Harmonic Coordinate Finite Element Method for Acoustic Waves Xin Wang The Rice Inversion Project Mar. 30th, 2012
Outline Introduction Harmonic Coordinates FEM Mass Lumping Numerical Results Conclusion 2
Motivation Scalar acoustic wave equation ∂ 2 u 1 ∂ t 2 − ∇ · 1 ρ ∇ u = f κ with appropriate boundary, initial conditions Typical setting in seismic applications: ◮ heterogeneous κ, ρ with low contrast O (1) ◮ model data κ, ρ defined on regular Cartesian grids ◮ large scale ⇒ waves propagate O (10 2 ) wavelengths; solutions for many different f ◮ f smooth in time 3
Motivation For piecewise constant κ, ρ with interfaces ◮ FDM: first order interface error, time shift, incorrect arrival time, no obvious way to fix (Brown 84, Symes & Vdovina 09) ◮ Accuracy of standard FEM (eg specFEM3D) relies on adaptive, interface fitting meshes ◮ Exception: FDM derived from mass-lumped FEM on regular grid for constant density acoustics has 2nd order convergence even with interfaces (Symes & Terentyev 2009) ◮ Aim of this project: modify FD/FEM with full accuracy for variable density acoustics offset (km) 0 2 4 6 8 0 depth (km) 1 2 1.5 2.0 2.5 3.0 3.5 4.0 km/s Figure: Velocity model 4
Motivation For highly oscillating coefficient (rough medium) e.g., coefficient varies on scale 1 m ⇒ accurate regular FD simulations of 30 Hz waves may require 1 m grid though the corresponding wavelength is about 100 m at velocity of 3 km/s Can we create an accurate FEM on coarse ( wavelength) grid? Depth (km) 2.5 2.0 1.5 1.0 0.5 2 Velocity �(km/sec) 3 4 5 A log of compressional wave velocity from a well in West Texas, supplied to TRIP by Total E&P USA and used by permission. 5
Motivation Ideal numerical method: ◮ high order ◮ no special meshing, i.e., on regular grids ◮ works for problems with heterogeneous media Owhadi and Zhang 2007: 2D harmonic coordinate finite element method, sub-optimal convergence on regular grids Binford 11: showed using the true support recovers the optimal order of convergence on triangular meshes for 2D static interface problems, and mesh of diameter h 2 is used to approximate the harmonic coordinates Result of project: modification of Owhadi and Zhang’s method with full 2nd order convergence for variable density acoustic WE 6
Outline Introduction Harmonic Coordinates FEM Mass Lumping Numerical Results Conclusion 7
Harmonic Coordinates Global C -harmonic coordinates F in 2D, its components F 1 ( x 1 , x 2 ) , F 2 ( x 1 , x 2 ) are weak sols of ∇ · C ( x ) ∇ F i = 0 in Ω F i = x i on ∂ Ω F : Ω → Ω C -harmonic coordinates e.g., x 2 (1 , 1) C 2 = 1 C 1 = 20 x 1 1 r 0 = √ 2 π ( − 1 , − 1) 8
Harmonic Coordinates ◮ physical regular grid ( x 1 , x 2 ) = ( jh x , kh y ) (left), � � ◮ harmonic grid ( F 1 , F 2 ) = F 1 ( jh x , kh y ) , F 2 ( jh x , kh y ) (right) 9
