基于高阶双渐近透射边界的重力坝-层状地基动力相互作用分析
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摘要
时域高阶双渐近透射边界能够同时模拟层状介质中行波和快衰波的传播,具有很高的计算精度和计算效率.本文将高阶双渐近透射边界推广应用到多层层状地基系统弹性波传播问题的模拟,采用广义特征值分解分析该透射边界的数值稳定性,通过移谱法消除导致数值不稳定的虚假模态.将高阶双渐近透射边界以超单元的形式直接嵌入到近场有限元方程,建立了有限元-高阶双渐近透射边界时域耦合分析模型,并将其应用到重力坝-层状地基动力相互作用分析.数值算例分析结果表明,该时域耦合分析模型具有很高的精度和计算效率,适用于实际重力坝工程的地震响应分析.
The high-order doubly asymptotic open boundary in the time domain is capable of accurately modeling traveling and evanescent waves in layered medium with high computational accuracy and efficiency.In this paper,the high-order doubly asymptotic open boundary is extended to the analysis of elastic wave propagation in a semi-infinite multiple layered foundation systems.The numerical stability of this open boundary is analyzed by the generalized eigenvalue decomposition.Possible spurious modes are eliminated by using the spectral shifting technique.Embedding the high-order doubly asymptotic open boundary as a super element into the near-field finite element equation,a time-domain coupled analysis model is established.This model is applied to the analysis of dynamic gravity dam-layered foundation interaction.Numerical results demonstrate that this proposed model is of high accuracy and efficiency and applicable to seismic response analysis of real gravity dams.
The high-order doubly asymptotic open boundary in the time domain is capable of accurately modeling traveling and evanescent waves in layered medium with high computational accuracy and efficiency.In this paper,the high-order doubly asymptotic open boundary is extended to the analysis of elastic wave propagation in a semi-infinite multiple layered foundation systems.The numerical stability of this open boundary is analyzed by the generalized eigenvalue decomposition.Possible spurious modes are eliminated by using the spectral shifting technique.Embedding the high-order doubly asymptotic open boundary as a super element into the near-field finite element equation,a time-domain coupled analysis model is established.This model is applied to the analysis of dynamic gravity dam-layered foundation interaction.Numerical results demonstrate that this proposed model is of high accuracy and efficiency and applicable to seismic response analysis of real gravity dams.
引文
Bazyar M H,Song C M.2008.A continued-fraction-based highorder transmitting boundary for wave propagation in unbounded domains of arbitrary geometry.International Journal for Numerical Methods in Engineering,74(2):209-237.
Birk C,Behnke R.2012.A modified scaled boundary finite element method for three-dimensional dynamic soil-structure interaction in layered soil.International Journal for Numerical Methods in Engineering,89(3):371-402.
Birk C,Prempramote S,Song C M.2012.An improved continuedfraction-based high-order transmitting boundary for time-domain analyses in unbounded domains.International Journal for Numerical Methods in Engineering,89(3):269-298.
Birk C,Song C.2009.A continued-fraction approach for transient diffusion in unbounded medium.ComputerMethodsinApplied MechanicsandEngineering,198(33-36):2576-2590.
Chen J Y,Bording R P,Liu E R,et al.2010.The application of the nearly optimal sponge boundary conditions for seismic wave propagation in poroelastic media.Journal of Seismic Exploration,19(1):1-9.
Chen X J,Birk C,Song C M.2015.Transient analysis of wave propagation in layered soil by using the scaled boundary finite element method.Computers and Geotechnics,63:1-12.
Feng D S,Chen C S,Wang H H.2012.Finite element method GPR forward simulation based on mixed boundary condition.Chinese Journal of Geophysics(in Chinese),55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
Feng D S,Wang X.2017.Convolution perfectly matched layer for the Finite-Element Time thod modeling of Ground Penetrating Radar.Chinese Journal of Geophysics(in Chinese),60(1):413-423,doi:10.6038/cjg20170134.
Gao Y C,Xu Y J,Jin F,et al.2013.The direct coupled model for dam-reservoir dynamic interaction analysis based on high-order doubly asymptotic open boundary.Chinese Journal of Geophysics(in Chinese),56(12):4189-4196,doi:10.6038/cjg20131221.
