Skip to main content
SpringerLink
Log in

A novel plate element based on absolute nodal coordinate formulation with collocation strategy

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper, a novel multi-node plate element with absolute node coordinate formulation (ANCF) is proposed. The nodes of the element are collocated to coincide with the in-plane integral quadrature points, which are used to calculate the elastic and inertia functions. The unevenly distributed nodes of the element are the zero points of the second-order derivative of Legendre polynomial and the boundary ends of the element. The tensor product of the two-direction univariate Lagrange interpolation is used to define the displacement field. To alleviate the locking problem, the gradient deficient setup and the second-order gradient of the thickness direction are used as the nodal coordinates. The standard continuum mechanic formulation is used to deduce the elastic forces. The proposed plate element based on the ANCF with collocated nodes is denoted as ANCF\(\_\)C element. The performance of the ANCF\(\_\)C element is verified by static, eigenfrequency and dynamic examples. The results show that the ANCF\(\_\)C element is more accurate and computationally efficient.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Crisfield, M.: A fast incremental/iterative solution procedure that handles ‘snap-through’. Comput. Methods Nonlinear Struct. Solid Mech. 13, 55–62 (1981)

    Article  MATH  Google Scholar 

  2. Ebel, H., Matikainen, M.K., Hurskainen, V.V., Mikkola, A.: Analysis of high-order quadrilateral plate elements based on the absolute nodal coordinate formulation for three-dimensional elasticity. Adv. Mech. Eng. 9(6), 1–12 (2017)

    Article  Google Scholar 

  3. Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Sys.Dyn. 20(4), 359–384 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordinate formulation for large deformation analysis of multibody systems. J. Comput. Nonlinear Dyn. 8(3), 031016 (2013)

    Article  Google Scholar 

  5. Gu, Y., Lan, P., Cui, Y., Li, K., Yu, Z.: Dynamic interaction between the transmission wire and cross-frame. Mech. Mach. Theory 155, 104068 (2021)

    Article  Google Scholar 

  6. Li, P., Liu, C., Tian, Q., Hu, H., Song, Y.: Dynamics of a deployable mesh reflector of satellite antenna: form-finding and modal analysis. J. Comput. Nonlinear Dyn. 11(4), 041017 (2016)

    Article  Google Scholar 

  7. Nachbagauer, K., Gruber, P., Gerstmayr, J.: Structural and continuum mechanics approaches for a 3d shear deformable ANCF beam finite element: application to static and linearized dynamic examples. J. Comput. Nonlinear Dyn. 8(2), 92–110 (2013)

    Google Scholar 

  8. Obrezkov, L.P., Mikkola, A., Matikainen, M.K.: Performance review of locking alleviation methods for continuum ANCF beam elements. Nonlinear Dyn. 109(2), 531–546 (2022)

    Article  Google Scholar 

  9. Olshevskiy, A., Dmitrochenko, O., Kim, C.W.: A plate element with second-order absolute nodal coordinate slopes: numerical computation of shape functions. In: Asme International Design Engineering Technical Conferences & Computers & Information in Engineering Conference (2014)

  10. Otsuka, K., Makihara, K.: Deployment simulation using absolute nodal coordinate plate element for next-generation aerospace structures. AIAA J. 56(3), 1266–1276 (2018)

    Article  Google Scholar 

  11. Otsuka, K., Makihara, K., Sugiyama, H.: Recent advances in the absolute nodal coordinate formulation: literature review from 2012 to 2020. J. Comput. Nonlinear Dyn. 17(8), 080803 (2022)

    Article  Google Scholar 

  12. Patel, M., Shabana, A.A.: Locking alleviation in the large displacement analysis of beam elements: the strain split method. Acta Mech. 229(7), 2923–2946 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Richard, L.B., Douglas, F.J.: Numerical Analysis, ninth ed. Richard Stratton (2010)

  14. Schwab, A.L., Gerstmayr, J., Meijaard, J.P.: Comparison of three-dimensional flexible thin plate elements for multibody dynamic analysis: finite element formulation and absolute nodal coordinate formulation. In: Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. Las Vegas, Nevada (2007)

  15. Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1(3), 339–348 (1997)

    Article  MATH  Google Scholar 

  16. Shabana, A.A., Desai, C.J., Grossi, E., Patel, M.: Generalization of the strain-split method and evaluation of the nonlinear ANCF finite elements. Acta Mech. 231(4), 1365–1376 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shen, Z., Tian, Q., Liu, X., Hu, G.: Thermally induced vibrations of flexible beams using absolute nodal coordinate formulation. Aerosp. Sci. Technol. 29(1), 386–393 (2013)

    Article  Google Scholar 

  18. Sun, D., Chen, G., Sun, R.: Model reduction of a multibody system including a very flexible beam element. J. Mech. Sci. Technol. 28(8), 2963–2969 (2014)

    Article  Google Scholar 

  19. Tang, L., Baeder, J.D.: Uniformly accurate finite difference schemes for p-refinement. SIAM J. Sci. Comput. (1998). https://doi.org/10.1137/S1064827596308354

    Article  MATH  Google Scholar 

  20. Tang, Y., Hu, H., Tian, Q.: Model order reduction based on successively local linearizations for flexible multibody dynamics. Int. J. Numer. Methods Eng. (2018). https://doi.org/10.1002/nme.6011

    Article  Google Scholar 

  21. Valkeapää, A.I., Yamashita, H., Jayakumar, P., Sugiyama, H.: On the use of elastic middle surface approach in the large deformation analysis of moderately thick shell structures using absolute nodal coordinate formulation. Nonlinear Dyn. 80(3), 1133–1146 (2015)

    Article  MathSciNet  Google Scholar 

  22. Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numer. Math. 118(2), 271–305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, J., Wang, T., Bian, H., Liu, W.: A novel collocation beam element based on absolute nodal coordinate formulation. Acta Mech. 234, 2695–2707 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, T.: Two new triangular thin plate/shell elements based on the absolute nodal coordinate formulation. Nonlinear Dyn. 99, 2707–2725 (2020)

    Article  MATH  Google Scholar 

  25. Wang, T., Mikkola, A., Matikainen, M.K.: An overview of higher-order beam elements based on the absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 17(9), 091001 (2022)

    Article  Google Scholar 

  26. Yamashita, H., Valkeapaa, A.I., Jayakumar, P., Sugiyama, H.: Continuum mechanics based bi-linear shear deformable shell element using absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 10(5), 051012 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Shenyang University of Technology, No. 111, Shenliao West Road, Shenyang, China

    Jia Wang, Yulong Zhang, Hongyou Bian & Weijun Liu

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China

    Tengfei Wang

Authors
  1. Jia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tengfei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yulong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongyou Bian
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Weijun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weijun Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Wang, T., Zhang, Y. et al. A novel plate element based on absolute nodal coordinate formulation with collocation strategy. Acta Mech 234, 6677–6690 (2023). https://doi.org/10.1007/s00707-023-03730-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03730-z

Search

Navigation