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Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

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Abstract

In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF tetrahedral elements have twelve coordinates per node that include three position and nine gradient coordinates. The fundamental differences between the coordinate parametrizations used for conventional finite elements and the coordinate parametrizations employed for the proposed ANCF tetrahedral elements are discussed. Two different parametric definitions are introduced: a volume parametrization based on coordinate lines along the sides of the tetrahedral element in the straight (un-deformed) configuration and a Cartesian parametrization based on coordinate lines directed along the global axes. The volume parametrization facilitates the development of a concise set of shape functions in a closed form, and the Cartesian parametrization serves as a unique standard for the element assembly. A linear mapping based on the Bezier geometry is used to systematically define the cubic position fields of ANCF tetrahedral elements: the complete polynomial-based eight-node mixed-coordinate and the incomplete polynomial-based four-node ANCF tetrahedral elements. An element transformation matrix that defines the relationship between the volume and Cartesian parametrizations is developed and used to convert the parametric gradients to structure gradients, thereby allowing for the use of a standard FE assembly procedure. A general computational approach is employed to formulate the generalized inertia, external, and elastic forces. The performance of the proposed ANCF tetrahedral elements is evaluated by comparison with the conventional linear and quadratic tetrahedral elements and also with the ANCF brick element. In the case of small deformations, the numerical results obtained show that all the tetrahedral elements considered can correctly produce rigid body motion. In the case of large deformations, on the other hand, the solutions of all the elements considered are in good agreement, provided that appropriate mesh sizes are used.

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References

  1. Magnus, K.: Dynamics of Multibody Systems. Springer, Berlin (1978)

    Book  Google Scholar 

  2. Huston, R.L.: Multibody Dynamics. Butterworth-Heineman, Boston (1990)

    MATH  Google Scholar 

  3. Roberson, R.E., Schwertassek, R.: Dynamics of Multibody Systems. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  4. Udwadia, F.E., Schutte, A.D.: Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech. 213, 111–129 (2010)

    Article  MATH  Google Scholar 

  5. Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs (1988)

    Google Scholar 

  6. Guida, D., Pappalardo, C.M.: Forward and inverse dynamics of nonholonomic mechanical systems. Meccanica 49, 1547–1559 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guida, D., Pappalardo, C.M.: Control design of an active suspension system for a quarter-car model with hysteresis. J. Vib. Eng. Technol. 3, 277–299 (2015)

    Google Scholar 

  8. Pappalardo, C.M.: A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems. J. Nonlinear Dyn. 81, 1841–1869 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Shabana, A.A.: Dynamics of Multibody Systems, 4th edn. Cambridge University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  10. Tian, Q., Sun, Y., Liu, C., Hu, H., Flores, P.: Elastohydrodynamic lubricated cylindrical joints for rigid-flexible multibody dynamics. J. Comput. Struct. 114, 106–120 (2013)

    Article  Google Scholar 

  11. Kubler, L., Eberhard, P., Geisler, J.: Flexible multibody systems with large deformations and nonlinear structural damping using absolute nodal coordinates. J. Nonlinear Dyn. 34, 31–52 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mikkola, A.M., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. J. Multibody Syst. Dyn. 9, 283–309 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nachbagauer, K.: State of the art of ANCF elements regarding geometric description, interpolation strategies, definition of elastic forces, validation and the locking phenomenon in comparison with proposed beam finite elements. Arch. Comput. Methods Eng. 21, 293–319 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shabana, A.A.: Computational Continuum Mechanics, 2nd edn. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

  15. Dmitrochenko, O., Mikkola, A.: Two simple triangular plate elements based on the absolute nodal coordinate formulation. J. Multibody Syst. Dyn. 3, 1–8 (2008)

    Google Scholar 

  16. Hu, W., Tian, Q., Hu, H.Y.: Dynamics simulation of the liquid-filled flexible multibody system via the absolute nodal coordinate formulation and SPH method. Nonlinear Dyn. 75, 653–671 (2014)

    Article  MathSciNet  Google Scholar 

  17. Liu, C., Tian, Q., Hu, H.Y.: Dynamics of large scale rigid-flexible multibody system composed of composite laminated plates. Multibody Syst. Dyn. 26, 283–305 (2011)

    Article  MATH  Google Scholar 

  18. Liu, C., Tian, Q., Yan, D., Hu, H.Y.: Dynamic analysis of membrane systems undergoing overall motions, large deformations, and wrinkles via thin shell elements of ANCF. J. Comput. Methods Appl. Mech. Eng. 258, 81–95 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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, 1–7 (2012)

    Google Scholar 

  20. Nachbagauer, K., Gerstmayr, J.: Structural and continuum mechanics approaches for a 3D shear deformable ANCF beam finite element: application to buckling and nonlinear dynamic examples. J. Comput. Nonlinear Dyn. 9, 1–8 (2013)

    Google Scholar 

  21. He, G., Patel, M. D., Shabana, A. A.: Integration of localized surface geometry in fully parameterized ANCF finite elements. J. Comput. Methods Appl. Mech. Eng. 313, 966–985 (2017)

  22. Kulkarni, S., Pappalardo, C.M., Shabana, A.A.: Pantograph/catenary contact formulations. ASME J. Vib. Acoust. 139, 1–12 (2016)

