American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020276

Time-domain analysis of forward obstacle scattering for elastic wave

Lu Zhao , Heping Dong , and  Fuming Ma

School of Mathematics, Jilin University, Changchun, 130012, China

* Corresponding author: Heping Dong

Received  April 2020 Revised  July 2020 Published  September 2020

This paper concerns a time-domain scattering problem of elastic plane wave by a rigid obstacle, which is immersed in an open space filled with homogeneous and isotropic elastic medium in two dimensions. A new compressed coordinate transformation is developed to reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval. The well-posednesss is established for the reduced problem. This paper adopts Galerkin method to prove the uniqueness results and employs energy method to derive stability results for the scattering problem. Furthermore, we achieve a priori estimate with explicit time dependence.

Keywords: Time-domain, elastic wave equation, compressed coordinate transformation, well-posedness, priori estimates.
Mathematics Subject Classification: 35A15, 35P25.
Citation: Lu Zhao, Heping Dong, Fuming Ma. Time-domain analysis of forward obstacle scattering for elastic wave. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020276
References:
[1]

G. Bao, B. Hu, P. Li and J. Wang, Analysis of time-domain Maxwell's equations in biperiodic structures, Discrete Cont Dyn-B, 25 (2020), 259-286.  doi: 10.3934/dcdsb.2019181.  Google Scholar

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Z. Chen, Convergence of the time-domain perfectly matched layer method for acoustic scattering problems, Int J Numer Anal Mod, 6 (2009), 124–146. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=61E76534114A89A0D109AAE9DF588AC5?doi=10.1.1.407.8691&rep=rep1&type=pdf Google Scholar

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[9]

Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems, SIAM J. Numer Anal, 50 (2012), 2632-2655.  doi: 10.1137/110835268.  Google Scholar

[10]

Z. Chen, X. Xiang and X. Zhang, Convergence of the PML method for elastic wave scattering problems, Math Comput, 85 (2016), 2687-2714.  doi: 10.1090/mcom/3100.  Google Scholar

[11]

H. Dong, J. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased or phaseless far-field data, SIAM J. Imaging Sci, 12 (2019), 809-838.  doi: 10.1137/18M1227263.  Google Scholar

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Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differ Equations, 261 (2016), 5094-5118.  doi: 10.1016/j.jde.2016.07.020.  Google Scholar

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Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math Mod Meth Appl S, 27 (2017), 1843-1870.  doi: 10.1142/S0218202517500336.  Google Scholar

[15]

Y. Gao, P. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J Math Anal, 49 (2017), 3951-3972.  doi: 10.1137/16M1090326.  Google Scholar

[16]

Y. Gao, P. Li and Y. Li, Analysis of time-domain elastic scattering by an unbounded structure, Math Method Appl Sci, 41 (2018), 7032-7054.  doi: 10.1002/mma.5214.  Google Scholar

[17]

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[18]

P. Li, Y. Wang, Z. Wang and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Probl, 32 (2016), 115018, 24pp. doi: 10.1088/0266-5611/32/11/115018.  Google Scholar

[19]

P. Li and X. Yuan, Inverse obstacle scattering for elastic waves in three dimensions, Inverse Probl Imag, 13 (2019), 545-573.  doi: 10.3934/ipi.2019026.  Google Scholar

[20]

P. Li and L. Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun Math Sci, 17 (2019), 1671-1698.  doi: 10.4310/CMS.2019.v17.n6.a8.  Google Scholar

[21]

D. J. Riley and J.-M. Jin, Finite-element time-domain analysis of electrically and magnetically dispersive periodic structures, IEEE T Antenn Propag, 56 (2008), 3501-3509.  doi: 10.1109/TAP.2008.2005454.  Google Scholar

[22]

C. Wei and J. Yang, Analysis of a time-dependent fluid-solid interaction problem above a local rough surface, Sci China Math, 63 (2020), 887-906.  doi: 10.1007/s11425-017-9364-3.  Google Scholar

show all references

References:
[1]

G. Bao, B. Hu, P. Li and J. Wang, Analysis of time-domain Maxwell's equations in biperiodic structures, Discrete Cont Dyn-B, 25 (2020), 259-286.  doi: 10.3934/dcdsb.2019181.  Google Scholar

