Concurrent optimization of material spatial distribution and material anisotropy repartition for two-dimensional structures


An optimization methodology to find concurrently material spatial distribution and material anisotropy repartition is proposed for orthotropic, linear and elastic two-dimensional membrane structures. The shape of the structure is parameterized by a density variable that determines the presence or absence of material. The polar method is used to parameterize a general orthotropic material by its elasticity tensor invariants by change of frame. A global structural stiffness maximization problem written as a compliance minimization problem is treated, and a volume constraint is applied. The compliance minimization can be put into a double minimization of complementary energy. An extension of the alternate directions algorithm is proposed to solve the double minimization problem. The algorithm iterates between local minimizations in each element of the structure and global minimizations. Thanks to the polar method, the local minimizations are solved explicitly providing analytical solutions. The global minimizations are performed with finite element calculations. The method is shown to be straightforward and efficient. Concurrent optimization of density and anisotropy distribution of a cantilever beam and a bridge are presented.

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  1. 1.

    Allaire, G., Bonnetier, E., Francfort, G., Jouve, F.: Shape optimization by the homogenization method. Numer. Math. 76(1), 27–68 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Allaire, G., Kohn, R.V.: Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur. J. Mech. A/Solids 6, 839–878 (1993)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Allaire, G.: Conception optimale des structures. Springer, New York (2002)

    Google Scholar 

  4. 4.

    Andreaus, U., Colloca, M., Iacoviello, D.: Optimal bone density distributions: numerical analysis of the osteocyte spatial influence in bone remodeling. Comput. Methods Prog. Biomed. 113(1), 80–91 (2014)

    Article  Google Scholar 

  5. 5.

    Arora, J.S., Belegundu, A.D.: Structural optimization by mathematical programming methods. AIAA J. 22(6), 854–856 (1984)

    ADS  Article  Google Scholar 

  6. 6.

    Bendsœ, M.P.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin, Heidelberg (1995)

    Google Scholar 

  7. 7.

    Bendsœ, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1, 193–202 (1989)

    Article  Google Scholar 

  8. 8.

    Bendsœ, M.P., Sigmund, O.: Topology Optimization. Theory, Methods and Applications. Springer, Berlin, Heidelberg (2004)

  9. 9.

    Berrehili, Y., Marigo, J.-J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst. 2(2), 181–207 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bordogna, M. T., Macquart, T., Bettebghor, D., De Breuker, R.: Aeroelastic Optimization of Variable Stiffness Composite Wing with Blending Constraints. American Institute of Aeronautics and Astronautics (2016)

  11. 11.

    Bourdin, B., Chambolle, A.: Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9, 19–48 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Bourdin, B., Chambolle, A.: The phase-field method in optimal design. IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, vol. 137, pp. 207–215. Springer, Dordrecht (2006)

    Google Scholar 

  13. 13.

    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Bruns, T.E.: A reevaluation of the SIMP method with filtering and an alternative formulation for solidvoid topology optimization. Struct. Multidiscip. Optim. 30(6), 428–436 (2005)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Burger, M., Osher, S.J.: A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math. 16(2), 263–301 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Catapano, A., Desmorat, B., Vannucci, P.: Stiffness and strength optimization of the anisotropy distribution for laminated structures. J. Optim. Theory Appl. 167(1), 118–146 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Deaton, J.D., Grandhi, R.V.: A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49(1), 1–38 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift für angewandte Mathematik und Physik 66(6), 3473–3498 (2015)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 21 (2016).

    Article  Google Scholar 

  20. 20.

    Desmorat, B.: Structural rigidity optimization with frictionless unilateral contact. Int. J. Solids Struct. 44(3), 1132–1144 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Desmorat, B.: Structural rigidity optimization with an initial design dependent stress field. Application to thermo-elastic stress loads. Eur. J. Mech. A. Solids 37, 150–159 (2013)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Fuchs, M.B., Jiny, S., Peleg, N.: The SRV constraint for 0/1 topological design. Struct. Multidiscip. Optim. 30(4), 320–326 (2005)

    Article  Google Scholar 

  24. 24.