Givoli D.1991.Non-reflecting boundary conditions.Journal of Computational Physics,94(1):1-29.
Givoli D.2004.High-order local non-reflecting boundary conditions:a review.Wave Motion,39(4):319-326.
Han Z J,Lin G.2012.A continued-fraction algorithm for dynamic stiffness matrix of foundation located or embedded in threedimensional layered subgrade.Journal of Dalian University of Technology(in Chinese),52(6):862-869.
Hu S Z,Fu L Y,Pei Z L.2009.A boundary element method for the 2-D wave equation in fluid-saturated porous media.Chinese Journal of Geophysics(in Chinese),52(9):2364-2369,doi:10.3969/j.issn.0001-5733.2009.09.022.
Kausel E,Peek R.1982.Dynamic loads in the interior of a layered stratum:an explicit solution.Bulletin of the Seismological Society of America,72(5):1459-1481.
Lan H Q,Liu J,Bai Z M.2011.Wave-field simulation in VTI media with irregular free surface.Chinese Journal of Geophysics(in Chinese),54(8):2072-2084,doi:10.3969/j.issn.0001-5733.2011.08.014.
Lan H Q,Zhang Z J.2011.Three-dimensional wave-field simulation in heterogeneous transversely isotropic medium with irregular free surface.Bulletin of the Seismological Society of America,101(3):1354-1370.
Li W H,Liu Q H,Zhao C G.2010.Three-dimensional viscousspring boundaries in time domain and dynamic analysis using explicit finite element method of saturated porous medium.Chinese Journal of Geophysics(in Chinese),53(10):2460-2469,doi:10.3969/j.issn.0001-5733.2010.10.020.
Liao Z P,Zhou Z H,Zhang Y H.2002.Stable implementation of transmitting boundary in numerical simulation of wave motion.Chinese Journal of Geophysics(in Chinese),45(4):533-545.
Lin G,Hu Z Q.2005.Earthquake safety assessment of concrete arch and gravity dams.Earthquake Engineering and Engineering Vibration,4(2):251-264.
Long Y C,Zhang C H,Chi F D,et al.2008.Study of steel reinforcement effects on concrete gravity dams under earthquake.Journal of Hydroelectric Engineering(in Chinese),27(4):77-82.
Lu S,Liu J,Lin G.2016.A time domain solution for complex multilayered soil model with circular inhomogeneity by the SBFEM.Computers&Mathematics with Applications,71(2):652-675.
Padrón L A,Aznárez J J,Maeso O.2008.Dynamic analysis of piled foundations in stratified soils by a BEM-FEM model.Soil Dynamics and Earthquake Engineering,28(5):333-346.
Park J,Kausel E.2004.Numerical dispersion in the thin-layer method.Computers&Structures,82(7-8):607-625.
Prempramote S.2011.Development of high-order doubly asymptotic open boundaries for wave propagation in unbounded domains by extending the scaled boundary finite element[Ph.D.thesis].Sydney:School of Civil and Environmental Engineering,University of New South Wales.
Prempramote S,Song C M,Tin-Loi F,et al.2009.High-order doubly asymptotic open boundaries for scalar wave equation.International Journal for Numerical Methods in Engineering,79(3):340-374.
Shao X M,Lan Z L.1995.Absorbing boundary conditions for anisotropic elastic wave equations.Chinese Journal of Geophysics(in Chinese),38(S1):56-73.
Song C M,Wolf J P.1997.The scaled boundary finite-element method--alias consistent infinitesimal finite-element cell method--for elastodynamics.Computer Methods in Applied Mechanics and Engineering,147(3-4):329-355.
Song C M,Wolf J P.2000.The scaled boundary finite-element method-aprimer:solution procedures.Computers&Structures,78(1-3):211-225.
Spyrakos C C,Xu C J.2004.Dynamic analysis of flexible massive strip-foundations embedded in layered soils by hybrid BEM-FEM.Computers&Structures,82(29-30):2541-2550.
Wang X,Jin F,Prempramote S,et al.2011.Time-domain analysis of gravity dam-reservoir interaction using high-order doubly asymptotic open boundary.Computers&Structures,89(7-8):668-680.
Xu S Z.1994.The Finite Element Method in Geophysics(in Chinese).Beijing:Science Press.