    Article  Google Scholar 

  23. Pappalardo, C.M., Patel, M.D., Tinsley, B., Shabana, A.A.: Contact force control in multibody pantograph/catenary systems. In: Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, pp. 1–22 (2015)

  24. Pappalardo, C.M., Yu, Z., Zhang, X., Shabana, A.A.: Rational ANCF thin plate finite element. ASME J. Comput. Nonlinear Dyn. 11, 1–15 (2016)

    Google Scholar 

  25. Pappalardo, C.M., Wallin, M., Shabana, A.A.: ANCF/CRBF Fully parametrized plate finite element. ASME J. Comput. Nonlinear Dyn. 12(3), 031008 (2017)

  26. Patel, M.D., Orzechowski, G., Tian, Q., Shabana, A.A.: A new multibody system approach for tire modeling using ANCF finite elements. In: Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, vol. 230, pp. 69–84 (2015)

  27. Shabana, A.A.: Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics. Nonlinear Dyn. 16, 293–306 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. McGraw-Hill, London (1977)

    MATH  Google Scholar 

  29. Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  30. Shabana, A.A.: Definition of ANCF Finite Elements. ASME J. Comput. Nonlinear Dyn. 10, 1–5 (2015)

    Google Scholar 

  31. Dmitrochenko, O., Mikkola, A.: Digital nomenclature code for topology and kinematics of finite elements based on the absolute nodal coordinate formulation. J. Multibody Syst. Dyn. 225, 229–252 (2011)

    Google Scholar 

  32. Olshevskiy, A., Dmitrochenko, O., Kim, C.W.: Three-dimensional solid brick element using slopes in the absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 9, 1–10 (2014)

  33. Wei, C., Wang, L., Shabana, A.A.: A total Lagrangian ANCF liquid sloshing approach for multibody system applications. J. Comput. Nonlinear Dyn. 10, 1–10 (2015)

    Google Scholar 

  34. Mohamed, A.N.A.: ANCF tetrahedral solid element using shape functions based on cartesian coordinates. In: ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC2016, August 21–24, 2016, Charlotte, North Carolina, pp. 1–6 (2016)

  35. Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, New York (1978)

    MATH  Google Scholar 

  36. Pappalardo, C.M., Wang, T., Shabana, A.A.: On the formulation of the planar ANCF triangular finite elements. Nonlinear Dyn. 89, 1019–1045 (2017)

  37. Bathe, J.K.: Finite Element Procedures. Prentice Hall, Upper Saddle River (2007)

    MATH  Google Scholar 

  38. Belytschko, T., Liu, W.K., Moran, B., Elkhodary, K.I.: Nonlinear Finite Elements for Continua and Structures, 2nd edn. Wiley, Hoboken (2013)

    MATH  Google Scholar 

Download references

Acknowledgements

This research was supported, in part, by the National Science Foundation (Project # 1632302).

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Correspondence to Ahmed A. Shabana.

Appendix A

Appendix A

The linear transformations between the Bezier tetrahedral patch basis functions \(g_{k} ,k=1,2,\ldots ,20\), and the shape functions of the ANCF/ENMC tetrahedral element \(\bar{{s}}_{k} ,k=1,2,\ldots ,20\), are given by

$$\begin{aligned} \left. {\begin{array}{ll} \bar{{s}}_1 &{}=g_1 +g_5 +g_9 +g_{11} -\frac{7}{6}\left( {g_{17} +g_{18} +g_{20} } \right) ,\\ \bar{{s}}_2 &{}=\frac{1}{3}g_5 -\frac{1}{6}\left( {g_{17} +g_{18} } \right) ,\\ \bar{{s}}_3 &{}=\frac{1}{3}g_9 -\frac{1}{6}\left( {g_{17} +g_{20} } \right) , \\ \bar{{s}}_4 &{}=\frac{1}{3}g_{11} -\frac{1}{6}\left( {g_{18} +g_{20} } \right) ,\\ \bar{{s}}_5 &{}=g_2 +g_6 +g_7 +g_{13} -\frac{7}{6}\left( {g_{17} +g_{18} +g_{19} } \right) ,\\ \bar{{s}}_6 &{}=\frac{1}{3}g_7 -\frac{1}{6}\left( {g_{17} +g_{19} } \right) , \\ \bar{{s}}_7 &{}=\frac{1}{3}g_{13} -\frac{1}{6}\left( {g_{18} +g_{19} } \right) ,\\ \bar{{s}}_8 &{}=\frac{1}{3}g_6 -\frac{1}{6}\left( {g_{17} +g_{18} } \right) ,\\ \bar{{s}}_9 &{}=g_3 +g_8 +g_{10} +g_{15} -\frac{7}{6}\left( {g_{17} +g_{19} +g_{20} } \right) , \\ \bar{{s}}_{10} &{}=\frac{1}{3}g_{15} -\frac{1}{6}\left( {g_{19} +g_{20} } \right) ,\\ \bar{{s}}_{11} &{}=\frac{1}{3}g_{10} -\frac{1}{6}\left( {g_{17} +g_{20} } \right) ,\\ \bar{{s}}_{12} &{}=\frac{1}{3}g_8 -\frac{1}{6}\left( {g_{17} +g_{19} } \right) , \\ \bar{{s}}_{13} &{}=g_4 +g_{12} +g_{14} +g_{16} -\frac{7}{6}\left( {g_{18} +g_{19} +g_{20} } \right) ,\\ \bar{{s}}_{14} &{}=\frac{1}{3}g_{12} -\frac{1}{6}\left( {g_{18} +g_{20} } \right) ,\\ \bar{{s}}_{15} &{}=\frac{1}{3}g_{14} -\frac{1}{6}\left( {g_{18} +g_{19} } \right) , \\ \bar{{s}}_{16} &{}=\frac{1}{3}g_{16} -\frac{1}{6}\left( {g_{19} +g_{20} } \right) ,\quad \bar{{s}}_{17} =\frac{9}{2}g_{17} ,\\ \bar{{s}}_{18} &{}=\frac{9}{2}g_{18} ,\quad \bar{{s}}_{19} =\frac{9}{2}g_{19} ,\quad \bar{{s}}_{20} =\frac{9}{2}g_{20} \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.1)

