Abstract
A pseudo-spectral approach to 2D vibrational problems arising in linear elasticity is considerede using differentiation matrices. The governing partial differential equations and associated boundary conditions on regular domains can be translated into matrix eigenvalue problems. Accurate results are obtained to the precision expected in spectral-type methods. However, we show that it is necessary to apply an additional “pole” condition to deal with ther=0 coordinate singularity arising in the case of a 2D disc.
Similar content being viewed by others
References
H. Chen, Y. Su and B.D. Shizgal, A direct spectral collocation Poisson solver in polar and cylindrical coordinates. J. Comput. Phys. 160 (2000) 453–469.
A.C. Eringen and E.S. Suhubi,Elastodynamics, Vol. II (Academic Press, New York, 1975).
B. Fornberg, The pseudospectral method: comparisons with finite differences for the elastic wave equation, Geophys. 52 (1987) 483–501.
B. Fornberg, The pseudospectral method: accurate representation of interfaces in elastic wave calculations, Geophys. 53 (1988) 625–637.
B. Fornberg,A Practical Guide to Pseudospectral Methods (Cambridge Univ Press, Cambridge, 1996).
W. Huang and D.M. Sloan, Pole conditions for singular problems: the pseudospectral approximation, J. Comput. Phys. 107 (1993) 254–261.
M.-C. Lai, W.-W. Lin and W. Wang, A fast spectral/difference method without pole conditions for Poisson-type equations in cylindrical and spherical geometries, IMA J. Numer. Anal. 22 (2002) 537–548.
J.C. Mason and D.C. Handscomb,Chebyshev Polynomials—Theory and Applications (Chapman and Hall/CRC Press, London, 2003).
E. Tessmer and D. Kosloff, 3-D elastic modeling with surface topography by a Chebyshev spectral method, Geophys. 59 (1994) 464–473.
L.N. Trefethen,Spectral Methods in MATLAB (SIAM, Philadelphia, PA, 2000).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Talbot, C.J., Crampton, A. Application of the pseudo-spectral method to 2D eigenvalue problems in elasticity. Numer Algor 38, 95–110 (2005). https://doi.org/10.1007/BF02810618
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02810618