基于球面波势函数基本解方法的弹性波三维散射与动应力求解
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摘要
针对弹性波三维散射和动应力集中问题,提出一种基于球面波势函数的基本解方法(SWP-MFS)。方法基于单层位势理论,采用膨胀波和矢量剪切波球面波势函数构造散射波场,根据边界条件建立边界积分方程并配点求解。精度检验表明,该方法具有良好的数值精度及稳定性。以无限空间中三维夹杂体及空洞对平面P、SV波的散射为例,进行方法展示,并揭示了三维夹杂体周围弹性波散射的一些重要规律。结果表明:三维夹杂体随其内部介质刚度降低,位移谱曲线震荡越加强烈。三维球形空洞在P波和SV波水平入射下应力集中规律不同,前者在顶部和底部及附近更明显,后者在纵截面两45°角线附近更明显。与以往的集中力源函数相比,新的波场构造基本解更为简洁易用,为三维弹性波动分析提供了一种新型无网格数值方法。
A new method of fundamental solution based on complete spherical wave potential(SWP-MFS) is proposed to solve 3-D elastic-wave scattering and dynamic stress concentration. The method established the boundary integral equation according to the boundary condition. The solution was solved by placing the spherical wave sources of compressional wave and shear wave on a virtual boundary based on the single layer potential theory. Through comparing with other available results, the excellent numerical accuracy and stability of the SWP-MFS are validated. The method was demonstrated in an example of the 3-D scattering of P or SV waves around an inclusion and a cavity in elastic full-space. Several important conclusions about scattering of 3-D elastic waves around an inclusion were obtained. As the modulus ratio decreases(the inclusion becomes softer), the displacement amplitude spectrums oscillate more rapidly with large amplitude. For horizontal incident P waves, it seems that the dynamic stress concentration effect is more pronounced near the top and bottom of the spherical cavity, while it is more obvious near the two 45-degree angles of the longitudinal section for horizontal incident SV waves. Compared with employing the Green's functions of concentrated force, the fundamental solution of SWP-MFS is more concise and easy to use, and provides a new meshless boundary-type method for elastic waves analysis.
A new method of fundamental solution based on complete spherical wave potential(SWP-MFS) is proposed to solve 3-D elastic-wave scattering and dynamic stress concentration. The method established the boundary integral equation according to the boundary condition. The solution was solved by placing the spherical wave sources of compressional wave and shear wave on a virtual boundary based on the single layer potential theory. Through comparing with other available results, the excellent numerical accuracy and stability of the SWP-MFS are validated. The method was demonstrated in an example of the 3-D scattering of P or SV waves around an inclusion and a cavity in elastic full-space. Several important conclusions about scattering of 3-D elastic waves around an inclusion were obtained. As the modulus ratio decreases(the inclusion becomes softer), the displacement amplitude spectrums oscillate more rapidly with large amplitude. For horizontal incident P waves, it seems that the dynamic stress concentration effect is more pronounced near the top and bottom of the spherical cavity, while it is more obvious near the two 45-degree angles of the longitudinal section for horizontal incident SV waves. Compared with employing the Green's functions of concentrated force, the fundamental solution of SWP-MFS is more concise and easy to use, and provides a new meshless boundary-type method for elastic waves analysis.
引文
[1]梁建文,魏新磊,Vincent W LEE.圆弧形沉积谷地对平面P波的三维散射解析解[J].岩土力学,2010,31(2):461―470.LIANG Jian-wen,WEI Xin-lei,VINCENT W LEE.3-Dscattering of plane P waves by a circular-arc alluvial valley[J].Rock and Soil Mechanics,2010,31(2):461―470.
[2]ZHANG N,Gao Y F,PAK R Y.Soil and topographic effects on ground motion of a surficially inhomogeneous semi-cylindrical canyon under oblique incident SHwaves[J].Soil Dynamics and Earthquake Engineering,2017,95:17―28.
[3]LIU Q J,ZHAO M J,WANG L H.Scattering of plane P,SV or Rayleigh waves by a shallow lined tunnel in an elastic half space[J].Soil Dynamics and Earthquake Engineering,2013,49(6):52―63.
[4]韩铮,赵成刚.半球形沉积谷场地对入射平面Rayleigh波的三维散射解析解[J].岩土力学,2007,28(12):2607―2613.HAN Zheng,ZHAO Cheng-gang.Analytical solution of three-dimensional scattering and diffraction of plane Rayleigh-waves by a hemispherical alluvial valley with saturated soil deposit[J].Rock and Soil Mechanics,2007,28(12):2607―2613.