    Gersborg-Hansen, A., Bendsœ, M.P., Sigmund, O.: Topology optimization of heat conduction problems using the finite volume method. Struct. Multidiscip. Optim. 31(4), 251–259 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Ghiasi, H., Fayazbakhsh, K., Pasini, D., Lessard, L.: Optimum stacking sequence design of composite materials part II: variable stiffness design. Compos. Struct. 93(1), 1–13 (2010)

    Article  Google Scholar 

  26. 26.

    Ghiasi, H., Pasini, D., Lessard, L.: Optimum stacking sequence design of composite materials part I: constant stiffness design. Compos. Struct. 90(1), 1–11 (2009)

    Article  Google Scholar 

  27. 27.

    Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.J.: Buckling modes in pantographic lattices. C. R. Mech. 344(7), 487–501 (2016)

    Article  Google Scholar 

  28. 28.

    Haftka, R.T., Gürdal, Z.: Elements of Structural Optimization, volume 11 of Solid Mechanics And Its Applications, vol. 11. Springer, Netherlands (1992)

  29. 29.

    Han, J., Bertram, A., Olschewski, J., Hermann, W., Sockel, H.-G.: Identification of elastic constants of alloys with sheet and fibre textures based on resonance measurements and finite element analysis. Mater. Sci. Eng. A 191(1–2), 105–111 (1995)

    Article  Google Scholar 

  30. 30.

    IJsselmuiden, S. T.: Optimal design of variable stiffness composite structures using lamination parameters. Ph. D. Thesis. Delft University of Technology, Delft, Netherlands (2011)

  31. 31.

    Irisarri, F.-X., Bassir, D.Hm, Carrere, N., Maire, J.-F.: Multiobjective stacking sequence optimization for laminated composite structures. Compos. Sci. Technol. 69(7), 983–990 (2009)

    Article  Google Scholar 

  32. 32.

    Irisarri, F.-X., Peeters, D.M.J., Abdalla, M.: Optimisation of ply drop order in variable stiffness laminates. Compos. Struct. 152, 791–799 (2016)

    Article  Google Scholar 

  33. 33.

    Jia, H., Misra, A., Poorsolhjouy, P., Liu, C.: Optimal structural topology of materials with micro-scale tension-compression asymmetry simulated using granular micromechanics. Mater. Des. 115, 422–432 (2017)

    Article  Google Scholar 

  34. 34.

    Jibawy, A., Julien, C., Desmorat, B., Vincenti, A., Lén’e, F.: Hierarchical structural optimization of laminated plates using polar representation. Int. J. Solids Struct. 48(18), 2576–2584 (2011)

    Article  Google Scholar 

  35. 35.

    Julien, C.: Conception Optimale de l’Anisotropie dans les Structures Stratifiées à Rigidité Variable par la Méthode Polaire-Génétique. Ph.D. thesis, UPMC (2010)

  36. 36.

    Kirsch, U.: Optimum Structural Design: Concepts, Methods, and Applications. McGraw-Hill, New York (1981)

    Google Scholar 

  37. 37.

    Lekszycki, T.: Functional adaptation of bone as an optimal control problem. J. Theor. Appl. Mech. 43(3), 555–574 (2005)

    Google Scholar 

  38. 38.

    Lekszycki, T., Bucci, S., Del Vescovo, D., Turco, E., Rizzi, N.L.: A comparison between different approaches for modelling media with viscoelastic properties via optimization analyses. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 97(5), 515–531 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Lund, E.: Buckling topology optimization of laminated multi-material composite shell structures. Compos. Struct. 91(2), 158–167 (1989)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Meddaikar, Y.M., Irisarri, F.-X., Abdalla, M.: Laminate optimization of blended composite structures using a modified Shepard’s method and stacking sequence tables. Struct. Multidiscip. Optim. 55(2), 535–546 (2017)

    Article  Google Scholar 

  41. 41.

