C1 Discretizations for the Application to Gradient Elasticity

  • Paul Fischer
  • Julia Mergheim
  • Paul Steinmann
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 21)


For the numerical solution of gradient elasticity, the appearance of strain gradients in the weak form of the equilibrium equation leads to the need for C 1-continuous discretization methods. In the present work, the performances of a variety of C 1-continuous elements as well as the C 1 Natural Element Method are investigated for the application to nonlinear gradient elasticity. In terms of subparametric triangular elements the Argyris, Hsieh–Clough–Tocher and Powell–Sabin split elements are utilized. As an isoparametric quadrilateral element, the Bogner–Fox–Schmidt element is used. All these methods are applied to two different numerical examples and the convergence behavior with respect to the L 2, H 1 and H 2 error norms is examined.


Delaunay Triangulation Uniaxial Loading Gradient Elasticity Thick Hollow Cylinder Natural Element Method 
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  1. 1.
    Amanatidou, E., Aravas, N.: Mixed finite element formulations of strain-gradient elasticity problems. Comput. Methods Appl. Mech. Eng. 191, 1723–1751 (2002) MATHCrossRefGoogle Scholar
  2. 2.
    Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of elements for the matrix displacement method. Aeronaut. J. 72, 701–709 (1968) Google Scholar
  3. 3.
    Askes, H., Bennett, T., Aifantis, E.C.: A new formulation and C 0-implementation of dynamically consistent gradient elasticity. Int. J. Numer. Methods Eng. 72, 111–126 (2007) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Askes, H., Gutiérrez, M.A.: Implicit gradient elasticity. Int. J. Numer. Methods Eng. 67, 400–416 (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Bogner, F.K., Fox, R.L.: The generation of inter-element-compatible stiffness and mass matrices by the use of interpolation formulas. In: Proceedings of the Conference held at Wright-Patterson Air Force Base, pp. 397–443. Wright-Patterson Air Force Base, Ohio (1965) Google Scholar
  6. 6.
    Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plate bending. In: Proceedings of the Conference held at Wright-Patterson Air Force Base, pp. 515–545. Wright-Patterson Air Force Base, Ohio (1965) Google Scholar
  7. 7.
    Cueto, E., Sukumar, N., Calvo, B., Cegonio, J., Doblare, M.: Overview and recent advantages in natural neighbor Galerkin methods. Arch. Comput. Methods Eng. 10, 307–384 (2002) CrossRefGoogle Scholar
  8. 8.
    Fischer, P., Mergheim, J., Steinmann, P.: On the C 1 continuous discretization of nonlinear gradient elasticity: a comparison of NEM and FEM based on Bernstein–Bézier patches. Int. J. Numer. Methods Eng. (2009). doi: 10.1002/nme.2802 Google Scholar
  9. 9.
    Hirschberger, C.B., Kuhl, E., Steinmann, P.: On deformational and configurational mechanics for micromorphic hyperelasticity—theory and computation. Comput. Methods Appl. Mech. Eng. 196, 4027–4044 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kirchner, N., Steinmann, P.: A unifying treatise on variational principles for gradient and micro-morphic continua. Philos. Mag. 85, 3875–3895 (2005) CrossRefGoogle Scholar
  11. 11.
    Papanicolopulos, S.A., Zervos, A., Vardoulakis, I.: A three-dimensional C 1 finite element for gradient elasticity. Int. J. Numer. Methods Eng. 77, 1396–1415 (2009) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Petera, J., Pittman, J.F.T.: Isoparametric Hermite elements. Int. J. Numer. Methods Eng. 37, 3489–3519 (1994) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Powell, M.J., Sabin, M.A.: Piecewise quadratic approximation on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rajagopal, A., Scherer, M., Steinmann, P., Sukumar, N.: Smooth conformal α-nem for gradient elasticity. Int. J. Struct. Changes Solids 1, 83–109 (2009) Google Scholar
  15. 15.
    Shu, J.Y., King, W.E., Fleck, N.E.: Finite elements for materials with strain gradient effects. Int. J. Numer. Methods Eng. 44, 373–391 (1999) MATHCrossRefGoogle Scholar
  16. 16.
    Sibson, R.A.: A vector identity for the Dirichlet tessellation. Math. Proc. Camb. Philos. Soc. 87, 151–155 (1980) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Sukumar, N., Moran, B., Belytschko, T.: The natural element method in solid mechanics. Int. J. Numer. Methods Mech. 43, 839–887 (1998) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zervos, A., Papanicolopulos, S.A., Vardoulakis, I.: Two finite-element discretizations for gradient elasticity. J. Eng. Mech. 135, 203–213 (2009) CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University Erlangen-NurembergErlangenGermany

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