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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Earth Sci.</journal-id>
<journal-title>Frontiers in Earth Science</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Earth Sci.</abbrev-journal-title>
<issn pub-type="epub">2296-6463</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">1056583</article-id>
<article-id pub-id-type="doi">10.3389/feart.2022.1056583</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Earth Science</subject>
<subj-group>
<subject>Methods</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Strategy for eliminating high-frequency instability caused by multi-transmitting boundary in numerical simulation of seismic site effect</article-title>
<alt-title alt-title-type="left-running-head">Yang et al.</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/feart.2022.1056583">10.3389/feart.2022.1056583</ext-link>
</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Yang</surname>
<given-names>Yu</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<uri xlink:href="https://loop.frontiersin.org/people/1835367/overview"/>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Li</surname>
<given-names>Xiaojun</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
<xref ref-type="aff" rid="aff3">
<sup>3</sup>
</xref>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Rong</surname>
<given-names>Mianshui</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Yang</surname>
<given-names>Zhibo</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
</contrib>
</contrib-group>
<aff id="aff1">
<sup>1</sup>
<institution>Nuclear and Radiation Safety Center MEE</institution>, <addr-line>Beijing</addr-line>, <country>China</country>
</aff>
<aff id="aff2">
<sup>2</sup>
<institution>Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education</institution>, <institution>Beijing University of Technology</institution>, <addr-line>Beijing</addr-line>, <country>China</country>
</aff>
<aff id="aff3">
<sup>3</sup>
<institution>Institute of Geophysics</institution>, <institution>China Earthquake Administration</institution>, <addr-line>Beijing</addr-line>, <country>China</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1692474/overview">Kun Ji</ext-link>, Hohai University, China</p>
</fn>
<fn fn-type="edited-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1898672/overview">Qingzhi Hou</ext-link>, Tianjin University, China</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1157585/overview">Maryam Khosravi</ext-link>, Isfahan University of Technology, Iran</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Xiaojun Li, <email>64482261@qq.com</email>, <email>beerli@vip.sina.com</email>
</corresp>
<fn fn-type="other">
<p>This article was submitted to Structural Geology and Tectonics, a section of the journal Frontiers in Earth Science</p>
</fn>
</author-notes>
<pub-date pub-type="epub">
<day>22</day>
<month>12</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="collection">
<year>2022</year>
</pub-date>
<volume>10</volume>
<elocation-id>1056583</elocation-id>
<history>
<date date-type="received">
<day>29</day>
<month>09</month>
<year>2022</year>
</date>
<date date-type="accepted">
<day>08</day>
<month>12</month>
<year>2022</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2022 Yang, Li, Rong and Yang.</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Yang, Li, Rong and Yang</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<p>A multi-transmitting boundary is a local artificial boundary widely used for numerically simulating seismic site effects. However, similar to other artificial boundaries, the multi-transmitting boundary has instability issue in numerical simulation. Based on the concept of multi-directional transmitting formula, a strategy for eliminating the high-frequency instability of the transmitting boundary is studied and a measure is proposed using a neighbour node of a boundary node to realize smoothing filtering. The proposed measure is verified through numerical analysis. The smoothing coefficient chosen for this measure provides a reference for deriving the coefficient of multidirectional transmitting formula in the time domain.</p>
</abstract>
<kwd-group>
<kwd>seismic site effect</kwd>
<kwd>wave propagating simulation</kwd>
<kwd>multi-transmitting boundary</kwd>
<kwd>high-frequency instability</kwd>
<kwd>multi-direction transmitting formula</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="s1">
<title>1 Introduction</title>
<p>The influence of local topography on ground motion is fundamentally a wave scattering problem. Hence, simulating near-field waves is crucial to the numerical simulation of seismic site effects. The accuracy of near-field wave numerical simulations directly depends on whether artificial boundary conditions can accurately simulate an infinite domain. Since the 1960s, several achievements have been attained in the study of artificial boundaries (<xref ref-type="bibr" rid="B9">Liao, 1984</xref>, <xref ref-type="bibr" rid="B10">2002</xref>; <xref ref-type="bibr" rid="B16">Wolf, 1988</xref>; <xref ref-type="bibr" rid="B2">Givoli, 1992</xref>; <xref ref-type="bibr" rid="B1">Cheng et al., 1995</xref>; <xref ref-type="bibr" rid="B15">Wolf, 1996</xref>; <xref ref-type="bibr" rid="B21">Xu et al., 2018</xref>; <xref ref-type="bibr" rid="B20">Xing et al., 2021</xref>). Among the established artificial boundary conditions, the multi-transmitting boundary (<xref ref-type="bibr" rid="B11">Liao et al., 1984a</xref>; <xref ref-type="bibr" rid="B12">Liao et al., 1984b</xref>; <xref ref-type="bibr" rid="B18">Xing et al., 2017a</xref>; <xref ref-type="bibr" rid="B19">Xing et al., 2017b</xref>) has a wide application range and high precision. Moreover, combined with the finite element method, the multi-transmitting boundary can facilitate decoupling.</p>
