Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures

doi:10.1007/s42967-024-00450-6

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (3): 1122-1145.doi: 10.1007/s42967-024-00450-6

• ORIGINAL PAPERS • 上一篇下一篇

Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures

Gang Bao¹, Peijun Li², Xiaokai Yuan³

1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, Zhejiang, China;
2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3 School of Mathematics, Jilin University, Changchun 130012, Jilin, China

收稿日期:2023-11-20 修回日期:2024-06-30 接受日期:2024-07-15 出版日期:2025-09-20 发布日期:2025-05-23
通讯作者: Peijun Li, lipeijun@lsec.cc.ac.cn;Gang Bao, baog@zju.edu.cn;Xiaokai Yuan, yuanxk@jlu.edu.cn E-mail:lipeijun@lsec.cc.ac.cn;baog@zju.edu.cn;yuanxk@jlu.edu.cn
基金资助:
The first author is supported partially by the National Natural Science Foundation of China under Grant No. U21A20425 and a Key Laboratory of Zhejiang Province. The third author is supported by the NSFC under Grant Nos. 12201245 and 12171017.

Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures

Gang Bao¹, Peijun Li², Xiaokai Yuan³

1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, Zhejiang, China;
2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3 School of Mathematics, Jilin University, Changchun 130012, Jilin, China

Received:2023-11-20 Revised:2024-06-30 Accepted:2024-07-15 Online:2025-09-20 Published:2025-05-23
Supported by:
The first author is supported partially by the National Natural Science Foundation of China under Grant No. U21A20425 and a Key Laboratory of Zhejiang Province. The third author is supported by the NSFC under Grant Nos. 12201245 and 12171017.

摘要/Abstract

摘要： This paper investigates the scattering of biharmonic waves by a one-dimensional periodic array of cavities embedded in an infinite elastic thin plate. The transparent boundary conditions (TBCs) are introduced to formulate the problem from an unbounded domain to a bounded one. The well-posedness of the associated variational problem is demonstrated utilizing the Fredholm alternative theorem. The perfectly matched layer (PML) method is employed to reformulate the original scattering problem, transforming it from an unbounded domain to a bounded one. The TBCs for the PML problem are deduced, and the wellposedness of its variational problem is established. Moreover, the exponential convergence is achieved between the solution of the PML problem and that of the original scattering problem.

关键词: Biharmonic wave equation, Transparent boundary condition (TBC), Perfectly matched layer (PML), Variational problem, Well-posedness, Convergence analysis

Abstract: This paper investigates the scattering of biharmonic waves by a one-dimensional periodic array of cavities embedded in an infinite elastic thin plate. The transparent boundary conditions (TBCs) are introduced to formulate the problem from an unbounded domain to a bounded one. The well-posedness of the associated variational problem is demonstrated utilizing the Fredholm alternative theorem. The perfectly matched layer (PML) method is employed to reformulate the original scattering problem, transforming it from an unbounded domain to a bounded one. The TBCs for the PML problem are deduced, and the wellposedness of its variational problem is established. Moreover, the exponential convergence is achieved between the solution of the PML problem and that of the original scattering problem.

Key words: Biharmonic wave equation, Transparent boundary condition (TBC), Perfectly matched layer (PML), Variational problem, Well-posedness, Convergence analysis

中图分类号:

78A45
65N30

Gang Bao, Peijun Li, Xiaokai Yuan. Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1122-1145.

