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An implicit strong \(\mathrm {G}^{1}\)-conforming formulation for the analysis of the Kirchhoff plate model

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Abstract

In this work, we present a quadrilateral plate element for the Kirchhoff plate bending model that satisfies the continuity requirements in implicit way. The element is designed on the basis of the rational Gregory’s enhancement of the bi-cubic Coons patch. This Coons-Gregory patch is based on the boundary data set of a surface, that accounts for both the displacement and the edge rotation along the sides of the element. In this way, an implicitly conforming interpolation with 20-dofs per element is obtained. The Coons-Gregory patch ensures \(\mathrm {G^1}\)-conformity only for the case of structured meshes. Numerical examples show that the proposed formulation is highly efficient with respect to accuracy, rate of convergence and robustness.

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References

  1. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, London (2009)

    Book  MATH  Google Scholar 

  2. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, Fifth Edition, Vol 2: Solid Mechanics, 5th edn. Butterworth-Heinemann, London (2002)

    Google Scholar 

  3. Bogner, F.K., Fox, R.L., Schmit, L.A.: The generation of interelement-compatible stiffness and mass matrices by the use of of interpolation fomulae. In: Przemienicki, J.S., Bader, R.M., Bozich, W.F., Johnson, J.R., Mykytow, W.J. (eds.) Proceeding of 1st Conference of Matrix Methods in Structural Mechanics. volume AFFDITR-66-80, pp. 397–433. Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH (1966)

  4. de Veubeke, B.F.: A conforming finite element for plate bending. Int. J. Solids Struct. 4, 95–108 (1968)

    Article  MATH  Google Scholar 

  5. Petera, J., Pittman, J.F.T.: Isoparametric Hermite elements. Int. J. Numer. Method Eng. 37, 3489–3519 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 701–709 (1968)

    Google Scholar 

  7. Andreaus, U., dell’Isola, F., Giorgio, I., Placidi, L., Lekszycki, T., Rizzi, N.L.: Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int. J. Eng. Sci. 108, 34–50 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fischer, P., Mergheim, J., Steinmann, P.: On the C1 continuous discretization of non-linear gradient elasticity: a comparison of NEM and FEM based on Bernstein–Bézier patches. Int. J. Numer. Methods Eng. 82, 1282–1307 (2010)

    MATH  Google Scholar 

  9. Wempner, G.A., Oden, J.T., Cross, D.K.: Finite element analysis of thin shells. Proc. ASCE 94, 1273–1294 (1968)

    Google Scholar 

  10. Batoz, J.L., Dhatt, G.: Development of two simple shell elements. AIAA J. 10(2), 237–238 (1972)

    Article  ADS  Google Scholar 

  11. Batoz, J.L., Bathe, K.-J., Ho, L.-W.: A study of three-node triangular plate bending elements. Int. J. Numer. Methods Eng. 15(12), 1771–1812 (1980)

    Article  MATH  Google Scholar 

  12. Guo, Y.Q., Gati, W., Naceur, H., Batoz, J.L.: An efficient DKT rotation free shell element for springback simulation in sheet metal forming. Comput. Struct. 80, 2299–2312 (2002)

    Article  Google Scholar 

  13. Irons, B.M.: The semi-Loof shell element. In: Ashwell, D.G., Gallagher, R.H. (eds.) Finite Elements for Thin Shells and Curved Members, pp. 197–222. Wiley, Chichester (1976)

    Google Scholar 

  14. Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plates in bending. In: Przemienicki, J.S., Bader, R.M., Bozich, W.F., Johnson, J.R., Mykytow, W.J. (eds.) Proceedings of Conference on Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pp. 515–545, Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH (1965)

  15. Fraejis de Veubeke, B.: An equilibrium model for plate bending. Int. J. Solids Struct. 4, 447–468 (1967)

    Article  Google Scholar 

  16. Herrmann, L.R.: Finite element bending analysis of plates. In: Proceedings of ASCE, EM-5:13–25 (1968)

  17. Pian, T.H.H., Sze, K.Y.: Hybrid stress finite element methods for plate and shell structures. Adv. Struct. Eng. 4, 13–18 (2001)

    Article  Google Scholar 

  18. Zienkiewicz, O.C., Xu, Z., Zeng, L.F., Samuelsson, A., Wiberg, N.-E.: Linked interpolation for reissner-mindlin plate elements: Part I: A simple quadrilateral. Int. J. Numer. Methods Eng. 36, 3043–3056 (1993)

    Article  MATH  Google Scholar 

  19. Garusi, E., Tralli, A., Cazzani, A.: An unsymmetric stress formulation for reissner-mindlin plates: a simple and locking-free rectangular element. Int. J. Comput. Eng. Sci. 5(3), 589–618 (2004)

    Article  Google Scholar 

  20. Bathe, K.-J., Dvorkin, E.N.: A formulation ofa general shell elements—the use of mixed interpolation of tensorial components. Int. J. Numer. Methods Eng. 22, 697–722 (1986)

    Article  MATH  Google Scholar 

  21. Bathe, K.-J., Brezzi, F.: A simplified analysis of two plate bending elements—the MITC4 and MITC9 elements. In: Pande, G.N., Middleton, J. (eds.) Numerical Techniques for Engineering Analysis and Design, pp. 407–417. Martinus Nijhoff, Amsterdam (1987)

    Chapter  Google Scholar 

  22. Bathe, K.-J., Brezzi, F., Cho, S.W.: The MITC7 and MITC9 plate bending elements. Comput. Struct. 32(3/4), 797–814 (1989)

