Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

A superconvergent nonconforming quadrilateral spline element for biharmonic equation using the B-net method

  • 2 Accesses


In this paper, we construct a new nonconforming quadrilateral element with 12 degrees of freedom to solve the biharmonic problems. \({\mathscr {T}}_{h}\) is a triangulated quadrangulation of the domain \(\varOmega \). For a quadrilateral element \(Q_{T}\), the finite element space, which contains \({\mathbb {P}}_{3}(Q_{T})\), is a subspace of the bivariate spline space \({\mathbf {S}}_{3}^{1}(Q_{T})\). The degrees of freedom are chosen as the four point values, the four edge integrals of the shape functions and the edge integrals of their normal derivatives such that the weak continuity between elements can be satisfied. Accordingly, we explicitly establish 12 spline interpolation bases in the B-net form. Error estimates are given with optimal convergence order in both discrete \(H^{2}\) and \(H^{1}\) seminorms. The proposed element NCQS12 can get the superconvergence results with theoretical proof when \({\mathscr {T}}_{h}\) is an uniform parallelogram mesh. Some degenerate meshes are considered subsequently. Finally, we do some numerical experiments to verify the theoretical analysis. Numerical results show that the proposed element performs well over the asymptotically regular parallelogram meshes, which is same as over the uniform parallelogram meshes.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. Adini A, Clough R (1960) Analysis of plate bending by the finite element method. University of California, California

  2. Argyris J, Fried I, Scharpf D (1968) The TUBA family of plate elements for the matrix displacement method. Aeronaut J 72(692):701–709

  3. Batoz JL, Tahar MB (1982) Evaluation of a new quadrilateral thin plate bending element. Int J Numer Methods Eng 18(11):1655–1677

  4. Bell K (1969) A refined triangular plate bending finite element. Int J Numer Methods Eng 1(1):101–122

  5. Bogner F, Fox R, Schmit L (1965) The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulae. In: Przemieniecki J (ed) Proceedings of the conference on matrix methods in structural mechanics, pp 397–444. Wright Patterson Air Force Base, Ohio

  6. Chen S (1996) 12-parameter rectangular plate elements ith geometric symmetry. Numer Math A J Chin Univ 3:233–238 (In Chinese)

  7. Chen J, Li C (2013) Development of quadrilateral spline thin plate elements using the B-net method. Acta Mech Sin 29(4):567–574

  8. Chen J, Li C (2015) The cubic spline Hermite interpolation bases for thin plate bending quadrilateral elements. Sci Sin 45(9):1523

  9. Chen J, Li CJ, Chen WJ (2010) A family of spline finite elements. Comput Struct 88(11–12):718–727

  10. Chen S, Shi D, I chiro H (2003) Trapezoidal plate blending element with double set parameters. J Comput Math 21(4):513–518

  11. Che W, Cheung YK (1997) Refined quadrilateral discrete Kirchhoff thin plate bending element. Int J Numer Methods Eng 40(21):3937–3953

  12. Chen X, Cen S, Long Y (2005) Two thin plate elements developed by assuming rotations and using quadrilateral area coordinates. Eng Mech 22(4):1–5,30

  13. Clough R, Tocher J (1965) Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the 1st conference on matrix methods in structural mechanics. Wright Patterson AFB, Ohio

  14. Farin G (1986) Triangular Bernstein–Bézier patches. Comput Aided Geom Des 3(2):83–127

  15. Ciarlet PG (1978) The finite element method for elliptic problems, vol 4. North-Holland, Amsterdam

  16. Grisvard P (1985) Elliptic problems in nonsmooth domains. Pitman Advanced Publishing Program, Boston

  17. Hrabok M, Hrudey T (1984) A review and catalogue of plate bending finite elements. Comput Struct 19(3):479–495

  18. Jeyachandrabose C, Kirkhope J, Meekisho L (1987) An improved discrete Kirchhoff quadrilateral thin-plate bending element. Int J Numer Methods Eng 24(3):635–654

  19. Lascaux P, Lesaint P (1975) Some nonconforming finite elements for the plate bending problem. Revue française d’automatique, informatique, recherche opérationnelle. Anal Numér 9(R1):9–53

  20. Li C, Wang R (2006) A new 8-node quadrilateral spline finite element. J Comput Appl Math 195(1):54–65

  21. Ming P, Shi Z (2002) Quadrilateral mesh. Chin Ann Math B 23(2):235–252

  22. Park C, Sheen D (2013) A quadrilateral Morley element for biharmonic equations. Numer Math 124(2):395–413

  23. Sander G (1964) Bornes supérieures et inférieures dans l’analyse matricielle des plaques en flexion–torsion. Bull Soc R Sci Liège 33:456–494 (In French)

  24. Shi Z (1984) A convergence condition for the quadrilateral Wilson element. Numer Math 44(3):349–361

  25. Shi Z (1986) On the convergence of the incomplete biquadratic nonconforming plate element. Math Numer Sin 8(8):53–62 (In Chinese)

  26. Shi Z (1990a) On the accuracy of the quasi-conforming and generalized conforming finite elements. Chin Ann Math 11(2):148–155

  27. Shi Z (1990b) On the error estimates of Morley element. Math Numer Sin 12(2):113–118

  28. Shi Z, Wang M (1988) Finite element methods. Science Press, Beijing

  29. Soh AK, Long Z, Cen S (2000) Development of a new quadrilateral thin plate element using area coordinates. Comput Methods Appl Mech Eng 190(8–10):979–987

  30. Suire G, Cederbaum G (1995) Periodic and chaotic behavior of viscoelastic nonlinear (elastica) bars under harmonic excitations. Int J Mech Sci 37(7):753–772

  31. Veubeke BFD (1968) A conforming finite element for plate bending. Int J Solids Struct 4(1):95–108

  32. Wang RH (1975) The structural characterization and interpolation for multivariate splines. Acta Math Sin 18(2):91–106 (In Chinese)

  33. Wang RH (2001) Multivariate spline functions and their applications. Science Press, Beijing

  34. Zhao Z, Xiao L, Chen S (2012) 13-parameter quadrilateral plate blending element with double set parameters. Math Numer Sin 34(3):285–296 (In Chinese)

Download references

Author information

Correspondence to Chong-Jun Li.

Additional information

Publisher's Note

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

This work was supported by the National Natural Science Foundation of China (nos. 11572081, 11871137, 11471066).

Communicated by Philippe Heluy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, C., Jia, Y. A superconvergent nonconforming quadrilateral spline element for biharmonic equation using the B-net method. Comp. Appl. Math. 39, 70 (2020).

Download citation


  • Biharmonic problem
  • Quadrilateral element
  • Superconvergence
  • Spline interpolation bases
  • The B-net method

Mathematics Subject Classification

  • 65D07
  • 65N30