Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1115–1127 | Cite as

Multiobjective optimization of laminated composite parts with curvilinear fibers using Kriging-based approaches

  • A. G. Passos
  • M. A. Luersen


This paper describes the multiobjective optimization of parts made with curvilinear fiber composites. Two structures are studied: a square plate and a fuselage-like section. The square plate is designed in two ways. First, classical lamination theory (CLT) is used to obtain the structural response for a plate with straight fibers designed for maximum buckling load and maximum stiffness. The same plate is then designed with curved fibers using finite element analysis (FEA) to determine the structural response. Next, the fuselage-like section is designed using the same FEA approach. The problems have three to twelve variables. To enable the resulting Pareto front to be visualized more clearly, only two objectives are considered. The first two optimization problems are unconstrained, while the last one is constrained by two project requirements. To overcome the problem of long computational run time when using FEA, Kriging-based approaches are used. Three such approaches suitable for multiobjective problems are compared: (i) the efficient global optimization algorithm (EGO) is applied to a single-objective function consisting of a weighted combination of the objectives, (ii) a technique that involves sequential maximization of the expected hypervolume improvement, and (iii) a novel approach proposed here based on sequential minimization of the variance of the predicted Pareto front. Comparison of the results using the inverted generational distance (IGD) metric revealed that the approach (iii) had the best performance (mean) and best robustness (standard deviation) for all the cases studied.