<p>Similar to other local artificial boundaries, the transmitting boundary&#x2019;s computational stability is a key issue that requires further study. High-frequency oscillation and low-frequency drift are two types of numerical instability phenomena that may occur when the multi-transmission boundary is combined with the finite element method (<xref ref-type="bibr" rid="B5">Li et al., 2012</xref>; <xref ref-type="bibr" rid="B22">Yang et al., 2014</xref>). In this paper, a strategy for eliminating the instability of high-frequency oscillations of the multi-transmission boundary is suggested.</p>
<p>Smoothing factor filtering is an effective measure for restraining high-frequency instability of transmission boundary (<xref ref-type="bibr" rid="B6">Liao et al., 1989</xref>; <xref ref-type="bibr" rid="B7">Liao et al., 1992</xref>; <xref ref-type="bibr" rid="B8">Liao et al., 2002</xref>). Another measure to restrain high-frequency instability is utilizing the energy consumption characteristics of explicit integration scheme. This measure inhibits high-frequency instability by increasing damping in proportion to strain velocity (<xref ref-type="bibr" rid="B3">Li et al., 1992</xref>; <xref ref-type="bibr" rid="B4">Li et al., 2007</xref>; <xref ref-type="bibr" rid="B14">Tang et al., 2010</xref>). Modifying the internal node motion equation and stiffness of the finite element is also an effective measure for stabilizing the high-frequency of multi-transmission boundary (<xref ref-type="bibr" rid="B17">Xie et al., 2012</xref>; <xref ref-type="bibr" rid="B23">Zhang et al., 2021</xref>).</p>
<p>This paper proposes an improved measure for existing strategies to restrain high-frequency instability using a smoothing factor. When considering only the high-frequency error oscillation perpendicular to the artificial boundary and ignoring the high-frequency oscillation parallel to the artificial boundary, the current method only smooths the points perpendicular to the boundary. Based on the concept of a multidirectional transmitting formula, this paper proposes smoothing the points on the artificial boundary to restrain high-frequency instability.</p>
</sec>
<sec id="s2">
<title>2 Multi-transmitting formula and its instability of high-frequency oscillation for restraining instability</title>
<sec id="s2-1">
<title>2.1 Multi-transmitting formula (MTF)</title>
<p>The multi-transmitting boundary is also called Multi-transmitting formula (MTF), which is a boundary condition using the general expression of a one-sided traveling wave solution to simulate an external wave crossing the boundary at a point on the artificial boundary. It uses internal point displacement to represent the boundary point displacement. MTF was proposed by Liao et al. (<xref ref-type="bibr" rid="B9">Liao, 1984</xref>, <xref ref-type="bibr" rid="B10">2002</xref>). In the finite element discrete model (<xref ref-type="fig" rid="F1">Figure 1</xref>), the MTF of the arbitrary artificial boundary point <bold>
<italic>J</italic>
</bold> can be expressed as<disp-formula id="e1">
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<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">T</mml:mi>
<mml:mi mathvariant="bold-italic">K</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf8">
<mml:math id="m10">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mi mathvariant="bold-italic">n</mml:mi>
</mml:msub>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> are dimensionless parameters. Here, <inline-formula id="inf9">
<mml:math id="m11">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">T</mml:mi>
<mml:mi mathvariant="bold-italic">K</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is obtained by summing <inline-formula id="inf10">
<mml:math id="m12">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mi mathvariant="bold-italic">n</mml:mi>
</mml:msub>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> to satisfy the following:<disp-formula id="e3">
<mml:math id="m13">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mo>&#x22ef;</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mi mathvariant="bold-italic">n</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mi mathvariant="bold-italic">n</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x22ef;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">K</mml:mi>
<mml:mi mathvariant="bold-italic">n</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn mathvariant="bold">1,2,3</mml:mn>
</mml:mrow>
</mml:math>
<label>(3)</label>
</disp-formula>
<disp-formula id="e4">
<mml:math id="m14">
<mml:mrow>
<mml:mfenced open="{" close="" separators="|">
<mml:mrow>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mfrac>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:mn mathvariant="bold">3</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mfrac>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">c</mml:mi>
<mml:mi mathvariant="bold-italic">a</mml:mi>
</mml:msub>
<mml:mo>&#x394;</mml:mo>
<mml:mi mathvariant="bold-italic">t</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x394;</mml:mo>
<mml:mi mathvariant="bold-italic">x</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
