Anisotropy and Shape Optimal Design of Shells by the Polar–Isogeometric Approach

  • Dosso Felix Kpadonou
  • Christian Fourcade
  • Paul de Nazelle
  • Paolo VannucciEmail author


We focus in this paper on the simultaneous shape and material optimal design of shells by an isogeometric-like approach of a new kind, in which geometry and material properties of the structure are defined by spline functions and the design variables are the polar parameters at the control points. Different kinds of constraints on the regularity, the admissibility of the elastic moduli etc, are taken into account and some numerical examples, proving the effectiveness of the approach, are given.


Shell design Shape Anisotropy Polar formalism Isogeometric method Naghdi’s shell 

Mathematics Subject Classification

49J53 49K99 



The authors sincerely acknowledge RENAULT SA for its support to this research through the granting of the Ph.D. thesis of D. F. Kpadonou.


  1. 1.
    Banichuk, N.V.: Problems and Methods of Optimal Structural Design. Springer, New York (1983). CrossRefGoogle Scholar
  2. 2.
    Allaire, G.: Shape Optimization by the Homogenization Method. Springer, New York (2002). CrossRefzbMATHGoogle Scholar
  3. 3.
    Bendsøe, M.P., Sigmund, O.: Topology Optimization. Springer, Berlin (2004). CrossRefzbMATHGoogle Scholar
  4. 4.
    Gurdal, Z., Haftka, R.T., Hajela, P.: Design and Optimization of Laminated Composite Materials. Wileys, New York (1999)Google Scholar
  5. 5.
    Vannucci, P.: Anisotropic Elasticity. Springer, Singapore (2018). CrossRefzbMATHGoogle Scholar
  6. 6.
    Montemurro, M., Vincenti, A., Vannucci, P.: Design of the elastic properties of laminates with a minimum number of plies. Mech. Compos. Mater. 48(4), 369–390 (2012). CrossRefGoogle Scholar
  7. 7.
    Vannucci, P.: Designing the elastic properties of laminates as an optimisation problem: a unified approach based on polar tensor invariants. Struct. Multidiscip. Optim. 31(5), 378–387 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Vannucci, P., Vincenti, A.: The design of laminates with given thermal/hygral expansion coefficients: a general approach based upon the polar-genetic method. Compos. Struct. 79(3), 454–466 (2007). CrossRefGoogle Scholar
  9. 9.
    Vannucci, P., Barsotti, R., Bennati, S.: Exact optimal flexural design of laminates. Compos. Struct. 90(3), 337–345 (2009). CrossRefGoogle Scholar
  10. 10.
    Vincenti, A., Desmorat, B.: Optimal orthotropy for minimum elastic energy by the polar method. J. Elast. 102(1), 55–78 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Vincenti, A., Vannucci, P., Ahmadian, M.R.: Optimization of laminated composites by using genetic algorithm and the polar description of plane anisotropy. Mech. Adv. Mater. Struct. 20(3), 242–255 (2013). CrossRefGoogle Scholar
  12. 12.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jibawy, A., Julien, C., Desmorat, B., Vincenti, A., Léné, F.: Hierarchical structural optimization of laminated plates using polar representation. Int. J. Solids Struct. 48(18), 2576–2584 (2011). CrossRefGoogle Scholar
  14. 14.
    Montemurro, M., Vincenti, A., Vannucci, P.: A two-level procedure for the global optimum design of composite modular structures—application to the design of an aircraft wing. Part 1: theoretical formulation. J. Optim. Theory Appl. 155(1), 1–23 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Montemurro, M., Vincenti, A., Vannucci, P.: A two-level procedure for the global optimum design of composite modular structures—application to the design of an aircraft wing. Part 2: numerical aspects and examples. J. Optim. Theory Appl. 155, 24–53 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Vannucci, P.: Strange laminates. Math. Methods Appl. Sci. 35(13), 1532–1546 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hughes, T., Cottrell, J., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput. Mech. 43(1), 3–37 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bazilevs, Y., Calo, V.M., Zhang, Y., Hughes, T.J.R.: Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput. Mech. 38(4), 310–322 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bazilevs, Y., Hsu, M.C., Scott, M.: Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput. Methods Appl. Mech. Eng. 249–252, 28–41 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cho, S., Ha, S.H.: Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct. Multidiscip. Optim. 