高阶双渐近透射边界研究及应用
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摘要
基于比例边界有限单元法发展的高阶透射边界是具有良好计算性能的无限域数值模拟方法。本文从比例边界有限单元法的控制方程出发，致力于发展可用于标量波问题和矢量波问题模拟的高阶双渐近透射边界，并同有限单元法结合，在应用方面进行了探索和研究。论文的工作主要包括以下几个方面：
1.基于开源软件框架OpenSees数值实现了动水压力波高阶双渐近透射边界与有限单元法的直接耦合分析模型和分离耦合分析模型，并用于大坝-库水动力相互作用分析。数值分析结果表明两种耦合分析模型均具有很高的计算精度和计算效率，在实际应用中可以选取较少的透射边界阶数、减少近场有限单元离散的范围，进一步提高求解效率。
2.通过引入简单的反射系数考虑库底柔性，基于比例边界坐标变换和伽辽金加权余量法推导了三维层状库水的比例边界有限单元方程，并构建了动力刚度的改进连分式双渐近解。通过引入辅助变量发展了可以考虑库底柔性的高阶双渐近透射边界，并将其嵌入到近场有限单元方程中，用于大坝-库水动力相互作用分析。数值算例表明，该透射边界具有很高的计算精度和计算效率。
3.针对二维半无限水平层状介质，将弹性动力学方程强行解耦得到两个与标量波方程形式相同的波动方程，发展了一种简化的模拟层状介质矢量波的高阶双渐近透射边界。数值算例表明，该高阶透射边界对于底部为固定边界条件的水平层状介质具有良好的计算精度；它具有与标量波高阶双渐近透射边界相同的计算效率以及良好的数值稳定性。
4.从弹性动力学的比例边界有限单元方程出发推导了动力刚度的改进连分式双渐近解，发展了高阶双渐近透射边界。为了获得准确的计算结果，论文利用基函数的性质，将比例边界有限单元方程拆分成高阶模态方程和低阶模态方程两部分，然后分别采用连分式双渐近解和高频渐近解求解。相比于高频连分式单向渐近解，该方法在低频段能够更快地收敛到准确解，因此在应用中可以选取较少的渐近阶数，从而提高计算效率。
The high-order transmitting boundary based on the scaled boundary finite elementmethod is of excellent computational performance. The objective of this paper is todevelop high-order doubly asymptotic transmitting boundaries for modeling theunbounded domains. These high-order transmitting boundaries are applicable to bothscalar and vector waves and can be coupled seamlessly with finite element method. Themain contents are summarized as follows:
1. Two coupled numerical methods for dam-reservoir interaction analysis aredeveloped by incorporating the excellent high-order doubly asymptotic transmittingboundary and the finite element method, namely the direct coupled method and thepartitioned coupled method. These two coupled methods are numerically implementedin the open-source finite element code OpenSees. Numerical experiments demonstratedthe high efficiency and accuracy of both coupled methods. As for the high accuracy ofthe high-order transmitting boundary, its order and the region discretized by finiteelement can be reduced in the analysis.
2. The reservoir absorption effect is taken into consideration by introducing thereflection coefficient. Based on the scale boundary transformation of geometry and theGalerkin's weighted residual technique, the scaled boundary finite element (SBFE)equation for three-dimensional layered reservoir is derived. Then, the improved doublyasymptotic continued fraction solution of the dynamic stiffness is constructed. Thecoefficient matrices are determined recursively from the SBFE equation in dynamicstiffness. By introducing the auxiliary variables, a high-order doubly asymptotictransmitting boundary is developed for modeling the unbounded reservoir. Thishigh-order transmitting boundary is incorporated with finite element method to analyzethe dam-reservoir interaction. Numerical examples demonstrate the high accuracy andefficiency of the proposed method.
3. To simply the analysis of the two-dimensional semi-infinite layer with constantdepth, the governing elastic wave equation is decoupled into two scalar wave equationsby neglecting the coupling terms. As a result, the high-order doubly asymptotictransmitting open boundary developed for modeling the scalar wave can be applied tomodel the propagation of elastic waves in the semi-infinite layer. Numerical examples show that this method is applicable to the semi-infinite layer with fixed boundarycondition at the bottom. Besides, this method is of high computational efficiency andnumerical stability.