<label>(4)</label>
</disp-formula>
</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Discrete model of multi-transmitting boundary.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g001.tif"/>
</fig>
<p>In Eq. <xref ref-type="disp-formula" rid="e4">4</xref>, <inline-formula id="inf11">
<mml:math id="m15">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">c</mml:mi>
<mml:mi mathvariant="bold-italic">a</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the artificial wave velocity. The wave unilaterally travels at velocity <inline-formula id="inf12">
<mml:math id="m16">
<mml:mrow>
<mml:mi mathvariant="bold-italic">c</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and transmits in the direction of angle <inline-formula id="inf13">
<mml:math id="m17">
<mml:mrow>
<mml:mi mathvariant="bold-italic">&#x3b8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> with a boundary surface (<inline-formula id="inf14">
<mml:math id="m18">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">c</mml:mi>
<mml:mi mathvariant="bold-italic">a</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold-italic">c</mml:mi>
<mml:mo>/</mml:mo>
<mml:mi mathvariant="bold-italic">cos</mml:mi>
<mml:mo>&#x2061;</mml:mo>
<mml:mi mathvariant="bold-italic">&#x3b8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>). As shown in <xref ref-type="fig" rid="F1">Figure 1</xref>, <bold>
<italic>&#x394;x</italic>
</bold> is the spatial step of the discrete grid in the direction perpendicular to the artificial boundary; <bold>
<italic>&#x394;t</italic>
</bold> is the time step of the finite element calculation; and <inline-formula id="inf15">
<mml:math id="m19">
<mml:mrow>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is a dimensionless parameter.</p>
<p>For first-order transmission (<bold>
<italic>N</italic>
</bold>&#x3d;1), under the condition that Eq. <xref ref-type="disp-formula" rid="e3">3</xref> is satisfied, Eq. <xref ref-type="disp-formula" rid="e1">1</xref> can be written as follows:<disp-formula id="e5">
<mml:math id="m20">
<mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">t</mml:mi>
<mml:mn mathvariant="bold">3</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi mathvariant="bold-italic">T</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
<label>(5)</label>
</disp-formula>
</p>
<p>By substituting Eq. <xref ref-type="disp-formula" rid="e4">4</xref> into Eq. <xref ref-type="disp-formula" rid="e5">5</xref>, the first-order MTF can be derived as<disp-formula id="e6">
<mml:math id="m21">
<mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold-italic">s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi mathvariant="bold-italic">s</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
<label>(6)</label>
</disp-formula>
</p>
</sec>
<sec id="s2-2">
<title>2.2 Analysis of the instability of high-frequency oscillations</title>
<p>The most intuitive explanation for high-frequency oscillation instability of MTF is the reflection amplification of high-frequency wave component in the artificial boundary. An amplification error wave is reflected to the artificial boundary in the finite calculation area and then amplified again, resulting in the instability of the artificial boundary. Such error wave amplification only occurs in high-frequency waves approaching the cut-off frequency. These high-frequency fluctuations causing oscillation instability are outside the scope of the frequency components considered in numerical simulation; and they do not benefit the computational stability of numerical simulation. In the numerical simulation, the high-frequency waves approaching the cut-off frequency have an insignificant effect on the accuracy of frequency bands. These high-frequency fluctuations exist perpendicular and parallel to the artificial boundary. Therefore, the elimination of useless high-frequency fluctuations in all directions can stabilize high-frequency oscillation without affecting the calculation accuracy.</p>
</sec>
<sec id="s2-3">
<title>2.3 Fundamental ideas of stabilization measures</title>
<p>In the meaningful frequency band of the finite element (or finite difference) simulation of the wave, the transmission boundary does not produce oscillation instability. Oscillation instability only occurs in the high-frequency band approaching the cut-off frequency. Therefore, the guiding principle of stabilization is to eliminate meaningless high-frequency components without affecting the low-frequency components meaningful for wave simulation.</p>
<p>In this paper, the proposed measure for suppressing oscillation instability is inspired by the concept of a multi-directional transmitting formula (<xref ref-type="bibr" rid="B13">Liao et al., 1993</xref>). The fundamental concept of the multi-direction transmitting formula is that the scattering wave from various directions radiates to the artificial boundary. This abandons the assumption that the scattering wave is based on a single direction and only uses the motion information of the node in the normal direction of the boundary. Instead, the transmission boundary formula is established using the motion information of all nodes adjacent to the artificial boundary node (including those on the artificial boundary and normal line).</p>
<p>The node position is shown in <xref ref-type="fig" rid="F2">Figure 2</xref> (I is the target node, and smooth filtering is performed using the nodes adjacent to point I on the boundary). When smoothing using three points, I, I &#x2212; 1, and I &#x2b; 1 are involved. When five points are used, I &#x2212; 2, I &#x2212; 1, I, I &#x2b; 1, and I&#x2b;2 are involved. In this regard, the following three considerations are emphasized.<list list-type="simple">
