1887
Volume 49, Issue 4
  • ISSN: 0812-3985
  • E-ISSN: 1834-7533

Abstract

[

Flexible computational domains with irregular absorbing boundaries can improve the efficiency of seismic modelling. We propose a hybrid absorbing boundary condition for piecewise smooth curved boundary in mesh-free discretisation based on radial-basis-function-generated finite difference modelling. Modelling examples in both homogeneous and heterogeneous media with flexibly shaped computational domain demonstrate the effectiveness of our proposed method.

,

Flexible computational domains and their corresponding irregular absorbing boundary conditions (ABCs) have previously been shown to improve the efficiency of finite difference (FD) modelling. However, these proposed ABCs for irregular boundaries are based on irregular grid methods. Although they can improve geometric flexibility of FD modelling, irregular grid methods are still complex to implement due to computationally expensive meshing process. To avoid complex mesh generation, FDs in mesh-free discretisation have been developed for nontrivial geometric settings, where scattered nodes can be placed suitably with respect to irregular boundaries and arbitrarily shaped anomalies without coordinate mapping or mesh-element forming. Radial-basis-function-generated FD (RBF-FD) has been proven successful in modelling seismic wave propagation based on mesh-free discretisation. Using RBF-FD, we develop a hybrid ABC for piecewise smooth curved boundary in 2D acoustic wave modelling based on straightforward expanding strategies for sampled boundary and corner nodes. The whole irregular computational domain is combined by an objective zone and a transition zone separated by piecewise smooth curved boundary. Nodal distribution in the objective zone is adaptive to the model structure through special alignment and varying density of nodes, while the transition zone are designed for solving one way wave equation more easily and accurately in the presence of piecewise smooth curved boundary. Modelling examples for homogenous and heterogeneous models with differently shaped computational domains demonstrate the effectiveness of our proposed method.

]
ASEG
Loading

Article metrics loading...

