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An Adaptive Finite Element PML Method for the Acoustic-Elastic Interaction in Three Dimensions

Part of: Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems Optics, electromagnetic theory - Basic methods

Published online by Cambridge University Press:  31 October 2017

Xue Jiang  and
Peijun Li
Xue Jiang*
Affiliation:
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Peijun Li*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
*
*Corresponding author. Email addresses:jxue@lsec.cc.ac.cn(X. Jiang), lipeijun@math.purdue.edu(P. Li)
*Corresponding author. Email addresses:jxue@lsec.cc.ac.cn(X. Jiang), lipeijun@math.purdue.edu(P. Li)
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Abstract

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

Keywords

Acoustic-elastic interactionperfectly matched layeradaptive finite element methodtransparent boundary condition

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 78M10: Finite element methods 35Q99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1486 - 1507
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Babuška, I. and Aziz, A., Survey Lectures onMathematical Foundations of the Finite Element Method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, ed. by Aziz, A., Academic Press, New York, 1973, 5–359.Google Scholar
[2] Bao, G., Li, P., and Wu, H., An adaptive edge element method with perfectly matched absorbing layers for wave scattering by periodic structures, Math. Comp., 79 (2010), 1–34.CrossRefGoogle Scholar
[3] Bao, G. and Wu, H., On the convergence of the solutions of PML equations for Maxwell's equations, SIAM J. Numer. Anal., 43 (2005), 2121–2143.CrossRefGoogle Scholar
[4] Bérenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185–200.CrossRefGoogle Scholar
[5] Bramble, J. H. and Pasciak, J. E., Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems, Math. Comp., 76 (2007), 597–614.Google Scholar
[6] Bramble, J. H., Pasciak, J. E., and Trenev, D., Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem, Math. Comp., 79 (2010), 2079–2101.CrossRefGoogle Scholar
[7] Chen, J. and Chen, Z., An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp., 77 (2008), 673–698.CrossRefGoogle Scholar
[8] Chen, Z. and Wu, H., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799–826.Google Scholar
[9] Chen, Z. and Wu, X., An adaptive uniaxial perfectlymatched layermethod for time-harmonic scattering problems, Numer. Math. Theor. Meth. Appl., 1 (2008), 113–137.Google Scholar
[10] Chen, Z. and Liu, X., An adptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645–671.Google Scholar
[11] Chen, Z., Xiang, X., and Zhang, X., Convergence of the PML method for elasticwave scattering problems, Math. Comp., to appear.Google Scholar
[12] Collino, F. and Monk, P., The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061–1090.Google Scholar
[13] Collino, F. and Tsogka, C., Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics, 66 (2001), 294–307.CrossRefGoogle Scholar
[14] Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.Google Scholar
[15] Chew, W. and Weedon, W., A 3D perfectly matched medium for modified Maxwell's equations with stretched coordinates, Microwave Opt. Techno. Lett., 13 (1994), 599–604.CrossRefGoogle Scholar
[16] Estorff, O. V. and Antes, H., On FEM-BEM coupling for fluid-structure interaction analyses in the time domain, Internat. J. Numer. Methods Engrg., 31 (1991), 1151–1168.Google Scholar
[17] Flemisch, B., Kaltenbacher, M., and Wohlmuth, B. I., Elasto-acoustic and acoustic-acoustic coupling on non-matching grids, Internat. J. Numer. Methods Engrg., 67 (2006), 1791–1810.CrossRefGoogle Scholar
[18] Gao, Y. and Li, P., Time-domain analysis of an acoustic-elastic interaction problem, preprint.Google Scholar
[19] Gao, Y., Li, P., and Zhang, B., Analysis of transient acoustic-elastic interaction in an unbounded structure, preprint.Google Scholar
[20] Hastings, F. D., Schneider, J. B., and Broschat, S. L., Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation, J. Acoust. Soc. Am., 100 (1996), 3061–3069.CrossRefGoogle Scholar
[21] Hohage, T., Schmidt, F., and Zschiedrich, L., Solving time-harmonic scattering problems based on the pole condition. II: Convergence of the PML method, SIAM J. Math. Anal., 35 (2003), 547–560.CrossRefGoogle Scholar
[22] Hsiao, G. C., On the boundary-field equation methods for fluid-structure interactions, In Problems and methods in mathematical physics (Chemnitz, 1993), vol. 134, Teubner-Texte Math., 79–88, Teubner, Stuttgart, 1994.Google Scholar
[23] Hsiao, G. C., Kleinman, R. E., and Schuetz, L. S., On variational formulations of boundary value problems for fluid-solid interactions, In Elastic wave propagation (Galway, 1988), vol. 35, North-Holland Ser. Appl. Math. Mech., 321–326, North-Holland, Amsterdam, 1989.Google Scholar
[24] Hsiao, G. C., Sánchez-Vizuet, T., and Sayas, F.-J., Boundary and coupled boundary-finite element methods for transient wave-structure interaction, IMA J. Numer. Anal., 37 (2017), 237–265.CrossRefGoogle Scholar
[25] Jiang, X., Li, P., Lv, J., and Zheng, W., An adaptive finite element PML method for the elastic wave scattering problem in periodic structure, ESAIM: Math. Model. Numer. Anal., to appear.Google Scholar
[26] Jiang, X., Li, P., Lv, J., and Zheng, W., Convergence of the PML solution for elastic wave scattering by biperiodic structures, preprint.Google Scholar
[27] Lassas, M. and Somersalo, E., On the existence and convergence of the solution of PML equations, Computing, 60 (1998), 229–241.Google Scholar
[28] Luke, C. J. and Martin, P. A., Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math., 55 (1995), 904–922.Google Scholar
[29] PHG (Parallel Hierarchical Grid), http://lsec.cc.ac.cn/phg/.Google Scholar
[30] Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483–493.Google Scholar
[31] Soares, D. and Mansur, W., Dynamic analysis of fluid-soil-structure interaction problems by the boundary element method, J. Comput. Phys., 219 (2006), 498–512.Google Scholar
[32] Turkel, E. and Yefet, A., Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27 (1998), 533–557.Google Scholar