SpringerLink
Log in

The Polar-Isogeometric Method for the Simultaneous Optimization of Shape and Material Properties of Anisotropic Shell Structures

  • Chapter
  • First Online:
Variational Views in Mechanics

Part of the book series: Advances in Mechanics and Mathematics ((ACM,volume 46))

  • 566 Accesses

Abstract

A new kind of structural optimization problem is tackled in this study: the simultaneous optimal design of the shape and of the material properties distribution of an anisotropic shell. Practically, we consider laminated shells made of composite materials, whose elastic properties can vary locally, a possibility now allowed by the recent techniques of additive manufacturing like fiber placing. Then, we ponder on how to design the shape of the shell and the distribution of the material so as to maximize the stiffness of the structure. The approach proposed is an isogeometric-like one while the description of the locally variable anisotropic properties is done using the polar method. That is why the proposed method is called polar-isogeometric approach.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 93.08
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 117.69
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
EUR 117.69
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Banichuk, N.V.: Problems and Methods of Optimal Structural Design. Springer US, Berlin (1983). https://doi.org/10.1007/978-1-4613-3676-1

  2. Allaire, G.: Shape Optimization by the Homogenization Method. Springer, New York (2002). https://doi.org/10.1007/978-1-4684-9286-6

  3. Bendsøe, M.P., Sigmund, O.: Topology Optimization. Springer, Berlin (2004). https://doi.org/10.1007/978-3-662-05086-6

  4. Gurdal, Z., Haftka, R.T., Hajela, P.: Design and Optimization of Laminated Composite Materials. Wiley, New York (1999)

    Google Scholar 

  5. Vannucci, P.: Anisotropic Elasticity. Springer, Singapore (2018). https://doi.org/10.1007/978-981-10-5439-6

  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). https://doi.org/10.1007/s11029-012-9284-4

    Article  Google Scholar 

  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). https://doi.org/10.1007/s00158-005-0566-5

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.compstruct.2006.02.004

    Article  Google Scholar 

  9. Vannucci, P., Barsotti, R., Bennati, S.: Exact optimal flexural design of laminates. Compos. Struct. 90(3), 337–345 (2009). https://doi.org/10.1016/j.compstruct.2009.03.017

    Article  Google Scholar 

  10. Vincenti, A., Desmorat, B.: Optimal orthotropy for minimum elastic energy by the polar method. J. Elast. 102(1), 55–78 (2010). https://doi.org/10.1007/s10659-010-9262-9

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1080/15376494.2011.563415

    Article  Google Scholar 

  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). https://doi.org/10.1007/s10957-014-0693-5.

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ijsolstr.2011.05.015

    Article  Google Scholar 

  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). https://doi.org/10.1007/s10957-012-0067-9

  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)

    Google Scholar 

  16. Vannucci, P.: Strange laminates. Math. Methods Appl. Sci. 35(13), 1532–1546 (2012). https://doi.org/10.1002/mma.2539

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.cma.2004.10.008

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s00466-008-0315-x

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s00466-006-0084-3.

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.cma.2012.03.028

    Article  MathSciNet  MATH  Google Scholar 

  21. Cho, S., Ha, S.H.: Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct. Multidiscip. Optim. 38(1), 53–70 (2008). https://doi.org/10.1007/s00158-008-0266-z

    Article  MathSciNet  MATH  Google Scholar 

  22. Qian, X.: Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput. Methods Appl. Mech. Eng. 199(29–32), 2059–2071 (2010). https://doi.org/10.1016/j.cma.2010.03.005

    Article  MathSciNet  MATH  Google Scholar 

  23. De Nazelle, P.: Paramétrage de formes surfaciques pour l’optimisation. Ph.D. thesis, Ecole Centrale Lyon (2013)

    Google Scholar 

  24. Julisson, S.: Shape optimization of thin shell structures for complex geometries. Ph.D. thesis, Paris-Saclay University (2016). https://tel.archives-ouvertes.fr/tel-01503061

  25. Kpadonou, D.F.: Shape and anisotropy optimization by an isogeometric-polar approach. Ph.D. thesis, Paris-Saclay University (2017)

    Google Scholar 

  26. Montemurro, M., Catapano, A.: A new paradigm for the optimum design of variable angle tow laminates. In: Variational analysis and aerospace engineering: mathematical challenges for the aerospace of the future. Springer Optimization and Its Applications, vol. 116, pp. 375–400. Springer, Berlin (2016). http://dx.doi.org/10.1007/978-3-319-45680-5