38(1), 53–70 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Qian, X.: Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput. Methods Appl. Mech. Eng. 199(29–32), 2059–2071 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    De Nazelle, P.: Paramétrage de formes surfaciques pour l’optimisation. Ph.D. thesis, Ecole Centrale Lyon (2013)Google Scholar
  24. 24.
    Julisson, S.: Shape optimization of thin shell structures for complex geometries. Ph.D. thesis, Paris-Saclay University. (2016)
  25. 25.
    Kpadonou, D.F.: Shape and anisotropy optimization by an isogeometric-polar approach. Ph.D. thesis, Paris-Saclay University (2017)Google Scholar
  26. 26.
    Montemurro, M., Catapano, A.: A new paradigm for the optimum design of variable angle tow laminates. In: Frediani, A., Mohammadi, B., Pironneau, O., Cipolla, V. (eds.) Variational Analysis and Aerospace Engineering: Mathematical Challenges for the Aerospace of the Future, Springer Optimization and Its Applications, vol. 116, pp. 375–400. Springer (2016).
  27. 27.
    Montemurro, M., Catapano, A.: On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates. Compos. Struct. 161, 145–159 (2017)CrossRefGoogle Scholar
  28. 28.
    Morgan, K.: The finite element method for elliptic problems, phillipe g. ciarlet, north-holland, amsterdam, 1978. no. of pages 530. price \({\$}57.75\). Int. J. Numer. Methods Eng. 14(5), 786–786 (1979). CrossRefGoogle Scholar
  29. 29.
    Banichuk, N.V.: Introduction to Optimization of Structures. Springer, New York (1990). CrossRefzbMATHGoogle Scholar
  30. 30.
    Ciarlet, P.G.: An Introduction to Differential Geometry with Applications to Elasticity. Springer, New York (2005). CrossRefzbMATHGoogle Scholar
  31. 31.
    Love, A.E.H.: The small free vibrations and deformation of a thin elastic shell. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 179, 491–546 (1888). CrossRefzbMATHGoogle Scholar
  32. 32.
    Koiter, W.: Foundations and Basic Equations of Shell Theory: A Survey of Recent Progress. Afdeling der Werktuigbouwkunde: WTHD. Labor. voor Techn. Mechanica (1968)Google Scholar
  33. 33.
    Naghdi, P.: Foundations of Elastic Shell Theory. North-Holland Publishing CO., Amsterdam (1963)Google Scholar
  34. 34.
    Reissner, E.: On the theory of transverse bending of elastic plates. Int. J. Solids Struct. 12(8), 545–554 (1976). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bézier, P.: Essai de définition numérique des courbes et des surfaces expérimentales: contribution à l’étude des propriétés des courbes et des surfaces paramétriques polynomiales à coefficients vectoriels, vol. 1. Ph.D. thesis, University Paris 6 (1977)Google Scholar
  36. 36.
    Rogers, D.F.: An Introduction to NURBS with Historical Perspective. Elsevier, Amsterdam (2001). CrossRefGoogle Scholar
  37. 37.
    de Boor, C.: On the evaluation of box splines. Numer. Algorithms 5(1), 5–23 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Verchery, G.: Les invariants des tenseurs d’ordre 4 du type de l’élasticité. In: Boehler, J.P. (ed.) Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, pp. 93–104. Springer, Berlin (1982). CrossRefGoogle Scholar
  39. 39.
    Vannucci, P.: Plane anisotropy by the polar method. Meccanica 40, 437–454 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Vannucci, P.: A note on the elastic and geometric bounds for composite laminates. J. Elast. 112, 199–215 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Vannucci, P., Verchery, G.: Anisotropy of plane complex elastic bodies. Int. J. Solids Struct. 47, 1154–1166 (2010). CrossRefzbMATHGoogle Scholar
  42. 42.
    Valot, E., Vannucci, P.: Some exact solutions for fully orthotropic laminates. Compos. Struct. 69(2), 157–166 (2005). CrossRefGoogle Scholar
  43. 43.