The linear transformations between the shape functions of the ANCF/ENMC tetrahedral element \(\bar{{s}}_{k} ,k=1,2,\ldots ,20\), and the shape functions of the ANCF four-node (FN) tetrahedral element \(s_{k} ,k=1,2,\ldots ,16\), are given by:

$$\begin{aligned} \left. {\begin{array}{ll} s_1 &{}=\bar{{s}}_1 +\frac{1}{3}\bar{{s}}_{17} +\frac{1}{3}\bar{{s}}_{18} +\frac{1}{3}\bar{{s}}_{20} ,\\ s_2 &{}=\bar{{s}}_2 +\frac{2}{81}\bar{{s}}_{17} +\frac{2}{81}\bar{{s}}_{18} ,\\ s_3 &{}=\bar{{s}}_3 +\frac{2}{81}\bar{{s}}_{17} +\frac{2}{81}\bar{{s}}_{20} , \\ s_4 &{}=\bar{{s}}_4 +\frac{2}{81}\bar{{s}}_{18} +\frac{2}{81}\bar{{s}}_{20} ,\\ s_5 &{}=\bar{{s}}_5 +\frac{1}{3}\bar{{s}}_{17} +\frac{1}{3}\bar{{s}}_{18} +\frac{1}{3}\bar{{s}}_{19} ,\\ s_6 &{}=\bar{{s}}_6 +\frac{2}{81}\bar{{s}}_{17} +\frac{2}{81}\bar{{s}}_{19} , \\ s_7 &{}=\bar{{s}}_7 +\frac{2}{81}\bar{{s}}_{18} +\frac{2}{81}\bar{{s}}_{19} ,\\ s_8 &{}=\bar{{s}}_8 +\frac{2}{81}\bar{{s}}_{17} +\frac{2}{81}\bar{{s}}_{18} ,\\ s_9 &{}=\bar{{s}}_9 +\frac{1}{3}\bar{{s}}_{17} +\frac{1}{3}\bar{{s}}_{19} +\frac{1}{3}\bar{{s}}_{20} , \\ s_{10} &{}=\bar{{s}}_{10} +\frac{2}{81}\bar{{s}}_{19} +\frac{2}{81}\bar{{s}}_{20} ,\\ s_{11} &{}=\bar{{s}}_{11} +\frac{2}{81}\bar{{s}}_{17} +\frac{2}{81}\bar{{s}}_{20} ,\\ s_{12} &{}=\bar{{s}}_{12} +\frac{2}{81}\bar{{s}}_{17} +\frac{2}{81}\bar{{s}}_{19} , \\ s_{13} &{}=\bar{{s}}_{13} +\frac{1}{3}\bar{{s}}_{18} +\frac{1}{3}\bar{{s}}_{19} +\frac{1}{3}\bar{{s}}_{20} ,\\ s_{14} &{}=\bar{{s}}_{14} +\frac{2}{81}\bar{{s}}_{18} +\frac{2}{81}\bar{{s}}_{20} ,\\ s_{15} &{}=\bar{{s}}_{15} +\frac{2}{81}\bar{{s}}_{18} +\frac{2}{81}\bar{{s}}_{19} , \\ s_{16} &{}=\bar{{s}}_{16} +\frac{2}{81}\bar{{s}}_{19} +\frac{2}{81}\bar{{s}}_{20} \end{array}} \right\} \end{aligned}$$
(A.2)

The shape functions of the ENMC and FN elements associated with the Cartesian gradient vectors \(\mathbf{r}_x^k \), \(\mathbf{r}_y^k \), and \(\mathbf{r}_z^k \) can be systematically evaluated. The ENMC shape functions \(\bar{{s}}_k^{*} ,k=1,2,\ldots ,20\), associated with the Cartesian gradients can be explicitly written in terms of the tetrahedral volume parameters as

