Abstract

This paper addresses a constrained optimization problem in the context of Topological Optimum Design (TOD): the aim is to find the optimal shape of a structure (i.e a repartition of material in a given design domain) such that the mechanical behavior of that structure meets some requirement (e.g. a bound on the maximal displacement under a prescribed loading). We restrict to stochastic optimization methods such as Evolutionary Algorithms (EAs): they do not require any a priori assumption about the function to optimize (or about the constraints) and they are able to tackle optimization problems on different kinds of search spaces. The most crucial step when constructing an EA is the choice of representation, which determines the search space. In order to overcome limitation of previous works, a new representation is presented, termed Voronoi representation, which is independent of any priori discretization. Moreover, constraints are accounted for through penalty function, and a new adaptive penalty method is proposed to explore the neighborhood of the boundary of the feasible region. The results of TOD of standard benchmark 2-D cantilever problems are improved. Further, this approach allows to address 3-D problems, on which it demonstrates its ability to find multiple quasi-optimal solutions

Author information

Contact details are reproduced from the original publication and may be historical.

Marc Schoenauer

CMAP - UMR CNRS 7641, École Polytechnique, 91128 Palaiseau, France

References

  1. G. Allaire, Z. Belhachmi, F. Jouve: The homogenization method for topology and shape optimization. Single and multiple loads case, European Journal of Finite Elements 15(5–6) (1996) 649–672.https://doi.org/10.1080/12506559.1996.10511241Google Scholar
  2. G. Allaire, E. Bonnetier, G. Francfort, F. Jouve: Shape optimization by the homogenization method, Numerische Mathematik 76 (1997) 27–68.https://doi.org/10.1007/s002110050253Google Scholar
  3. G. Allaire, R. V. Kohn: Optimal design for minimum weight and compliance in plane stress using extremal microstructures, European Journal of Mechanics, A/Solids 12(6) (1993) 839–878.Google Scholar
  4. G. Anagnostou, E. Ronquist, A. Patera: A computational procedure for part design, Computer Methods in Applied Mechanics and Engineering 97 (1992) 33–48.https://doi.org/10.1016/0045-7825(92)90106-TGoogle Scholar
  5. T. Bäck: Evolutionary Algorithms in Theory and Practice, Oxford University Press, New York (1995).https://doi.org/10.1093/oso/9780195099713.001.0001Google Scholar
  6. M. Bendsoe, N. Kikuchi: Generating Optimal Topologies in Structural Design Using a Homogenization Method, Computer Methods in Applied Mechanics and Engineering 71 (1988) 197–224.https://doi.org/10.1016/0045-7825(88)90086-2Google Scholar
  7. S. BenHamida, M. Schoenauer: An adaptive algorithm for constrained optimization problems, in: Proceedings of the 6th Conference on Parallel Problems Solving from Nature, M. Schoenauer et al. (eds.), Springer-Verlag, LNCS (2000) 529–539https://doi.org/10.1007/3-540-45356-3_52Google Scholar
  8. O. Bohnenberger, J. Hesser, R. Männer: Automatic design of truss structures using evolutionary algorithms, in: Proceedings of the Second IEEE International Conference on Evolutionary Computation 1, D. B. Fogel (ed.), IEEE Press (1995) 143–149.https://doi.org/10.1109/ICEC.1995.489132Google Scholar
  9. J. Cea: Problems of shape optimum design, in: Optimization of Distributed Parameter Structures - Vol. II, E. J. Haug, J. Cea (eds.), NATO Series, Series E 50 (1981) 1005–1088.https://doi.org/10.1007/978-94-009-8606-0_10Google Scholar
  10. R. Cerf: An asymptotic theory of genetic algorithms, in: Artificial Evolution, J.-M. Alliot, E. Lutton, E. Ronald, M. Schoenauer, D. Snyers (eds.), Springer-Verlag, LNCS 1063 (1996) 37–53.Google Scholar
  11. C. D. Chapman, K. Saitou, M. J. Jakiela: Genetic algorithms as an approach to configuration and topology design, Journal of Mechanical Design 116 (1994) 1005–1012.https://doi.org/10.1115/1.2919480Google Scholar
  12. P. G. Ciarlet: Mathematical Elasticity, Vol I: Three-Dimensional Elasticity, North-Holland, Amsterdam (1978).Google Scholar
