The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle

Affiliation(s)

Institute of Computational Modeling, Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia.

Institute of Computational Modeling, Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia;Vladimir Shaydurov.

Siberian Federal University, Krasnoyarsk, Russia.

Institute of Computational Modeling, Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia.

Institute of Computational Modeling, Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia;Vladimir Shaydurov.

Siberian Federal University, Krasnoyarsk, Russia.

ABSTRACT

The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.

KEYWORDS

Continuously Differentiable Finite Elements; Bogner-Fox-Schmit Rectangle; Triangular Hermite Element

Continuously Differentiable Finite Elements; Bogner-Fox-Schmit Rectangle; Triangular Hermite Element

Cite this paper

L. Gileva, V. Shaydurov and B. Dobronets, "The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle,"*Applied Mathematics*, Vol. 4 No. 12, 2013, pp. 50-56. doi: 10.4236/am.2013.412A006.

L. Gileva, V. Shaydurov and B. Dobronets, "The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle,"

References

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http://dx.doi.org/10.1002/nme.1620010108

[3] J. Morgan and L. R. Scott, “A Nodal Basis for C1 Piecewise Polynomials of Degree n,” Mathematics of Computation, Vol. 29, No. 131, 1975, pp. 736-740.

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http://dx.doi.org/10.1137/0713011

[6] J. Douglas Jr., T. Dupont, P. Percell and R. Scott, “A Family of C1 Finite Elements with Optimal Approximation Properties for Various Galerkin Methods for 2nd and 4th Order Problems”, RAIRO Analise Numérique, Vol. 13, No. 3, 1979, pp. 227-255.

[7] M. J. D. Powell and M. A. Sabin, “Piecewise Quadratic Approximations on Triangles,” ACM Transactions on Mathematical Software, Vol. 3-4, No. 4, 1977, pp. 316325.

http://dx.doi.org/10.1145/355759.355761

[8] B. Fraeijs de Veubeke, “Bending and Stretching of Plates”, Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, October 1965, pp. 863-886.

[9] B. Fraeijs de Veubeke, “A Conforming Finite Element for Plate Bending,” In: J. C. Zienkievicz and G. S. Holister, Eds., Stree Analysis, Wiley, New York, 1965, pp. 145197.

[10] F. K. Bogner, R. L. Fox and L. A. Schmit, “The Generation of Interelement Compatible Stiffness and Mass Matrices by the Use of Interpolation Formulas,” Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, October 1965, pp. 397-444.

[11] S. Zhang, “On the full C1-Qk Finite Element Spaces on Rectangles and Cuboids,” Advances in Applied Mathematics and Mechanics, Vol. 2, No. 6, 2010, pp. 701-721.

[12] P. G. Ciarlet, “The Finite Element Method for Elliptic Problems,” North-Holland, Amsterdam, 1978.

[13] Z. C. Li and N. Yan, “New Error Estimates of Bi-Cubic Hermite Finite Element Methods for Biharmonic Equations,” Journal of Computational and Applied Mathematics, Vol. 142, No. 2, 2002, pp. 251-285.

http://dx.doi.org/10.1016/S0377-0427(01)00494-0

[14] S. C. Brenner and L. R. Scott, “The Mathematical Theory of Finite Element Methods,” Springer-Verlag, New York, 1994. http://dx.doi.org/10.1007/978-1-4757-4338-8

[15] B. M. Irons, “A Conforming Quadratic Triangular Element for Plate Bending,” International Journal for Numerical Methods in Engineering, Vol. 1, No. 1, 1969, pp. 101-122.

[16] D. L. Logan, “A First Course of Finite Element Method,” SI Edition, 2011.

[17] J. Blaauwendraad, “Plates and FEM: Surprises and Pitfalls,” Springer, New York, 2010.

http://dx.doi.org/10.1007/978-90-481-3596-7

[18] J. Zhao, “Convergence of Vand F-cycle Multigrid Methods for Biharmonic Problem Using the HsiehClough-Tocher Element,” Numerical Methods for Partial Differential Equations, Vol. 21, No. 3, 2005, pp. 451-471.

http://dx.doi.org/10.1002/num.20048

[19] J. Petera and J. F. T. Pittman, “Isoparametric Hermite elements,” Intenational Journal for Numerical Methods in Engineering, Vol. 37, No. 20, 1994, pp. 3480-3519.

[20] R. A. Adams and J. J. F. Fournier, “Sobolev Spaces,” Academic Press, New York, 2003.