[5]丁海平,于彦彦,郑志法.P波斜入射陡坎地形对地面运动的影响[J].岩土力学,2017,38(6):1716―1724.DING Hai-ping,DING Yan-yan,ZHENG Zhi-fa.Effects of scarp topography on seismic ground motion under inclined P waves[J].Rock and Soil Mechanics,2017,38(6):1716―1724.
[6]范留明.非均匀层状介质一维波动方程精确解的有限差分算法[J].岩土力学,2013,34(9):2715―2720.FAN Liu-ming.A new kind of finite difference scheme for exact solutions of one-dimensional wave equation in heterogeneous layer media[J].Rock and Soil Mechanics,2013,34(9):2715―2720.
[7]MORENCY C,LUO Y,TROMP J.Spectral-element simulations of wave propagation in porous media[J].Geophysical Journal of the Royal Astronomical Society,2010,175(1):301―345.
[8]刘中宪,武凤娇,王冬,等.弹性半空间夹杂群对平面SH波散射的快速多极多域边界元法模拟[J].岩土力学,2017,38(4):1154―1163.LIU Zhong-xian,WU Feng-jiao,WANG Dong,et al.Multi-domain FMM-IBEM simulation of plane SH wave scattering by inclusions in elastic half-space[J].Rock and Soil Mechanics,2017,38(4):1154―1163.
[9]赵成刚,杜修力,李小军.用三维半解析边界元的子结构法分析凸起山包对地震波的散射[J].中国地震,1993,9(2):154―162.ZHAO Cheng-gang,DU Xiu-li,LI Xiao-jun.Substructure method of semi-analysis boundary element and application to seismic scattering by three-dimensional topography of a hill[J].Earthquake Research in China,1993,9(2):154―162.
[10]KAWASE H,KEIITI A K.A study on the response of a soft basin for incident P,S,Rayleigh waves with special reference to the long duration observed in Mexico City[J].Bulletin of Seismological Society of America,1989,79(5):1361―1382.
[11]GUARINZAPATA N,GOMEZ J,JARAMILLO J.Seismic wave scattering through a compressed hybrid BEM/FEM method[J].Physics,2014(5):1―19.
[12]KUPRADZE V D,ALEKSIDZE M A.The method of functional equations for the approximate solution of certain boundary value problems[J].Computational Mathematics and Mathematical Physics,1964,4(4):82―126.
[13]CHEN J T,LEE Y T,YU S R,et al.Equivalence between the Trefftz method and the method of fundamental solution for the annular Green’s function using the addition theorem and image concept[J].Engineering Analysis with Boundary Elements,2009,33(5):678―688.
[14]WONG H L.Effect of surface topography on the diffraction of P,SV,and Rayleigh waves[J].Bulletin of the Seismological Society of America,1982,72(4):1167―1183.
[15]梁建文,胡淞淋,刘中宪,等.平面SV波入射下弹性半空间中三维球形洞室的动力响应[J].岩土工程学报,2016,38(9):1559―1568.LIANG Jian-wen,HU Song-lin,LIU Zhong-xian,et al.Dynamic response of 3D spherical cavity in elastic half-space under plane SV waves[J].Chinese Journal of Geotechnical Engineering,2016,38(9):1559―1568.
[16]LAMB H.On the Propagation of tremors over the surface of an elastic solid[J].Philosophical Transactions of the Royal Society of London,1904,203(359―371):1―42.
[17]MOON F C,PAO Y H.The influence of the curvature of spherical waves on dynamic stress concentration[J].Journal of Applied Mechanics,1967,34(2):373―379.
[18]KANAUN S,LEVIN V.Scattering of elastic waves on a heterogeneous inclusion of arbitrary shape:an efficient numerical method for 3D-problems[J].Wave Motion,2013,50(4):687―707.
[2]ZHANG N,Gao Y F,PAK R Y.Soil and topographic effects on ground motion of a surficially inhomogeneous semi-cylindrical canyon under oblique incident SHwaves[J].Soil Dynamics and Earthquake Engineering,2017,95:17―28.
