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Hybrid-Trefftz finite element model for antisymmetric laminated composite plates using a high order shear deformation theory

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Abstract

A novel hybrid-Trefftz finite element (HTFE) has been developed for the static analysis of thick and thin antisymmetric cross-ply and angle-ply laminated composite plates. For the first time a high order shear deformation theory is used for both the internal and auxiliary displacement fields for constructing the Trefftz functions. The usual tedious approach of converting the system of governing partial differential equations into a single governing equation has been avoided in the present development. The Trefftz functions have been derived from the exact solutions of the homogeneous governing equations of equlibrium. The current HTFE formulation does not need to find the particular solutions of the system of governing partial differential equations. Comparison with exact and analytical solutions reveals that the HTFE developed here is excellently capable of predicting the responses of the antisymmetric laminated composite plates. The Trefftz functions developed here can be used for deriving the polygonal HTFE with any number of sides.

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References

  • Choo, Y.S., Choi, N., Lee, B.C.: A new hybrid-Trefftz triangular and quadrilateral plate elements. Appl. Math. Model. 34(1), 14–23 (2010)

    Article  MathSciNet  Google Scholar 

  • De Freitas, J.A.T.: Formulation of elastostatic hybrid-Trefftz stress elements. Comput. Methods Appl. Mech. Eng. 153(1–2), 127–151 (1998)

    Article  MathSciNet  Google Scholar 

  • de Freitas, J.A.T., Ji, Z.-Y.: Hybrid-Trefftz finite element formulation for simulation of singular stress fields. Int. J. Numer. Methods Eng. 39(2), 281–308 (1996)

    Article  Google Scholar 

  • Dvorkin, E.N., Cuitiño, A.M., Gioia, G.: Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int. J. Numer. Methods Eng. 30(3), 541–564 (1990)

    Article  Google Scholar 

  • Goudarzi, T.: Iterative and variational homogenization methods for filled elastomers (2014)

  • Ha, S.K., Keilers, C., Chang, F.-K.: Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators. AIAA J. 30(3), 772–780 (1992)

    Article  Google Scholar 

  • Jin, W.G., Cheung, Y.K., Zienkiewicz, O.C.: Application of the Trefftz method in plane elasticity problems. Int. J. Numer. Methods Eng. 30(6), 1147–1161 (1990)

    Article  MathSciNet  Google Scholar 

  • Jin, W.G., Cheung, Y.K., Zienkiewicz, O.C.: Trefftz method for Kirchhoff plate bending problems. Int. J. Numer. Methods Eng. 36(5), 765–781 (1993)

    Article  Google Scholar 

  • Jirousek, J., Guex, L.: The hybrid-Trefftz finite element model and its application to plate bending. Int. J. Numer. Methods Eng. 23(4), 651–693 (1986)

    Article  MathSciNet  Google Scholar 

  • Kant, T.: A higher-order theory for free vibration of unsymmetrically laminated composite and sandwich plates—finite element evaluations. Comput. Struct. 32(5), 1125–1132 (1989)

    Article  Google Scholar 

  • Karkon, M., Rezaiee-Pajand, M.: Hybrid-Trefftz formulation for analysis of thick orthotropic plates. Aerosp. Sci. Technol. 50, 234–244 (2016)

    Article  Google Scholar 

  • Kawai, T.: My challenge in the development of a mixed variational method in solid mechanics. Appl. Mech. Rev. 60(March 2007), 51–64 (2007)

    Article  Google Scholar 

  • Kita, E., Kamiya, N.: Trefftz method: an overview. Adv. Eng. Softw. 24(1–3), 3–12 (1995)

    Article  Google Scholar 

  • Li, Y., Wang, W., Ling, R., Tu, C.: Shape optimization of quad mesh elements. Comput. Graph. 35(3), 444–451 (2011)

    Article  Google Scholar 

  • Pagano, N.J.: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Compos. Mater. 4(1), 20–34 (1970)

    Article  Google Scholar 

  • Pian, T.H.H.: State-of-the-art development of hybrid/mixed finite element method. Finite Elem. Anal. Des. 21(1–2), 5–20 (1995)

    Article  MathSciNet  Google Scholar 

  • Piltner, R.: The derivation of special purpose element functions using complex solution representations. Comput. Assist. Mech. Eng. Sci. 10, 1–15 (2003)

    MATH  Google Scholar 

  • Qin, Q.H.: Dual variational formulation for Trefftz finite element method of elastic materials. Mech. Res. Commun. 31(3), 321–330 (2004)