$$\begin{aligned} \left. {\begin{array}{ll} \bar{{s}}_1^{*} &{}=\xi \left( {\xi ^{2}+3\xi \left( {\eta +\zeta +\chi } \right) -7\left( {\eta \zeta +\chi \left( {\eta +\zeta } \right) } \right) } \right) \\ \bar{{s}}_2^{*} &{}=a_{2,1} \xi \eta \left( {\xi -\zeta -\chi } \right) +a_{3,1} \xi \zeta \left( {\xi -\chi -\eta } \right) \\ &{}\quad +\,a_{4,1} \xi \chi \left( {\xi -\eta -\zeta } \right) \\ \bar{{s}}_3^{*} &{}=b_{2,1} \xi \eta \left( {\xi -\zeta -\chi } \right) +b_{3,1} \xi \zeta \left( {\xi -\chi -\eta } \right) \\ &{}\quad +\,b_{4,1} \xi \chi \left( {\xi -\eta -\zeta } \right) \\ \bar{{s}}_4^{*} &{}=c_{2,1} \xi \eta \left( {\xi -\zeta -\chi } \right) +c_{3,1} \xi \zeta \left( {\xi -\chi -\eta } \right) \\ &{}\quad +\,c_{4,1} \xi \chi \left( {\xi -\eta -\zeta } \right) \\ \bar{{s}}_5^{*} &{}=\eta \left( {\eta ^{2}+3\eta \left( {\zeta +\chi +\xi } \right) -7\left( {\zeta \chi +\xi \left( {\zeta +\chi } \right) } \right) } \right) \\ \bar{{s}}_6^{*} &{}=a_{3,2} \eta \zeta \left( {\eta -\chi -\xi } \right) +a_{4,2} \eta \chi \left( {\eta -\xi -\zeta } \right) \\ &{}\quad +\,a_{1,2} \eta \xi \left( {\eta -\zeta -\chi } \right) \\ \bar{{s}}_7^{*} &{}=b_{3,2} \eta \zeta \left( {\eta -\chi -\xi } \right) +b_{4,2} \eta \chi \left( {\eta -\xi -\zeta } \right) \\ &{}\quad +\,b_{1,2} \eta \xi \left( {\eta -\zeta -\chi } \right) \\ \bar{{s}}_8^{*} &{}=c_{3,2} \eta \zeta \left( {\eta -\chi -\xi } \right) +c_{4,2} \eta \chi \left( {\eta -\xi -\zeta } \right) \\ &{}\quad +\,c_{1,2} \eta \xi \left( {\eta -\zeta -\chi } \right) \\ \bar{{s}}_9^{*} &{}=\zeta \left( {\zeta ^{2}+3\zeta \left( {\chi +\xi +\eta } \right) -7\left( {\chi \xi +\eta \left( {\chi +\xi } \right) } \right) } \right) \\ \bar{{s}}_{10}^{*} &{}=a_{4,3} \zeta \chi \left( {\zeta -\xi -\eta } \right) +a_{1,3} \zeta \xi \left( {\zeta -\eta -\chi } \right) \\ &{}\quad +\,a_{2,3} \zeta \eta \left( {\zeta -\chi -\xi } \right) \\ \bar{{s}}_{11}^{*} &{}=b_{4,3} \zeta \chi \left( {\zeta -\xi -\eta } \right) +b_{1,3} \zeta \xi \left( {\zeta -\eta -\chi } \right) \\ &{}\quad +\,b_{2,3} \zeta \eta \left( {\zeta -\chi -\xi } \right) \\ \bar{{s}}_{12}^{*} &{}=c_{4,3} \zeta \chi \left( {\zeta -\xi -\eta } \right) +c_{1,3} \zeta \xi \left( {\zeta -\eta -\chi } \right) \\ &{}\quad +\,c_{2,3} \zeta \eta \left( {\zeta -\chi -\xi } \right) \\ \bar{{s}}_{13}^{*} &{}=\chi \left( {\chi ^{2}+3\chi \left( {\xi +\eta +\zeta } \right) -7\left( {\xi \eta +\zeta \left( {\xi +\eta } \right) } \right) } \right) \\ \bar{{s}}_{14}^{*} &{}=a_{1,4} \chi \xi \left( {\chi -\eta -\zeta } \right) +a_{2,4} \chi \eta \left( {\chi -\zeta -\xi } \right) \\ &{}\quad +\,a_{3,4} \chi \zeta \left( {\chi -\xi -\eta } \right) \\ \bar{{s}}_{15}^{*} &{}=b_{1,4} \chi \xi \left( {\chi -\eta -\zeta } \right) +b_{2,4} \chi \eta \left( {\chi -\zeta -\xi } \right) \\ &{}\quad +\,b_{3,4} \chi \zeta \left( {\chi -\xi -\eta } \right) \\ \bar{{s}}_{16}^{*} &{}=c_{1,4} \chi \xi \left( {\chi -\eta -\zeta } \right) +c_{2,4} \chi \eta \left( {\chi -\zeta -\xi } \right) \\ &{}\quad +\,c_{3,4} \chi \zeta \left( {\chi -\xi -\eta } \right) \\ \bar{{s}}_{17}^{*} &{}=27\xi \eta \zeta ,\quad \bar{{s}}_{18}^{*} =27\xi \eta \chi ,\quad \bar{{s}}_{19}^{*} =27\eta \zeta \chi ,\\ \bar{{s}}_{20}^{*} &{}=27\xi \zeta \chi \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.3)

