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Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

1
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China
2
Key Laboratory of Advanced Material of Ship and Mechanics, Ministry of Industry and Information Technology, Harbin Engineering University, Harbin 150001, China
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(2), 202; https://doi.org/10.3390/sym12020202
Received: 28 December 2019 / Revised: 16 January 2020 / Accepted: 18 January 2020 / Published: 1 February 2020
In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations. View Full-Text
Keywords: elastic wave equation; symmetric matrix form; perfectly matched layer; finite difference method; SBP-SAT; stability elastic wave equation; symmetric matrix form; perfectly matched layer; finite difference method; SBP-SAT; stability
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Sun, C.; Yang, Z.; Jiang, G. Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain. Symmetry 2020, 12, 202.

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