    Article  Google Scholar 

  • Qin, Q.H.: Formulation of hybrid Trefftz finite element method for elastoplasticity. Appl. Math. Model. 29(3), 235–252 (2005)

    Article  Google Scholar 

  • Ray, M.C.: Zeroth-order shear deformation theory for laminated composite plates. J. Appl. Mech. 70(3), 374 (2003)

    Article  Google Scholar 

  • Ray, M.C.: A novel hybrid-Trefftz finite element for symmetric laminated composite plates. Int. J. Mech. Mater. Des. 15(3), 629–646 (2019)

    Article  Google Scholar 

  • Ray, M.C., Mallik, N.: Finite element analysis of smart structures containing piezoelectric fiber-reinforced composite actuator. AIAA J. 42(7), 1398–1405 (2004)

    Article  Google Scholar 

  • Ray, M.C., Pradhan, A.K.: The performance of vertically reinforced 1–3 piezoelectric composites in active damping of smart structures. Smart Mater. Struct. 15(2), 631–641 (2006)

    Article  Google Scholar 

  • Reddy, J.N.: Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, Boca Raton (2003)

    Google Scholar 

  • Rezaiee-Pajand, M., Karkon, M.: Two efficient hybrid-Trefftz elements for plate bending analysis. Lat. Am. J. Solids Struct. 9(1), 43–67 (2012)

    Article  Google Scholar 

  • Rezaiee-Pajand, M., Yaghoobi, M., Karkon, M.: Hybrid Trefftz formulation for thin plate analysis. Int. J. Comput. Methods 09(04), 1250053 (2012)

    Article  MathSciNet  Google Scholar 

  • Wang, K.Y., Li, P.C., Wang, D.Z.: Trefftz-type FEM for solving orthotropic potential problems. Lat. Am. J. Solids Struct. 11(14), 2537–2554 (2014)

    Article  Google Scholar 

  • Zieliński, A.P.: On trial functions applied in the generalized Trefftz method. Adv. Eng. Softw. 24(1–3), 147–155 (1995)

    Article  Google Scholar 

  • Zielinski, A.P., Zienkiewicz, O.C.: Generalized finite element analysis with T-complete boundary solution functions. Int. J. Numer. Methods Eng. 21(3), 509–528 (1985)

    Article  Google Scholar 

  • Zienkiewicz, O.C.: Trefftz type approximation and the generalized finite element method—history and development. Comput. Assist. Mech. Eng. Sci. 4, 305–316 (1997)

    MATH  Google Scholar 

  • Zienkiewicz, O.C., Kelly, D.W., Bettess, P.: The coupling of the finite element method and boundary solution procedure. Int. J. Numer. Methods Eng. 11(March 1976), 355–375 (1977)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Rowan University, Glassboro, NJ, USA

    M. C. Ray

  2. Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    Subhasankar Dwibedi

Authors
  1. M. C. Ray
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  2. Subhasankar Dwibedi
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Correspondence to M. C. Ray.

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M. C. Ray: Currently on leave from Indian Institute of Technology, Kharagpur-721302, India.

Appendices

Appendix 1

The matrices appearing in Eq. (20) are as follows:

