Analysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method

Document Type: Research Paper


1 Federal University of Alagoas, Laboratory of Scientific Computing and Visualization Technology Center, Campus A.C. Simões, Maceió-AL, 57092-970, Brazil

2 Federal University of Sergipe, Department of Civil Engineering, Campus São Cristovão, Aracaju-SE, 49100-000, Brazil


The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation nodes are positioned in the zeros of orthogonal polynomials (Legendre, Lobatto, or Chebychev) or equally spaced nodal bases. A comparative study between the bases in the recovery of solutions to 1D and 2D elastostatic problems are performed. Examples are evaluated, and a significant improvement is observed when the SFEM, particularly the Lobatto approach, is used in comparison to the equidistant base interpolation.


Main Subjects

[1] Babuska, I., Szabo, B.A., Katz, I.N., The p-version of the finite element method. SIAM Journal on Numerical Analysis, 18(3), 1981, 515-545.

[2] Rocha, F.C., Kzam, A.K.L., Análise das aproximações de alta ordem por meio da interpolação espectral aplicadas ao MEC potencial. In proceeding XXXIV Iberian Latin-American Congress in Computational Methods in Engineering, 2013.

[3] Zak, A., Krawczuk, M., Certain numerical issues of wave propagation modelling in rods by the Spectral Finite Element Method. Finite Elements in Analysis and Design, 47, 2011, 1036-1046.

[4] Kudela, P., et al. Wave propagation modelling in 1D structures using spectral finite elements. Journal of Sound and Vibration, 300(1-2), 2007, 88-100.

[5] Karniadaki, G. E., Sherwin, S. J., Spectral/Hp Element Methods for CFD. Oxford: Oxford University Press, 1999.

[6] Nogueira, A.C. Jr., Bittencourt, M.L., Spectral/HP finite elements applied to linear and non-linear structural elastic problems. Latin American Journal of Solids and Structures, 4, 2007, 61-85.

[7] Willberg, C., Duczek, S., Vivar Perez, J.M., Schmicker, D., Gabbert, U., Comparison of different higher order finite element schemes for the simulation of lamb waves. Computer Methods in Applied Mechanics and Engineering, 241, 2012, 246-261.

[8] Fornberg, B., and Julia, Z., The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Computers & Mathematics with Applications, 54(3), 2007, 379-398.

[9] Brutman, L., Lebesgue functions for polynomial interpolation, a survey. Annals of Numerical Mathematics, 4, 1997, 111-127.

[10] Vos, P.E.J., Sherwin, S.J., Kirby, R.M., From h to p efficiently: implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations. Journal of Computational Physics,229(13), 2010, 5161-5181.

[11] Sherwin, S.J., Karniadakis, G.E., A triangular element method; applications to the imcompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 123(1–4), 1995, 189-229.

[12] Sherwin, S.J., Karniadakis, G.E., Tetrahedral hp finite elements: algorithms and flow simulations. Journal of Computational Physics, 124, 1996, 14-45.

[13] Tai, C.-Y., & Chan, Y. J. A hierarchic high-order Timoshenko beam finite element. Computers & Structures, 165, 2016, 48-58.

[14] W. Dauksher, A.F. Emery, The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements, Computer Methods in Applied Mechanics and Engineering, 188, 2000, 217-33.

[15] Khaji, N., & Zakian, P. Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method. Mechanics of Advanced Materials and Structures, 24(12), 2016, 1030-1042.

[16] Man, H., Song, C., Xiang, T., Gao, W., & Tin-Loi, F. High-order plate bending analysis based on the scaled boundary finite element method. International Journal for Numerical Methods in Engineering, 95(4), 2013, 331-360.

[17] Man, H., Song, C., Gao, W., & Tin-Loi, F. Semi-analytical analysis for piezoelectric plate using the scaled boundary finite-element method. Computers & Structures, 137, 2014, 47-62.

[18] Lin, G., Zhang, P., Liu, J., & Li, J. Analysis of laminated composite and sandwich plates based on the scaled boundary finite element method. Composite Structures, 187, 2018, 579-592.

[19] Zak, A., & Krawczuk, M. Static and dynamic analysis of isotropic shell structures by the spectral finite element method. Journal of Physics: Conference Series, 382, 2012, 012054.

[20] Wang, Q., Sprague, M.A., Legendre spectral finite element implementation of geometrically exact beam theory, in: Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, National Harbor, USA, 2013.

[21] Proriol, J., Sur une famille de polynomes à deux variables orthogonaux dans um triangle. Comptes Rendus Mathematique Academie des Sciences, 245(26), 1957, 2459-2461.

[22] Fejer, L. Lagrangesche interpolation und die zugehorigen konjugierten punkte. Mathematische Annalen, 106, 1932, 1-55.

[23] Blyth, M. G., & Pozrikidis, C. A Lobatto interpolation grid over the triangle. IMA Journal of Applied Mathematics, 71(1), 2016, 153-169.

[24] Pozrikidis, C., Introduction to Finite and Spectral Element Methods Using Matlab. Chapman & Hall/CRC, 2005.

[25] Franco, N. B., Cálculo numérico, São Paulo: Pearson Prentice Hall, 2007.

[26] Oñate, E., Structural Analysis with the Finite Element Method. Linear Statics. 1st ed. Barcelona: Springer, 2013.

[27] Ostachowicz, W., Kudela, P., Krawczuk, M., Zak, A., Guided Waves in Structures for SHM: The Time-domain Spectral Element Method. John Wiley & Sons Ltd: Chichester, UK, 2011.

[28] Blyth, M.G., Pozrikidis, C. A., Lobatto interpolation grid over the triangle. IMA Journal of Applied Mathematics, 71, 2006, 153-169.

[29] Pozrikidis, C.A., Spectral-element method for particulate Stokes flow. Journal of Computational Physics, 156, 1999, 360-381.     

[30] Rivlin, T.J., An introduction to the approximation of functions. Dover Publications; Revised edition, 1969.

[31] Reddy, J.N., On locking-free shear deformable beam finite elements. Computer Methods in Applied Mechanics and Engineering, 149(1-4), 1997, 113-132.

[32] Brebbia, C. A., The Boundary element Method for Engineers. London: Pentech Press, 1978.

[33] Hughes, T. J. R., The Finite Element Analysis. Prentice-Hall. New Jersey, 1987.

[34] Szabi, B., Babuska, I. Finite element analysis. Wiley. New York, 1991.

[35] Zienkiewicz, O. C., Taylor, R. L., The Finite Element Method. Fourth Ed. McGraw-Hill. New York, 1989.