August  2021, 26(8): 4111-4130. doi: 10.3934/dcdsb.2020276

Time-domain analysis of forward obstacle scattering for elastic wave

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

* Corresponding author: Heping Dong

Received  April 2020 Revised  July 2020 Published  August 2021 Early access  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.

Citation: Lu Zhao, Heping Dong, Fuming Ma. Time-domain analysis of forward obstacle scattering for elastic wave. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4111-4130. doi: 10.3934/dcdsb.2020276
References:
[1]

G. BaoB. HuP. 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.

[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.

[3]

G. BaoY. 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.

[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.

[5]

J. H. BrambleJ. 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.

[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.

[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

[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. 

[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.

[10]

Z. ChenX. 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.

[11]

H. DongJ. 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.

[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.

[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.

[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.

[15]

Y. GaoP. 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.

[16]

Y. GaoP. 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.

[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

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

G. BaoB. HuP. 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.

[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.

[3]

G. BaoY. 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.

[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.

[5]

J. H. BrambleJ. 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.

[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.

[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

[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. 

[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.

[10]

Z. ChenX. 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.

[11]

H. DongJ. 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.

[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.

[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.

[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.

[15]

Y. GaoP. 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.

[16]

Y. GaoP. 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.

[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

[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.

[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.

[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.

[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.

[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.

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