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Effect of Element Distortion on the Numerical Dispersion of Spectral Element Methods

Published online by Cambridge University Press:  20 August 2015

S.P. Oliveira  and
G. Seriani
S.P. Oliveira*
Affiliation:
Departamento de Matemática, Universidade Federal do Paraná, Curitiba-PR, 81531-980, Brazil
G. Seriani*
Affiliation:
Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, Borgo Grotta Gigante, 42/c, Sgonico (TS), 34010, Italy
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

Spectral element methods are well established in the field of wave propagation, in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error. The latter is experimentally acknowledged, but has been theoretically shown only in limited cases, such as Cartesian meshes. It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions. In the present work, we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance. We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.

Keywords

65C2065M6065M7065M1586A1565Z05Spectral element methoddispersion analysisslivering
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 937 - 958
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Abboud, N. and Pinsky, P., Finite element dispersion analysis for the three-dimensional second-order scalar wave equation, Int. J. Numer. Meth. Eng., 35(6) (1992), 1183–1218.Google Scholar
[2]Burnett, D., Finite Element Analysis: From Concepts to Applications, Addison-Wesley, 1987.Google Scholar
[3]Canuto, C., Hussaini, M., Quarteroni, A., and Zang, T., Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, NY, 1987.Google Scholar
[4]Chaljub, E., Capdeville, Y., and Vilotte, J. P., Solving elastodynamics in a fluid-solid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids, J. Comput. Phys., 187(2) (2003), 457–491.Google Scholar
[5]Chaljub, E., Komatitsch, D., Vilotte, J.-P., Capdeville, Y., Valette, B., and Festa, G., Spectral element analysis in seismology, in Wu, Ru-Shan and Maupin, Valérie, editors, Advances in Wave Propagation in Heterogeneous Media, volume 48 of Advances in Geophysics, pages 365–419, Elsevier, 2007.Google Scholar
[6]Christon, M., The influence of the mass matrix on the dispersive nature of the semi-discrete, second-order wave equation, Comput. Methods. Appl. Mech. Engrg., 173(1-2) (1999), 147–166.Google Scholar
[7]Cohen, G., Higher-Order Numerical Methods for Transient Wave Equations, Springer-Verlag, New York, NY, 2002.CrossRefGoogle Scholar
[8]Cohen, G., Joly, P., and Tordjman, N., Higher-order finite elements with mass-lumping for the 1D wave equation, Finite. Elem. Anal. Des., 16(3-4) (1994), 329–336.CrossRefGoogle Scholar
[9]Dauksher, W. and Emery, A., The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements, Comput. Methods. Appl. Mech. Engrg., 188(1-3) (2000), 217–233.Google Scholar
[10]De Basabe, J. and Sen, M., Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations, Geophysics., 72(6) (2007), T81–T95.Google Scholar
[11]De Basabe, J. and Sen, M., Comment on “dispersion analysis of spectral element methods for elastic wave propagation”, Wave. Motion., 46(1) (2009), 92–93.CrossRefGoogle Scholar
[12]Deville, M. O., Fischer, P., and Mund, E., High-Order Methods for Incompressible Fluid Flow, Cambridge University Press, 2002.Google Scholar
[13]Hesthaven, J. S. and Warburton, T., Nodal high-order methods on unstructured grids, I: timedomain solution of Maxwell’s equations, J. Comput. Phys., 181(1) (2002), 186–221.Google Scholar
[14]Hochstenbach, M. E. and van der Vorst, H. A., Alternatives to the Rayleigh quotient for the quadratic eigenvalue problem, SIAM J. Sci. Comput., 25(2) (2003), 591–603.Google Scholar
[15]Hughes, T., The Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1987.Google Scholar
[16]Komatitsch, D. and Vilotte, J. P., The spectral-element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures, Bull. Seismol. Soc. Am., 88(2) (1998), 368–392.Google Scholar