Harmonic Coordinate FEM Workflow of HCFEM: 1 prepare a regular mesh on physical domain, T H ; 2 approximate F on a fine mesh T h by F h T H = F h ( T H ); 3 construct the harmonic triangulation � 4 construct the HCFE space S H = span { ˜ φ H i ◦ F h : i = 0 , · · · , N h } , where S H = span { ˜ ˜ φ H i : i = 0 , · · · , N h } is Q 1 FEM space on harmonic grid � T H ; 5 solve the original problem by Galerkin method on S H . ⇒ solve n ( ≤ 3) harmonic problems to obtain harmonic coordinates F ; resulting stiffness and mass matrices of HCFEM have the same sparsity pattern as in standard FEM 10
Harmonic Coordinate FEM Galerkin approximation u h ∈ S h (harmonic coordinate FE space) to u (weak solution of −∇ · C ∇ u = f ) has optimal error: � � |∇ u − ∇ u h | 2 d x = O ( h 2 ) Ω Why this is so: see WWS talk in 2008 TRIP review meeting 11
Harmonic Coordinate FEM For wave equation: ∂ 2 u 1 ∂ t 2 − ∇ · 1 in Ω ⊂ R 2 ρ ∇ u = f κ Dirichlet BC, u ≡ 0 , t < 0, f ∈ L 2 ([0 , T ] , L 2 (Ω)) Assume f is finite bandwidth ⇒ Galerkin approximation u h in HCFE space S h on mesh of diam h has optimal order approximation in energy e [ u − u h ]( t ) = O ( h 2 ) , 0 ≤ t ≤ T ��� � with e [ u ]( t ) := 1 Ω d x | 1 ∂ u ∂ t | 2 + | 1 √ ρ ∇ u | 2 √ κ ( t ) 2 12
1D Illustration 1D elliptic interface problem ( β u x ) x = f 0 ≤ x ≤ 1 , u (0) = u (1) = 0 f ∈ L 2 (0 , 1), β has discontinuity at x = α � β − x < α β ( x ) = β + x > α displacement u is continuous as well as normal stress β u x at α 1D ’linear’ HCFE basis: 1 0.8 0.6 0.4 0.2 0 0.6 0.65 0.7 13
Outline Introduction Harmonic Coordinates FEM Mass Lumping Numerical Results Conclusion 14
FEM Discretization for Acoustic Waves For acoustic wave equation: ∂ 2 u 1 ∂ t 2 − ∇ · 1 ρ ∇ u = f κ N h � FE space S = span { ψ j ( x ) } N h j =0 , FE solution u h = u j ( t ) ψ j ( x ). j =0 FEM semi-discretization: M h d 2 U h d t 2 + N h U h = F h � � 1 U h ( t ) = [ u 0 , ... u N h ] T , F h f ψ i d x , M h i = ij = κψ i ψ j d x , � Ω Ω 1 N h ij = ρ ∇ ψ i · ∇ ψ j d x Ω 15
Mass Lumping 2nd order time discretization: M h U h ( t + ∆ t ) − 2 U h ( t ) + U h ( t − ∆ t ) + N h U h ( t ) = F h ( t ) ∆ t 2 ⇒ every time update involves solving a linear system M h U h =RHS Replace M h by a diagonal matrix ˜ M h , � ˜ M h M h ii = ij j Can achieve optimal rate of convergence if the solution u is smooth (e.g., H 2 (Ω)) My thesis has the details for validation of mass lumping 16
Outline Introduction Harmonic Coordinates FEM Mass Lumping Numerical Results Conclusion 17
Implementation and Computation Implementation: based on deal.II , a C++ program library targeted at the computational solution of partial differential equations using adaptive finite elements ◮ built-in quadrilateral mesh generation and mesh adaptivity ◮ various finite element spaces, DG ◮ interfaces for parallel linear system solvers (eg PETSc) ◮ data output format for quick view (eg paraview, opendx, gnuplot, ps) Computation: using DAVinCI@RICE cluster, 2304 processor cores in 192 Westmere nodes (12 processor cores per node) at 2.83 GHz with 48 GB of RAM per node (4 GB per core). 18