The ANCF/FN shape functions \(s_k^{*} ,k=1,2,\ldots ,16\), associated with the Cartesian gradients can be written as

$$\begin{aligned} \left. {\begin{array}{ll} s_1^{*} &{}=\xi \left( {\xi ^{2}+3\xi \left( {\eta +\zeta +\chi } \right) +2\left( {\eta \zeta +\chi \left( {\eta +\zeta } \right) } \right) } \right) \\ s_2^{*} &{}=a_{2,1} \frac{1}{3}\xi \eta \left( {3\xi -\zeta -\chi } \right) +a_{3,1} \frac{1}{3}\xi \zeta \left( {3\xi -\chi -\eta } \right) \\ &{}\quad +\,a_{4,1} \frac{1}{3}\xi \chi \left( {3\xi -\eta -\zeta } \right) \\ s_3^{*} &{}=b_{2,1} \frac{1}{3}\xi \eta \left( {3\xi -\zeta -\chi } \right) +b_{3,1} \frac{1}{3}\xi \zeta \left( {3\xi -\chi -\eta } \right) \\ &{}\quad +\,b_{4,1} \frac{1}{3}\xi \chi \left( {3\xi -\eta -\zeta } \right) \\ s_4^{*} &{}=c_{2,1} \frac{1}{3}\xi \eta \left( {3\xi -\zeta -\chi } \right) +c_{3,1} \frac{1}{3}\xi \zeta \left( {3\xi -\chi -\eta } \right) \\ &{}\quad +\,c_{4,1} \frac{1}{3}\xi \chi \left( {3\xi -\eta -\zeta } \right) \\ s_5^{*} &{}=\eta \left( {\eta ^{2}+3\eta \left( {\zeta +\chi +\xi } \right) +2\left( {\zeta \chi +\xi \left( {\zeta +\chi } \right) } \right) } \right) \\ s_6^{*} &{}=a_{3,2} \frac{1}{3}\eta \zeta \left( {3\eta -\chi -\xi } \right) +a_{4,2} \frac{1}{3}\eta \chi \left( {3\eta -\xi -\zeta } \right) \\ &{}\quad +\,a_{1,2} \frac{1}{3}\eta \xi \left( {3\eta -\zeta -\chi } \right) \\ s_7^{*} &{}=b_{3,2} \frac{1}{3}\eta \zeta \left( {3\eta -\chi -\xi } \right) +b_{4,2} \frac{1}{3}\eta \chi \left( {3\eta -\xi -\zeta } \right) \\ &{}\quad +\,b_{1,2} \frac{1}{3}\eta \xi \left( {3\eta -\zeta -\chi } \right) \\ s_8^{*} &{}=c_{3,2} \frac{1}{3}\eta \zeta \left( {3\eta -\chi -\xi } \right) +c_{4,2} \frac{1}{3}\eta \chi \left( {3\eta -\xi -\zeta } \right) \\ &{}\quad +\,c_{1,2} \frac{1}{3}\eta \xi \left( {3\eta -\zeta -\chi } \right) \\ s_9^{*} &{}=\zeta \left( {\zeta ^{2}+3\zeta \left( {\chi +\xi +\eta } \right) +2\left( {\chi \xi +\eta \left( {\chi +\xi } \right) } \right) } \right) \\ s_{10}^{*} &{}=a_{4,3} \frac{1}{3}\zeta \chi \left( {3\zeta -\xi -\eta } \right) +a_{1,3} \frac{1}{3}\zeta \xi \left( {3\zeta -\eta -\chi } \right) \\ &{}\quad +\,a_{2,3} \frac{1}{3}\zeta \eta \left( {3\zeta -\chi -\xi } \right) \\ s_{11}^{*} &{}=b_{4,3} \frac{1}{3}\zeta \chi \left( {3\zeta -\xi -\eta } \right) +b_{1,3} \frac{1}{3}\zeta \xi \left( {3\zeta -\eta -\chi } \right) \\ &{}\quad +\,b_{2,3} \frac{1}{3}\zeta \eta \left( {3\zeta -\chi -\xi } \right) \\ s_{12}^{*} &{}=c_{4,3} \frac{1}{3}\zeta \chi \left( {3\zeta -\xi -\eta } \right) +c_{1,3} \frac{1}{3}\zeta \xi \left( {3\zeta -\eta -\chi } \right) \\ &{}\quad +\,c_{2,3} \frac{1}{3}\zeta \eta \left( {3\zeta -\chi -\xi } \right) \\ s_{13}^{*} &{}=\chi \left( {\chi ^{2}+3\chi \left( {\xi +\eta +\zeta } \right) +2\left( {\xi \eta +\zeta \left( {\xi +\eta } \right) } \right) } \right) \\ s_{14}^{*} &{}=a_{1,4} \frac{1}{3}\chi \xi \left( {3\chi -\eta -\zeta } \right) +a_{2,4} \frac{1}{3}\chi \eta \left( {3\chi -\zeta -\xi } \right) \\ &{}\quad +\,a_{3,4} \frac{1}{3}\chi \zeta \left( {3\chi -\xi -\eta } \right) \\ s_{15}^{*} &{}=b_{1,4} \frac{1}{3}\chi \xi \left( {3\chi -\eta -\zeta } \right) +b_{2,4} \frac{1}{3}\chi \eta \left( {3\chi -\zeta -\xi } \right) \\ &{}\quad +\,b_{3,4} \frac{1}{3}\chi \zeta \left( {3\chi -\xi -\eta } \right) \\ s_{16}^{*} &{}=c_{1,4} \frac{1}{3}\chi \xi \left( {3\chi -\eta -\zeta } \right) +c_{2,4} \frac{1}{3}\chi \eta \left( {3\chi -\zeta -\xi } \right) \\ &{}\quad +\,c_{3,4} \frac{1}{3}\chi \zeta \left( {3\chi -\xi -\eta } \right) \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.4)