    Article  MATH  Google Scholar 

  23. Bathe, K.-J., Chapelle, D., Lee, P.S.: A shell problem highly sensitive to thickness changes. Int. J. Numer. Meth. Eng. 57, 1039–1052 (2003)

    Article  MATH  Google Scholar 

  24. Farin, G.: Curve and Surfaces for Computer Aided Geometric Design, A Practical Guide, 5th edn. Morgan Kaufmann Publishers, Los Altos (1999)

    Google Scholar 

  25. Gregory, J.A.: Smooth interpolation without twist constraint. Computer Aided Geometric Design. R.E. Barnhill and R. F. Riesenfeld, Academic Press, New York (1974)

    Chapter  Google Scholar 

  26. Areias, P.M.A., Song, J.-H., Belytschko, T.: A finite-strain quadrilateral shell element based on discrete Kirchhoff–Love constraints. Int. J. Numer. Methods Eng. 64(9), 1166–1206 (2005)

    Article  MATH  Google Scholar 

  27. Greco, L., Cuomo, M.: B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Greco, L., Cuomo, M.: An implicit \(g^1\) multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Greco, L., Cuomo, M.: An isogeometric implicit \(g^1\) mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Ciancio, D., Carol, I., Cuomo, M.: On inter-element forces in the FEM-displacement formulation, and implications for stress recovery. Int. J. Numer. Methods Eng. 66(3), 502–528 (2006)

    Article  MATH  Google Scholar 

  31. Ciancio, D., Carol, I., Cuomo, M.: Crack opening conditions at ’corner nodes’ in FE analysis with cracking along mesh lines. Eng. Fract. Mech. 74(13), 1963–1982 (2007)

    Article  Google Scholar 

  32. Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press, Oxford (1954)

    MATH  Google Scholar 

  33. Chapelle, D., Bathe, K.-J.: The Finite Elment Analysis of Shells-Fundamentals, 2nd edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  34. Pietraszkiewicz, W.: Lagrangian description and incremental formulation in the non-linear theory of thin shell. Int. J. Non-Linear Mech. 19(2), 115–140 (1984)

    Article  ADS  MATH  Google Scholar 

  35. Eremeyev, V.A., Pietraszkiewicz, W.: The non linear theory of elastic shells with phase transition. J. Elast. 74(1), 67–86 (2004)

    Article  MATH  Google Scholar 

  36. Placidi, L., Andreaus, U., Della Corte, A., Lekszyck, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für Angew. Math. und Phys. 66(6), 3699–3725 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of cauchy cuts in n-th gradient continua: approach á la D’Alembert. ZAMP 63, 1119–1141 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  38. Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics á la Hamilton–Piola: least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Coons, S.: Surface for computer aided design of space forms. Project MAC-TR 41. Technical report, MIT (1967)

  40. Farin, G., Hansford, D.: Agnostic G1 Gregory surfaces. Graph. Models 74(6), 346–350 (2012)

    Article  Google Scholar 

  41. Barsky, B.A., DeRose, T.D.: Geometric continuity of parametric curves: three equivalent characterizations. IEEE Comput. Gr. Appl. 9(6), 60–68 (1989)

    Article  Google Scholar 

  42. Greco, L., Cuomo, M., Contrafatto, L., Gazzo, S.: An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng. 324, 476–511 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Greco, L., Cuomo, M., Contrafatto, L.: A reconstructed local \(\bar{B}\) formulation for isogeometric Kirchhoffc–Love shells. Comput. Methods Appl. Mech. Eng. 332, 462–487 (2018)

    Article  ADS  MATH  Google Scholar 

  44. Cuomo, M.: Continuum model of microstructure induced softening for strain gradient materials. Math. Mech. Solids (2018). https://doi.org/10.1177/1081286518755845

    Article  MathSciNet  Google Scholar 

  45. Quang, Wang, Da-ju, Wang: Singularity under a concentrated force in elasticity. Appl. Math. Mech. 14(8), 707–711 (1993)

    Article  Google Scholar 

  46. dell’Isola, F., Corte, A.D., Giorgio, I.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22(4), 852–872 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. 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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. Lond. A 473(2207), 21 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Steigmann, D.J., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Meccanica Sinica 31(3), 373–382 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.J.: Buckling modes in pantographic lattices. Comptes Rendus - Mécanique 344, 487–501 (2016)

    Article  ADS  Google Scholar 

  51. Misra, A., Lekszycki, T., Giorgio, I., Ganzosch, G., Wolfgang, H.M., dell’Isola, F.: Pantographic metamaterials show atypical poynting effect reversal. Mech. Res. Commun. 89, 6–10 (2018)

    Article  Google Scholar 

  52. Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta)material with controlled elastic properties. R. Soc. Open Sci. 4(10), 171153 (2017)

    Article  ADS  Google Scholar 

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  1. Dipartimento di Ingegneria Civile e Architettura (DICAR), Universitá degli Studi di Catania, Via Santa Sofia 54, 95125, Catania, Italy

    M. Cuomo & L. Greco

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  1. M. Cuomo
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Correspondence to M. Cuomo.

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Communicated by Francesco dell’Isola.

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Cuomo, M., Greco, L. An implicit strong \(\mathrm {G}^{1}\)-conforming formulation for the analysis of the Kirchhoff plate model. Continuum Mech. Thermodyn. 32, 621–645 (2020). https://doi.org/10.1007/s00161-018-0701-3

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