The spectral element method for elastic wave ... - Dimitri Komatitsch

The method is also sho wn to provide an efficient tool for studying the diffraction by ... lem Bouchon et al., 1996 are often very accurate but unfortunately limited to ...
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The Spectral Element method for elastic wa v eequations: application to 2D and 3D seismic problems

Dimitri Komatitsch , Jer oen Tromp, EPS, Harvard University and Jean-Pierre Vilotte, IPGP

Summary We presen t a spectral element method to simulate elastic w ave propagation in realistic geological structures involving interfaces and steep topography for 2D and 3D geometries. The spectral element method is a high-order variational appro ximationof the elastic w aveequation. The mass matrix is diagonal by construction, whic h drastically reduces the computational cost. The time discretization is based on a Newmark scheme written in a predictor/multi-corrector format. A spatial sampling of appro ximately 4or 5 poin tsper wavelength is found to be very accurate. This fact is demonstrated by comparing the computed solution to the analytical solution of the classical 2D problem of an explosiv e source in a half-space. The exibilit yof the method is illustrated by studying a realistic tw o-dimensional model with steep topography (mountain ranges). The method is also sho wn to provide an ecient tool for studying the di raction by 3D topography and the associated e ects on ground motion.

puts stringent constrains on the time-step that cannot be easily removed, and as in any global method, only smooth topography can be handled. Therefore, here w e use a spectral element method (P atera, 1984; Priolo et al., 1994; Komatitsch and Vilotte, 1998) to solve the 2D and 3D elastic wave propagation in complex geometries. The method, which deriv es from a w eak v ariational formulation, allows a exible treatment of boundaries and interfaces, and deals with free-surface boundary conditions naturally. It combines the geometrical exibility of a low-order method with the exponential con vergence rate associated with spectral techniques, and su ers from minimal numerical dispersion and di usion.

The Spectral Element Method

We consider the variational formulation of the elastodynamic equation. The most commonly used formulation is based on the principle of virtual work and can be written using the displacement vector u and the test function w as (Hughes, 1987): In troduction (w; u) + a(w; u) = (w; f ) (1) The use of elastic w aveequations to model the seismic where a(; ) denotes the bilinear form that expresses the response of heterogeneous geophysical media with topog- virtual work of the internal stresses, de ned as : raph y and complex interfaces is a subject that has been Z Z intensiv ely in vestigated.The challenge is to develop high a(w; u) =  : rw dV = rw : c : ru dV (2) performance methods capable of solving the elastic wave



equations accurately, and that are capable of dealing with large and complicated domains as encountered in realistic where :rw =  @w [email protected] and c is the sti ness tensor. Lik e in a standard nite element method, the original do3D applications. main is discretized n non-o verlapping quadrilateral S into Finite di erence methods (Virieux, 1986) ha ve been elements: el

 

= helement  is mapped =1 widely implemented with a varying degree of sophisti- on to a reference volume .2 Eac that is de ned, in a local cation. Unfortunately, con ventional sc hemes su er from system of coordinates, as a square a cube  d with grid dispersion near large gradien tsof the w ave eldor  = [,1; 1]. Eac h element integral, or de ned over the dowhen too coarse computational grids are used. Although main  in the ph ysical space, is pulled bac k, using a more suited to heterogeneous media with complicated ge- local mapping F , on the parent domain 2 and numerometries, nite element methods have attracted less in- ically in tegrated using the numerical quadrature de ned terest in geophysics due to the fact that low-order nite as the tensor-product of the 1D Gauss-Lobatto-Legendre element methods exhibit poor dispersion properties (Mar- formulas. The (N + 1) d basis points for the polynomial furt, 1984), while higher order classical nite elements basis taken to be the same as the quadrature points lead to some problems such as the occurrence of spurious on eacarehelement  , and de ne a local collocation grid w aves. Boundary integral represen tations of the prob-  = f ;  ;  g that is the n -tensor product of the lem (Bouchon et al., 1996) are often very accurate but N + 1 Gauss-Lobatto-Legendre integration points. unfortunately limited to linear and homogeneous problems. Moreover, the resulting linear systems of equations The piecewise-polynomial approximation w of w is deare very large, non-symmetric and dense, whic hmakes ned using the Lagrange interpolation operator I on their application to 3D problems dicult. Pseudospec- the Gauss-Lobatto grid  : I (wj  e ) is the unique tral methods have also been proposed for elastodynamics polynomial of P (2) whic hcoincides with wj  e at the (Carcione and Wang, 1993), but su er from important (N + 1) d poin ts of  . If l () denotes the characterislimitations: non-uniform spacing of the collocation points tic Lagrange polynomial of degree N associated with the ij

i

j

el

n

e

e

e

n

e

e

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e N

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h N

e N

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n

1998 SEG Expanded Abstracts

e N

N i

N

N

Spectral Elements for 2D and 3D seismic problems

Gauss-Lobatto point i of the 1D quadrature formula, the appro ximation ofwj  e is de ned as :

w j  e (x) = h N

X N

i;j;k=0

li ( )lj ( )lk ( ) wijk N

N

N

(3)

e

with x = F ( ;  ;  ) and w = w j  e  F ( ;  ;  ). This procedure leads, like in the nite element method, to a coupled system of ordinary di erential equations : e

i

j

e ijk

k

h N

e

i

j

M u = F , F (u ): ext t

t

int t

k

(4)