/content/journals/10.1071/EG17012
2018-08-01
2026-01-21

  • From This Site

    /content/journals/10.1071/EG17012
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Bayona V. Flyer N. Lucas G. M. Baumgaertner A. J. G. 2015 A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GECRBFFD v1.0): Geoscientific Model Development 8 3007 3020 10.5194/gmd‑8‑3007‑2015
    https://doi.org/10.5194/gmd-8-3007-2015 [Google Scholar]
  2. Bayona V. Flyer N. Fornberg B. Barnett G. A. 2017 On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs: Journal of Computational Physics 332 257 273 10.1016/j.jcp.2016.12.008
    https://doi.org/10.1016/j.jcp.2016.12.008 [Google Scholar]
  3. Bentley J. L. 1975 Multidimensional binary search trees used for associative searching: Communications of the ACM 18 509 517 10.1145/361002.361007
    https://doi.org/10.1145/361002.361007 [Google Scholar]
  4. Bérenger J. P. 1994 A perfectly matched layer for the absorption of electromagnetic waves: Journal of Computational Physics 114 185 200 10.1006/jcph.1994.1159
    https://doi.org/10.1006/jcph.1994.1159 [Google Scholar]
  5. Cerjan C. Kosloff D. Kosloff R. Resef M. 1985 A nonreflecting boundary condition for discrete acoustic and elastic wave equations: Geophysics 50 705 708 10.1190/1.1441945
    https://doi.org/10.1190/1.1441945 [Google Scholar]
  6. Clayton R. W. Engquist B. 1977 Absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seismological Society of America 6 1529 1540
    [Google Scholar]
  7. Compani-Tabrizi B. 1986 k-t scattering formulation of the absorptive acoustic wave equation: wraparound and edge-effect elimination: Geophysics 51 2185 2192 10.1190/1.1442071
    https://doi.org/10.1190/1.1442071 [Google Scholar]
  8. Flyer N. Lehto E. Blaise S. Wright G. B. St-Cyr A. 2012 A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere: Journal of Computational Physics 231 4078 4095 10.1016/j.jcp.2012.01.028
    https://doi.org/10.1016/j.jcp.2012.01.028 [Google Scholar]
  9. Flyer, N., Wright, G. B., and Fornberg, B., 2014, Radial basis function-generated finite differences: a mesh-free method for computational geosciences, in W. Freeden, M. Z. Nashed, and T. Sonar, eds., Handbook of Geomathematics: Springer, 1–30.
  10. Flyer N. Fornberg B. Bayona V. Barnett G. A. 2016 On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy: Journal of Computational Physics 321 21 38 10.1016/j.jcp.2016.05.026
    https://doi.org/10.1016/j.jcp.2016.05.026 [Google Scholar]
  11. Fornberg B. Flyer N. 2015 a Solving PDEs with radial basis functions: Acta Numerica 24 215 258 10.1017/S0962492914000130
    https://doi.org/10.1017/S0962492914000130 [Google Scholar]
  12. Fornberg B. Flyer N. 2015 b Fast generation of 2-D node distributions for mesh-free PDE discretizations: Computers & Mathematics with Applications 69 531 544 10.1016/j.camwa.2015.01.009
    https://doi.org/10.1016/j.camwa.2015.01.009 [Google Scholar]
  13. Fornberg B. Piret C. 2008 A stable algorithm for flat radial basis functions on a sphere: SIAM Journal on Scientific Computing 30 60 80 10.1137/060671991
    https://doi.org/10.1137/060671991 [Google Scholar]
  14. Fornberg B. Wright G. 2004 Stable computation of multiquadric interpolants for all values of the shape parameters: Computers & Mathematics with Applications 48 853 867 10.1016/j.camwa.2003.08.010
    https://doi.org/10.1016/j.camwa.2003.08.010 [Google Scholar]
  15. Fries, T., and Matthies, H., 2004, Classification and overview of meshfree methods: Informatikbericht Nr. 2003–3, Institute of Scientific Computing.
  16. Gao H. Zhang J. 2008 Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modeling: Geophysical Journal International 174 1029 1036 10.1111/j.1365‑246X.2008.03883.x
    https://doi.org/10.1111/j.1365-246X.2008.03883.x [Google Scholar]
  17. Gao Y. Song H. Zhang J. Yao Z. 2016 Comparison of artificial absorbing boundaries for acoustic wave equation modeling: Exploration Geophysics 48 76 93 10.1071/EG15068
    https://doi.org/10.1071/EG15068 [Google Scholar]
  18. Gedney S. D. Zhao B. 2010 An auxiliary differential equation formulation for the complex-frequency shifted PML: IEEE Transactions on Antennas and Propagation 58 838 847 10.1109/TAP.2009.2037765
    https://doi.org/10.1109/TAP.2009.2037765 [Google Scholar]
  19. Heidari A. H. Guddati M. N. 2006 Highly accurate absorbing boundary conditions for wide-angle wave equations: Geophysics 71 S85 S97 10.1190/1.2192914
    https://doi.org/10.1190/1.2192914 [Google Scholar]
  20. Higdon R. L. 1991 Absorbing boundary conditions for elastic waves: Geophysics 56 231 241 10.1190/1.1443035
    https://doi.org/10.1190/1.1443035 [Google Scholar]
  21. Hu W. Abubakar A. Habashy T. M. 2007 Application of the nearly perfectly matched layer in acoustic wave modeling: Geophysics 72 SM169 SM175 10.1190/1.2738553
    https://doi.org/10.1190/1.2738553 [Google Scholar]
  22. Kindelan M. Moscoso M. González-Rodríguez P. 2016 Radial basis function interpolation in the limit of increasingly flat basis functions: Journal of Computational Physics 307 225 242 10.1016/j.jcp.2015.12.015
    https://doi.org/10.1016/j.jcp.2015.12.015 [Google Scholar]
  23. Komatitsch D. Martin R. 2007 An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation: Geophysics 72 SM155 SM167 10.1190/1.2757586
    https://doi.org/10.1190/1.2757586 [Google Scholar]
  24. Kosloff R. Kosloff D. 1986 Absorbing boundaries for wave propagation problems: Journal of Computational Physics 63 363 376 10.1016/0021‑9991(86)90199‑3
    https://doi.org/10.1016/0021-9991(86)90199-3 [Google Scholar]
  25. Lee G. H. Chung H. J. Choi C. K. 2003 Adaptive crack propagation analysis with the element-free Galerkin method: International Journal for Numerical Methods in Engineering 56 331 350 10.1002/nme.564