<list-item>
<p>1) Smoothing is performed after calculating the artificial boundary point at time P &#x2b; 1.</p>
</list-item>
<list-item>
<p>2) Three or five points are selected to be used in smoothing; all points use their P &#x2b; 1 values of time. For example, if the smoothing target point is I on the boundary, the participating points include point I on the boundary and the points adjacent to the boundary.</p>
</list-item>
<list-item>
<p>3) Smoothing is performed not only for displacement but also for the velocity values of the boundary point. This is implemented after calculating the displacement and velocity of the artificial boundary point at time P &#x2b; 1.</p>
</list-item>
</list>
</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Position of the boundary point involvedin filtering.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g002.tif"/>
</fig>
<p>After calculating the movement of the artificial boundary point at P &#x2b; 1 using MTF (Eq. <xref ref-type="disp-formula" rid="e1">1</xref>), the displacement and velocity values of the artificial boundary point I at P &#x2b; 1 are smoothed. For point I on the boundary shown in <xref ref-type="fig" rid="F2">Figure 2</xref>, three-point smoothing involves I &#x2212; 1, I, and I &#x2b; 1, and five-point smoothing involves I &#x2212; 2, I &#x2212; 1, I, I &#x2b; 1, and I &#x2b; 2. If three-point smoothing is used, the displacement and velocity can be calculated using Eqs <xref ref-type="disp-formula" rid="e7">7</xref>, <xref ref-type="disp-formula" rid="e9">9</xref>, respectively. If five-point smoothing is used, the displacement and velocity can be calculated using Eqs <xref ref-type="disp-formula" rid="e8">8</xref>, <xref ref-type="disp-formula" rid="e10">10</xref>, respectively. The displacement and velocity of point I after smoothing at P&#x2b;1 are <inline-formula id="inf16">
<mml:math id="m22">
<mml:mrow>
<mml:msubsup>
<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf17">
<mml:math id="m23">
<mml:mrow>
<mml:msubsup>
<mml:mover accent="true">
<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x2d9;</mml:mo>
</mml:mover>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula>, respectively. Coefficients &#x3b2;<sub>1</sub>, &#x3b2;<sub>2</sub>, and &#x3b2;<sub>3</sub> in Eqs <xref ref-type="disp-formula" rid="e7">7</xref>, <xref ref-type="disp-formula" rid="e9">9</xref> are three-point smoothing coefficients, and coefficients &#x3b2;<sub>1</sub>, &#x3b2;<sub>2</sub>, &#x3b2;<sub>3</sub>, &#x3b2;<sub>4</sub>, and &#x3b2;<sub>5</sub> in Eqs <xref ref-type="disp-formula" rid="e8">8</xref>, <xref ref-type="disp-formula" rid="e10">10</xref> are five-point smoothing coefficients. The values of the smoothing coefficients in Eqs <xref ref-type="disp-formula" rid="e7">7</xref>&#x2013;<xref ref-type="disp-formula" rid="e10">10</xref> are presented in <xref ref-type="sec" rid="s2-4">Section 2.4</xref> of this paper.<disp-formula id="e7">
<mml:math id="m24">
<mml:mrow>
<mml:msubsup>
<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mrow>
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<mml:mo>&#x2b;</mml:mo>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
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<mml:mn mathvariant="bold">1</mml:mn>
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<label>(7)</label>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
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<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">i</mml:mi>
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<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
<mml:mn mathvariant="bold">5</mml:mn>
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<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<label>(8)</label>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x2d9;</mml:mo>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x2b;</mml:mo>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
<mml:mn mathvariant="bold">3</mml:mn>
</mml:msub>
<mml:msubsup>
<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x2d9;</mml:mo>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:msubsup>
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<label>(9)</label>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x3d;</mml:mo>
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<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
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<mml:mover accent="true">
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<mml:mo>&#x2d9;</mml:mo>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x2b;</mml:mo>
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<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x2d9;</mml:mo>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
<mml:mn mathvariant="bold">3</mml:mn>
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<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x2d9;</mml:mo>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
<mml:mn mathvariant="bold">4</mml:mn>
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<mml:mover accent="true">
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<mml:mo>&#x2d9;</mml:mo>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x2b;</mml:mo>
