Structural and Multidisciplinary Optimization

, Volume 55, Issue 1, pp 257–277 | Cite as

An evaluation of constraint aggregation strategies for wing box mass minimization

  • Andrew B. Lambe
  • Graeme J. Kennedy
  • Joaquim R. R. A. Martins


Constraint aggregation makes it feasible to solve large-scale stress-constrained mass minimization problems efficiently using gradient-based optimization where the gradients are computed using adjoint methods. However, it is not always clear which constraint aggregation method is more effective, and which values to use for the aggregation parameters. In this work, the accuracy and efficiency of several aggregation methods are compared for an aircraft wing design problem. The effect of the type of aggregation function, the number of constraints, and the value of the aggregation parameter are studied. Recommendations are provided for selecting a constraint aggregation scheme that balances computational effort with the accuracy of the computed optimal design. Using the recommended aggregation method and associated parameters, a mass of within 0.5 % of the true optimal design was obtained.


Structural optimization Constraint aggregation Stress constraints Kreisselmeier– Steinhauser function Induced aggregation 



The authors would like to thank Gaetan K. W. Kenway for his assistance in setting up the CRM wing geometry used in this paper. Computations were performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; and the University of Toronto.


  1. Akgün M A, Haftka R T, Wu K C, Walsh J L, Garcelon J H (2001) Efficient Structural Optimization for Multiple Load Cases Using Adjoint Sensitivities. AIAA J 39(3):511–516. doi: 10.2514/2.1336 CrossRefGoogle Scholar
  2. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43:1453–1478MathSciNetCrossRefzbMATHGoogle Scholar
  3. Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distribution. In: Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, vol 1Google Scholar
  4. Gill P E, Murray W, Saunders M A (2002) SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J Optim 12(4):979–1006MathSciNetCrossRefzbMATHGoogle Scholar
  5. Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48:33–47. doi: 10.1007/s00158-012-0880-7 MathSciNetCrossRefzbMATHGoogle Scholar
  6. Kennedy G J (2015) Strategies for adaptive optimization with aggregation constraints using interior-point methods. Comput Struct 153:217–229. doi: 10.1016/j.compstruc.2015.02.024 CrossRefGoogle Scholar
  7. Kennedy G J, Hicken J E (2015) Improved constraint-aggregation methods. Comput Methods Appl Mech Eng 289:332–354. doi: 10.1016/j.cma.2015.02.017 MathSciNetCrossRefGoogle Scholar
  8. Kennedy G J, Martins J R R A (2014a) A parallel finite-element framework for large-scale gradient-based design optimization of high-performance structures. Finite Elements in Analysis and Design 87:56–73. doi: 10.1016/j.finel.2014.04.011
  9. Kennedy GJ, Martins JRRA (2014b) A parallel aerostructural optimization framework for aircraft design studies. Struct Multidiscip Optim 50(6):1079–1101. doi: 10.1007/s00158-014-1108-9
  10. Kenway G K W, Martins J R R A (2014) Multipoint high-fidelity aerostructural optimization of a transport aircraft configuration. J Aircr 51(1):144–160. doi: 10.2514/1.C032150 CrossRefGoogle Scholar
  11. Kenway G K W, Kennedy G J, Martins J R R A (2010) A CAD-Free Approach to High-Fidelity Aerostructural Optimization. In: 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Fort Worth,TX. doi: 10.2514/6.2010-9231
  12. Kenway G K W, Kennedy G J, Martins J R R A (2014a) Aerostructural optimization of the Common Research Model configuration. In: 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. doi: 10.2514/6.2014-3274, Atlanta
  13. Kenway G K W, Kennedy G J, Martins J R R A (2014b) Scalable parallel approach for high-fidelity steady-state aeroelastic analysis and adjoint derivative computations. AIAA J 52(5):935–951. doi: 10.2514/1.J052255
  14. Kreisselmeier G, Steinhauser R (1979) Systematic Control Design by Optimizing a Vector Performance Indicator. In: Symposium on Computer-Aided Design of Control Systems, IFAC, Zurich, Switzerland, pp 113-117Google Scholar
  15. Kreisselmeier G, Steinhauser R (1983) Application of Vector Performance Optimization to a Robust Control Loop Design for a Fighter Aircraft. Int J Control 37(2):251–284. doi: 10.1080/00207179.1983.9753066  10.1080/00207179.1983.9753066 CrossRefzbMATHGoogle Scholar
  16. Lambe AB, Martins JRRA (2016) Matrix-free aerostructural optimization of aircraft wings. Struct Multidiscip Optim 53(3):589–603MathSciNetCrossRefGoogle Scholar
  17. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Structural and Multidisciplinary Optimization 41(4):605–620. doi: 10.1007/s00158-009-0440-y CrossRefGoogle Scholar
  18. Lyu Z, Kenway G K W, Martins J R R A (2014) Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark. AIAA J 53(4):968–985. doi: 10.2514/6.2014-0567  10.2514/6.2014-0567 CrossRefGoogle Scholar
  19. París J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidiscip Optim 39:419–437. doi: 10.1007/s00158-008-0336-2  10.1007/s00158-008-0336-2 MathSciNetCrossRefzbMATHGoogle Scholar
  20. París J, Navarrina F, Colominas I, Casteleiro M (2010) Block aggregation of stress constraints in topology optimization of structures. Adv Eng Softw 41:433–441. doi: 10.1016/j.advengsoft.2009.03.006  10.1016/j.advengsoft.2009.03.006 CrossRefzbMATHGoogle Scholar
  21. Perez RE, Jansen PW, Martins JRRA (2012) pyOpt: A Python-Based Object-Oriented Framework for Nonlinear Constrained Optimization. Struct Multidiscip Optim 45(1):101–118. doi: 10.1007/s00158-011-0666-3 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Poon NMK, Martins JRRA (2007) An adaptive approach to constraint aggregation using adjoint sensitivity analysis. Struct Multidiscip Optim 34:61–73. doi: 10.1007/s00158-006-0061-7 CrossRefGoogle Scholar
  23. Qiu GY, Li XS (2010) A note on the derivation of global stress constraints. Struct Multidiscip Optim 40:625–628. doi: 10.1007/s00158-009-0397-x MathSciNetCrossRefzbMATHGoogle Scholar
  24. Raspanti CG, Bandoni JA, Biegler LT (2000) New strategies for flexibility analysis and design under uncertainty. Comput Chem Eng 24:2193–2209. doi: 10.1016/S0098-1354(00)00591-3 CrossRefGoogle Scholar
  25. Vassberg J C, DeHaan M A, Rivers S M, Wahls R A (2008) Development of a Common Research Model for applied CFD validation studies. AIAA:2008–6919Google Scholar
  26. van der Weide E, Kalitzin G, Schluter J, Alonso J (2006) Unsteady Turbomachinery Computations Using Massively Parallel Platforms. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings, American Institute of Aeronautics and Astronautics. doi: 10.2514/6.2006-421
  27. Wrenn G A (1989) An Indirect Method for Numerical Optimization Using the Kreisselmeier–Steinhauser Function. Tech. rep. NASA Langley Research Center, Hampton, VAGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Andrew B. Lambe
    • 1
  • Graeme J. Kennedy
    • 2
  • Joaquim R. R. A. Martins
    • 3
  1. 1.Department of Mechanical EngineeringYork University TorontoOntarioCanada
  2. 2.School of Aerospace EngineeringGeorgia Institute of Technology AtlantaGeorgiaUSA
  3. 3.Department of Aerospace EngineeringUniversity of Michigan Ann ArborMichiganUSA

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