Xu Y J,Mu H L,Zhang C H,et al.2012.3Dfinite element modeling of seismic responses of Baozhusi gravity dam in MS8.0Wenchuan Earthquake.Chinese Journal of Geophysics(in Chinese),55(1):293-303,doi:10.6038/j.issn.0001-5733.2012.01.029.
Yang D H,Wang S Q,Zhang Z J,et al.2003.n-Times absorbing boundary conditions for compact finite-difference modeling of acoustic and elastic wave propagation in the 2D TI medium.Bulletin of the Seismological Society of America,93(6):2389-2401.
Yuan J W,Du C B,Liu Z M.2011.Time-domain seismic response for gravity dam to obliquely incident and seismic waves.Journal of Vibration and Shock(in Chinese),30(7):120-126.
Zhang B Q,Zhou H,Chen H M,et al.2016.Time-space domain high-order finite-difference methods for seismic wave numerical simulation based on new stencils.Chinese Journal of Geophysics(in Chinese),59(5):1804-1814,doi:10.6038/cjg20160523.
冯德山,陈承申,王洪华.2012.基于混合边界条件的有限单元法GPR正演模拟.地球物理学报,55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
冯德山,王珣.2017.基于卷积完全匹配层的非规则网格时域有限元探地雷达数值模拟.地球物理学报,60(1):413-423,doi:10.6038/cjg20170134.
高毅超,徐艳杰,金峰等.2013.基于高阶双渐近透射边界的大坝-库水动力相互作用直接耦合分析模型.地球物理学报,56(12):4189-4196,doi:10.6038/cjg20131221.
韩泽军,林皋.2012.三维层状地基动力刚度矩阵连分式算法.大连理工大学学报,52(6):862-869.
胡善政,符力耘,裴正林.2009.流体饱和多孔隙介质弹性波方程边界元解法研究.地球物理学报,52(9):2364-2369,doi:10.3969/j.issn.0001-5733.2009.09.022.
兰海强,刘佳,白志明.2011.VTI介质起伏地表地震波场模拟.地球物理学报,54(8):2072-2084,doi:10.3969/j.issn.0001-5733.2011.08.014.
李伟华,刘清华,赵成刚.2010.饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法.地球物理学报,53(10):2460-2469,doi:10.3969/j.issn.0001-5733.2010.10.020.
廖振鹏,周正华,张艳红.2002.波动数值模拟中透射边界的稳定实现.地球物理学报,45(4):533-545.
龙渝川,张楚汉,迟福东等.2008.混凝土重力坝抗震配筋加固措施的效果研究.水力发电学报,27(4):77-82.
邵秀民,蓝志凌.1995.各向异性弹性介质中波动方程的吸收边界条件.地球物理学报,38(S1):56-73.
徐世浙.1994.地球物理中的有限单元法.北京:科学出版社.
徐艳杰,牟海磊,张楚汉等.2012.汶川地震中宝珠寺重力坝地震响应的三维有限元模拟.地球物理学报,55(1):293-303,doi:10.6038/j.issn.0001-5733.2012.01.029.
苑举卫,杜成斌,刘志明.2011.地震波斜入射条件下重力坝动力响应分析.振动与冲击,30(7):120-126.
张保庆,周辉,陈汉明等.2016.基于新的差分结构的时-空域高阶有限差分波动方程数值模拟方法.地球物理学报,59(5):1804-1814,doi:10.6038/cjg20160523.
Birk C,Behnke R.2012.A modified scaled boundary finite element method for three-dimensional dynamic soil-structure interaction in layered soil.International Journal for Numerical Methods in Engineering,89(3):371-402.
Birk C,Prempramote S,Song C M.2012.An improved continuedfraction-based high-order transmitting boundary for time-domain analyses in unbounded domains.International Journal for Numerical Methods in Engineering,89(3):269-298.
Birk C,Song C.2009.A continued-fraction approach for transient diffusion in unbounded medium.ComputerMethodsinApplied MechanicsandEngineering,198(33-36):2576-2590.
Chen J Y,Bording R P,Liu E R,et al.2010.The application of the nearly optimal sponge boundary conditions for seismic wave propagation in poroelastic media.Journal of Seismic Exploration,19(1):1-9.
