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Multi-axial unsplit frequency-shifted perfectly matched layer for displacement-based anisotropic wave simulation in infinite domain

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Abstract

Multi-axial perfectly matched layer (M-PML), known to have lost the perfect-matching property owing to multi-axial coordinate stretching, has been numerically validated to be long-time stable and it is thus used extensively in linear anisotropic wave simulation and in isotropic cases where the PML becomes unstable. We are concerned with the construction of the M-PML for anisotropic wave simulation based on a second order wave equation implemented with the displacement-based numerical method. We address the benefit of the incorrect chain rule, which is implicitly adopted in the previous derivation of the M-PML. We show that using the frequency-shifted stretching function improves the absorbing efficiency of the M-PML for near-grazing incident waves. Then, through multi-axial complex-coordinate stretching the second order anisotropic wave equation in a weak form, we derive a time-domain multi-axial unsplit frequency-shifted PML (M-UFSPML) using the frequency-shifted stretching function and the incorrect chain rule. A new approach is provided to reduce the number of memory variables needed for computing convolution terms in the M-UFSPML. The obtained M-UFSPML is well suited for implementation with a finite element or the spectral element method. By providing several typical examples, we numerically verify the accuracy and long-time stability of the implementation of our M-UFSPML by utilizing the Legendre spectral element method.

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Acknowledgement

This study was funded by the Scientific Research Fund of the Institute of Engineering Mechanics, China Earthquake Administration (No. 2021EEEVL0102); the National Natural Science Foundation of China (Nos. U2039209; 51808516); the National Key RandD Program of China (No. 2018YFC1504004); and the Distinguished Young Scholars Program of the Natural Science Foundation of Heilongjiang province, China (No. YQ2020E005). Zhinan mourns for Dimitri Komatitsch, a visionary computational geophysicist and a great teacher.

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Authors and Affiliations

  1. Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, 29 Xuefu Road, Harbin, 150080, China

    Zhinan Xie, Yonglu Zheng & Xubin Zhang

  2. Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin, 150080, China

    Zhinan Xie, Yonglu Zheng & Xubin Zhang

  3. CNRS, Centrale Marseille, LMA, Aix Marseille University, Marseille, F-13353, France

    Paul Cristini

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  1. Zhinan Xie
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  2. Yonglu Zheng
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  3. Paul Cristini
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  4. Xubin Zhang
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Correspondence to Zhinan Xie.

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Supported by: Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration under Grant No. 2021EEEVL0102, National Natural Science Foundation of China under Grant Nos. U2039209 and 51808516, the National Key R&D Program of China under Grant No. 2018YFC1504004, and Distinguished Young Scholars Program of the Natural Science Foundation of Heilongjiang province, China under Grant No. YQ2020E005

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Xie, Z., Zheng, Y., Cristini, P. et al. Multi-axial unsplit frequency-shifted perfectly matched layer for displacement-based anisotropic wave simulation in infinite domain. Earthq. Eng. Eng. Vib. 22, 407–421 (2023). https://doi.org/10.1007/s11803-023-2170-3

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  • DOI: https://doi.org/10.1007/s11803-023-2170-3

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