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<mml:mi mathvariant="bold-italic">&#x3b2;</mml:mi>
<mml:mn mathvariant="bold">5</mml:mn>
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<mml:mover accent="true">
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mo>&#x2d9;</mml:mo>
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<mml:mi mathvariant="bold-italic">i</mml:mi>
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<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
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<label>(10)</label>
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</p>
</sec>
<sec id="s2-4">
<title>2.4 Derivation of smoothing formula coefficient</title>
<p>For the foregoing smoothing formula, the key problem is the means for determining the value of the smoothing coefficient. The values of the smoothing coefficients are discussed as follows.</p>
<p>The relationship between the wavelength that may cause high-frequency instability at the boundary point and the mesh size of the finite element calculation is simplified into four cases, as shown in <xref ref-type="fig" rid="F3">Figure 3</xref>.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Wavelength and mesh size. <bold>(A)</bold> Wavelength and mesh size of case <bold>(a)</bold>. <bold>(B)</bold> Wavelength and mesh size of case <bold>(b</bold>). <bold>(C)</bold> Wavelength and mesh size of case <bold>(c)</bold>. <bold>(D)</bold> Wavelength and mesh size of case <bold>(d)</bold>.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g003.tif"/>
</fig>
<p>The smoothing effect of the coefficients considering four wavelengths shown in the figure is evaluated. Consider three-point smoothing as an example. The following four situations are discussed:<list list-type="simple">
<list-item>
<p>1) For case (a), the amplitude at point 1 in <xref ref-type="fig" rid="F3">Figure 3A</xref> represents all points under the case. At point 1, the amplitudes before and after smoothing are &#x2212;1 and &#xbd; &#xd7; (&#x2212;1) &#x2b; &#xbc; &#xd7;1 &#x2b; &#xbc; &#xd7; 1 &#x3d; 0, respectively. The smoothed amplitude is found to be 0% of the original amplitude.</p>
</list-item>
<list-item>
<p>2) For case (b), the amplitudes at points 1 and 2 in <xref ref-type="fig" rid="F3">Figure 3B</xref> represent those at all points. The amplitudes before and after smoothing at point 1 are 0 and &#xbd; &#xd7; 0 &#x2b; &#xbc; &#xd7; (&#x2212;1) &#x2b; &#xbc; &#xd7; 1 &#x3d; 0, respectively. The smoothed amplitude is found to be 0% of the original amplitude. At point 2, the amplitudes before and after smoothing are &#x2212;1 and &#xbd; &#xd7; (&#x2212;1) &#x2b; &#xbc; &#xd7; 0 &#x2b; &#xbc; &#xd7; 0 &#x3d; &#x2212;&#xbd;, respectively. The smoothed amplitude is observed to be 50% of the original amplitude.</p>
</list-item>
<list-item>
<p>3) In case (c), the amplitudes at points 1 and 2 in <xref ref-type="fig" rid="F3">Figure 3C</xref> represent those at all points in the case. At point 1, the amplitudes before and after smoothing are -1/2 and &#xbd; &#xd7; (&#x2212;&#xbd;) &#x2b; &#xbc; &#xd7; &#xbd; &#x2b; &#xbc; &#xd7; (&#x2212;1) &#x3d; &#x2212;3/8, respectively. The smoothed amplitude is observed to be 75% of the original amplitude. At point 2, the amplitudes before and after smoothing are &#x2212;1 and &#xbd; &#xd7; (&#x2212;1) &#x2b; &#xbc;&#xd7; (&#x2212;&#xbd;) &#x2b; &#xbc; &#xd7; (&#x2212;&#xbd;) &#x3d; &#x2212;3/4, respectively. The smoothed amplitude is 75% of the original amplitude.</p>
</list-item>
<list-item>
<p>4) For case (d), the amplitudes at points 1, 2, and 3 in <xref ref-type="fig" rid="F3">Figure 3D</xref> represent those at all points. At point 1, the amplitudes before and after smoothing are 0 and &#xbd; &#xd7; 0 &#x2b; &#xbc; &#xd7; &#xbd; &#x2b; &#xbc; &#xd7; (&#x2212;&#xbd;) &#x3d; 0, respectively. The smoothed amplitude is 0% of the original amplitude. At point 2, the amplitudes before and after smoothing are &#x2212;1 and &#xbd; &#xd7; (&#x2212;1) &#x2b; &#xbc; &#xd7; (&#x2212;&#xbd;) &#x2b; &#xbc; &#xd7; (&#x2212;&#xbd;) &#x3d; &#x2212;3/4, respectively. The smoothed amplitude is observed to be 75% of the original amplitude. At point 3, the amplitudes before and after smoothing are &#x2212;&#xbd; and &#xbd; &#xd7; (&#x2212;&#xbd;) &#x2b; &#xbc;&#xd7; 0 &#x2b; &#xbc;&#xd7; (&#x2212;1) &#x3d; &#x2212;&#xbd;, respectively. The smoothed amplitude is 100% of the original amplitude.</p>
</list-item>
</list>
</p>
<p>
<xref ref-type="table" rid="T1">Table 1</xref> summarizes the smoothing values of using three coefficients in the four wavelength cases. The values in the table are amplitude percentages after smoothing relative to the original amplitude.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Smoothed amplitude percentage.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="3" colspan="2" align="center"/>
<th align="center">Three-point smoothing</th>
<th align="center">Five-point smoothing (1)</th>
<th align="center">Five-point smoothing (2)</th>
</tr>
<tr>
<th align="center">(1/2, 1/4,1/4)</th>
<th align="center">(1/3, 1/4, 1/4,1/12, 1/12)</th>
<th align="center">(1/2, 1/6, 1/6, 1/12, 1/12)</th>
</tr>
<tr>
<th align="center">(%)</th>
<th align="center">(%)</th>
<th align="center">(%)</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">case (a)</td>
<td align="center">Point 1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">30</td>
</tr>
<tr>
<td rowspan="2" align="center">case (b)</td>
<td align="center">Point 1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">Point 2</td>