$$ \begin{aligned} {\mathbf{D}}_{1} & = \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,1} \right)} \hfill & {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {1,7} \right)} \hfill \\ {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {2,2} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {2,7} \right)} \hfill \\ {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {3,7} \right)} \hfill \\ {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {4,4} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {4,7} \right)} \hfill \\ {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {5,5} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {5,7} \right)} \hfill \\ {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {6,6} \right)} \hfill & {\bar{A}\left( {6,7} \right)} \hfill \\ \end{array} } \right], \\ {\mathbf{N}}_{{{\text{s}}1}} & = - \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {1,7} \right)} \hfill \\ {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {2,2} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {2,7} \right)} \hfill \\ {\bar{A}\left( {3,3} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {3,7} \right)} \hfill \\ {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {4,4} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {4,7} \right)} \hfill \\ {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {5,5} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {5,7} \right)} \hfill \\ {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {6,6} \right)} \hfill & {\bar{A}\left( {6,7} \right)} \hfill \\ \end{array} } \right], \\ {\mathbf{N}}_{{{\text{s}}2}} & = - \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,1} \right)} \hfill & {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {1,7} \right)} \hfill \\ {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {2,7} \right)} \hfill \\ {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {3,3} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {3,7} \right)} \hfill \\ {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {4,4} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {4,7} \right)} \hfill \\ {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {5,5} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {5,7} \right)} \hfill \\ {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {6,6} \right)} \hfill & {\bar{A}\left( {6,7} \right)} \hfill \\ \end{array} } \right], \\ \end{aligned} $$
$$ {\mathbf{N}}_{{{\text{s}}3}} = - \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,1} \right)} \hfill & {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {1,7} \right)} \hfill \\ {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {2,2} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {2,7} \right)} \hfill \\ {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {3,3} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {3,7} \right)} \hfill \\ {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {4,7} \right)} \hfill \\ {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {5,5} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {5,7} \right)} \hfill \\ {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {6,6} \right)} \hfill & {\bar{A}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\mathbf{s}}4}} = - \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,1} \right)} \hfill & {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {1,7} \right)} \hfill \\ {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {2,2} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {2,7} \right)} \hfill \\ {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {3,3} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {3,7} \right)} \hfill \\ {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {4,4} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {4,7} \right)} \hfill \\ {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {5,7} \right)} \hfill \\ {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {6,6} \right)} \hfill & {\bar{A}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{s}}5}} = - \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,1} \right)} \hfill & {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {1,7} \right)} \hfill \\ {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {2,2} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {2,7} \right)} \hfill \\ {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {3,3} \right)} \hfill & {\bar{A}\left( {3,7} \right)} \hfill \\ {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {4,4} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {4,7} \right)} \hfill \\ {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {5,5} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {5,7} \right)} \hfill \\ {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{s}}6}} = - \left[ {\begin{array}{*{20}l} {\bar{A}\left( {1,1} \right)} \hfill & {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {1,3} \right)} \hfill \\ {\bar{A}\left( {1,2} \right)} \hfill & {\bar{A}\left( {2,2} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill \\ {\bar{A}\left( {1,3} \right)} \hfill & {\bar{A}\left( {2,3} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill & {\bar{A}\left( {3,3} \right)} \hfill \\ {\bar{A}\left( {1,4} \right)} \hfill & {\bar{A}\left( {2,4} \right)} \hfill & {\bar{A}\left( {4,4} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {3,4} \right)} \hfill \\ {\bar{A}\left( {1,5} \right)} \hfill & {\bar{A}\left( {2,5} \right)} \hfill & {\bar{A}\left( {4,5} \right)} \hfill & {\bar{A}\left( {5,5} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {3,5} \right)} \hfill \\ {\bar{A}\left( {1,6} \right)} \hfill & {\bar{A}\left( {2,6} \right)} \hfill & {\bar{A}\left( {4,6} \right)} \hfill & {\bar{A}\left( {5,6} \right)} \hfill & {\bar{A}\left( {6,6} \right)} \hfill & {\bar{A}\left( {3,6} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{D}}_{2} = \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,1} \right)} \hfill & {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {1,7} \right)} \hfill \\ {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {2,2} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {2,7} \right)} \hfill \\ {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {3,7} \right)} \hfill \\ {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {4,4} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {4,7} \right)} \hfill \\ {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {5,5} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {5,7} \right)} \hfill \\ {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {6,6} \right)} \hfill & {\bar{B}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{r}}1}} = - \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {1,7} \right)} \hfill \\ {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {2,2} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {2,7} \right)} \hfill \\ {\bar{B}\left( {3,3} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {3,7} \right)} \hfill \\ {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {4,4} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {4,7} \right)} \hfill \\ {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {5,5} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {5,7} \right)} \hfill \\ {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {6,6} \right)} \hfill & {\bar{B}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{r}}2}} = - \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,1} \right)} \hfill & {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {1,7} \right)} \hfill \\ {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {2,7} \right)} \hfill \\ {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {3,3} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {3,7} \right)} \hfill \\ {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {4,4} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {4,7} \right)} \hfill \\ {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {5,5} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {5,7} \right)} \hfill \\ {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {6,6} \right)} \hfill & {\bar{B}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{r}}3}} = - \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,1} \right)} \hfill & {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {1,7} \right)} \hfill \\ {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {2,2} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {2,7} \right)} \hfill \\ {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {3,3} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {3,7} \right)} \hfill \\ {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {4,7} \right)} \hfill \\ {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {5,5} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {5,7} \right)} \hfill \\ {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {6,6} \right)} \hfill & {\bar{B}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{r}}4}} = - \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,1} \right)} \hfill & {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {1,7} \right)} \hfill \\ {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {2,2} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {2,7} \right)} \hfill \\ {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {3,3} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {3,7} \right)} \hfill \\ {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {4,4} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {4,7} \right)} \hfill \\ {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {5,7} \right)} \hfill \\ {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {6,6} \right)} \hfill & {\bar{B}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{r}}5}} = - \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,1} \right)} \hfill & {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {1,7} \right)} \hfill \\ {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {2,2} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {2,7} \right)} \hfill \\ {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {3,3} \right)} \hfill & {\bar{B}\left( {3,7} \right)} \hfill \\ {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {4,4} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {4,7} \right)} \hfill \\ {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {5,5} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {5,7} \right)} \hfill \\ {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {6,7} \right)} \hfill \\ \end{array} } \right], $$
$$ {\mathbf{N}}_{{{\text{r}}6}} = - \left[ {\begin{array}{*{20}l} {\bar{B}\left( {1,1} \right)} \hfill & {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {1,3} \right)} \hfill \\ {\bar{B}\left( {1,2} \right)} \hfill & {\bar{B}\left( {2,2} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill \\ {\bar{B}\left( {1,3} \right)} \hfill & {\bar{B}\left( {2,3} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill & {\bar{B}\left( {3,3} \right)} \hfill \\ {\bar{B}\left( {1,4} \right)} \hfill & {\bar{B}\left( {2,4} \right)} \hfill & {\bar{B}\left( {4,4} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {3,4} \right)} \hfill \\ {\bar{B}\left( {1,5} \right)} \hfill & {\bar{B}\left( {2,5} \right)} \hfill & {\bar{B}\left( {4,5} \right)} \hfill & {\bar{B}\left( {5,5} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {3,5} \right)} \hfill \\ {\bar{B}\left( {1,6} \right)} \hfill & {\bar{B}\left( {2,6} \right)} \hfill & {\bar{B}\left( {4,6} \right)} \hfill & {\bar{B}\left( {5,6} \right)} \hfill & {\bar{B}\left( {6,6} \right)} \hfill & {\bar{B}\left( {3,6} \right)} \hfill \\ \end{array} } \right], $$
(47)