4. A high-order doubly asymptotic transmitting boundary is developed to modelthe propagation of elastic waves in unbounded domains. This transmitting boundary isconstructed based on the improved doubly asymptotic continued fraction solution of thedynamic stiffness. To achieve accurate results, a set of weighted block-orthogonal basefunctions is introduced. The SBFE equation is split into a high-order mode equation anda low-order mode equation. The former is approximated by the doubly asymptoticcontinued fraction solution and the latter by the high-frequency singly asymptoticcontinued fraction solution. Compared to the high-order transmitting boundary based onthe high-frequency singly asymptotic continued fraction solution, the proposed methodhas a faster convergence speed in the low-frequency range. Thus, the order oftransmitting boundary can be reduced in the analysis to achieve higher computationalefficiency.
1.基于开源软件框架OpenSees数值实现了动水压力波高阶双渐近透射边界与有限单元法的直接耦合分析模型和分离耦合分析模型，并用于大坝-库水动力相互作用分析。数值分析结果表明两种耦合分析模型均具有很高的计算精度和计算效率，在实际应用中可以选取较少的透射边界阶数、减少近场有限单元离散的范围，进一步提高求解效率。
2.通过引入简单的反射系数考虑库底柔性，基于比例边界坐标变换和伽辽金加权余量法推导了三维层状库水的比例边界有限单元方程，并构建了动力刚度的改进连分式双渐近解。通过引入辅助变量发展了可以考虑库底柔性的高阶双渐近透射边界，并将其嵌入到近场有限单元方程中，用于大坝-库水动力相互作用分析。数值算例表明，该透射边界具有很高的计算精度和计算效率。
3.针对二维半无限水平层状介质，将弹性动力学方程强行解耦得到两个与标量波方程形式相同的波动方程，发展了一种简化的模拟层状介质矢量波的高阶双渐近透射边界。数值算例表明，该高阶透射边界对于底部为固定边界条件的水平层状介质具有良好的计算精度；它具有与标量波高阶双渐近透射边界相同的计算效率以及良好的数值稳定性。
4.从弹性动力学的比例边界有限单元方程出发推导了动力刚度的改进连分式双渐近解，发展了高阶双渐近透射边界。为了获得准确的计算结果，论文利用基函数的性质，将比例边界有限单元方程拆分成高阶模态方程和低阶模态方程两部分，然后分别采用连分式双渐近解和高频渐近解求解。相比于高频连分式单向渐近解，该方法在低频段能够更快地收敛到准确解，因此在应用中可以选取较少的渐近阶数，从而提高计算效率。
The high-order transmitting boundary based on the scaled boundary finite elementmethod is of excellent computational performance. The objective of this paper is todevelop high-order doubly asymptotic transmitting boundaries for modeling theunbounded domains. These high-order transmitting boundaries are applicable to bothscalar and vector waves and can be coupled seamlessly with finite element method. Themain contents are summarized as follows:
1. Two coupled numerical methods for dam-reservoir interaction analysis aredeveloped by incorporating the excellent high-order doubly asymptotic transmittingboundary and the finite element method, namely the direct coupled method and thepartitioned coupled method. These two coupled methods are numerically implementedin the open-source finite element code OpenSees. Numerical experiments demonstratedthe high efficiency and accuracy of both coupled methods. As for the high accuracy ofthe high-order transmitting boundary, its order and the region discretized by finiteelement can be reduced in the analysis.
2. The reservoir absorption effect is taken into consideration by introducing thereflection coefficient. Based on the scale boundary transformation of geometry and theGalerkin's weighted residual technique, the scaled boundary finite element (SBFE)equation for three-dimensional layered reservoir is derived. Then, the improved doublyasymptotic continued fraction solution of the dynamic stiffness is constructed. Thecoefficient matrices are determined recursively from the SBFE equation in dynamicstiffness. By introducing the auxiliary variables, a high-order doubly asymptotictransmitting boundary is developed for modeling the unbounded reservoir. Thishigh-order transmitting boundary is incorporated with finite element method to analyzethe dam-reservoir interaction. Numerical examples demonstrate the high accuracy andefficiency of the proposed method.
3. To simply the analysis of the two-dimensional semi-infinite layer with constantdepth, the governing elastic wave equation is decoupled into two scalar wave equationsby neglecting the coupling terms. As a result, the high-order doubly asymptotictransmitting open boundary developed for modeling the scalar wave can be applied tomodel the propagation of elastic waves in the semi-infinite layer. Numerical examples show that this method is applicable to the semi-infinite layer with fixed boundarycondition at the bottom. Besides, this method is of high computational efficiency andnumerical stability.
