Summary - We review a recent methodology for estimating the

numerical dispersion of spectral element methods with arbitrary

order for 1D to 3D seismic wave propagation problems. This

approach circumvents the issue of spurious modes of propagation

and reduces to simple 1D calculations in Cartesian grids. We use

this approach to select the combination of consistent and lumped

mass matrices that yields the lowest dispersion error and to study

numerical dispersion caused by mesh distortion.

Accuracy of Spectral Element Methods for Wave Propagation Modeling

Saulo P. Oliveira and Géza Seriani Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, Trieste, Italy

References

1 Belytschko & Mullen (1978) On dispersive properties of finite element solutions. In Miklowitz et al, Modern problems in Elastic Wave Propagation, Wiley

2 Cohen (2002) Higher-Order Numerical Methods for Transient Wave Equations. Springer

3 Komatitsch & Tromp (1999) Introduction to the spectral-element method for 3-D seismic wave propagation. Geophys J Int 139:806-822

4 Marfurt (1990) Analysis of higher order finite-element methods. In Kelly & Marfurt, Numerical modeling of seismic wave propagation, SEG

5 Seriani & Oliveira (2007) Optimal blended spectral-element operators for acoustic wave modeling. Geophysics 72(5):SM95-SM106

Standard Analysis Rayleigh Quotient Approximation

We can’t always find !* because w may not be an eigenvector, but

its Rayleigh quotient approximation is well defined and unique

! = c|!|

Fig 1: dispersion of the 1D quadratic element [1] Fig 2: dispersion of the elastic 2D cubic element [4]

KA! = !MA

! = ("!/c)2

Fig 3: RQ dispersion of the 1D quadratic element

! =wT Kw

wT Mw!! = c

!wT Kw

wT Mw

Fig 4: RQ dispersion of the elastic 2D cubic element

Assume a uniform mesh with coordinates

Case 1: 1D Acoustic

e = 1 . . . ne: elementsxp = (e + !j)h, h =

1ne

Element-by-element computation

Restrict w to an element:

Element Matrices

wT Mw =h

2nev

T Av

The estimate reduces to

!! =2c

h

!vT Bv

vT Av

Chebyshev: !j = [1! cos("j/N)]/2

Fig 5: phase error: 1D elements of degree N=1,..,12 [5]

p = j + eN

Kw = !MwMu! + c2Ku! = 0

Mu! + c2Ku! = 0

u! c2!u = 0

Assume a square mesh with coordinates

Discrete system of equations:M e = (h2/4)A!A

Ke1 = EA!B + µB !A

Ke2 = !CT !C + µC !CT

Ke3 = µA!B + EB !A.

(xp, yp) = ( (e1 + !j1)h, (e2 + !j2)h ), p = p1 + p2Nne, p! = j! + e!N

Ci,j =! 1

!1!j(z)

"!i

"z(z) dz

Discrete plane-wave solution:

Rayleigh quotient appoximation:

Restrict w to an element:

A mesh of cubes structured as in Case 2 yields

Mu! + c2Ku! = 0

M e =h3

8A!A!A

Ke =h

2(A!A!B + B !A!B + B !A!A)

Assume an infinite, periodical mesh and homogeneous media

Plug plane wave into discrete equations; the amplitude depends

on the mesh nodes that define the mesh periodicity

Eigenvalue problem yields multiple dispersion relations

Let us now consider a plane wave with constant amplitude:

Case 2: 2D Elastic, Isotropic Case 3: 3D Acoustic

Fig 8: amplified dispersion / Chbyshev, N=2

Fig 7: S-phase error: 2D elements of degree N=4,8,12

e = e1 + e2ne

u! e!i(!t!!·x)

u!p ! Ape"i(!!t"!·xp)

u!p ! e"i(!!t"!·xp)

w!p ! ei!·xp

u!! ← R!!e"i"!tw

!d1 d2

d2 d3

" !R!

1

R!2

"= !

!R!

1

R!2

", di =

wT Kiw

wT Mw.

we = exp(i!eh)v, v = (exp(i!"0h), exp(i!"1h), . . . , exp(i!"Nh))

Ke =2h

B, Bi,j =! 1

!1

!"j

!z(z)

!"i

!z(z) dzM e =

h

2A, Ai,j =

! 1

!1!j(z)!i(z) dz

wT Kw =ne!1!

e=0

weTKewe =

2h

ne!1!

e=0

exp(!i!eh)vT B exp(i!eh)v =2h

nevT Bv

we = ei(!1e1+!2e2)v2 ! v1, v" = (exp(i!""0h), . . . , exp(i!""Nh))

d1 =4

!h2

!E

v1T Bv1

v1T Av1

+ µv2

T Bv2

v2T Av2

", d2 = . . .

u! e!i!!tei("1e1+...+"3e3)v3 " v2 " v1

!! =2c

h

!v1

T Bv1

v1T Av1

+v2

T Bv2

v2T Av2

+v3

T Bv3

v3T Av3

Optimal Blending

A! = !A + (1! !)AL, 0 " ! " 1,

Combination of consistent and lumped mass matrices [4,5]

Write 1D dispersion error as and solve!!(H), H = 1/G,

for a prescribed tolerance tol

max0!R!0.5

!min

0!!!1

" R

0!!(H) dH ; |!!(H)| < tol in [0, R]

#

Rmax: admissible domain of dispersion error

!min: optimal blending parameterFig 9: Optimal Blended x Consistent, N=2 [5]Fig 6: P-phase error: 2D elements of degree N=4,8,12

Legendre : P !N (!j) = 0 [3]

!j , j = 0 . . . N : collocation points in [0, 1]

!!Mu!

1 + K1u!1 + K2u!

2 = 0!Mu!

2 + KT2 u!

1 + K3u!2 = 0

Dispersion by Mesh DistortionConsider a periodical, non-rectangular mesh such that

the first M elements define the mesh periodicity

An element-by-element computation yields

Figs 10-11: phase error: 2D acoustic elements, N=4 Figs 11-12: phase error: 2D acoustic elements, N=4

Test 1 [2] Test 2

0.6

1.4

!! = c

!""#$M

e=1 weTKewe

$Me=1 weT

M ewe