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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 ω = c|κ| Fig 1: dispersion of the 1D quadratic element [1] Fig 2: dispersion of the elastic 2D cubic element [4] KA * = χ MA χ =(ω * /c) 2 Fig 3: RQ dispersion of the 1D quadratic element χ = w T Kw w T Mw ω * = c w T Kw w T Mw Fig 4: RQ dispersion of the elastic 2D cubic element Assume a uniform mesh with coordinates Case 1: 1D Acoustic e =1 ...ne: elements xp =(e + ζj )h, h = 1 ne Element-by-element computation Restrict w to an element: Element Matrices w T Mw = h 2 ne v T Av The estimate reduces to ω * = 2c h v T Bv v T Av Chebyshev: ζj = [1 - cos(πj/N)]/2 Fig 5: phase error: 1D elements of degree N=1,..,12 [5] p = j + eN Kw = χMw M ¨ u * + c 2 Ku * = 0 M ¨ u * + c 2 Ku * = 0 ¨ u - c 2 Δu =0 Assume a square mesh with coordinates Discrete system of equations: M e = (h 2 /4)A ⊗ A K e 1 = EA ⊗ B + μB ⊗ A K e 2 = λC T ⊗ C + μC ⊗ C T K e 3 = μA ⊗ B + EB ⊗ A. (xp,yp)=((e1 + ζj1 )h, (e2 + ζj2 )h ), p = p1 + p2Nne, pα = jα + eαN Ci,j = 1 -1 φj (z) ∂φi ∂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 M ¨ u * + c 2 Ku * = 0 M e = h 3 8 A ⊗ A ⊗ A K e = 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(ωt-κ·x) u * p ← Ape -i(ω * t-κ·xp) u * p ← e -i(ω * t-κ·xp) w * p ← e iκ·xp u * α ← R * α e -iω * t w d 1 d 2 ¯ d 2 d 3 R * 1 R * 2 = χ R * 1 R * 2 , d i = w T K i w w T Mw . w e = exp(iκeh)v, v = (exp(iκζ0h), exp(iκζ1h),..., exp(iκζNh)) K e = 2 h B, Bi,j = 1 -1 ∂φj ∂z (z) ∂φi ∂z (z) dz M e = h 2 A, Ai,j = 1 -1 φj (z)φi(z) dz w T Kw = ne-1 e=0 w e T K e w e = 2 h ne-1 e=0 exp(-iκeh) v T B exp(iκeh)v = 2 h ne v T Bv w e = e i(κ1e1+κ2e2) v2 ⊗ v1, vα = (exp(iκαζ0h),..., exp(iκαζNh)) d1 = 4 ρh 2 E v1 T Bv1 v1 T Av1 + μ v2 T Bv2 v2 T Av2 ,d2 = ... u ← e -iω * t e i(κ1e1+...+κ3e3) v3 ⊗ v2 ⊗ v1 ω * = 2c h v1 T Bv1 v1 T Av1 + v2 T Bv2 v2 T Av2 + v3 T Bv3 v3 T Av3 Optimal Blending A ϑ = ϑA + (1 - ϑ)A L , 0 ≤ ϑ ≤ 1, Combination of consistent and lumped mass matrices [4,5] Write 1D dispersion error as and solve εϑ(H),H =1/G, for a prescribed tolerance tol max 0≤R≤0.5 min 0≤ϑ≤1 R 0 εϑ(H) dH ; |εϑ(H)| < tol in [0,R] Rmax: admissible domain of dispersion error ϑmin: optimal blending parameter Fig 9: Optimal Blended x Consistent, N=2 [5] Fig 6: P-phase error: 2D elements of degree N=4,8,12 Legendre : P N (ζj ) = 0 [3] ζj ,j =0 ...N: collocation points in [0, 1] ρM ¨ u * 1 + K1u * 1 + K2u * 2 = 0 ρM ¨ u * 2 + K T 2 u * 1 + K3u * 2 = 0 Dispersion by Mesh Distortion Consider 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 ω * = c ∑ M e=1 w e T K e w e ∑ M e=1 w e T M e w e