[3]LIU Q J,ZHAO M J,WANG L H.Scattering of plane P,SV or Rayleigh waves by a shallow lined tunnel in an elastic half space[J].Soil Dynamics and Earthquake Engineering,2013,49(6):52―63.
[4]韩铮,赵成刚.半球形沉积谷场地对入射平面Rayleigh波的三维散射解析解[J].岩土力学,2007,28(12):2607―2613.HAN Zheng,ZHAO Cheng-gang.Analytical solution of three-dimensional scattering and diffraction of plane Rayleigh-waves by a hemispherical alluvial valley with saturated soil deposit[J].Rock and Soil Mechanics,2007,28(12):2607―2613.
[5]丁海平,于彦彦,郑志法.P波斜入射陡坎地形对地面运动的影响[J].岩土力学,2017,38(6):1716―1724.DING Hai-ping,DING Yan-yan,ZHENG Zhi-fa.Effects of scarp topography on seismic ground motion under inclined P waves[J].Rock and Soil Mechanics,2017,38(6):1716―1724.
[6]范留明.非均匀层状介质一维波动方程精确解的有限差分算法[J].岩土力学,2013,34(9):2715―2720.FAN Liu-ming.A new kind of finite difference scheme for exact solutions of one-dimensional wave equation in heterogeneous layer media[J].Rock and Soil Mechanics,2013,34(9):2715―2720.
[7]MORENCY C,LUO Y,TROMP J.Spectral-element simulations of wave propagation in porous media[J].Geophysical Journal of the Royal Astronomical Society,2010,175(1):301―345.
[8]刘中宪,武凤娇,王冬,等.弹性半空间夹杂群对平面SH波散射的快速多极多域边界元法模拟[J].岩土力学,2017,38(4):1154―1163.LIU Zhong-xian,WU Feng-jiao,WANG Dong,et al.Multi-domain FMM-IBEM simulation of plane SH wave scattering by inclusions in elastic half-space[J].Rock and Soil Mechanics,2017,38(4):1154―1163.
[9]赵成刚,杜修力,李小军.用三维半解析边界元的子结构法分析凸起山包对地震波的散射[J].中国地震,1993,9(2):154―162.ZHAO Cheng-gang,DU Xiu-li,LI Xiao-jun.Substructure method of semi-analysis boundary element and application to seismic scattering by three-dimensional topography of a hill[J].Earthquake Research in China,1993,9(2):154―162.
[10]KAWASE H,KEIITI A K.A study on the response of a soft basin for incident P,S,Rayleigh waves with special reference to the long duration observed in Mexico City[J].Bulletin of Seismological Society of America,1989,79(5):1361―1382.
[11]GUARINZAPATA N,GOMEZ J,JARAMILLO J.Seismic wave scattering through a compressed hybrid BEM/FEM method[J].Physics,2014(5):1―19.
[12]KUPRADZE V D,ALEKSIDZE M A.The method of functional equations for the approximate solution of certain boundary value problems[J].Computational Mathematics and Mathematical Physics,1964,4(4):82―126.
[13]CHEN J T,LEE Y T,YU S R,et al.Equivalence between the Trefftz method and the method of fundamental solution for the annular Green’s function using the addition theorem and image concept[J].Engineering Analysis with Boundary Elements,2009,33(5):678―688.
[14]WONG H L.Effect of surface topography on the diffraction of P,SV,and Rayleigh waves[J].Bulletin of the Seismological Society of America,1982,72(4):1167―1183.
[15]梁建文,胡淞淋,刘中宪,等.平面SV波入射下弹性半空间中三维球形洞室的动力响应[J].岩土工程学报,2016,38(9):1559―1568.LIANG Jian-wen,HU Song-lin,LIU Zhong-xian,et al.Dynamic response of 3D spherical cavity in elastic half-space under plane SV waves[J].Chinese Journal of Geotechnical Engineering,2016,38(9):1559―1568.
[16]LAMB H.On the Propagation of tremors over the surface of an elastic solid[J].Philosophical Transactions of the Royal Society of London,1904,203(359―371):1―42.
[17]MOON F C,PAO Y H.The influence of the curvature of spherical waves on dynamic stress concentration[J].Journal of Applied Mechanics,1967,34(2):373―379.
[18]KANAUN S,LEVIN V.Scattering of elastic waves on a heterogeneous inclusion of arbitrary shape:an efficient numerical method for 3D-problems[J].Wave Motion,2013,50(4):687―707.