Implementation of PML in the Depth-oriented Extended Forward Modeling

Transcription

1 Implementation of PML in the Depth-oriented Extended Forward Modeling Lei Fu, William W. Symes The Rice Inversion Project (TRIP) April 19, 2013 Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

2 Outline 1 Introduction 2 Formulation 3 Numerical experiment 4 Summary Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

3 Introduction Objective Numerically solving the wave propagation problem in a bounded region. Solution Absorbing Boundary Conditions (ABCs) Perfectly Marched Layers (PML) Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

4 History of PML method 1994, Berenger, split-field PML for use with Maxwell s equations. 1996, Gedney, uniaxial PML or UPML, described as an artificial anisotropic absorbing material. 2010, Marcus J. Grote, Efficient PML for the wave equation. (fewer auxiliary variables) Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

5 Formulation Denote û as the Laplace transform of u û(x, s) = 0 e st u(x, t)dt (1) Outside of the computational domain Ω, û then satisfies the Helmholtz equation, s 2 û = ( c 2 û ) + ( c 2 û ) + ( c 2 û ) (2) x 1 x 1 x 2 x 2 x 3 x 3 Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

6 Formulation Next, we introduce the coordinate transformation x i x i := x i + 1 s where ζ i is the damping profile. Partial differentiation: x i = xi 0 s s + ζ i ζ i (x)dx, i = 1, 2, 3 (3) x i (4) Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

7 Formulation If û satisfies the modeified Helmholtz equation, then s 2 û = ( c 2 û ) + ( c 2 ũ ) + ( c 2 û ) x 1 x 1 x 2 x 2 x 3 x 3 (5) Then, by replacing partial derivatives x i by x i, and transforming back to time domain, we get the PML modified wave equation: u tt +(ζ 1 +ζ 2 +ζ 3 )u t +(ζ 1 ζ 2 +ζ 2 ζ 3 +ζ 3 ζ 1 )u = (c 2 u)+ Φ ζ 1 ζ 2 ζ 3 ψ Φ t = Γ 1 Φ + c 2 Γ 2 u + c 2 Γ 3 ψ (6) ψ t = u Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

8 Formulation ζ ζ 2 + ζ 3 ζ Γ 1 = 0 ζ 2 0, Γ 2 = 0 ζ 3 + ζ 1 ζ 2 0, 0 0 ζ ζ 1 + ζ 2 ζ 3 ζ 2 ζ Γ 3 = 0 ζ 3 ζ ζ 1 ζ 2 Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

9 Formulation The choice of damping profiles ζ i (x i ): ζ i (x i ) = ζ i x i a i L i sin ( 2π xi a i L i ) 2π, a i x i a i + L i (7) The relative reflection, R, is given by ζ i = c L i log ( ) 1 R (8) Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

10 Numerical experiment Constant velocity c = 3 Grid spacing: x = z = 0.01 Domain size: 5 5 Thickness of PMLs: L = 0.5 Source: Ricker wavelet at the center.(f peak = 10Hz) Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

11 t=0.2 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

12 t=0.5 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

13 t=0.9 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

14 t=1.0 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

15 t=1.1 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

16 t=1.2 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

17 t=1.3 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

18 t=1.4 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

19 t=1.5 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

20 t=1.6 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

21 t=1.7 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

22 t=1.8 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

23 t=1.9 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

24 t=10.0 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

25 t=0.2 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

26 t=0.5 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

27 t=0.9 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

28 t=1.0 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

29 t=1.1 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

30 t=1.2 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

31 t=1.3 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

32 t=1.4 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

33 t=1.5 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

34 t=1.6 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

35 t=1.7 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

36 t=1.8 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

37 t=1.9 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

38 t=10.0 s Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

39 Summary Implement PML in depth-oriented extended forward modeling. The PML does suppress the reflections exfficiently. Simplicity and the fewer auxiliary variables. Furture work R vs. ζ, x i, θ,... Choice of ζ(x) function ABCs vs. PML 2D 3D Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40

40 Acknowledgement Thanks to Dr. William Symes, TRIP Sponsors and members Lei Fu, William W. Symes (TRIP) PML in Extended modeling April 19, / 40