Dispersion by Mesh Distortion - Geodynamics · 2014-09-24 · 2 Cohen (2002) Higher-Order Numerical...
Transcript of Dispersion by Mesh Distortion - Geodynamics · 2014-09-24 · 2 Cohen (2002) Higher-Order Numerical...
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