The above FN shape functions can be written in terms of the ENMC shape functions as

$$\begin{aligned} \left. {\begin{array}{ll} s_1^{*} &{}=\bar{{s}}_1^{*} +\frac{1}{3}\bar{{s}}_{17}^{*} +\frac{1}{3}\bar{{s}}_{18}^{*} +\frac{1}{3}\bar{{s}}_{20}^{*} ,\\ s_2^{*} &{}=\bar{{s}}_2^{*} +\frac{2}{81}\alpha _{2,3,1} \bar{{s}}_{17}^{*} +\frac{2}{81}\alpha _{2,4,1} \bar{{s}}_{18}^{*} +\frac{2}{81}\alpha _{3,4,1} \bar{{s}}_{20}^{*} , \\ s_3^{*} &{}=\bar{{s}}_3^{*} +\frac{2}{81}\beta _{2,3,1} \bar{{s}}_{17}^{*} +\frac{2}{81}\beta _{2,4,1} \bar{{s}}_{18}^{*} +\frac{2}{81}\beta _{3,4,1} \bar{{s}}_{20}^{*} ,\\ s_4^{*} &{}=\bar{{s}}_4^{*} +\frac{2}{81}\gamma _{2,3,1} \bar{{s}}_{17}^{*} +\frac{2}{81}\gamma _{2,4,1} \bar{{s}}_{18}^{*} +\frac{2}{81}\gamma _{3,4,1} \bar{{s}}_{20}^{*} , \\ s_5^{*} &{}=\bar{{s}}_5^{*} +\frac{1}{3}\bar{{s}}_{17}^{*} +\frac{1}{3}\bar{{s}}_{18}^{*} +\frac{1}{3}\bar{{s}}_{19}^{*} ,\\ s_6^{*} &{}=\bar{{s}}_6^{*} +\frac{2}{81}\alpha _{1,3,2} \bar{{s}}_{17}^{*} +\frac{2}{81}\alpha _{1,4,2} \bar{{s}}_{18}^{*} +\frac{2}{81}\alpha _{3,4,2} \bar{{s}}_{19}^{*} , \\ s_7^{*} &{}=\bar{{s}}_7^{*} +\frac{2}{81}\beta _{1,3,2} \bar{{s}}_{17}^{*} +\frac{2}{81}\beta _{1,4,2} \bar{{s}}_{18}^{*} +\frac{2}{81}\beta _{3,4,2} \bar{{s}}_{19}^{*} ,\\ s_8^{*} &{}=\bar{{s}}_8^{*} +\frac{2}{81}\gamma _{1,3,2} \bar{{s}}_{17}^{*} +\frac{2}{81}\gamma _{1,4,2} \bar{{s}}_{18}^{*} +\frac{2}{81}\gamma _{3,4,2} \bar{{s}}_{19}^{*} , \\ s_9^{*} &{}=\bar{{s}}_9^{*} +\frac{1}{3}\bar{{s}}_{17}^{*} +\frac{1}{3}\bar{{s}}_{19}^{*} +\frac{1}{3}\bar{{s}}_{20}^{*} ,\\ s_{10}^{*} &{}=\bar{{s}}_{10}^{*} +\frac{2}{81}\alpha _{1,2,3} \bar{{s}}_{17}^{*} +\frac{2}{81}\alpha _{2,4,3} \bar{{s}}_{19}^{*} +\frac{2}{81}\alpha _{1,4,3} \bar{{s}}_{20}^{*} , \\ s_{11}^{*} &{}=\bar{{s}}_{11}^{*} +\frac{2}{81}\beta _{1,2,3} \bar{{s}}_{17}^{*} +\frac{2}{81}\beta _{2,4,3} \bar{{s}}_{19}^{*} +\frac{2}{81}\beta _{1,4,3} \bar{{s}}_{20}^{*} ,\\ s_{12}^{*} &{}=\bar{{s}}_{12}^{*} +\frac{2}{81}\gamma _{1,2,3} \bar{{s}}_{17}^{*} +\frac{2}{81}\gamma _{2,4,3} \bar{{s}}_{19}^{*} +\frac{2}{81}\gamma _{1,4,3} \bar{{s}}_{20}^{*} , \\ s_{13}^{*} &{}=\bar{{s}}_{13}^{*} +\frac{1}{3}\bar{{s}}_{18}^{*} +\frac{1}{3}\bar{{s}}_{19}^{*} +\frac{1}{3}\bar{{s}}_{20}^{*} ,\\ s_{14}^{*} &{}=\bar{{s}}_{14}^{*} +\frac{2}{81}\alpha _{1,2,4} \bar{{s}}_{18}^{*} +\frac{2}{81}\alpha _{2,3,4} \bar{{s}}_{19}^{*} +\frac{2}{81}\alpha _{1,3,4} \bar{{s}}_{20}^{*} , \\ s_{15}^{*} &{}=\bar{{s}}_{15}^{*} +\frac{2}{81}\beta _{1,2,4} \bar{{s}}_{18}^{*} +\frac{2}{81}\beta _{2,3,4} \bar{{s}}_{19}^{*} +\frac{2}{81}\beta _{1,3,4} \bar{{s}}_{20}^{*} ,\\ s_{16}^{*} &{}=\bar{{s}}_{16}^{*} +\frac{2}{81}\gamma _{1,2,4} \bar{{s}}_{18}^{*} +\frac{2}{81}\gamma _{2,3,4} \bar{{s}}_{19}^{*} +\frac{2}{81}\gamma _{1,3,4} \bar{{s}}_{20}^{*} \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.5)