Chen X J,Birk C,Song C M.2015.Transient analysis of wave propagation in layered soil by using the scaled boundary finite element method.Computers and Geotechnics,63:1-12.
Feng D S,Chen C S,Wang H H.2012.Finite element method GPR forward simulation based on mixed boundary condition.Chinese Journal of Geophysics(in Chinese),55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
Feng D S,Wang X.2017.Convolution perfectly matched layer for the Finite-Element Time thod modeling of Ground Penetrating Radar.Chinese Journal of Geophysics(in Chinese),60(1):413-423,doi:10.6038/cjg20170134.
Gao Y C,Xu Y J,Jin F,et al.2013.The direct coupled model for dam-reservoir dynamic interaction analysis based on high-order doubly asymptotic open boundary.Chinese Journal of Geophysics(in Chinese),56(12):4189-4196,doi:10.6038/cjg20131221.
Givoli D.1991.Non-reflecting boundary conditions.Journal of Computational Physics,94(1):1-29.
Givoli D.2004.High-order local non-reflecting boundary conditions:a review.Wave Motion,39(4):319-326.
Han Z J,Lin G.2012.A continued-fraction algorithm for dynamic stiffness matrix of foundation located or embedded in threedimensional layered subgrade.Journal of Dalian University of Technology(in Chinese),52(6):862-869.
Hu S Z,Fu L Y,Pei Z L.2009.A boundary element method for the 2-D wave equation in fluid-saturated porous media.Chinese Journal of Geophysics(in Chinese),52(9):2364-2369,doi:10.3969/j.issn.0001-5733.2009.09.022.
Kausel E,Peek R.1982.Dynamic loads in the interior of a layered stratum:an explicit solution.Bulletin of the Seismological Society of America,72(5):1459-1481.
Lan H Q,Liu J,Bai Z M.2011.Wave-field simulation in VTI media with irregular free surface.Chinese Journal of Geophysics(in Chinese),54(8):2072-2084,doi:10.3969/j.issn.0001-5733.2011.08.014.
Lan H Q,Zhang Z J.2011.Three-dimensional wave-field simulation in heterogeneous transversely isotropic medium with irregular free surface.Bulletin of the Seismological Society of America,101(3):1354-1370.
Li W H,Liu Q H,Zhao C G.2010.Three-dimensional viscousspring boundaries in time domain and dynamic analysis using explicit finite element method of saturated porous medium.Chinese Journal of Geophysics(in Chinese),53(10):2460-2469,doi:10.3969/j.issn.0001-5733.2010.10.020.
Liao Z P,Zhou Z H,Zhang Y H.2002.Stable implementation of transmitting boundary in numerical simulation of wave motion.Chinese Journal of Geophysics(in Chinese),45(4):533-545.
Lin G,Hu Z Q.2005.Earthquake safety assessment of concrete arch and gravity dams.Earthquake Engineering and Engineering Vibration,4(2):251-264.
Long Y C,Zhang C H,Chi F D,et al.2008.Study of steel reinforcement effects on concrete gravity dams under earthquake.Journal of Hydroelectric Engineering(in Chinese),27(4):77-82.
Lu S,Liu J,Lin G.2016.A time domain solution for complex multilayered soil model with circular inhomogeneity by the SBFEM.Computers&Mathematics with Applications,71(2):652-675.
Padrón L A,Aznárez J J,Maeso O.2008.Dynamic analysis of piled foundations in stratified soils by a BEM-FEM model.Soil Dynamics and Earthquake Engineering,28(5):333-346.
Park J,Kausel E.2004.Numerical dispersion in the thin-layer method.Computers&Structures,82(7-8):607-625.
Prempramote S.2011.Development of high-order doubly asymptotic open boundaries for wave propagation in unbounded domains by extending the scaled boundary finite element[Ph.D.thesis].Sydney:School of Civil and Environmental Engineering,University of New South Wales.
Prempramote S,Song C M,Tin-Loi F,et al.2009.High-order doubly asymptotic open boundaries for scalar wave equation.International Journal for Numerical Methods in Engineering,79(3):340-374.
Shao X M,Lan Z L.1995.Absorbing boundary conditions for anisotropic elastic wave equations.Chinese Journal of Geophysics(in Chinese),38(S1):56-73.