  13. D. Dasgupta, Z. Michalewicz (eds.): Evolutionary Computation in Engineering, Springer-Verlag (1997).Google Scholar
  14. D. Delahaye, J.-M. Alliot, M. Schoenauer, J.-L. Farges: Genetic algorithms for automatic regrouping of air traffic control sectors, in: Proceedings of the 4 th Annual Conference on Evolutionary Programming, J. R. McDonnell, R. G. Reynolds, D. B. Fogel (eds.), MIT Press, March (1995) 657–672.https://doi.org/10.7551/mitpress/2887.003.0057Google Scholar
  15. A. E. Eiben, Z. Ruttkay: Self-adaptivity for constraint satisfaction: learning penalty functions, in: Proceedings of the Third IEEE International Conference on Evolutionary Computation, T. Fukuda (ed.), IEEE Service Center (1996) 258–261.https://doi.org/10.1109/ICEC.1996.542371Google Scholar
  16. D. B. Fogel: Evolutionary Computation. Toward a New Philosophy of Machine Intelligence, IEEE Press, Piscataway, NJ (1995).Google Scholar
  17. C. M. Fonseca, P. J. Fleming: An overview of evolutionary algorithms in multiobjective optimization, Evolutionary Computation 3(1) (1995) 1–16.https://doi.org/10.1162/evco.1995.3.1.1Google Scholar
  18. S. Garreau, Ph. Guillaume, M. Masmoudi: The topological sensitivity for linear isotropic elasticity, in: Proceedings of European Conference on Computational Mechanics (1999).Google Scholar
  19. D. Grierson, W. Pak: Discrete optimal design using a genetic algorithm, in: Topology Design of Structures, M. Bendsoe, C. Soares (eds.), NATO Series (1993) 117–133.https://doi.org/10.1007/978-94-011-1804-0_8Google Scholar
  20. A. B. Hadj-Alouane, J. C. Bean: A Genetic Algorithm for the Multiple-Choice Integer Program, Department of Industrial and Operations Engineering, The University of Michigan, TR 92-50 (1992).Google Scholar
  21. H. Hamda, M. Schoenauer: Adaptive techniques for evolutionary topological optimum design, in: Evolutionary Design and Manufacture, I. Parmee (ed.) (2000) 123–136.https://doi.org/10.1007/978-1-4471-0519-0_10Google Scholar
  22. R. Hinterding, Z. Michalewicz, A. E. Eiben: Adaptation in evolutionary computation: a survey, in: Proceedings of the Fourth IEEE International Conference on Evolutionary Computation, T. Bäck, Z. Michalewicz, X. Yao (eds.), IEEE Press (1997) 65–69.https://doi.org/10.1109/ICEC.1997.592270Google Scholar
  23. E. Jensen: Topological Structural Design using Genetic Algorithms, Purdue University, November (1992).Google Scholar
  24. J. A. Joines, C. R. Houck: On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GAs, in: Proceedings of the First IEEE International Conference on Evolutionary Computation, Z. Michalewicz, J. D. Schaffer, H.-P. Schwefel, D. B. Fogel, H. Kitano (eds.), IEEE Press (1994) 579–584.https://doi.org/10.1109/ICEC.1994.349995Google Scholar
  25. F. Jouve: Modélisation mathématique de l’œil en élasticité non-linéaire, RMA 26, Masson Paris (1993).Google Scholar
  26. L. Kallel, M. Schoenauer: Alternative random initialization in genetic algorithms, in: Proceedings of the 7th International Conference on Genetic Algorithms, Th. Bäck (ed.), Morgan Kaufmann (1997) 268–275.Google Scholar
  27. C. Kane: Algorithmes génétiques et optimisation topologique, Université de Paris VI, July (1996)Google Scholar
  28. C. Kane, M. Schoenauer: Genetic operators for two-dimensional shape optimization, in: Artificial Evolution, J.-M. Alliot, E. Lutton, E. Ronald, M. Schoenauer, D. Snyers (eds.), Springer-Verlag, LNCS 1063 (1995) 355–369.https://doi.org/10.1007/3-540-61108-8_50Google Scholar
  29. C. Kane, M. Schoenauer: Topological Optimum Design using Genetic Algorithms, Control and Cybernetics 25(5) (1996) 1059–1088.Google Scholar
  30. C. Kane, M. Schoenauer: Optimisation topologique de formes par algorithmes génétiques, Revue Française de Mécanique 4 (1997) 237–246.Google Scholar
  31. R. G. Leriche, R. T. Haftka: Optimization of laminate stacking sequence for buckling load maximization by genetic algorithms, AIAA Journal 31 (1993) 951–970.https://doi.org/10.2514/3.11710Google Scholar
  32. R. G. Leriche, C. Knopf-Lenoir, R. T. Haftka: A segregated genetic algorithm for constrained structural optimization, in: Proceedings of the 6 th International Conference on Genetic Algorithms, L. J. Eshelman (ed.), (1995) 558–565.Google Scholar