  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)

    Article  Google Scholar 

  28. Banichuk, N.V.: Introduction to Optimization of Structures. Springer, New York (1990). https://doi.org/10.1007/978-1-4612-3376-3

  29. 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(0), 491–546 (1888). https://doi.org/10.1098/rsta.1888.0016

    MATH  Google Scholar 

  30. Koiter, W.: Foundations and Basic Equations of Shell Theory: A Survey of Recent Progress. Afdeling der Werktuigbouwkunde: WTHD. Labor. voor Techn. Mechanica (1968). https://books.google.fr/books?id=74REHQAACAAJ

  31. Naghdi, P.: Foundations of Elastic Shell Theory. North-Holland, Amsterdam (1963)

    Google Scholar 

  32. Reissner, E.: On the theory of transverse bending of elastic plates. Int. J. Solids Struct. 12(8), 545–554 (1976). https://doi.org/10.1016/0020-7683(76)90001-9

    Article  MathSciNet  MATH  Google Scholar 

  33. Ciarlet, P.: An Introduction to Differential Geometry with Application to Elasticity. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  34. Bernadou, M.: Mèthodes d’éléments finis pour les problèmes de coques minces. Recherches en Mathématiques Appliquées, Masson (1994)

    MATH  Google Scholar 

  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 (1977). https://books.google.fr/books?id=vK4POAAACAAJ

  36. Rogers, D.F.: An Introduction to NURBS with Historical Perspective. Elsevier (2001). https://doi.org/10.1016/b978-1-55860-669-2.x5000-3

  37. Risler, J.J.: Méthodes Mathématiques pour la CAO. Masson (1991)

    Google Scholar 

  38. Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  39. 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 Netherlands (1982). https://doi.org/10.1007/978-94-009-6827-1_7

  40. Vannucci, P.: Plane anisotropy by the polar method. Meccanica 40, 437–454 (2005). https://doi.org/10.1007/s11012-005-2132-z

    Article  MathSciNet  MATH  Google Scholar 

  41. Vannucci, P.: A note on the elastic and geometric bounds for composite laminates. J. Elast. 112, 199–215 (2013). https://doi.org/10.1007/s10659-012-9406-1

    Article  MathSciNet  MATH  Google Scholar 

  42. Vannucci, P., Verchery, G.: Anisotropy of plane complex elastic bodies. Int. J. Solids Struct. 47, 1154–1166 (2010). https://doi.org/10.1016/j.ijsolstr.2010.01.002. http://www.sciencedirect.com/science/article/pii/S002076831000003X

  43. Valot, E., Vannucci, P.: Some exact solutions for fully orthotropic laminates. Compos. Struct. 69(2), 157–166 (2005). https://doi.org/10.1016/j.compstruct.2004.06.007

    Article  Google Scholar 

  44. Vannucci, P., Pouget, J.: Laminates with given piezoelectric expansion coefficients. Mech. Adv. Mater. Struct. 13(5), 419–427 (2006). https://doi.org/10.1080/15376490600777699

    Article  Google Scholar 

  45. Vannucci, P.: Influence of invariant material parameters on the flexural optimal design of thin anisotropic laminates. Int. J. Mech. Sci. 51, 192–203 (2009). https://doi.org/10.1016/j.ijmecsci.2009.01.005. http://www.sciencedirect.com/science/article/pii/S0020740309000198

  46. Vannucci, P.: A new general approach for optimizing the performances of smart laminates. Mech. Adva. Mater. Struct. 18(7), 548–558 (2011). https://doi.org/10.1080/15376494.2011.605015

    Article  Google Scholar 

  47. Montemurro, M., Koutsawa, Y., Belouettar, S., Vincenti, A., Vannucci, P.: Design of dam** properties of hybrid laminates through a global optimisation strategy. Compos. Struct. 94(11), 3309–3320 (2012). https://doi.org/10.1016/j.compstruct.2012.05.003

    Article  Google Scholar 

  48. Vannucci, P.: The design of laminates as a global optimization problem. J. Optim. Theory Appl. 157(2), 299–323 (2012). https://doi.org/10.1007/s10957-012-0175-6

    Article  MathSciNet  MATH  Google Scholar 

  49. Montemurro, M., Vincenti, A., Koutsawa, Y., Vannucci, P.: A two-level procedure for the global optimization of the dam** behavior of composite laminated plates with elastomer patches. J. Vib. Control. 21(9), 1778–1800 (2013). https://doi.org/10.1177/1077546313503358