t

We use n to denote the total number of nodes of the global integration grid  de ned as the assembly of the S element domain integration grids  =  ; u denotes the global displacement vector at a giv en time t; F is the in ternal nodal force vector, and F the external source term. The mass matrix M is diagonal by construction. This system is then discretized in time using a classical second-order Newmark scheme (Hughes, 1987) written in predictor-multicorrector format. The spectral element method combines the geometric exibility of the nite element method with the fast convergence associated with spectral techniques. The discrete solution su ers from minimal numerical dispersion and di usion, a fact of primary importance in the solution of realistic geoph ysicalproblems. In practice, a spatial sampling of appro ximately4 or 5 poin tsper minimum w avelength is found very accurate when working with a polynomial degree N = 8. T ypically , for 2D sim ulations with a 100,000 points curvilinear grid, the memory occupation is of the order of 30 Megabytes and the CPU time, for a simulation over 2000 time steps, is of the order of 15 minutes on an UltraSparc-1. F or large 3D simulations, using a 5,000,000 points curvilinear grid, the memory occupation is of the order of 1 Gigabyte and the CPU time is of the order of 1 or 2 hours on a parallel computer. node

int

e

e N

t

ext

Garvin's problem Garvin's problem (Garvin, 1956) is a classical test to check the accuracy of a wave propagation code. An homogeneous elastic half-space is considered, with a compressiv e source (explosion) placed exactly at the surface. There exists an analytical solution to this problem. The main event is the propagation of a strong Rayleigh wave along the surface. The Rayleigh pulse is non dispersive, as the medium is homogeneous and the surface is at. Its amplitude remains constan tin the case of plane strain. The corresponding snapshots are presented on Figure 1, and the seismograms on Figure 2, the receivers being located at the free surface. One can observe the direct P w ave, the strong Rayleigh wave with its t ypical elliptical polarization, and the head w ave. Comparison with the analytical solution at receiver 100 at the end of the line is displa yed on Figure 3.V ery good agreement is found, the maximum relative error being less than 1 %.

1998 SEG Expanded Abstracts

Fig. 1: Snapshots for tilted Garvin's problem with an explosive source placed at the free surface. The strong Rayleigh w ave with its t ypical elliptical polarization can be clearly observed, as w ell as the head wave. x (m) 0

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Fig. 2: Seismograms for tilted Garvin's problem with an explosiv e source placed at the free surface. The main event is a strong non dispersive Rayleigh w ave. No signi cant numerical noise can be observed.

Realistic model in South America It is also interesting to study a more realistic example. We consider a geological structure in the Andes (courtesy Elf Aquitaine). The width of the model is 5500 m, and the \a verage" height of the topography is roughly 1300 m. The mesh is shown on Figure 4. It is composed of 60  12 elements, using a polynomial degree of 8. The total number of points is 46657. The source is an explosion inside the model at (x; z) = (1000; 670) m. Its time dependence is a Ric ker w avelet ha ving a cen tral frequency of 12 Hz. The line of receivers is placed at the free surface betw een x = 900 and x = 5000 m.

Ux residuals trace 100

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1998 SEG Expanded Abstract

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Fig. 7: Seismograms of the horizontal component of the disFig. 5: Seismograms obtained for the realistic model in the placement vector at receivers placed on the free surface, along Andes using the \real" velocity model. Di raction from the the minor (left) and major (right) axes of the 3D topography. topograph y is particularly clearly . In addition to the direct wave (a), a strong directivit y e ect is observ ed on the di racted P (b) and Rayleigh (c) w aves that are recorded mainly along the minor axis. Some artefacts due to the periodic boundary conditions (d) are also observed. Ux component

Uz component

References

Fig. 6: Three dimensional model : a Gaussian-shaped hill is superimposed on an homogeneous elastic half-space. The total num ber of collocation points is 4935953.

accuracy of the method. The discrete solution presen ts minimal numerical dispersion. High accuracy is obtained using only 4 or 5 poin tsper minimal w avelength. The capabilit y of the method to handle complex free-surface geometries and deformed internal interfaces ha ve been illustrated by solving a realistic 2D problem involving a mountain range. Finally, the spectral method has been sho wn to be an ecient tool to study the di raction of elastic w aves by 3D topograph y and its e ect on ground motion.

Acknowledgements The authors w ould like to thank F. J. Sanc hez-Sesma, R. Madariaga, E. Chaljub, E. Priolo and G. Seriani for numerous fruitful discussions. G. Moguilny pro vided an invaluable help for the graphics.

1998 SEG Expanded Abstracts

Bouchon, M., Schultz, C. A., and T oksoz,M. N., 1996, E ect of three-dimensional topography on seismic motion: J. Geophys. Res., 101, 5835{5846. Carcione, J. M., and Wang, P . J., 1993, A Chebyshev collocation method for the wave equation in generalized coordinates: Comp. Fluid Dyn. J., 2, 269{290. Garvin, W. W., 1956, Exact transien t solution of the buried line source problem: Proc. R. Soc. London Ser. A, 234, 528{541. Hughes, T. J. R., 1987, The nite element method, linear static and dynamic nite element analysis: Pren ticeHall International, Englewood Cli s, NJ. Komatitsch, D., and Vilotte, J. P., 1998, The spectral element method: an ecient tool to simulate the seismic response of 2D and 3D geological structures: Bull. Seis. Soc. Am., in press. Marfurt, K. J., 1984, Accuracy of nite-di erence and nite-element modeling of the scalar w aveequation: Geophysics, 49, 533{549. P atera, A. T., 1984, A spectral element method for uid dynamics: laminar o w in a channel expansion: J. Comput. Phys., 54, 468{488. Priolo, E., Carcione, J. M., and Seriani, G., 1994, Numerical simulation of interface w aves by high-order spectral modeling techniques: J. Acoust. Soc. Am., 95, no. 2, 681{693. Virieux, J., 1986, P-SV w avepropagation in heterogeneous media: velocity-stress nite-di erence method: Geophysics, 51, 889{901.