[21] Z. Chen and H. Wu, “Selected Topics in Finite Element Methods,” Science Press, Beijing, 2010.

[1] J. H. Argyris, I. Fried and D. W. Scharpf, “The TUBA Family of Plate Elements for the Matrix Displacement Method,” Journal of the Royal Aeronautical Society, Vol. 72, No. 692, 1968, pp. 701-709.

[2] K. Bell, “A Refined Triangular Plate Bending Element”, International Journal of Numerical Methods in Engineering, Vol. 1, No. 1, 1969, pp. 101-122.

http://dx.doi.org/10.1002/nme.1620010108

[3] J. Morgan and L. R. Scott, “A Nodal Basis for C1 Piecewise Polynomials of Degree n,” Mathematics of Computation, Vol. 29, No. 131, 1975, pp. 736-740.

[4] R. W. Clough and J. L. Tocher, “Finite Element Stiffness Matrices for Analysis of Plates in Bending,” Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, October 1965, pp. 515-545.

[5] P. Percell, “On Cubic and Quartic Clough-Tocher Finite Elements,” SIAM Journal on Numerical Analysis, Vol. 13, No. 1, 1976, pp. 100-103.

http://dx.doi.org/10.1137/0713011

[6] J. Douglas Jr., T. Dupont, P. Percell and R. Scott, “A Family of C1 Finite Elements with Optimal Approximation Properties for Various Galerkin Methods for 2nd and 4th Order Problems”, RAIRO Analise Numérique, Vol. 13, No. 3, 1979, pp. 227-255.

[7] M. J. D. Powell and M. A. Sabin, “Piecewise Quadratic Approximations on Triangles,” ACM Transactions on Mathematical Software, Vol. 3-4, No. 4, 1977, pp. 316325.

http://dx.doi.org/10.1145/355759.355761

[8] B. Fraeijs de Veubeke, “Bending and Stretching of Plates”, Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, October 1965, pp. 863-886.

[9] B. Fraeijs de Veubeke, “A Conforming Finite Element for Plate Bending,” In: J. C. Zienkievicz and G. S. Holister, Eds., Stree Analysis, Wiley, New York, 1965, pp. 145197.

[10] F. K. Bogner, R. L. Fox and L. A. Schmit, “The Generation of Interelement Compatible Stiffness and Mass Matrices by the Use of Interpolation Formulas,” Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, October 1965, pp. 397-444.

[11] S. Zhang, “On the full C1-Qk Finite Element Spaces on Rectangles and Cuboids,” Advances in Applied Mathematics and Mechanics, Vol. 2, No. 6, 2010, pp. 701-721.

[12] P. G. Ciarlet, “The Finite Element Method for Elliptic Problems,” North-Holland, Amsterdam, 1978.

[13] Z. C. Li and N. Yan, “New Error Estimates of Bi-Cubic Hermite Finite Element Methods for Biharmonic Equations,” Journal of Computational and Applied Mathematics, Vol. 142, No. 2, 2002, pp. 251-285.

http://dx.doi.org/10.1016/S0377-0427(01)00494-0

[14] S. C. Brenner and L. R. Scott, “The Mathematical Theory of Finite Element Methods,” Springer-Verlag, New York, 1994. http://dx.doi.org/10.1007/978-1-4757-4338-8

[15] B. M. Irons, “A Conforming Quadratic Triangular Element for Plate Bending,” International Journal for Numerical Methods in Engineering, Vol. 1, No. 1, 1969, pp. 101-122.

[16] D. L. Logan, “A First Course of Finite Element Method,” SI Edition, 2011.

[17] J. Blaauwendraad, “Plates and FEM: Surprises and Pitfalls,” Springer, New York, 2010.

http://dx.doi.org/10.1007/978-90-481-3596-7

[18] J. Zhao, “Convergence of Vand F-cycle Multigrid Methods for Biharmonic Problem Using the HsiehClough-Tocher Element,” Numerical Methods for Partial Differential Equations, Vol. 21, No. 3, 2005, pp. 451-471.

http://dx.doi.org/10.1002/num.20048

[19] J. Petera and J. F. T. Pittman, “Isoparametric Hermite elements,” Intenational Journal for Numerical Methods in Engineering, Vol. 37, No. 20, 1994, pp. 3480-3519.

[20] R. A. Adams and J. J. F. Fournier, “Sobolev Spaces,” Academic Press, New York, 2003.

[21] Z. Chen and H. Wu, “Selected Topics in Finite Element Methods,” Science Press, Beijing, 2010.