4. A high-order doubly asymptotic transmitting boundary is developed to modelthe propagation of elastic waves in unbounded domains. This transmitting boundary isconstructed based on the improved doubly asymptotic continued fraction solution of thedynamic stiffness. To achieve accurate results, a set of weighted block-orthogonal basefunctions is introduced. The SBFE equation is split into a high-order mode equation anda low-order mode equation. The former is approximated by the doubly asymptoticcontinued fraction solution and the latter by the high-frequency singly asymptoticcontinued fraction solution. Compared to the high-order transmitting boundary based onthe high-frequency singly asymptotic continued fraction solution, the proposed methodhas a faster convergence speed in the low-frequency range. Thus, the order oftransmitting boundary can be reduced in the analysis to achieve higher computationalefficiency.
引文
[1]陈厚群,徐泽平,李敏.汶川大地震和大坝抗震安全.水利学报,2008,(10):1158-1167.
[2] Brebbia C A, Telles J C F, Wrobel L C. Boundary element techniques: theory andapplications in engineering. Berlin and New York, Springer-Verlag,1984,478p,1984.
[3] Beskos D E. Boundary element methods in dynamic analysis. Applied Mechanics Reviews,1987,40(1):1-23.
[4] Beskos D E. Boundary element methods in dynamic analysis: Part II (1986-1996). AppliedMechanics Reviews,1997,50(3):149.
[5] Tai G R C, Shaw R P. Helmholtz-equation eigenvalues and eigenmodes for arbitrary domains.The Journal of the Acoustical Society of America,1974,56(3):796-804.
[6] Karabalis D L, Beskos D E. Dynamic response of3-D rigid surface foundations by timedomain boundary element method. Earthquake Engineering&Structural Dynamics,1984,12(1):73-93.
[7] Abascal R, Domínguez J. Vibrations of footings on zoned viscoelastic soils. Journal ofEngineering Mechanics,1986,112(5):433-447.
[8] Spyrakos C C, Beskos D E. Dynamic response of rigid strip-foundations by a time-domainboundary element method. International Journal for Numerical Methods in Engineering,1986,23(8):1547-1565.
[9] Sánchez-Sesma F J, Esquivel J A. Ground motion on alluvial valleys under incident plane SHwaves. Bulletin of the Seismological Society of America,1979,69(4):1107-1120.
[10] Mossessian T K, Dravinski M. Scattering of elastic waves by three-dimensional surfacetopographies. Wave Motion,1989,11(6):579-592.
[11] Sánchez-Sesma F J, Ramos-Martínez J, Campillo M. An indirect boundary element methodapplied to simulate the seismic response of alluvial valleys for incident P, S and Rayleighwaves. Earthquake Engineering&Structural Dynamics,1993,22(4):279-295.
[12] Bonnet M, Maier G, Polizzotto C. Symmetric Galerkin boundary element method. AppliedMechanics Reviews,1998,51(11):669-704.
[13] Beer G, Watson J O. Infinite boundary elements. International Journal for Numerical Methodsin Engineering,1989,28(6):1233-1247.
[14] Chuhan Z, Chongmin S, Guanglun W, et al.3-D infinite boundary elements and simulation ofmonolithic dam foundations. Communications in Applied Numerical Methods,1989,5(6):389-400.
[15] Chuhan Z, Chongmin S, Pekau O A. Infinite boundary elements for dynamic problems of3-Dhalf space. International Journal for Numerical Methods in Engineering,1991,31(3):447-462.
[16] Song C, Wolf J P. The scaled boundary finite-element method--alias consistent infinitesimalfinite-element cell method--for elastodynamics. Computer Methods in Applied Mechanicsand Engineering,1997,147(3-4):329-355.
[17] Lysmer J. Lumped mass method for Rayleigh waves. Bulletin of the Seismological Society ofAmerica,1970,60(1):89-104.
[18] Lysmer J, Waas G. Shear waves in plane infinite structures. Journal of EngineeringMechanics,1972.
[19] Kausel E, Waas G, Roesset J M. Dynamic analysis of footings on layered media. Journal ofthe Engineering Mechanics Division,1975,101(5):679-693.
[20] Kausel E, Roesset J M. Semianalytic hyperelement for layered strata. Journal of theEngineering Mechanics Division,1977,103(4):569-588.
[21] Kausel E, Peek R. Dynamic loads in the interior of a layered stratum: an explicit solution.Bulletin of the Seismological Society of America,1982,72(5):1459-1481.
[22] Bougacha S, Tassoulas J, Ro sset J. Analysis of Foundations on Fluid‐Filled poroelasticstratum. Journal of Engineering Mechanics,1993,119(8):1632-1648.