Appendix 2

Figure 2b illustrates that the degenerated HTFE consists of four inter-elements. The explicit forms of the shape function matrix \( {\mathbf{N}}_{i} \) appearing in Eq. (29) for the \( i \)th (\( i \) = 1, 2, 3, 4) inter-element are given by

$$ \begin{aligned} & {\mathbf{N}}_{1} = \left[ {\begin{array}{*{20}c} {\hat{n}_{1} {\mathbf{I}}} & {\hat{n}_{2} {\mathbf{I}}} & {\mathbf{O}} & {\mathbf{O}} \\ \end{array} } \right],{\mathbf{N}}_{2} = \left[ {\begin{array}{*{20}c} {\mathbf{O}} & {\hat{n}_{1} {\mathbf{I}}} & {\hat{n}_{2} {\mathbf{I}}} & {\mathbf{O}} \\ \end{array} } \right], \\ & {\mathbf{N}}_{3} = \left[ {\begin{array}{*{20}c} {\mathbf{O}} & {\mathbf{O}} & {\hat{n}_{2} {\mathbf{I}}} & {\hat{n}_{1} {\mathbf{I}}} \\ \end{array} } \right]\;{\text{and}}\;{\mathbf{N}}_{4} = \left[ {\begin{array}{*{20}c} {\hat{n}_{1} {\mathbf{I}}} & {\mathbf{O}} & {\mathbf{O}} & {\hat{n}_{2} {\mathbf{I}}} \\ \end{array} } \right] \\ \end{aligned} $$
(48)

in which \( {\mathbf{I}} \) is a (\( 7 \times 7 \)) identity matrix while \( \hat{n}_{1} \) and \( \hat{n}_{2} \) are the shape functions associated with the nodes of the inter-element corresponding to the values of the natural coordinate as \( \zeta = - 1 \) and \( \zeta = 1 \), respectively. These are given by

$$ \hat{n}_{1} = \frac{1}{2}\left( {1 - \zeta } \right)\,{\text{and}}\,\hat{n}_{2} = \frac{1}{2}\left( {1 + \zeta } \right) $$
(49)

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Ray, M.C., Dwibedi, S. Hybrid-Trefftz finite element model for antisymmetric laminated composite plates using a high order shear deformation theory. Int J Mech Mater Des 16, 817–837 (2020). https://doi.org/10.1007/s10999-020-09496-9

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