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A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates

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Abstract

We use a gap function in order to compare the torsional performances of different reinforced plates under the action of external forces. Then, we address a shape optimization problem, whose target is to minimize the torsional displacements of the plate: This leads us to set up a minimaxmax problem, which includes a new kind of worst-case optimization. Two kinds of reinforcements are considered: One aims at strengthening the plate and the other aims at weakening the action of the external forces. For both of them, we study the existence of optima within suitable classes of external forces and reinforcements. Our results are complemented with numerical experiments and with a number of open problems and conjectures.

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References

  1. Abrams, D.M., Eckhardt, B., McRobie, A., Ott, E., Strogatz, S.H.: Crowd synchrony on the Millennium Bridge. Nat. Brief Commun. 438, 43–44 (2005)

    Google Scholar 

  2. Macdonald, J.H.G.: Lateral excitation of bridges by balancing pedestrians. Proc. R. Soc. A Math. Phys. Eng. Sci. 465, 1055–1073 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Sanderson, K.: Millennium bridge wobble explained. Nature (2008). https://doi.org/10.1038/news.2008.1311

  4. Gazzola, F.: Mathematical Models for Suspension Bridges. MS&A, vol. 15. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  5. Ferrero, A., Gazzola, F.: A partially hinged rectangular plate as a model for suspension bridges. Discrete Control Dyn. Syst. A 35, 5879–5908 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berchio, E., Buoso, D., Gazzola, F.: A measure of the torsional performances of partially hinged rectangular plates. In: Integral Methods in Science and Engineering, Theoretical Techniques, vol. 1, pp. 35–46. Birkhäuser/Springer, Cham (2017)

  7. Chanillo, S., Kenig, C.E., To, T.: Regularity of the minimizers in the composite membrane problem in \(\mathbb{R}^2\). J. Funct. Anal. 255, 2299–2320 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chanillo, S., Kenig, C.E.: Weak uniqueness and partial regularity for the composite membrane problem. J. Eur. Math. Soc. 10, 705–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kawohl, B., Stará, J., Wittum, G.: Analysis and numerical studies of a problem of shape design. Arch. Ration. Mech. Anal. 114, 349–363 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Murat, F., Tartar, L.: Calculus of variations and homogenization. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 139–173. Birkhäuser Boston, Boston, MA (1997)

  11. Nazarov, S.A., Sweers, G.H., Slutskij, A.S.: Homogenization of a thin plate reinforced with periodic families of rigid rods. Sb. Math. 202, 1127–1168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Michell, A.G.M.: The limits of economy of material in framed structures. Philos. Mag. Ser. 6 8(47), 589–597 (1904)

    Article  MATH  Google Scholar 

  13. Bendsøe, M.P., Sigmund, O.: Topology Optimization. Springer, Berlin (2003)

    MATH  Google Scholar 

  14. Allaire, G., Dapogny, C.: A linearized approach to worst-case design in parametric and geometric shape optimization. Math. Models Methods Appl. Sci. 24, 2199–2257 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cherkaev, A., Cherkaeva, E.: Principal compliance and robust optimal design. J. Elast. 72, 71–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kirchhoff, G.R.: Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40, 51–88 (1850)

    Article  MathSciNet  Google Scholar 

  17. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927)

    MATH  Google Scholar 

  18. Mansfield, E.H.: The Bending and Stretching of Plates. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  19. Berchio, E., Ferrero, A., Gazzola, F.: Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions. Nonlinear Anal. Real World Appl. 28, 91–125 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pólya, G., Szegő, G.: Problems and Theorems in Analysis, vol. II. Springer, New York (1976). 216

    Book  MATH  Google Scholar 

  21. Berchio, E., Buoso, D., Gazzola, F.: On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate. ESAIM: COCV 24(1), 63–87 (2018)

    Article  Google Scholar 

  22. Gazzola, F.: Hexagonal design for stiffening trusses. Ann. Mat. Pura Appl. 194, 87–108 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Buttazzo, G., Oudet, E., Stepanov, E.: Optimal transportation problems with free Dirichlet regions. In: dal Maso, G., Tomarelli, F. (eds.) Variational Methods for Discontinuous Structures. Progress in Nonlinear Differential Equations Applications, vol. 51, pp. 41–65. Birkhäuser Verlag, Basel (2002)

  24. Buttazzo, G., Stepanov, E.: Optimal transportation networks as free Dirichlet regions for the Monge–Kantorovich problem. Ann. Sci. Norm. Super. Pisa Cl. Sci. 2, 631–678 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Buttazzo, G., Stepanov, E.: Minimization problems for average distance functionals. In: De Giorgi, E. (ed.) Calculus of Variations: Topics from the Mathematical Heritage. Quad. Mat., pp. 48–83. Department of Mathematics, Seconda Università, Napoli (2004)

    Google Scholar 

  26. Thomson, W.: On the division of space with minimum partitional area. Acta Math. 11, 121–134 (1887)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ambrosio, L., Tilli, P.: Analysis in Metric Spaces. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  28. Henrot, A., Pierre, M.: Variation et Optimisation de Formes. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  29. Berchio, E., Buoso, D., Gazzola, F., Zucco, D.: A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates. arXiv:1802.07230

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Acknowledgements

The authors are grateful to J. B. Kennedy for his kind revision of the use of the English Language within the present paper. Elvise Berchio, Davide Buoso and Davide Zucco are partially supported by the Research Project FIR (Futuro in Ricerca) 2013 Geometrical and qualitative aspects of PDEs. Filippo Gazzola is partially supported by the PRIN project Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni. Elvise Berchio, Davide Buoso, Filippo Gazzola and Davide Zucco are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Correspondence to Elvise Berchio.

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Communicated by Giuseppe Buttazzo.

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Berchio, E., Buoso, D., Gazzola, F. et al. A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates. J Optim Theory Appl 177, 64–92 (2018). https://doi.org/10.1007/s10957-018-1261-1

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  • DOI: https://doi.org/10.1007/s10957-018-1261-1

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