参考文献

1. Amara,M., Dabaghi, F.: An optimal c0 finite element algorithmfor the 2D biharmonic problem: theoretical analysis and numerical results. Numer. Math. 90, 19–46 (2001)
2. Antonietti, P., Manzini, G., Verani, M.: The fully nonconforming virtual element method for biharmonic problems. Math. Model. Methods Appl. Sci. 28(2), 387–407 (2018)
3. Bao, G., Li, P.: Convergence analysis in near-field imaging. Inverse Probl. 30, 085008 (2014)
4. Bao, G., Li, P.: Maxwell’s equations in periodic structures. In: Series on Applied Mathematical Sciences, vol. 208. Science Press, Beijing (2022)
5. Bao, G., Wu, H.: Convergence analysis of the PML problems for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal. 43, 2121–2143 (2005)
6. Bérenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)
7. Bourgeois, L., Chesnel, L., Fliss, S.: On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model. Commun. Math. Sci. 17, 487–1529 (2019)
8. Bourgeois, L., Hazard, C.: On well-posedness of scattering problems in a Kirchhoff-love infinite plate. SIAM J. Appl. Math. 80, 1546–1566 (2020)
9. Bramble, J.H., Pasciak, J.E.: Analysis of a finite PML approximation for the three dimensional timeharmonic Maxwell and acoustic scattering problems. Math. Comput. 76, 597–614 (2007)
10. Bramble, J.H., Pasciak, J.E., Trenev, D.: Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem. Math. Comput. 79, 2079–2101 (2010)
11. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2008)
12. Chen, Z., Liu, X.: An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal. 43, 645–671 (2005)
13. Chen, Z., Xiang, X., Zhang, X.: Convergence of the PML method for elastic wave scattering problems. Math. Comput. 85, 2687–2714 (2016)
14. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
15. Ciarlet, P.G., Glowinski, R.: Dual iterative techniques for solving a finite element approximation of the biharmonic equation. Comput. Methods Appl. Mech. Eng. 5, 277–295 (1975)
16. Ciarlet, P.G., Raviart, P.: A mixed finite element method for the biharmonic equation. In: Boor, C.D. (ed.) Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 125–143. Academic Press, New York (1974)
17. Darabi, A., Zareei, A., Alam, M., Leamy, M.: Experimental demonstration of an ultrabroadband nonlinear cloak for flexural waves. Phys. Rev. Lett. 121, 174301 (2018)
18. Dong, H., Li, P.: A novel boundary integral formulation for the biharmonic wave scattering problem. arXiv:2301.10142 (2023)
19. Farhat, M., Guenneau, S., Enoch, S.: Ultrabroadband elastic cloaking in thin plates. Phys. Rev. Lett. 103, 024301 (2009)
20. Farhat, M., Guenneau, S., Enoch, S.: Finite elements modelling of scattering problems for flexural waves in thin plates: application to elliptic invisibility cloaks, rotators and the mirage effect. J. Comput. Phys. 230, 2237–2245 (2011)
21. Haslinger, S.: Mathematical Modelling of Flexural Waves in Structured Elastic Plates. Ph.D. Thesis. University of Liverpool, Liverpool ( 2014)
22. Haslinger, S., Craster, R., Movchan, A., Movchan, N., Jones, I.: Dynamic interfacial trapping of flexural waves in structured plates. Proc. R. Soc. A. 472, 20150658 (2016)
23. Hsiao, G.C., Wendland, W.L.: Boundary integral equations. In: Applied Mathematical Sciences, 2nd edition, vol. 164. Springer, Switzerland (2021)
24. Li, P., Wu, H., Zheng, W.: Electromagnetic scattering by unbounded rough surfaces. SIAM J. Math. Anal. 43, 1205–1231 (2011)
25. Morvaridi, M., Brun, M.: Perfectly matched layers for flexural waves: an exact analytical model. Int. J. Solids Struct. 102, 1–9 (2016)
26. Morvaridi, M., Brun, M.: Perfectly matched layers for flexural waves in Kirchhoff-love plates. Int. J. Solids Struct. 134, 293–303 (2018)
27. Mu, L., Wang, J., Ye, X., Zhang, S.: A C⁰ weak Galerkin finite element methods for the biharmonic equation. J. Sci. Comput. 59, 437–495 (2014)
28. Pelat, A., Gautier, F., Conlon, S.C., Semperlotti, F.: The acoustic black hole: a review of theory and applications. J. Sound Vib. 476, 115316 (2020)
29. Smith, M.J.A.: Wave Propagation Through Periodic Structures in Thin Plates. Ph.D. Thesis. The University of Auckland, Auckland (2013)
30. Smith, M.J.A., Meylan, M.H., Mcphedran, R.C.: Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems. J. Sound Vib. 330, 4029–4046 (2011)
31. Teixeira, F., Chew, W.: Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates. IEEE Microw. Guided Wave Lett. 7, 371–373 (1997)
32. Ye, X., Zhang, S.: A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes. SIAM J. Numer. Anal. 58, 2572–2588 (2020)
33. Yue, J., Li, P.: Numerical solution of the cavity scattering problem for flexural waves on thin plates: linear finite element methods. J. Comput. Phys. 497, 112606 (2024)
34. Yue, J., Li, P., Yuan, X., Zhu, X.: A diffraction problem for the biharmonic wave equation in onedimensional periodic structures. Results Appl. Math. 17, 100350 (2023)
35. Zhang, R., Zhai, Q.: A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J. Sci. Comput. 64, 559–585 (2015)
36. Zhao, J., Chen, S., Zhang, B.: The nonconforming virtual element method for plate bending problems. Math. Model. Methods Appl. Sci. 26, 1671–1687 (2016)

[1]	Andreas Dedner, Tristan Pryer. Discontinuous Galerkin Methods for a Class of Nonvariational Problems[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 634-656.
[2]	Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.
[3]	Qingqing Wu, Zhongshu Wu, Xiaoyan Zeng. A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 509-526.
[4]	Mehdi Samiee, Ehsan Kharazmi, Mark M. Meerschaert, Mohsen Zayernouri. A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 61-90.
[5]	Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven. A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 689-709.
[6]	Mark Ainsworth, Zhiping Mao. Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 5-19.

Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures

Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

编辑推荐

Metrics

本文评价