<td align="center">50</td>
<td align="center">17</td>
<td align="center">30</td>
</tr>
<tr>
<td rowspan="2" align="center">case (c)</td>
<td align="center">Point 1</td>
<td align="center">75</td>
<td align="center">50</td>
<td align="center">50</td>
</tr>
<tr>
<td align="center">Point 2</td>
<td align="center">75</td>
<td align="center">50</td>
<td align="center">50</td>
</tr>
<tr>
<td rowspan="3" align="center">case (d)</td>
<td align="center">Point 1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">Point 2</td>
<td align="center">100</td>
<td align="center">83</td>
<td align="center">83</td>
</tr>
<tr>
<td align="center">Point 3</td>
<td align="center">100</td>
<td align="center">60</td>
<td align="center">70</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>With this filtering method, the amplitudes of the high-frequency and low-frequency waves are expected to decrease after smoothing. The foregoing eliminates meaningless high-frequency components without affecting the low-frequency part of the wave simulation. The percentage values corresponding to the calculation in this study after smoothing situations (a) and (b) are anticipated to be lower than those before smoothing. The percentage values after smoothing situations (c) and (d) must be higher than those before smoothing.</p>
<p>
<xref ref-type="table" rid="T1">Table 1</xref> indicates that the effect of values resulting from three-point smoothing is closest to that expected, followed by the effect of the five-point smoothing coefficient values (1/3, 1/4, 1/4, 1/12, and 1/12). The five-point smoothing coefficient values (1/2, 1/6, 1/6, 1/12, and 1/12) have the worst effect. Later, numerical tests are conducted to verify the effects.</p>
</sec>
</sec>
<sec id="s3">
<title>3 Modified formula of the MTF with stabilization measure proposed</title>
<p>First-order and three-point smoothing are considered as an example to discuss the MTF after smoothing. With point I on the boundary shown in <xref ref-type="fig" rid="F4">Figure 4</xref> as the target point, three points, I, J, and R, on the boundary are involved in smoothing point I. According to Eq. <xref ref-type="disp-formula" rid="e7">7</xref>, the motion expression of point I after smoothing at time P &#x2b; 1 is.<disp-formula id="e11">
<mml:math id="m28">
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<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">J</mml:mi>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
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<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">R</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
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<label>(11)</label>
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</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Discrete model of multi-transmitting boundary area.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g004.tif"/>
</fig>
<p>According to Eq. <xref ref-type="disp-formula" rid="e6">6</xref>, the motion expressions of I, J, and R at time P &#x2b; 1 are Eqs <xref ref-type="disp-formula" rid="e12">12</xref>&#x2013;<xref ref-type="disp-formula" rid="e14">14</xref>, respectively:<disp-formula id="e12">
<mml:math id="m29">
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<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">I</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
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<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mi mathvariant="bold-italic">u</mml:mi>
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<mml:mo>&#x2b;</mml:mo>
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<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">I</mml:mi>
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<mml:mi mathvariant="bold-italic">S</mml:mi>
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<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi mathvariant="bold-italic">S</mml:mi>
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<mml:mn mathvariant="bold">1</mml:mn>
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<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
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<mml:mi mathvariant="bold-italic">I</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
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<label>(12)</label>
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<mml:mi mathvariant="bold-italic">u</mml:mi>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
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<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mrow>
<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mi mathvariant="bold-italic">J</mml:mi>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
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<mml:mn mathvariant="bold">2</mml:mn>
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<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mrow>
<mml:mn mathvariant="bold">1</mml:mn>
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<label>(13)</label>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn mathvariant="bold">1</mml:mn>
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<mml:mi mathvariant="bold-italic">P</mml:mi>
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<label>(14)</label>