Song C M,Wolf J P.1997.The scaled boundary finite-element method--alias consistent infinitesimal finite-element cell method--for elastodynamics.Computer Methods in Applied Mechanics and Engineering,147(3-4):329-355.
Song C M,Wolf J P.2000.The scaled boundary finite-element method-aprimer:solution procedures.Computers&Structures,78(1-3):211-225.
Spyrakos C C,Xu C J.2004.Dynamic analysis of flexible massive strip-foundations embedded in layered soils by hybrid BEM-FEM.Computers&Structures,82(29-30):2541-2550.
Wang X,Jin F,Prempramote S,et al.2011.Time-domain analysis of gravity dam-reservoir interaction using high-order doubly asymptotic open boundary.Computers&Structures,89(7-8):668-680.
Xu S Z.1994.The Finite Element Method in Geophysics(in Chinese).Beijing:Science Press.
Xu Y J,Mu H L,Zhang C H,et al.2012.3Dfinite element modeling of seismic responses of Baozhusi gravity dam in MS8.0Wenchuan Earthquake.Chinese Journal of Geophysics(in Chinese),55(1):293-303,doi:10.6038/j.issn.0001-5733.2012.01.029.
Yang D H,Wang S Q,Zhang Z J,et al.2003.n-Times absorbing boundary conditions for compact finite-difference modeling of acoustic and elastic wave propagation in the 2D TI medium.Bulletin of the Seismological Society of America,93(6):2389-2401.
Yuan J W,Du C B,Liu Z M.2011.Time-domain seismic response for gravity dam to obliquely incident and seismic waves.Journal of Vibration and Shock(in Chinese),30(7):120-126.
Zhang B Q,Zhou H,Chen H M,et al.2016.Time-space domain high-order finite-difference methods for seismic wave numerical simulation based on new stencils.Chinese Journal of Geophysics(in Chinese),59(5):1804-1814,doi:10.6038/cjg20160523.
冯德山,陈承申,王洪华.2012.基于混合边界条件的有限单元法GPR正演模拟.地球物理学报,55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
冯德山,王珣.2017.基于卷积完全匹配层的非规则网格时域有限元探地雷达数值模拟.地球物理学报,60(1):413-423,doi:10.6038/cjg20170134.
高毅超,徐艳杰,金峰等.2013.基于高阶双渐近透射边界的大坝-库水动力相互作用直接耦合分析模型.地球物理学报,56(12):4189-4196,doi:10.6038/cjg20131221.
韩泽军,林皋.2012.三维层状地基动力刚度矩阵连分式算法.大连理工大学学报,52(6):862-869.
胡善政,符力耘,裴正林.2009.流体饱和多孔隙介质弹性波方程边界元解法研究.地球物理学报,52(9):2364-2369,doi:10.3969/j.issn.0001-5733.2009.09.022.
兰海强,刘佳,白志明.2011.VTI介质起伏地表地震波场模拟.地球物理学报,54(8):2072-2084,doi:10.3969/j.issn.0001-5733.2011.08.014.
李伟华,刘清华,赵成刚.2010.饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法.地球物理学报,53(10):2460-2469,doi:10.3969/j.issn.0001-5733.2010.10.020.
廖振鹏,周正华,张艳红.2002.波动数值模拟中透射边界的稳定实现.地球物理学报,45(4):533-545.
龙渝川,张楚汉,迟福东等.2008.混凝土重力坝抗震配筋加固措施的效果研究.水力发电学报,27(4):77-82.
邵秀民,蓝志凌.1995.各向异性弹性介质中波动方程的吸收边界条件.地球物理学报,38(S1):56-73.
徐世浙.1994.地球物理中的有限单元法.北京:科学出版社.
徐艳杰,牟海磊,张楚汉等.2012.汶川地震中宝珠寺重力坝地震响应的三维有限元模拟.地球物理学报,55(1):293-303,doi:10.6038/j.issn.0001-5733.2012.01.029.
苑举卫,杜成斌,刘志明.2011.地震波斜入射条件下重力坝动力响应分析.振动与冲击,30(7):120-126.
张保庆,周辉,陈汉明等.2016.基于新的差分结构的时-空域高阶有限差分波动方程数值模拟方法.地球物理学报,59(5):1804-1814,doi:10.6038/cjg20160523.