Manag Sci 22(4):455–460MathSciNetCrossRefGoogle Scholar
  24. Glover F (1984) An improved mip formulation for products of discrete and continuous variables. J Inf Optim Sci 5(1):69–71zbMATHGoogle Scholar
  25. Gurobi Optimization I (2016) Gurobi optimizer reference manualGoogle Scholar
  26. Haftka RT, Gürdal Z (2012) Elements of structural optimization, vol 11. Springer Science & Business MediaGoogle Scholar
  27. Hajela P, Lee E (1995) Genetic algorithms in truss topological optimization. Int J Solids Struct 32 (22):3341–3357MathSciNetCrossRefGoogle Scholar
  28. Hanafi S (2016) New variable neighbourhood search based 0-1 MIP heuristics. Yugoslav Journal of Operations Research 25(3):343–360MathSciNetCrossRefGoogle Scholar
  29. Hansen P, Mladenović N (2003) Variable neighborhood search. Springer, Boston, pp 145–184zbMATHGoogle Scholar
  30. Ho-Huu V, Nguyen-Thoi T, Vo-Duy T, Nguyen-Trang T (2016) An adaptive elitist differential evolution for optimization of truss structures with discrete design variables. Comput Struct 165:59–75CrossRefGoogle Scholar
  31. Kaveh A, Azar BF, Talatahari S (2008) Ant colony optimization for design of space trusses. Int J Space Struct 23(3):167–181CrossRefGoogle Scholar
  32. Kaveh A, Ghazaan MI (2015) A comparative study of CBO and ECBO for optimal design of skeletal structures. Comput Struct 153:137–147CrossRefGoogle Scholar
  33. Kaveh A, Kalatjari V (2004) Size/geometry optimization of trusses by the force method and genetic algorithm. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift fr Angewandte Mathematik und Mechanik 84 (5):347–357MathSciNetCrossRefGoogle Scholar
  34. Kaveh A, Mahdavi V (2014) Colliding bodies optimization method for optimum discrete design of truss structures. Comput Struct 139:43–53CrossRefGoogle Scholar
  35. Kaveh A, Talatahari S (2009) A particle swarm ant colony optimization for truss structures with discrete variables. J Constr Steel Res 65(8):1558–1568CrossRefGoogle Scholar
  36. Kripka M (2004) Discrete optimization of trusses by simulated annealing. J Braz Soc Mech Sci Eng 26:170–173CrossRefGoogle Scholar
  37. Lazić J (2010) New variants of variable neighbourhood search for 0-1 mixed integer programming and clustering. PhD thesis Brunel University, School of Information Systems, Computing and MathematicsGoogle Scholar
  38. Li L, Huang Z, Liu F (2009) A heuristic particle swarm optimization method for truss structures with discrete variables. Comput Struct 87(7):435–443CrossRefGoogle Scholar
  39. Mahfouz SY (1999) Design optimization of structural steelwork. PhD thesis, University of Bradford, United KingdomGoogle Scholar
  40. Mela K (2014) Resolving issues with member buckling in truss topology optimization using a mixed variable approach. Struct Multidiscip Optim 50(6):1037–1049MathSciNetCrossRefGoogle Scholar
  41. Mellaert RV, Mela K, Tiainen T, Heinisuo M, Lombaert G, Schevenels M (2017) Mixed-integer linear programming approach for global discrete sizing optimization of frame structures. Struct Multidiscip Optim 57(2):579–593MathSciNetCrossRefGoogle Scholar
  42. Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24(11):1097–1100MathSciNetCrossRefGoogle Scholar
  43. Petersen CC (1971) A note on transforming the product of variables to linear form in linear programs. Working Paper, Purdue UniversityGoogle Scholar
  44. Petrovic N, Kostic N, Marjanovic N (2017) Comparison of approaches to 10 bar truss structural optimization with included buckling constraints. Applied Engineering Letters 2(3):98–103Google Scholar
  45. Rahami H, Kaveh A, Gholipour Y (2008) Sizing, geometry and topology optimization of trusses via force method and genetic algorithm. Engineering Structures 30(9):2360–2369CrossRefGoogle Scholar
  46. Rajeev S, Krishnamoorthy CS (1992) Discrete optimization of structures using genetic algorithms. J Struct Eng 118(5):1233–1250CrossRefGoogle Scholar
  47. Rajeev S, Krishnamoorthy CS (1997) Genetic algorithms-based methodologies for design optimization of trusses. J Struct Eng 123(3):350–358CrossRefGoogle Scholar
  48. Rasmussen M, Stolpe M (2008) Global optimization of discrete truss topology design problems using a parallel cut-and-branch method. Comput Struct 86(13):1527 – 1538. Structural OptimizationCrossRefGoogle Scholar
  49. Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2012) Mine blast algorithm for optimization of truss structures with discrete variables. Comput Struct 102-103:49–63CrossRefGoogle Scholar
  50. Sadollah A, Eskandar H, Bahreininejad A, Kim JH (2015) Water cycle, mine blast and improved mine blast algorithms for discrete sizing optimization of truss structures. Comput Struct 149:1– 16CrossRefGoogle Scholar
  51. SeokLee K, Geem ZW (2004) A new structural optimization method based on the harmony search algorithm. Comput Struct 82(9–10):781–798Google Scholar
  52. Sonmez M (2011) Discrete optimum design of truss structures using artificial bee colony algorithm. Struct Multidiscip Optim 43(1):85–97CrossRefGoogle Scholar
  53. Stolpe M (2004) Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound. Int J Numer Methods Eng 61(8):1270–1309MathSciNetCrossRefGoogle Scholar
  54. Stolpe M (2007) On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs. Optim Eng 8(2):163–192MathSciNetCrossRefGoogle Scholar
  55. Stolpe M (2011) To bee or not to bee—comments on “discrete optimum design of truss structures using artificial bee colony algorithm”. Struct Multidiscip Optim 44(5):707–711MathSciNetCrossRefGoogle Scholar
  56. Stolpe M (2016) Truss optimization with discrete design variables: a critical review. Struct Multidiscip Optim 53(2):349–374MathSciNetCrossRefGoogle Scholar
  57. Svanberg K, Werme M (2005) A hierarchical neighbourhood search method for topology optimization. Struct Multidiscip Optim 29(5):325–340MathSciNetCrossRefGoogle Scholar
  58. Svanberg K, Werme M (2007) Sequential integer programming methods for stress constrained topology optimization. Struct Multidiscip Optim 34(4):277–299MathSciNetCrossRefGoogle Scholar
  59. Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57MathSciNetCrossRefGoogle Scholar
  60. Wu S-J, Chow P-T (1995) Steady-state genetic algorithms for discrete optimization of trusses. Comput Struct 56(6):979–991CrossRefGoogle Scholar
  61. Zeng S, Li L (2012) Particle swarm-group search algorithm and its application to spatial structural design with discrete variables. International Journal of Optimization in Civil Engineering 2(4)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringLehigh University, Harold S. Mohler LaboratoryBethlehemUSA
  2. 2.Department of Aerospace EngineeringUniversity of MichiganAnn ArborUSA
  3. 3.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan DiegoUSA

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