A Non-Splitting PML for Transient Analysis of Poroelastic Media and Its Finite Element Implementation

ZHOU Feng-xi; GAO Bei-bei

doi:10.3879/j.issn.1000-0887.2016.02.008

Volume 37 Issue 2

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Article Navigation > Applied Mathematics and Mechanics > 2016 > 37(2): 195-209

ZHOU Feng-xi, GAO Bei-bei. A Non-Splitting PML for Transient Analysis of Poroelastic Media and Its Finite Element Implementation[J]. Applied Mathematics and Mechanics, 2016, 37(2): 195-209. doi: 10.3879/j.issn.1000-0887.2016.02.008

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A Non-Splitting PML for Transient Analysis of Poroelastic Media and Its Finite Element Implementation

doi: 10.3879/j.issn.1000-0887.2016.02.008

School of Civil Engineering, Lanzhou University of Technology, Lanzhou 730050, P.R.China

Funds: The National Natural Science Foundation of China(11162008;51368038)

Received Date: 2015-05-25
Rev Recd Date: 2015-09-30
Publish Date: 2016-02-15

Abstract

Abstract

The perfectly matched layer (PML) absorbing boundary condition had been proved to be a highly effective absorption technique for the numerical simulation of wave propagation and therefore widely used. In order to solve the problems of absorbing boundary conditions in the numerical modeling of 2nd-order elastic wave equations for the infinite domain poroelastic media, a non-splitting perfectly matched layer (NPML) was proposed. Firstly, based on the theory of Biot’s wave equations and in view of the compressibility of solid particles and pore fluid, the inertia and the pore fluid viscosity, the 2nd-order dynamic governing equations were established in the form of solid and fluid displacements. Secondly, according to the complex coordinate stretching technique, the frequency domain formulations of the NPML were obtained by means of the Laplace transform. Afterwards, with the aid of auxiliary functions in the absorption layer, an effective NPML was built through the transform of the frequency domain formulations back to the time domain. Finally, the time domain finite element scheme of the NPML on the basis of Galerkin approximate method was provided. The effectiveness of the NPML in the dynamic response analysis of saturated poroelastic media is demonstrated with several numerical examples.
- non-splitting perfectly matched layer,
- absorbing boundary,
- Biot’s wave equation,
- finite element method

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References(19)

References

[1]	Mur G. Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic-field equations[J].IEEE Trans EMC,1981,23(4): 377-382.
[2]	Higdon R L. Absorbing boundary condition for elastic waves[J].Geophysics,1991,56(2): 231-241.
[3]	廖振鹏, 黄孔亮, 袁一凡. 暂态弹性波分析中人工边界的研究[J]. 地震工程与工程振动, 1982,2(1): 1-11.（LIAO Zhen-peng, HUANG Kong-liang, YUAN Yi-fan. Artificial boundary in analysis of transient elastic wave propagation[J].Journal of Earthquake Engineering and Engineering Vibration,1982,2(1): 1-11.（in Chinese））
[4]	Cerjan C, Kosloff D, Kosloff R, Reshef M. A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J].Geophysics,1985,50(4): 705-708.
[5]	Berenger J P. A perfectly matched layer for the absorption of electromagnetic waves[J].Journal of Computational Physics,1994,114(2): 185-200.
[6]	陈可洋. 声波完全匹配层吸收边界条件的改进算法[J]. 石油物探, 2009,48(1): 76-79.（CHEN Ke-yang. Improved algorithm for absorbing boundary condition of acoustic perfectly matched layer[J].Geophysical Prospecting for Petroleum,2009,48(1): 76-79.（in Chinese））
[7]	Basu U, Chopra A K. Perfectly matched layers for transient elastodynamics of unbounded domains[J].International Journal for Numerical Methods in Engineering,2004,59(8): 1039-1074.
[8]	Grote M J, Sim I. Efficient PML for the wave equation[J].arXiv:〖STBX〗1001.0319[math.NA],2010,2: 1-15.
[9]	Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation[J].Geophysical Journal International,2003,154(1): 146-153.
[10]	Basu U. Explicit finite element perfectly matched layer for transient three-dimensional elastic waves[J].International Journal for Numerical Methods in Engineering,2009,77(2): 151-176.
[11]	Kucukcoban S, Kallivokas L F. Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media[J].Computer Methods in Applied Mechanics and Engineering,2011,200(1): 57-76.
[12]	Fathi A, Poursartip B, Kallivokas L F. Time-domain hybrid formulations for wave simulations in three-dimensional PML-truncated heterogeneous media[J].International Journal for Numerical Methods in Engineering,2014,101(3): 165-198.
[13]	Matzen R. An efficient finite element time-domain formulation for the elastic second-order wave equation: a non-split complex frequency shifted convolutional PML[J].International Journal for Numerical Methods in Engineering,2011,88(10): 951-973.
[14]	赵海波, 王秀明, 王东, 陈浩. 完全匹配层吸收边界在孔隙介质弹性波模拟中的应用[J]. 地球物理学报, 2007,50(2): 581-591.（ZHAO Hai-bo, WANG Xiu-ming, WANG Dong, CHEN Hao. Applications of the boundary absorption using a perfectly matched layer for elastic wave simulation in poroelastic media[J].Chinese Journal of Geophysics,2007,50(2): 581-591.（in Chinese））
[15]	Zhao J G, Shi R Q. Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations[J].Applied Geophysics,2013,10(3): 323-336.
[16]	Abarbanel S, Gottlieb D, Hesthaven J S. Long time behavior of the perfectly matched layer equations in computational electromagnetics[J].Journal of Scientific Computing,2002,17(1/4): 405-422.
[17]	Zienkiewicz O C, Taylor R L.The Finite Element Method [M]. 5th ed. New York: Butterworth-Heinemann, 2000.
[18]	Becache E, Fauqueux S, Joly P. Stability of perfectly matched layers, group velocities and anisotropic waves[J].Journal of Computational Physics,2003,188(2): 399-433.
[19]	Schanz M, Cheng A H D. Transient wave propagation in a one-dimensional poroelastic column[J].Acta Mechanica,2000,145(1/4): 1-18.