Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 3: 1122 - 1145

DOI: https://doi.org/10.1007/s42967-024-00450-6

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

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  • 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 date: 2023-11-20

  Revised date: 2024-06-30

  Accepted date: 2024-07-15

  Online 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.

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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

Cite this article

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 . DOI: 10.1007/s42967-024-00450-6

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