    Melnik, A.V., Goriely, A.: Dynamic fiber reorientation in a fiber-reinforced hyperelastic material. Math. Mech. Solids 18, 634–648 (2013).

    MathSciNet  Article  Google Scholar 

  42. 42.

    Miki, M., Sugiyamat, Y.: Optimum design of laminated composite plates using lamination parameters. AIAA J. 31(5), 921–922 (1993)

    ADS  Article  Google Scholar 

  43. 43.

    Norato, J.A., Bendsœ, M.P., Haber, R.B., Tortorelli, D.A.: A topological derivative method for topology optimization. Struct. Multidiscip. Optim. 33(4), 375–386 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Peeters, D., van Baalen, D., Abdallah, M.: Combining topology and lamination parameter optimisation. Struct. Multidiscip. Optim. 52(1), 105–120 (2015)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Placidi, L., Barchiesi, E., Della Corte, A.: Identification of two-dimensional pantographic structures with a linear D4 orthotropic second gradient elastic model accounting for external bulk double forces. In: Mathematical Modelling in Solid Mechanics (pp. 211–232). Springer, Singapore (2017)

  46. 46.

    Prager, W.: Introduction to Structural Optimization. Springer, Vienna (1972)

    Google Scholar 

  47. 47.

    Rozvany, G.I.N., Bendsoe, M.P., Kirsch, U.: Layout optimization of structures. Appl. Mech. Rev. 48(2), 41 (1995)

    ADS  Article  Google Scholar 

  48. 48.

    Schittkowski, K.: Software for mathematical programming. In: Schittkowski, K. (ed.) Computational Mathematical Programming, pp. 383–451. Springer, Berlin, Heidelberg (1985)

    Google Scholar 

  49. 49.

    Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163(2), 489–528 (2000)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Sigmund, O., Maute, K.: Topology optimization approaches: a comparative review. Struct. Multidiscip. Optim. 48(6), 1031–1055 (2013)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Sorensen, S.N., Lund, E.: Topology and thickness optimization of laminated composites including manufacturing constraints. Struct. Multidiscip. Optim. 48(2), 249–265 (2013)

    MathSciNet  Article  Google Scholar 

  52. 52.

    Stolpe, M., Svanberg, K.: An alternative interpolation scheme for minimum compliance topology optimization. Struct. Multidiscip. Optim. 22(2), 116–124 (2001)

    Article  Google Scholar 

  53. 53.

    Svanberg, K.: The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Meth. Eng. 24(2), 359–373 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4), 1–28 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Vannucci, P.: Plane anisotropy by the polar method. Meccanica 40(4–6), 437–454 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Vannucci, P.: A note on the elastic and geometric bounds for composite laminates. J. Elasticity 199–215 (July 2013)

  57. 57.

    Verchery, G.: Les invariants des tenseurs dordre 4 du type de l’élasticité. In: Mechanical behavior of anisotropic solids/comportment Méchanique des Solides Anisotropes (pp. 93–104). Springer (1982)

  58. 58.

    Vincenti, A., Desmorat, B.: Optimal orthotropy for minimum elastic energy by the polar method. J. Elast. 102(1), 55–78 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Zillober, C.: A globally convergent version of the method of moving asymptotes. Struct. Optim. 6(3), 166–174 (1993)

    Article  Google Scholar 

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Ranaivomiarana, N., Irisarri, F., Bettebghor, D. et al. Concurrent optimization of material spatial distribution and material anisotropy repartition for two-dimensional structures. Continuum Mech. Thermodyn. 31, 133–146 (2019).

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  • Topology optimization
  • SIMP
  • Distributed orthotropy
  • Polar method
  • Material design