To obtain the above equations, the following constant coefficients need to be defined:

$$\begin{aligned}&\left. {\begin{array}{ll} V_1 &{}=\frac{1}{6}\left( \left( {x_3 y_4 -x_4 y_3 } \right) z_2 +\left( {x_4 y_2 -x_2 y_4 } \right) z_3 \right. \\ &{}\quad \left. +\left( {x_2 y_3 -x_3 y_2 } \right) z_4 \right) \\ V_2 &{}=\frac{1}{6}\left( \left( {x_4 y_3 -x_3 y_4 } \right) z_1 +\left( {x_1 y_4 -x_4 y_1 } \right) z_3 \right. \\ &{}\quad \left. +\left( {x_3 y_1 -x_1 y_3 } \right) z_4 \right) \\ V_3 &{}=\frac{1}{6}\left( \left( {x_2 y_4 -x_4 y_2 } \right) z_1 +\left( {x_4 y_1 -x_1 y_4 } \right) z_2 \right. \\ &{}\quad \left. +\left( {x_1 y_2 -x_2 y_1 } \right) z_4 \right) \\ V_4 &{}=\frac{1}{6}\left( \left( {x_3 y_2 -x_2 y_3 } \right) z_1 +\left( {x_1 y_3 -x_3 y_1 } \right) z_2 \right. \\ &{}\quad \left. +\left( {x_2 y_1 -x_1 y_2 } \right) z_3 \right) \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.6)
$$\begin{aligned}&\left. {\begin{array}{l} L_{1,1} =y_4 \left( {z_3 -z_2 } \right) +y_3 \left( {z_2 -z_4 } \right) +y_2 \left( {z_4 -z_3 } \right) \\ L_{1,2} =x_4 \left( {z_2 -z_3 } \right) +x_2 \left( {z_3 -z_4 } \right) +x_3 \left( {z_4 -z_2 } \right) \\ L_{1,3} =x_4 \left( {y_3 -y_2 } \right) +x_3 \left( {y_2 -y_4 } \right) +x_2 \left( {y_4 -y_3 } \right) \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.7)
$$\begin{aligned}&\left. {\begin{array}{l} L_{2,1} =y_4 \left( {z_1 -z_3 } \right) +y_1 \left( {z_3 -z_4 } \right) +y_3 \left( {z_4 -z_1 } \right) \\ L_{2,2} =x_4 \left( {z_3 -z_1 } \right) +x_3 \left( {z_1 -z_4 } \right) +x_1 \left( {z_4 -z_3 } \right) \\ L_{2,3} =x_4 \left( {y_1 -y_3 } \right) +x_1 \left( {y_3 -y_4 } \right) +x_3 \left( {y_4 -y_1 } \right) \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.8)
$$\begin{aligned}&\left. {\begin{array}{l} L_{3,1} =y_4 \left( {z_2 -z_1 } \right) +y_2 \left( {z_1 -z_4 } \right) +y_1 \left( {z_4 -z_2 } \right) \\ L_{3,2} =x_4 \left( {z_1 -z_2 } \right) +x_1 \left( {z_2 -z_4 } \right) +x_2 \left( {z_4 -z_1 } \right) \\ L_{3,3} =x_4 \left( {y_2 -y_1 } \right) +x_2 \left( {y_1 -y_4 } \right) +x_1 \left( {y_4 -y_2 } \right) \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.9)
$$\begin{aligned}&\left. {\begin{array}{l} L_{4,1} =y_3 \left( {z_1 -z_2 } \right) +y_1 \left( {z_2 -z_3 } \right) +y_2 \left( {z_3 -z_1 } \right) \\ L_{4,2} =x_3 \left( {z_2 -z_1 } \right) +x_2 \left( {z_1 -z_3 } \right) +x_1 \left( {z_3 -z_2 } \right) \\ L_{4,3} =x_3 \left( {y_1 -y_2 } \right) +x_1 \left( {y_2 -y_3 } \right) +x_2 \left( {y_3 -y_1 } \right) \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$
(A.10)
$$\begin{aligned}&a_{i,j} =x_i -x_j ,\quad b_{i,j} =y_i -y_j ,\quad c_{i,j} =z_i -z_j ,\nonumber \\&\;i=1,2,\ldots ,4,\;j=1,2,\ldots ,4,\;i\ne j \end{aligned}$$
(A.11)
$$\begin{aligned}&\left. {\begin{array}{l} \alpha _{i,j,h} =x_i +x_j -2x_h \\ \beta _{i,j,h} =y_i +y_j -2y_h \\ \gamma _{i,j,h} =z_i +z_j -2z_h \\ \end{array}} \right\} ,\nonumber \\&\quad i=1,2,\ldots ,4,\;j=1,2,\ldots ,4,\nonumber \\&\quad h=1,2,\ldots ,4,\;i\ne j,\;i\ne h \end{aligned}$$
(A.12)

where \(x_{k} \), \(y_{k} \), and \(z_{k} \) represent the Cartesian coordinates of the tetrahedron vertex k, and \(V=V_1 +V_2 +V_3 +V_4 \). The tetrahedron volume coordinates \(\xi ,\eta ,\zeta \), and \(\chi \) can be expressed as a linear combination of the tetrahedral Cartesian coordinates x, y, and z as:

$$\begin{aligned} \left. {\begin{array}{l} \xi =\frac{1}{6V}\left( {6V_1 +L_{1,1} x+L_{1,2} y+L_{1,3} z} \right) \\ \eta =\frac{1}{6V}\left( {6V_2 +L_{2,1} x+L_{2,2} y+L_{2,3} z} \right) \\ \zeta =\frac{1}{6V}\left( {6V_3 +L_{3,1} x+L_{3,2} y+L_{3,3} z} \right) \\ \chi =\frac{1}{6V}\left( {6V_4 +L_{4,1} x+L_{4,2} y+L_{4,3} z} \right) \end{array}} \right\} \end{aligned}$$
(A.13)