    Article  MathSciNet  Google Scholar 

  50. Vannucci, P.: The polar analysis of a third order piezoelectricity-like plane tensor. Int. J. Solids Struct. 44(24), 7803–7815 (2007). https://doi.org/10.1016/j.ijsolstr.2007.05.012

    Article  MATH  Google Scholar 

  51. Vannucci, P.: On special orthotropy of paper. J. Elast. 99(1), 75–83 (2009). https://doi.org/10.1007/s10659-009-9232-2

    Article  MathSciNet  MATH  Google Scholar 

  52. 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). https://doi.org/10.1002/mma.2530

    Article  MathSciNet  MATH  Google Scholar 

  53. Barsotti, R., Vannucci, P.: Wrinkling of orthotropic membranes: an analysis by the polar method. J. Elast. 113(1), 5–26 (2012). https://doi.org/10.1007/s10659-012-9408-z

    Article  MathSciNet  MATH  Google Scholar 

  54. Vannucci, P.: General theory of coupled thermally stable anisotropic laminates. J. Elast. 113(2), 147–166 (2012). https://doi.org/10.1007/s10659-012-9415-0

    Article  MathSciNet  MATH  Google Scholar 

  55. Desmorat, B., Vannucci, P.: An alternative to the kelvin decomposition for plane anisotropic elasticity. Math. Methods Appl. Sci. 38(1), 164–175 (2013). https://doi.org/10.1002/mma.3059

    Article  MathSciNet  MATH  Google Scholar 

  56. Vannucci, P., Desmorat, B.: Analytical bounds for damage induced planar anisotropy. Int. J. Solids Struct. 60–61, 96–106 (2015). https://doi.org/10.1016/j.ijsolstr.2015.02.017. http://www.sciencedirect.com/science/article/pii/S0020768315000578

  57. 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). https://doi.org/10.1016/j.amc.2015.08.025

    MathSciNet  MATH  Google Scholar 

  58. Vannucci, P., Desmorat, B.: Plane anisotropic rari-constant materials. Math. Methods Appl. Sci. 39(12), 3271–3281 (2015). https://doi.org/10.1002/mma.3770

    Article  MathSciNet  MATH  Google Scholar 

  59. Vannucci, P.: A special planar orthotropic material. J. Elast. 67, 81–96 (2002). https://doi.org/10.1023/A:1023949729395

    Article  MathSciNet  MATH  Google Scholar 

  60. Vannucci, P., Verchery, G.: Stiffness design of laminates using the polar method. Int. J. Solids Struct. 38(50–51), 9281–9294 (2001). https://doi.org/10.1016/S0020-7683(01)00177-9. http://www.sciencedirect.com/science/article/pii/S0020768301001779

  61. Patrikalakis, N.M., Maekawa, T.: Shape interrogation for computer aided design and manufacturing. Springer, Berlin (2009)

    MATH  Google Scholar 

  62. de Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences. Springer, New York (2001). https://books.google.fr/books?id=m0QDJvBI_ecC

  63. Hooke, R.: A Description of Helioscopes, and Some Other Instruments. John and Martyn Printer, London (1675)

    Google Scholar 

  64. Heyman, J.: The Stone Skeleton. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  65. Cowan, H.J.: The Masterbuilders. Wiley, New York (1977)

    Google Scholar 

Download references

Acknowledgment

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

Author information

Authors and Affiliations

  1. Renaut SAS, Guyancourt, France

    Christian Fourcade

  2. LMV - UMR8100 CNRS and University of Versailles Saint Quentin, Versailles, France

    Paolo Vannucci

  3. Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles, Versailles, France

    Dosso Felix Kpadonou

  4. Institut de Recherche Technologique SystemX, Palaiseau, France

    Paul de Nazelle

Authors
  1. Christian Fourcade
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Vannucci
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dosso Felix Kpadonou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paul de Nazelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Vannucci .

Editor information

Editors and Affiliations

  1. DICEA, University of Florence, Florence, Italy

    Paolo Maria Mariano

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Fourcade, C., Vannucci, P., Kpadonou, D.F., Nazelle, P.d. (2021). The Polar-Isogeometric Method for the Simultaneous Optimization of Shape and Material Properties of Anisotropic Shell Structures. In: Mariano, P.M. (eds) Variational Views in Mechanics. Advances in Mechanics and Mathematics(), vol 46. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90051-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation