Structural and Multidisciplinary Optimization

, Volume 31, Issue 1, pp 49–59 | Cite as

A double-distribution statistical algorithm for composite laminate optimization

  • Laurent GrossetEmail author
  • Rodolphe LeRiche
  • Raphael T. Haftka
Research Paper


The paper proposes a new evolutionary algorithm termed Double-Distribution Optimization Algorithm (DDOA). DDOA belongs to the family of estimation of distribution algorithms (EDA) that build a statistical model of promising regions of the design space based on sets of good points and use it to guide the search. The efficiency of these algorithms is heavily dependent on the model accuracy. In this work, a generic framework for enhancing the model accuracy by incorporating statistical variable dependencies is presented. The proposed algorithm uses two distributions simultaneously: the marginal distributions of the design variables, complemented by the distribution of physically meaningful auxiliary variables. The combination of the two generates more accurate distributions of promising regions at a low computational cost. The paper demonstrates the efficiency of DDOA for three laminate optimization problems where the design variables are the fiber angles, and the auxiliary variables are integral quantities called lamination parameters. The results show that the reliability of DDOA in finding the optima is greater than that of simple EDA and a standard genetic algorithm, and that its advantage increases with the problem dimension.


Stacking sequence optimization Composite laminate Evolutionary computation Estimation of distribution algorithms Lamination parameters 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baluja S (1994) Population-based incremental learning: a method for integrating genetic search based function optimization and competitive learning. Technical Report CMU-CS-94-163. Carnegie Mellon University, Pittsburgh, PAGoogle Scholar
  2. Bosman P, Thierens D (2000) Continuous iterated density estimation evolutionary algorithms within the IDEA framework. In: Optimization by building and using probabilistic models, Las Vegas, NV, USA, pp 197–200Google Scholar
  3. De Bonet JS, Isbell CL Jr, Viola P (1997) MIMIC: finding optima by estimating probability densities. In: Mozer MC, Jordan MI, Petsche T (eds) Advances in neural information processing systems, vol 9. The MIT Press, p 424Google Scholar
  4. Diaconu CG, Sato M, Sekine H (2002) Feasible region in general design space of lamination parameters of laminated composites. AIAA J 50(4):559–565Google Scholar
  5. Foldager J, Hansen JS, Olhoff N (1998) A general approach forcing convexity of ply angle optimization in composite laminates. Struct Optim 16:201–211Google Scholar
  6. Gallagher M, Frean M, Downs T (1999) Real-valued evolutionary optimization using a flexible probability density estimator. In: Banzhaf W, Daida J, Eiben AE, Garzon MH, Honavar V, Jakiela M, Smith RE (eds) Proceedings of the genetic and evolutionary computation conference, Orlando, FL, USA. Morgan Kaufmann, pp 840–846Google Scholar
  7. Gürdal Z, Haftka RT, Hajela P (1999) Design and optimization of laminated composite materials. WileyGoogle Scholar
  8. Haftka RT, Gürdal Z (1992) Elements of structural optimization, 3rd edn. Kluwer Academic PublishersGoogle Scholar
  9. Le Riche R, Haftka RT (1993) Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm. AIAA J 31(5):951–957CrossRefzbMATHGoogle Scholar
  10. McMahon MT, Watson LT, Soremekun GA, Gürdal Z, Haftka RT (1998) A Fortran 90 genetic algorithm module for composite laminate structure design. Eng Comput 14(3):260–273CrossRefGoogle Scholar
  11. Miki M (1986) Optimum design of laminated composite plates subject to axial compression. In: Kawabata K, Umekawaand S, Kobayashi A (eds) Proc. Japan–U.S. CCM-III. pp 673–680Google Scholar
  12. Mühlenbein H, Mahnig T (1999) FDA—A scalable evolutionary algorithm for the optimization of additively decomposed functions. Evol Comput 7(1):45–68Google Scholar
  13. Mühlenbein H, Mahnig T (2000) Evolutionary algorithms: from recombination to search distributions. Theor Asp Evol Comput :137–176Google Scholar
  14. Mühlenbein H, Mahnig T (2002) Evolutionary algorithms and the Boltzmann distribution. In: Proceedings of the foundations of genetic algorithms VII conferenceGoogle Scholar
  15. Mühlenbein H, Paaß G (1996) From recombination of genes to estimation of distributions. I. Binary parameters. Lecture notes in computer science 1411: parallel problem solving from nature—PPSN IV, pp 178–187Google Scholar
  16. Pelikan M, Mühlenbein H (1999) The bivariate marginal distribution algorithm. In: Roy R, Furuhashi T, Chawdhry PK (eds) Advances in soft computing–engineering design and manufacturing. Springer-Verlag, pp 521–535; ISBN 1-85233-062-7Google Scholar
  17. Pelikan M, Goldberg DE, Cantú-Paz E (1999) BOA: the Bayesian optimization algorithm. In: Banzhaf W, Daida J, Eiben AE, Garzon MH, Honavar V, Jakiela M, Smith RE (eds) Proceedings of the genetic and evolutionary computation conference GECCO-99, volume I, Orlando, FL, 13–17 1999. Morgan Kaufmann Publishers, San Francisco, CA, pp 525–532; ISBN 1-55860-611-4Google Scholar
  18. Punch WF, Averill RC, Goodman ED, Lin S-C, Ding Y (1995) Using genetic algorithms to design laminated composite structures. IEEE Intell Syst 10(1):42–49Google Scholar
  19. Sebag M, Ducoulombier A (1998) Extending population-based incremental learning to continuous search spaces. In: Bäck T, Eiben G, Schoenauer M, Schwefel H-P (eds) Proceedings of the 5th conference on parallel problems solving from nature. Springer Verlag, pp 418–427Google Scholar
  20. Todoroki A, Haftka RT (1998) Lamination parameters for efficient genetic optimization of the stacking sequence of composite panels. In: Proc. 7th AIAA/USAF/NASA/ISSMO multidisciplinary analysis and optimization symposium, pp 870–879Google Scholar
  21. Tsai SW, Pagano NJ (1968) Invariant properties of composite materials. In: Tsai SW, Halpin JC, Pagano NJ (eds) Composite materials workshopGoogle Scholar
  22. Turlach BA (1993) Bandwidth selection in kernel density estimation: a review. Technical Report9317, C.O.R.E. and Institut de Statistique, Université Catholique de LouvainGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Laurent Grosset
    • 1
    Email author
  • Rodolphe LeRiche
    • 3
  • Raphael T. Haftka
    • 2
  1. 1.SMSÉcole des Mines de Saint-ÉtienneFrance
  2. 2.Department of Mechanical and Aerospace EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.CNRS UMR 5146/SMSÉcole des Mines de Saint-ÉtienneSaint-ÉtienneFrance

Personalised recommendations