    https://doi.org/10.1002/nme.564 [Google Scholar]
  26. Liu Y. Sen M. K. 2009 A new time-space domain high-order finite-difference method for the acoustic wave equation: Journal of Computational Physics 228 8779 8806 10.1016/j.jcp.2009.08.027
    https://doi.org/10.1016/j.jcp.2009.08.027 [Google Scholar]
  27. Liu Y. Sen M. K. 2010 A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation: Geophysics 75 A1 A6 10.1190/1.3295447
    https://doi.org/10.1190/1.3295447 [Google Scholar]
  28. Liu Y. Sen M. K. 2012 A hybrid absorbing boundary condition for elastic staggered-grid modeling: Geophysical Prospecting 60 1114 1132 10.1111/j.1365‑2478.2011.01051.x
    https://doi.org/10.1111/j.1365-2478.2011.01051.x [Google Scholar]
  29. Marcinkovich C. Olsen K. 2003 On the implementation of perfectly matched layers in a three-dimensional fourth-order velocity-stress finite difference scheme: Journal of Geophysical Research 108 1978 2012 10.1029/2002JB002235
    https://doi.org/10.1029/2002JB002235 [Google Scholar]
  30. Martin R. Komatitsch D. Gedney S. D. Bruthiaux E. 2010 A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML): Computer Modeling in Engineering & Sciences 56 17 41
    [Google Scholar]
  31. Martin, B., Flyer, N., Fornberg, B., and St-Cyr, A., 2013, Development of mesh-less computational algorithms for seismic exploration: 83rd Annual International Meeting, SEG, Expanded Abstracts, 3543–3547.
  32. Martin B. Fornberg B. St-Cyr A. 2015 Seismic modeling with radial-basis-function-generated finite differences: Geophysics 80 T137 T146 10.1190/geo2014‑0492.1
    https://doi.org/10.1190/geo2014-0492.1 [Google Scholar]
  33. Rao Y. Wang Y. 2013 Seismic waveform simulation with pseudo-orthogonal grids for irregular topographic models: Geophysical Journal International 194 1778 1788 10.1093/gji/ggt190
    https://doi.org/10.1093/gji/ggt190 [Google Scholar]
  34. Ren Z. Liu Y. 2013 A hybrid absorbing boundary condition for frequency-domain finite-difference modeling: Journal of Geophysics and Engineering 10 054003 10.1088/1742‑2132/10/5/054003
    https://doi.org/10.1088/1742-2132/10/5/054003 [Google Scholar]
  35. Reynolds A. C. 1978 Boundary conditions for the numerical solution of wave propagation problems: Geophysics 43 1099 1110 10.1190/1.1440881
    https://doi.org/10.1190/1.1440881 [Google Scholar]
  36. Roden J. A. Gedney S. D. 2000 Convolutional PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media: Microwave and Optical Technology Letters 27 334 339 10.1002/1098‑2760(20001205)27:5<334::AID‑MOP14>3.0.CO;2‑A
    https://doi.org/10.1002/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A [Google Scholar]
  37. Sochacki J. Kubichek R. George J. Fletcher W. Smithson S. 1987 Absorbing boundary conditions and surface waves: Geophysics 52 60 71 10.1190/1.1442241
    https://doi.org/10.1190/1.1442241 [Google Scholar]
  38. Takekawa J. Mikada H. 2016 An absorbing boundary condition for acoustic wave propagation using a mesh-free method: Geophysics 81 T145 T154 10.1190/geo2015‑0315.1
    https://doi.org/10.1190/geo2015-0315.1 [Google Scholar]
  39. Takekawa J. Mikada H. Imamura N. 2015 A mesh-free method with arbitrary-order accuracy for acoustic wave propagation: Computers & Geosciences 78 15 25 10.1016/j.cageo.2015.02.006
    https://doi.org/10.1016/j.cageo.2015.02.006 [Google Scholar]
  40. Virieux J. Calandra H. Plessix R. É. 2011 A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging: Geophysical Prospecting 59 794 813 10.1111/j.1365‑2478.2011.00967.x
    https://doi.org/10.1111/j.1365-2478.2011.00967.x [Google Scholar]
  41. Wenterodt C. von Estorff O. 2009 Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation: International Journal for Numerical Methods in Engineering 77 1670 1689 10.1002/nme.2463
    https://doi.org/10.1002/nme.2463 [Google Scholar]
  42. Zeng Y. Q. He J. Q. Liu Q. H. 2001 The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media: Geophysics 66 1258 1266 10.1190/1.1487073
    https://doi.org/10.1190/1.1487073 [Google Scholar]
  43. Zhang J. Gao H. 2011 Irregular perfectly matched layers for 3D elastic wave modeling: Geophysics 76 T27 T36 10.1190/1.3533999
    https://doi.org/10.1190/1.3533999 [Google Scholar]
  44. Zhang J. Liu T. 1999 P-SV-wave propagation in heterogeneous media: grid method: Geophysical Journal International 136 431 438 10.1111/j.1365‑246X.1999.tb07129.x
    https://doi.org/10.1111/j.1365-246X.1999.tb07129.x [Google Scholar]
  45. Zhang J. Liu T. 2002 Elastic wave modelling in 3D heterogeneous media: 3D grid method: Geophysical Journal International 150 780 799 10.1046/j.1365‑246X.2002.01743.x
    https://doi.org/10.1046/j.1365-246X.2002.01743.x [Google Scholar]
  46. Zhang W. Shen Y. 2010 Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling: Geophysics 75 T141 T154 10.1190/1.3463431
    https://doi.org/10.1190/1.3463431 [Google Scholar]
  47. Zhang X. Liu Y. Cai X. Ren Z. 2015 The AWWE-based hybrid absorbing boundary condition for finite-difference modeling and its application in reverse-time migration: Journal of Applied Geophysics 123 93 101 10.1016/j.jappgeo.2015.09.018
    https://doi.org/10.1016/j.jappgeo.2015.09.018 [Google Scholar]
/content/journals/10.1071/EG17012
Loading
Hybrid absorbing boundary condition for piecewise smooth curved boundary in 2D acoustic finite difference modelling
Exploration Geophysics 49, 469 (2018); https://doi.org/10.1071/EG17012
/content/journals/10.1071/EG17012
/content/journals/10.1071/EG17012
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): acoustic wave equation; hybrid absorbing boundary condition; mesh-free method; numerical modelling; one way wave equation; piecewise smooth curved boundary; radial-basis-function-generated finite difference

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error