[17]Korczak, K. and Patera, A., An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry, J. Comput. Phys., 62(2) (1986), 361–382.Google Scholar
[18]Laurenzano, G., Priolo, E., and Tondi, E., 2D numerical simulations of earthquake ground motion: examples from the Marche Region, Italy, J. Seismol., 12(3) (2008), 395–412.Google Scholar
[19]Lütkepohl, H., Handbook of Matrices, John Wiley & Sons, Chichester, UK, 1996.Google Scholar
[20]Maday, Y. and Rønquist, E., Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries, Comput. Methods. Appl. Mech. Eng., 80(1-3) (1990), 91–115.Google Scholar
[21]Marfurt, K., Appendix-analysis of higher order finite-element methods, in Kelly, K. and Marfurt, K., editors, Numerical modeling of seismic wave propagation, number 13, in Geophysics Reprint Series, pages 516–520, Soc. Expl. Geophys., Tulsa, OK, 1990.Google Scholar
[22]Melenk, J. M., Gerdes, K., and Schwab, C., Fully discrete hp-finite elements: fast quadrature, Comput. Methods. Appl. Mech. Engrg., 190(32-33) (2001), 4339–4364.Google Scholar
[23]Mercerat, E. D., Vilotte, J. P., and Sanchez-Sesma, F. J., Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids, Geophys. J. Int., 166(2) (2006), 679–690.Google Scholar
[24]Mulder, W., Spurious modes in finite-element discretizations of the wave equation may not be all that bad, Appl. Numer. Math., 30(4) (1999), 425–445.Google Scholar
[25]Oliveira, S. P. and Seriani, G., DFT modal analysis of spectral element methods for the 2D elastic wave equation, J. Comput. Appl. Math., 234 (2010), 1717–1724.Google Scholar
[26]Patera, A., A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54(3) (1984), 468–488.Google Scholar
[27]Priolo, E., 2-D spectral element simulations of destructive ground shaking in Catania (Italy), J. Seismol., 3(3) (1999), 289–309.Google Scholar
[28]Priolo, E., Carcione, J., and Seriani, G., Numerical simulation of interface waves by high-order spectral modeling techniques, J. Acoust. Soc. Am., 95(2) (1994), 681–693.Google Scholar
[29]Proot, M. and Gerritsma, M., Application of the least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier-Stokes equations, Numer. Algorithms., 38(1-3) (2005), 155–172.Google Scholar
[30]Seriani, G., 3-D large-scale wave propagation by spectral element method on Cray T3E multiprocessor, Comput. Methods. Appl. Mech. Engrg., 164(1-2) (1998), 235–247.Google Scholar
[31]Seriani, G., Double-grid Chebyshev spectral elements for acoustic wave modeling, Wave. Motion., 39(4) (2004), 351–360.CrossRefGoogle Scholar
[32]Seriani, G. and Oliveira, S. P., Optimal blended spectral element operators for forward modeling, Geophys., 72(5) (2007), SM95–SM106.Google Scholar
[33]Seriani, G. and Oliveira, S. P., Dispersion analysis of spectral element methods for acoustic wave propagation, J. Comput. Acoust., 16(4) (2008), 531–561.Google Scholar
[34]Seriani, G. and Oliveira, S. P., Dispersion analysis of spectral element methods for elastic wave propagation, Wave. Motion., 45(6) (2008), 729–744.Google Scholar
[35]Seriani, G. and Oliveira, S. P., Reply to comment on “dispersion analysis of spectral element methods for elastic wave propagation”, Wave. Motion., 46(1) (2009), 94–95.Google Scholar
[36]Seriani, G. and Priolo, E., Spectral element method for acoustic wave simulation in heterogeneous media, Finite. Elem. Anal. Des., 16(3-4) (1994), 337–348.CrossRefGoogle Scholar
[37]Stanescu, D., Ait-Ali-Yahia, D., Habashi, W. G., and Robichaud, M. P., Spectral element method for linear fan tone noise radiation, AIAA J., 42(4) (2004), 696–705.Google Scholar
[38]White, D., Numerical dispersion of a vector finite element method on skewed hexahedral grids, Commun. Numer. Methods. Eng., 16(1) (2000), 47–55.Google Scholar
[39]Zak, A., A novel formulation of a spectral plate element for wave propagation in isotropic structures, Finite. Elem. Anal. Des., 45(10) (2009), 650–658.Google Scholar
[40]Zhang, S., Numerical integration with Taylor truncations for the quadrilateral and hexahedral finite elements, J. Comput. Appl. Math., 205(1) (2007), 325–342.Google Scholar