    Vannucci, P., Pouget, J.: Laminates with given piezoelectric expansion coefficients. Mech. Adv. Mater. Struct. 13(5), 419–427 (2006). CrossRefGoogle Scholar
  44. 44.
    Vannucci, P.: Influence of invariant material parameters on the flexural optimal design of thin anisotropic laminates. Int. J. Mech. Sci. 51, 192–203 (2009). CrossRefzbMATHGoogle Scholar
  45. 45.
    Vannucci, P.: A new general approach for optimizing the performances of smart laminates. Mech. Adv. Mater. Struct. 18(7), 548–558 (2011). CrossRefGoogle Scholar
  46. 46.
    Montemurro, M., Koutsawa, Y., Belouettar, S., Vincenti, A., Vannucci, P.: Design of damping properties of hybrid laminates through a global optimisation strategy. Compos. Struct. 94(11), 3309–3320 (2012). CrossRefGoogle Scholar
  47. 47.
    Vannucci, P.: The design of laminates as a global optimization problem. J. Optim. Theory Appl. 157(2), 299–323 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Montemurro, M., Vincenti, A., Koutsawa, Y., Vannucci, P.: A two-level procedure for the global optimization of the damping behavior of composite laminated plates with elastomer patches. J. Vib. Control 21(9), 1778–1800 (2013). MathSciNetCrossRefGoogle Scholar
  49. 49.
    Vannucci, P.: The polar analysis of a third order piezoelectricity-like plane tensor. Int. J. Solids Struct. 44(24), 7803–7815 (2007). CrossRefzbMATHGoogle Scholar
  50. 50.
    Vannucci, P.: On special orthotropy of paper. J. Elast. 99(1), 75–83 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Catapano, A., Desmorat, B., Vannucci, P.: Invariant formulation of phenomenological failure criteria for orthotropic sheets and optimisation of their strength. Math. Methods Appl. Sci. 35(15), 1842–1858 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Barsotti, R., Vannucci, P.: Wrinkling of orthotropic membranes: an analysis by the polar method. J. Elast. 113(1), 5–26 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Vannucci, P.: General theory of coupled thermally stable anisotropic laminates. J. Elast. 113(2), 147–166 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Desmorat, B., Vannucci, P.: An alternative to the Kelvin decomposition for plane anisotropic elasticity. Math. Methods Appl. Sci. 38(1), 164–175 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Vannucci, P., Desmorat, B.: Analytical bounds for damage induced planar anisotropy. Int. J. Solids Struct. 60–61, 96–106 (2015). CrossRefGoogle Scholar
  56. 56.
    Vannucci, P.: A note on the computation of the extrema of young’s modulus for hexagonal materials: an approach by planar tensor invariants. Appl. Math. Comput. 270, 124–129 (2015). MathSciNetCrossRefGoogle Scholar
  57. 57.
    Vannucci, P., Desmorat, B.: Plane anisotropic rari-constant materials. Math. Methods Appl. Sci. 39(12), 3271–3281 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Vannucci, P.: A special planar orthotropic material. J. Elast. 67, 81–96 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Vannucci, P., Verchery, G.: Stiffness design of laminates using the polar method. Int. J. Solids Struct. 38(50–51), 9281–9294 (2001). CrossRefzbMATHGoogle Scholar
  60. 60.
    de Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences. Springer, New York (2001)zbMATHGoogle Scholar
  61. 61.
    Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Berlin (2009)zbMATHGoogle Scholar
  62. 62.
    Hooke, R.: A Description of Helioscopes, and Some Other Instruments. John and Martyn Printer, London (1675)Google Scholar
  63. 63.
    Heyman, J.: The Stone Skeleton. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  64. 64.
    Cowan, H.J.: The Masterbuilders. Wiley, New York (1977)Google Scholar

Copyright information

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Authors and Affiliations

  • Dosso Felix Kpadonou
    • 1
  • Christian Fourcade
    • 2
  • Paul de Nazelle
    • 3
  • Paolo Vannucci
    • 1
    Email author
  1. 1.Laboratoire de Mathématiques de Versailles, UMR 8100Université de VersaillesVersaillesFrance
  2. 2.Renault SASGuyancourtFrance
  3. 3.Institut de Recherche Technologique SystemXPalaiseauFrance

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