Elliptic BVP - Square Circle Model −∇ · C ( x ) ∇ u = − 9 r in Ω x 2 + y 2 � where r = For piecewise const C ( x ) shown in the figure below, analytical solution: 1 C ( x )( r 3 − r 3 u = 0 ) x 2 (1 , 1) C 2 C 1 x 1 1 r 0 = √ 2 π ( − 1 , − 1) 19
High Contrast: C 1 = 20 , C 2 = 1 1 0 10 10 −1 10 0 10 −2 10 semi−H 1 error L 2 error −1 −3 10 10 −4 10 −2 10 −5 10 HCFEM HCFEM O(H 2 ) O(H) −3 −6 10 10 −3 −2 −1 0 −3 −2 −1 0 10 10 10 10 10 10 10 10 grid size H grid size H ◮ HCFEM is applied on the physical grid of diameter H ◮ Harmonic coordinates are approximated on the locally refined grid, in which the grid size is O ( h ) ( h = H 2 ) near interfaces. 20
High Contrast: C 1 = 20 , C 2 = 1 0 1 10 10 −1 10 0 10 −2 10 semi−H 1 error L 2 error −3 −1 10 10 −4 10 −2 10 −5 10 Q 1 FEM Q 1 FEM O(H 2 ) O(H) −6 −3 10 10 −3 −2 −1 0 −3 −2 −1 0 10 10 10 10 10 10 10 10 grid size H grid size H ◮ Standard FEM is applied on the physical grid of diameter H 21
Low Contrast: C 1 = 2 , C 2 = 1 1 0 10 10 0 10 −2 10 semi−H 1 error L 2 error −1 10 −4 10 −2 10 HCFEM HCFEM O(H 2 ) O(H) −3 −6 10 10 −3 −2 −1 0 −3 −2 −1 0 10 10 10 10 10 10 10 10 grid size H grid size H ◮ HCFEM is applied on the physical grid of diameter H ◮ Harmonic coordinates are approximated on the locally refined grid, in which the grid size is O ( h ) ( h = H 2 ) near interfaces. 22
Low Contrast: C 1 = 2 , C 2 = 1 1 0 10 10 0 10 −2 10 semi−H 1 error L 2 error −1 10 −4 10 −2 10 Q 1 FEM Q 1 FEM O(H 2 ) O(H) −3 −6 10 10 −3 −2 −1 0 −3 −2 −1 0 10 10 10 10 10 10 10 10 grid size H grid size H ◮ Standard FEM is applied on the physical grid of diameter H 23
2D Acoustic Wave Tests Acoustic wave equation: � 1 � κ − 1 ∂ 2 u ∂ t 2 − ∇ ρ ∇ u = 0 u ( x , 0) = g ( x , 0) , u t ( x , 0) = g t ( x , 0) � � with g ( x , t ) = 1 t − r r f and c s � 1 − 2 ( π f 0 ( t + t 0 )) 2 � e − ( π f 0 ( t + t 0 )) 2 , f 0 central frequency, f ( t ) = � κ ( x s ) ρ ( x s ), t 0 = 1 . 45 c s = f 0 The following examples similar to those in Symes and Terentyev, SEG Expanded Abstracts 2009 24
Dip Model √ Central frequency f 0 = 10 Hz, x s = [ − 300 3 m , − 300 m ] x 2 (2 km,2 km) [ ρ 2 , c 2 ] = [1500 kg / m 3 , 3 m / s ] x 1 x s [ ρ 1 , c 1 ] = [3000 kg / m 3 , 1 . 5 m / s ] (-2 km,-2 km) 25
Dip Model Q 1 FEM solution, regular grid quadrature (= FDM) - this is equivalent to using ONLY the node values on the regular grid to define mass, stiffness matrices Entire domain h e p 7 . 8 m 3.62e-1 - 3 . 9 m 9.30e-2 1.96 1 . 9 m 2.31e-2 2.01 Figure: T = 0 . 75 s 26
Dip Model Q 1 FEM solution, accurate quadrature for mass and stiffness matrices’ computation, Q 1 FEM with accurate quadrature is also (2,2) FDM but with different coefficients - this is Igor’s result (constant density acoustics). Entire domain h e p 7 . 8 m 3.56e-1 - 3 . 9 m 9.13e-2 1.96 1 . 9 m 2.27e-2 2.01 27
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