[2]

G. Bao, G. Hu, J. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J Math Pures Appl, 117 (2018), 263–301, arXiv: 1612.06604. doi: 10.1016/j.matpur.2018.01.007.  Google Scholar

[3]

G. Bao, Y. Gao and P. Li, Time domain analysis of an acoustic-elastic interaction problem, Arch Ration Mech An, 229 (2018), 835-884.  doi: 10.1007/s00205-018-1228-2.  Google Scholar

[4]

M. J. Bluck and S. P. Walker, Time-domain BIE analysis of large three-dimensional electromagnetic scattering problems, IEEE T Antenn Propag, 45 (1997), 894-901.  doi: 10.1109/8.575643.  Google Scholar

[5]

J. H. Bramble, J. E. Pasciak and D. Trenev, Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem, Math Comput, 79 (2010), 2079-2101.  doi: 10.1090/S0025-5718-10-02355-0.  Google Scholar

[6]

Q. Chen and P. Monk, Discretization of the time domain CFIE for acoustic scattering problems using convolution quadrature, SIAM J. Math Anal, 46 (2016), 3107-3130.  doi: 10.1137/110833555.  Google Scholar

[7]

Z. Chen, Convergence of the time-domain perfectly matched layer method for acoustic scattering problems, Int J Numer Anal Mod, 6 (2009), 124–146. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=61E76534114A89A0D109AAE9DF588AC5?doi=10.1.1.407.8691&rep=rep1&type=pdf Google Scholar

[8]

Z. Chen and J.-C. N$\acute{e}$d$\acute{e}$lec, On Maxwell equations with the transparent boundary condition, J Comput Math, 26 (2008), 284-296.   Google Scholar

[9]

Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems, SIAM J. Numer Anal, 50 (2012), 2632-2655.  doi: 10.1137/110835268.  Google Scholar

[10]

Z. Chen, X. Xiang and X. Zhang, Convergence of the PML method for elastic wave scattering problems, Math Comput, 85 (2016), 2687-2714.  doi: 10.1090/mcom/3100.  Google Scholar

[11]

H. Dong, J. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased or phaseless far-field data, SIAM J. Imaging Sci, 12 (2019), 809-838.  doi: 10.1137/18M1227263.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, vol. 19, Graduate Studies in Mathematics, AMS, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[13]

Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differ Equations, 261 (2016), 5094-5118.  doi: 10.1016/j.jde.2016.07.020.  Google Scholar

[14]

Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math Mod Meth Appl S, 27 (2017), 1843-1870.  doi: 10.1142/S0218202517500336.  Google Scholar

[15]

Y. Gao, P. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J Math Anal, 49 (2017), 3951-3972.  doi: 10.1137/16M1090326.  Google Scholar

[16]

Y. Gao, P. Li and Y. Li, Analysis of time-domain elastic scattering by an unbounded structure, Math Method Appl Sci, 41 (2018), 7032-7054.  doi: 10.1002/mma.5214.  Google Scholar

[17]

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1959. http://gen.lib.rus.ec/book/index.php?md5=16af3489becf3ea6fb1d585b40658fcd  Google Scholar

[18]

P. Li, Y. Wang, Z. Wang and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Probl, 32 (2016), 115018, 24pp. doi: 10.1088/0266-5611/32/11/115018.  Google Scholar

[19]

P. Li and X. Yuan, Inverse obstacle scattering for elastic waves in three dimensions, Inverse Probl Imag, 13 (2019), 545-573.  doi: 10.3934/ipi.2019026.  Google Scholar

[20]

P. Li and L. Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun Math Sci, 17 (2019), 1671-1698.  doi: 10.4310/CMS.2019.v17.n6.a8.  Google Scholar

[21]

D. J. Riley and J.-M. Jin, Finite-element time-domain analysis of electrically and magnetically dispersive periodic structures, IEEE T Antenn Propag, 56 (2008), 3501-3509.  doi: 10.1109/TAP.2008.2005454.  Google Scholar

[22]

C. Wei and J. Yang, Analysis of a time-dependent fluid-solid interaction problem above a local rough surface, Sci China Math, 63 (2020), 887-906.  doi: 10.1007/s11425-017-9364-3.  Google Scholar

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