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<p>By substituting Eqs <xref ref-type="disp-formula" rid="e12">12</xref>&#x2013;<xref ref-type="disp-formula" rid="e14">14</xref> into Eq. <xref ref-type="disp-formula" rid="e11">11</xref>, the motion expression of point I after smoothing at time P &#x2b; 1 is derived as follows:<disp-formula id="e15">
<mml:math id="m32">
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</mml:msub>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">u</mml:mi>
<mml:mrow>
<mml:mi mathvariant="bold-italic">R</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn mathvariant="bold">2</mml:mn>
</mml:mrow>
<mml:mi mathvariant="bold-italic">P</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(15)</label>
</disp-formula>
</p>
<p>Eq. <xref ref-type="disp-formula" rid="e15">15</xref> can also be regarded as a multi-directional transmitting formula constructed using the information of all nodes (including I &#x2212; 1, I &#x2212; 2, J, J &#x2212; 1, J &#x2212; 2, R, R &#x2212; 1, and R &#x2212;2) around boundary node I, as shown in <xref ref-type="fig" rid="F5">Figure 5</xref>. Coefficients &#x3b2;<sub>1</sub>, &#x3b2;<sub>2</sub>, and &#x3b2;<sub>3</sub> in Eq.<xref ref-type="disp-formula" rid="e15">15</xref> can be considered as the share coefficients of node participation in transmission.</p>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Multi-directional transmitting boundary.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g005.tif"/>
</fig>
<p>Next, to verify the effectiveness of the proposed measure in suppressing high-frequency instability, numerical tests are conducted.</p>
</sec>
<sec id="s4">
<title>4 Numerical test</title>
<p>As an example, the wave propagation is simulated for a semi-infinite space model, as shown in <xref ref-type="fig" rid="F6">Figure 6</xref>. The coordinates of the observation points are as follows: point 1 (0&#xa0;m, 0&#xa0;m); point 2 (&#x2212;500&#xa0;m, 0&#xa0;m); point 3 (&#x2212;500&#xa0;m, &#x2212;500&#xa0;m); point 4 (&#x2212;500&#xa0;m, &#x2212;1,000&#xa0;m); point 5 (0&#xa0;m, &#x2212;1,000&#xa0;m); and point 6 (0&#xa0;m, &#x2212;500&#xa0;m). The input SH wave pulse&#x2013;time history is shown in <xref ref-type="fig" rid="F7">Figure 7</xref>. The incident angle is 0&#xb0;, and the wave velocity is 2000 m/s. The mesh size is &#x394;x &#x3d; 10&#xa0;m and &#x394;y &#x3d; 5&#xa0;m. The calculated time step is &#x394;t &#x3d; 0.0025&#xa0;s.</p>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Calculation model.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g006.tif"/>
</fig>
<fig id="F7" position="float">
<label>FIGURE 7</label>
<caption>
<p>Displacement pulse.</p>
</caption>
<graphic xlink:href="feart-10-1056583-g007.tif"/>
</fig>
<p>
<xref ref-type="fig" rid="F8">Figure 8</xref> shows the comparison results between implementing and not implementing the proposed measures for eliminating high-frequency instability. As shown in <xref ref-type="fig" rid="F8">Figure 8</xref>, the coefficients are as follows: in three-point smoothing, &#x3b2;<sub>1</sub> &#x3d; 1/2 and &#x3b2;<sub>2</sub> &#x3d; &#x3b2;<sub>3</sub> &#x3d; &#xbc;; in five-point smoothing (1), &#x3b2;<sub>1</sub> &#x3d; 1/3, &#x3b2;<sub>2</sub> &#x3d; &#x3b2;<sub>3</sub> &#x3d; &#xbc;, and &#x3b2;<sub>4</sub> &#x3d; &#x3b2;<sub>5</sub> &#x3d;1/12; and in five-point smoothing (2), &#x3b2;<sub>1</sub> &#x3d; 1/2, &#x3b2;<sub>2</sub> &#x3d; &#x3b2;<sub>3</sub> &#x3d; 1/6, and &#x3b2;<sub>4</sub> &#x3d; &#x3b2;<sub>5</sub> &#x3d;1/12.</p>
<fig id="F8" position="float">
<label>FIGURE 8</label>
<caption>
<p>Displacement time history, <bold>(A)</bold> Observation point 1 <bold>(B)</bold> Observation point 2, <bold>(C)</bold> Observation point 3 <bold>(D)</bold> Observation point 4, <bold>(E)</bold> Observation point 5 <bold>(F)</bold> Observation point 6</p>
</caption>
<graphic xlink:href="feart-10-1056583-g008.tif"/>
</fig>
<p>By analysing the results of the displacement&#x2013;time history comparison of observation points in <xref ref-type="fig" rid="F8">Figure 8</xref>, the following are deduced.<list list-type="simple">
<list-item>
<p>1) The processing method of adjacent nodes participating in filtering smoothing on the artificial boundary is effective for suppressing the instability of high-frequency oscillations.</p>
</list-item>
<list-item>
<p>2) The corresponding curve of the three-point smoothing measure does not exhibit high-frequency oscillations, indicating that the measure has a satisfactory effect on suppressing high-frequency instability.</p>
</list-item>
<list-item>
<p>3) The time history curve of the observation point obtained using the five-point smoothing measure exhibits slight oscillations. Between the two values yielded by five-point smoothing, the following coefficients is the worst: 1/2, 1/6, 1/6, 1/12, and 1/12. In <xref ref-type="fig" rid="F8">Figure 8B, C, E</xref>, the time history curves corresponding to the foregoing set of values have small high-frequency oscillations, indicating that this group of values cannot completely eliminate high-frequency instability.</p>
</list-item>
<list-item>
<p>4) In <xref ref-type="fig" rid="F8">Figure 8E, F</xref>, the curves corresponding to the two five-point smoothing measures have distinct abnormal fluctuations between 2 and 3&#xa0;s. No abnormal fluctuations are observed in the curves corresponding to those in which no measure for eliminating high-frequency oscillation is applied and the curves corresponding to the three-point smoothing measure. This shows that the abnormal fluctuation is caused by the disturbance from numerous low-frequency components introduced by the five-point smoothing method while filtering high-frequency components. The disturbance due to numerous low-frequency components causes abnormal fluctuations. This also demonstrates that the effect of the three-point smoothing measure is superior to that of the five-point smoothing one.</p>