Multiobjective optimization Kriging Curvilinear fiber composites 


  1. ANSYS (2015a) ANSYS Composite PrepPost User’s Guide, Canonsburg, United States of AmericaGoogle Scholar
  2. ANSYS (2015b) ANSYS Mechanical user’s guide, Canonsburg, United States of AmericaGoogle Scholar
  3. Bathe KJ, Dvorkin EN (1986) A formulation of general shell elements—the use of mixed interpolation of tensorial components. Int J Numer Methods Eng 22(3):697–722CrossRefzbMATHGoogle Scholar
  4. Beume N, Naujoks B, Emmerich M (2007) Sms-emoa: Multiobjective selection based on dominated hypervolume. Eur J Oper Res 181(3):1653–1669CrossRefzbMATHGoogle Scholar
  5. Binois M, Picheny V (2016) GPareto: Gaussian Processes for Pareto Front Estimation and Optimization., r package version 1.0.2
  6. Carnell R (2012) LHS: Latin Hypercube Samples,, r package version 0.10
  7. Chen B, Zeng W, Lin Y, Zhang D (2015) A new local search-based multiobjective optimization algorithm. IEEE Trans Evolutionary Comput 19(1):50–73CrossRefGoogle Scholar
  8. Corne DW, Jerram NR, Knowles JD, Oates MJ et al (2001) PESA-II: Region-Based selection in evolutionary multiobjective optimization. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2001)Google Scholar
  9. Couckuyt I, Deschrijver D, Dhaene T (2014) Fast calculation of multiobjective probability of improvement and expected improvement criteria for pareto optimization. J Glob Optim 60(3):575–594MathSciNetCrossRefzbMATHGoogle Scholar
  10. Deb K (2014) Multi-objective optimization. In: Search methodologies. Springer, pp 403–449Google Scholar
  11. Deb K, Thiele L, Laumanns M, Zitzler E (2002a) Scalable multi-objective optimization test problems. In: Proceedings of the 2002 congress on evolutionary computation, 2002. CEC’02, IEEE, vol 1, pp 825–830Google Scholar
  12. Deb K, Pratap A, Agarwal S, Meyarivan T (2002b) A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  13. Emmerich M, Beume N, Naujoks B (2005) An emo algorithm using the hypervolume measure as selection criterion. In: Evolutionary multi-criterion optimization. Springer, pp 62–76Google Scholar
  14. Emmerich M, Deutz AH, Klinkenberg JW (2011) Hypervolume-based expected improvement: Monotonicity properties and exact computation. In: 2011 IEEE Congress of Evolutionary Computation (CEC). IEEE, pp 2147–2154Google Scholar
  15. Fang J, Sun G, Qiu N, Kim NH, Li Q (2017) On design optimization for structural crashworthiness and its state of the art. Struct Multidiscip Optim 55(3):1091–1119MathSciNetCrossRefGoogle Scholar
  16. Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. Wiley, PondicherryCrossRefGoogle Scholar
  17. Forrester AI, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45 (1):50–79CrossRefGoogle Scholar
  18. Fu G, Khu ST, Butler D (2008) Multiobjective optimisation of urban wastewater systems using parego: a comparison with nsga ii. In: 11th International Conference on Urban Drainage, Edinburgh, ScotlandGoogle Scholar
  19. Ghiasi H, Pasini D, Lessard L (2010) Pareto frontier for simultaneous structural and manufacturing optimization of a composite part. Struct Multidiscip Optim 40(1–6):497–511CrossRefzbMATHGoogle Scholar
  20. Ginsbourger D, Picheny V, Roustant O, Chevalier with contributions by Clément, Wagner T (2013) DiceOptim: Kriging-based optimization for computer experiments., r package version 1.4
  21. Gürdal Z, Olmedo R (1993) In-plane response of laminates with spatially varying fiber orientations-variable stiffness concept. AIAA J 31(4):751–758CrossRefzbMATHGoogle Scholar
  22. Gürdal Z, Haftka RT, Hajela P (1999) Design and optimization of laminated composite materials. Wiley, New YorkGoogle Scholar
  23. Gürdal Z, Tatting BF, Wu C (2008) Variable stiffness composite panels: effects of stiffness variation on the in-plane and buckling response. Compos A: Appl Sci Manuf 39(5):911–922CrossRefGoogle Scholar
  24. Honda S, Igarashi T, Narita Y (2013) Multi-objective optimization of curvilinear fiber shapes for laminated composite plates by using nsga-ii. Composites Part B: Engineering 45(1):1071–1078CrossRefGoogle Scholar
  25. Hupkens I, Emmerich M, Deutz A (2014) Faster computation of expected hypervolume improvement. Tech. rep., LIACSGoogle Scholar
  26. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetCrossRefzbMATHGoogle Scholar
  27. Knowles J (2006) Parego: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans Evol Comput 10(1):50–66CrossRefGoogle Scholar
  28. Knowles J, Hughes EJ (2005) Multiobjective optimization on a budget of 250 evaluations. In: International Conference on Evolutionary Multi-Criterion Optimization. Springer, pp 176–190Google Scholar
  29. Li M (2011) An improved kriging-assisted multi-objective genetic algorithm. J Mech Des 133(7):071, 008CrossRefGoogle Scholar
  30. Lopes CS, Gürdal Z, Camanho P (2010) Tailoring for strength of composite steered-fibre panels with cutouts. Compos A: Appl Sci Manuf 41(12):1760–1767CrossRefGoogle Scholar
  31. Martínez-Frutos J, Herrero-Pérez D (2016) Kriging-based infill sampling criterion for constraint handling in multi-objective optimization. J Glob Optim 64(1):97–115MathSciNetCrossRefzbMATHGoogle Scholar
  32. Mersmann O (2014) MCO: Multiple Criteria Optimization Algorithms and Related Functions., r package version 1.0-15.1
  33. Murugan S, Friswell M (2013) Morphing wing flexible skins with curvilinear fiber composites. Compos Struct 99(May):69–75CrossRefGoogle Scholar
  34. Nik MA, Fayazbakhsh K, Pasini D, Lessard L (2012) Surrogate-based multi-objective optimization of a composite laminate with curvilinear fibers. Compos Struct 94(8):2306–2313CrossRefGoogle Scholar
  35. Parr J, Keane A, Forrester AI, Holden C (2012) Infill sampling criteria for surrogate-based optimization with constraint handling. Eng Optim 44(10):1147–1166CrossRefzbMATHGoogle Scholar
  36. Passos A (2016) moko: Multi-Objective Kriging Optimization., r package version 1.0.0
  37. Passos AG, Luersen MA, Steeves CA (2017) Optimal curved fibre orientations of a composite panel with cutout for improved buckling load using the efficient global optimization algorithm. Eng Optim 49(8):1354–1372CrossRefGoogle Scholar
  38. Pelletier JL, Vel SS (2006) Multi-objective optimization of fiber reinforced composite laminates for strength, stiffness and minimal mass. Comput Struct 84(29):2065–2080CrossRefGoogle Scholar
  39. Raju G, Wu Z, Kim BC, Weaver PM (2012) Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions. Compos Struct 94(9):2961–2970CrossRefGoogle Scholar
  40. Roustant O, Ginsbourger D, Deville Y (2012) DiceKriging, DiceOptim: Two R packages for the analysis of computer experiments by kriging-based metamodeling and optimization. Journal of Statistical Software 51(1):1–55CrossRefGoogle Scholar
  41. Sasena MJ, Papalambros P, Goovaerts P (2002) Exploration of metamodeling sampling criteria for constrained global optimization. Eng Optim 34(3):263–278CrossRefGoogle Scholar
  42. Shimoyama K, Jeong S, Obayashi S (2013) Kriging-surrogate-based optimization considering expected hypervolume improvement in non-constrained many-objective test problems. In: 2013 IEEE Congress on Evolutionary computation (CEC), IEEE, pp 658–665Google Scholar
  43. Stodieck O, Cooper JE, Weaver P, Kealy P (2015) Optimization of tow-steered composite wing laminates for aeroelastic tailoring. AIAA J 53(8):2203–2215CrossRefGoogle Scholar
  44. Tabatabaei M, Hakanen J, Hartikainen M, Miettinen K, Sindhya K (2015) A survey on handling computationally expensive multiobjective optimization problems using surrogates: non-nature inspired methods. Struct Multidiscip Optim 52(1):1–25MathSciNetCrossRefGoogle Scholar
  45. Van Veldhuizen DA, Lamont GB (1998) Multiobjective evolutionary algorithm research: a history and analysis. Tech. rep. , CiteseerGoogle Scholar
  46. Wu Z, Weaver PM, Raju G, Kim BC (2012) Buckling analysis and optimisation of variable angle tow composite plates. Thin-Walled Struct 60:163–172CrossRefGoogle Scholar
  47. Xiang Y, Gubian S, Suomela B, Hoeng J (2013) Generalized simulated annealing for efficient global optimization: the GenSA package for R. The R Journal Volume 5/1, June 2013
  48. Zitzler E (1999) Evolutionary algorithms for multiobjective optimization methods and applications, vol 63. CiteseerGoogle Scholar
  49. Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithms – a comparative case study. In: Parallel problem solving from nature. Springer, pp 292–301Google Scholar
  50. Zitzler E, Laumanns M, Thiele L, Zitzler E, Zitzler E, Thiele L, Thiele L (2001) SPEA2: Improving The strength pareto evolutionary algorithm, Tech. rep., TIKGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFederal University of TechnologyCuritibaBrazil

Personalised recommendations