It can also be shown that the ENMC and FN shape functions written in terms of the Cartesian gradients \(\bar{{s}}_k^{*} \) and \(s_k^{*} \) can be expressed in terms of the ENMC and FN shape functions associated with the volume gradients \(\bar{{s}}_{k} \) and \(s_{k} \) as \({\bar{\mathbf{S}}}^{{*}}={\bar{\mathbf{S}}\bar{\mathbf{T}}}\) and \(\mathbf{S}^{{*}}=\mathbf{ST}\), where the transformation matrices \({\bar{\mathbf{T}}}\) and \(\mathbf{T}\) are previously defined in the paper.

In developing new ANCF finite elements with certain number and type of nodal coordinates, the use of incomplete polynomials may be necessary. In order to avoid trials and errors in identifying such incomplete polynomials and obtain symmetric structure in xy, and z, the method of algebraic constraint equations used in developing the FN element presented in this paper can be systematically used. For the FN element, one can use from the outset, the algebraic constraint equations to reduce the number of each polynomial basis function from 20 to 16 and to systematically define the incomplete polynomial which has the following basis expressed in terms of the volume coordinates:

$$\begin{aligned} \left. {\begin{array}{ll} h_1 &{}=\xi \left( {\xi ^{2}+4\eta \chi +4\zeta \left( {\eta +\chi } \right) } \right) ,\\ h_2 &{}=\eta \left( {\eta ^{2}+4\xi \chi +4\zeta \left( {\xi +\chi } \right) } \right) , \\ h_3 &{}=\zeta \left( {\zeta ^{2}+4\xi \chi +4\eta \left( {\xi +\chi } \right) } \right) ,\\ h_4 &{}=\chi \left( {\chi ^{2}+4\eta \xi +4\zeta \left( {\eta +\xi } \right) } \right) , \\ h_5 &{}=\eta \xi \left( {3\xi -\zeta -\chi } \right) ,\\ h_6 &{}=\eta \xi \left( {3\eta -\zeta -\chi } \right) ,\\ h_7 &{}=\zeta \eta \left( {3\eta -\xi -\chi } \right) , \\ h_8 &{}=\zeta \eta \left( {3\zeta -\xi -\chi } \right) ,\\ h_9 &{}=\zeta \xi \left( {3\xi -\eta -\chi } \right) ,\\ h_{10} &{}=\zeta \xi \left( {3\zeta -\eta -\chi } \right) , \\ h_{11} &{}=\xi \chi \left( {3\xi -\zeta -\eta } \right) ,\\ h_{12} &{}=\xi \chi \left( {3\chi -\zeta -\eta } \right) ,\\ h_{13} &{}=\eta \chi \left( {3\eta -\zeta -\xi } \right) , \\ h_{14} &{}=\eta \chi \left( {3\chi -\zeta -\xi } \right) ,\\ h_{15} &{}=\zeta \chi \left( {3\zeta -\eta -\xi } \right) ,\\ h_{16} &{}=\zeta \chi \left( {3\chi -\eta -\xi } \right) \\ \end{array}} \right\} \end{aligned}$$
(A.14)

Using the relationship between the Cartesian and volume coordinates (Eq. A.13), the basis functions can be written in terms of the Cartesian coordinates xy, and z as \(h_i =\sum _{j=1}^{20} {a_{i,j} b_j } \), where

$$\begin{aligned} \begin{array}{l} \left[ {{\begin{array}{llll} {b_1 }&{} {b_2 }&{} \ldots &{} {b_{20} } \\ \end{array} }} \right] = \\ {{\begin{array}{llllllllllllllllllll} \big [1&{} x&{} y&{} z&{} {x^{2}}&{} {y^{2}}&{} {z^{2}}&{} {xy}&{} {yz}&{} {xz}&{} {x^{3}}&{} {y^{3}}&{} \\ {z^{3}}&{} {x^{2}y}&{} {y^{2}x}&{} {y^{2}z}&{} {z^{2}y}&{} {x^{2}z}&{} {z^{2}x}&{} {xyz} \big ]\\ \end{array} }} \\ \end{array}\nonumber \\ \end{aligned}$$
(A.15)

and the coefficients \(a_{i,j} ,i=1,2,\ldots ,16,j=1,2,\ldots ,20\), can be systematically defined. One can show that the incomplete polynomial defined by the basis functions \(h_i =\sum _{j=1}^{20} {a_{i,j} b_j } \) has the linear terms 1, xy, and z that ensure that the rigid body motion can be correctly described, and such a polynomial will have a symmetric structure in xy, and z.

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Pappalardo, C.M., Wang, T. & Shabana, A.A. Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures. Nonlinear Dyn 89, 2905–2932 (2017). https://doi.org/10.1007/s11071-017-3635-6

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