</list-item>
</list>
</p>
<p>
<xref ref-type="table" rid="T2">Table 2</xref> lists the peak displacement&#x2013;time histories of each observation point shown in <xref ref-type="fig" rid="F8">Figure 8</xref>. The data in <xref ref-type="table" rid="T2">Table 2</xref> indicate that the peak value of point 4 significantly differs. The peak value error obtained by the three-point smoothing measure is only 0.83%, whereas the errors obtained by the five-point smoothing one are 4.5% and 5.6%. This further demonstrates that three-point smoothing measure is better than five-point smoothing one. By considering the results in <xref ref-type="fig" rid="F8">Figure 8</xref>; <xref ref-type="table" rid="T2">Table 2</xref>, the three-point smoothing measure is found to resolve the high-frequency instability, and the peak value of the observation point is least disturbed. This verifies the observation presented in <xref ref-type="sec" rid="s1">Section 1.4</xref>. In terms of practical implementation, three-point smoothing is simpler than five-point smoothing. Accordingly, the use of the three-point smoothing measure is recommended.</p>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Displacement peak of observation point.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left"/>
<th align="center">Point 1</th>
<th align="center">Point 2</th>
<th align="center">Point 3</th>
<th align="center">Point 4</th>
<th align="center">Point 5</th>
<th align="center">Point 6</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">No measures</td>
<td align="center">1.9999</td>
<td align="center">1.9999</td>
<td align="center">0.9999</td>
<td align="center">0.9999</td>
<td align="center">0.9999</td>
<td align="center">1.0000</td>
</tr>
<tr>
<td align="center">Three-point smoothing</td>
<td align="center">2.0000</td>
<td align="center">1.9974</td>
<td align="center">0.9991</td>
<td align="center">1.0083</td>
<td align="center">1.0000</td>
<td align="center">1.0001</td>
</tr>
<tr>
<td align="center">Five-point smoothing (1)</td>
<td align="center">2.0002</td>
<td align="center">1.9944</td>
<td align="center">0.9979</td>
<td align="center">1.0452</td>
<td align="center">1.0000</td>
<td align="center">1.0001</td>
</tr>
<tr>
<td align="center">Five-point smoothing (2)</td>
<td align="center">2.0002</td>
<td align="center">1.9937</td>
<td align="center">0.9974</td>
<td align="center">1.0565</td>
<td align="center">1.0000</td>
<td align="center">1.0001</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec sec-type="conclusion" id="s5">
<title>5 Conclusion</title>
<p>Inspired by the multi-directional transmitting formula, and considering the high-frequency wave oscillation in the vertical and parallel directions with the artificial boundary, a strategy for filtering and smoothing adjacent nodes on the artificial boundary is proposed in this paper to suppress the instability of high-frequency oscillations of the multi-transmitting boundary. A reasonable smoothing coefficient value was obtained, and the effectiveness of the measure was verified through numerical tests. The main findings of the study are summarized as follows.<list list-type="simple">
<list-item>
<p>1) The smoothing filtering strategy using the adjacent nodes of the artificial boundary is effective in suppressing the instability of high-frequency oscillations of the multi-transmitting boundary.</p>
</list-item>
<list-item>
<p>2) This paper presents three types of smoothing coefficient value combinations. Both three-point and five-point smoothing measures are effective in suppressing high-frequency instability of the multi-transmitting boundary; however, the three-point smoothing measure exhibits better performance. This is because low-frequency components are inevitably introduced when high-frequency components are filtered. Five-point smoothing measure introduces more low-frequency interference factors than three-point smoothing one. Consequently, excessive low-frequency disturbances cause the time history curve to fluctuate and affect calculation accuracy.</p>
</list-item>
<list-item>
<p>3) This study analyses the conceptual similarity between the smoothing of the motion calculated by the boundary point and multi-directional transmitting formulas. Hence, it provides a reference for establishing the coefficient value of the multi-directional transmitting formula in the time domain.</p>
</list-item>
</list>
</p>
</sec>
</body>
<back>
<sec sec-type="data-availability" id="s6">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec id="s7">
<title>Author contributions</title>
<p>YY is the main author of this paper. XL made an important contribution to the innovation of this paper. MR and ZY gave good suggestions in the completion of the paper.</p>
</sec>
<sec id="s8">
<title>Funding</title>
<p>This study is supported by the National Natural Science Foundation of China (U1839202) and National Key Research and Development Program (2019YFB1900900).</p>
</sec>
<sec sec-type="COI-statement" id="s9">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s10">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
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