  33. C. Lin, P. Hajela: Genetic search strategies in large scale optimization, in: Structures, Structural Dynamics, and Materials Conference, AIAA paper #93-1585, La Jolla, CA, April (1993).https://doi.org/10.2514/6.1993-1585Google Scholar
  34. S. Martin, J. Rivory, M. Schoenauer: Synthesis of optical multi-layer systems using genetic algorithms, Applied Optics 34 (1995) 2267.https://doi.org/10.1364/AO.34.002247Google Scholar
  35. Z. Michalewicz: A survey of constraint handling techniques in evolutionary computation methods, in: Proceedings of the 4th Annual Conference on Evolutionary Programming, J. R. McDonnell, R. G. Reynolds, D. B. Fogel (1995).https://doi.org/10.7551/mitpress/2887.003.0018Google Scholar
  36. Z. Michalewicz, C. Z. Janikow: Handling constraints in genetic algorithms, in: Proceedings of the 4th International Conference on Genetic Algorithms, R. K. Belew, L. B. Booker (eds.), Morgan Kaufmann (1991) 151–157.Google Scholar
  37. Z. Michalewicz, M. Schoenauer: Evolutionary algorithms for constrained parameter optimization problems, Evolutionary Computation 4 (1996) 1–32.https://doi.org/10.1162/evco.1996.4.1.1Google Scholar
  38. S. Oussedik, D. Delahaye, M. Schoenauer: Flight alternative routes generation by genetic algorithms, in: Proceedings of the 2000 Congress on Evolutionary Computation, IEEE (2000) 896–901.https://doi.org/10.1109/CEC.2000.870735Google Scholar
  39. F. P. Preparata, M. I. Shamos: Computational Geometry: An Introduction, Springer-Verlag (1985).https://doi.org/10.1007/978-1-4612-1098-6Google Scholar
  40. I. Rechenberg: Evolutionstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution, Fromman-Hozlboog Verlag, Stuttgart (1973).Google Scholar
  41. M. Schoenauer, F. Jouve, L. Kallel: Identification of mechanical inclusions, in: Evolutionary Computation in Engineering, D. Dasgupta, Z. Michalewicz (eds.), Springer-Verlag (1997) 477–494https://doi.org/10.1007/978-3-662-03423-1_26Google Scholar
  42. M. Schoenauer, L. Kallel, F. Jouve: Mechanical inclusions identification by evolutionary computation, European Journal of Finite Elements 5(5–6) (1996) 619–648.https://doi.org/10.1080/12506559.1996.10511240Google Scholar
  43. M. Schoenauer, Z. Michalewicz: Evolutionary computation at the edge of feasibility, in: Proceedings of the 4th Conference on Parallel Problems Solving from Nature, H.-M. Voigt, W. Ebeling, I. Rechenberg, H.-P. Schwefel (eds.), Springer-Verlag, LNCS 1141 (1996) 245–254.https://doi.org/10.1007/3-540-61723-X_989Google Scholar
  44. M. Schoenauer, Z. Michalewiczkey: Boundary operators for constrained parameter optimization problems, in: Proceedings of the 7 th International Conference on Genetic Algorithms, Th. Bäck (ed.), Morgan Kaufmann (1997) 322–329.Google Scholar
  45. M. Schoenauer, Z. Michalewicz: Evolutionary computation, Control and Cybernetics 26(3) (1997) 307–338.https://doi.org/10.1007/978-1-4615-6135-4_1Google Scholar
  46. M. Schoenauer, Z. Wu: Discrete optimal design of structures by genetic algorithms, in: Conférence Nationale sur le Calcul de Structures, Bernadou et al. (eds.) Hermes (1993) 833–842.Google Scholar
  47. M. Schoenauer: Representations for evolutionary optimization and identification in structural mechanics, in: Genetic Algorithms in Engineering and Computer Sciences, J. Périaux, G. Winter (eds.), John Wiley (1995) 443–464.Google Scholar
  48. H.-P. Schwefel: Numerical Optimization of Computer Models, John Wiley & Sons, New York (1985), 2nd edition (1995).Google Scholar
  49. A. Smith, D. Tate: Genetic optimization using a penalty function, in: Proceedings of the 5th International Conference on Genetic Algorithms, S. Forrest (ed.), Morgan Kaufmann (1993) 499–503.Google Scholar
  50. D. Thierens, D. E. Goldberg: Mixing in genetic algorithms, in: Proceedings of the 5 th International Conference on Genetic Algorithms, S. Forrest (ed.), Morgan Kaufmann (1993) 38–55.Google Scholar

Suggested citation

H. Hamda, M. Schoenauer. “Topological Optimum Design with Evolutionary Algorithms.” Journal of Convex Analysis 9 (2002), No. 2, 503–517. https://doi.org/10.68381/jca09029

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.