[23] Kausel E. Thin‐layer method: Formulation in the time domain. International Journal forNumerical Methods in Engineering,1994,37(6):927-941.
[24] Park J, Kausel E. Numerical dispersion in the thin-layer method. Computers&Structures,2004,82(7–8):607-625.
[25] Kausel E, Park J. Response of Layered Half-Space Obtained Directly in the Time Domain,Part II: SV-P and Three-Dimensional Sources. Bulletin of the Seismological Society ofAmerica,2006,96(5):1810-1826.
[26] Park J, Kausel E. Response of Layered Half-Space Obtained Directly in the Time Domain,Part I: SH Sources. Bulletin of the Seismological Society of America,2006,96(5):1795-1809.
[27] Tsynkov S V. Numerical solution of problems on unbounded domains. A review. AppliedNumerical Mathematics,1998,27(4):465-532.
[28] Keller J B, Givoli D. Exact non-reflecting boundary conditions. Journal of ComputationalPhysics,1989,82(1):172-192.
[29] Givoli D, Keller J B. Non-reflecting boundary conditions for elastic waves. Wave Motion,1990,12(3):261-279.
[30] Givoli D. Non-reflecting boundary conditions. Journal of Computational Physics,1991,94(1):1-29.
[31] Givoli D, Keller J B. A finite element method for large domains. Computer Methods inApplied Mechanics and Engineering,1989,76(1):41-66.
[32] Grote M J, Keller J B. On Nonreflecting Boundary Conditions. Journal of ComputationalPhysics,1995,122(2):231-243.
[33] Harari I, Patlashenko I, Givoli D. Dirichlet-to-Neumann Maps for Unbounded Wave Guides.Journal of Computational Physics,1998,143(1):200-223.
[34] Givoli D, Cohen D. Nonreflecting Boundary Conditions Based on Kirchhoff-Type Formulae.Journal of Computational Physics,1995,117(1):102-113.
[35] Ting L, Miksis M J. Exact boundary conditions for scattering problems. The Journal of theAcoustical Society of America,1986,80(6):1825-1827.
[36] Alpert B, Greengard L, Hagstrom T. Nonreflecting Boundary Conditions for theTime-Dependent Wave Equation. Journal of Computational Physics,2002,180(1):270-296.
[37] Grote M, Keller J. Exact Nonreflecting Boundary Conditions for the Time Dependent WaveEquation. SIAM Journal on Applied Mathematics,1995,55(2):280-297.
[38] Grote M J, Keller J B. Nonreflecting Boundary Conditions for Maxwell's Equations. Journalof Computational Physics,1998,139(2):327-342.
[39] Keller J, Grote M. Exact Nonreflecting Boundary Condition For Elastic Waves. SIAMJournal on Applied Mathematics,2000,60(3):803-819.
[40] Grote M J. Local nonreflecting boundary condition for Maxwell’s equations. ComputerMethods in Applied Mechanics and Engineering,2006,195(29–32):3691-3708.
[41] Sofronov I L. Artificial boundary conditions of absolute transparency for two-andthree-dimensional external time-dependent scattering problems. European Journal of AppliedMathematics,1998,9(06):561-588.
[42] Geers T L, Sprague M A. A residual-potential boundary for time-dependent, infinite-domainproblems in computational acoustics. The Journal of the Acoustical Society of America,2010,127(2):675-682.
[43]赵崇斌,张楚汉,张光斗.用无穷元模拟半无限平面弹性地基.清华大学学报(自然科学版),1986,(03):51-64.
[44] Astley R J. Infinite elements for wave problems: a review of current formulations and anassessment of accuracy. International Journal for Numerical Methods in Engineering,2000,49(7):951-976.
[45] Bettess P. More on infinite elements. International Journal for Numerical Methods inEngineering,1980,15(11):1613-1626.
[46] Zienkiewicz O C, Bando K, Bettess P, et al. Mapped infinite elements for exterior waveproblems. International Journal for Numerical Methods in Engineering,1985,21(7):1229-1251.
[47] Zienkiewicz O C, Emson C, Bettess P. A novel boundary infinite element. InternationalJournal for Numerical Methods in Engineering,1983,19(3):393-404.
[48] Bettess P. Infinite elements. International Journal for Numerical Methods in Engineering,1977,11(1):53-64.
[49] Bettess P, Zienkiewicz O C. Diffraction and refraction of surface waves using finite andinfinite elements. International Journal for Numerical Methods in Engineering,1977,11(8):1271-1290.
[50] Astley R J. Wave envelope and infinite elements for acoustical radiation. International Journalfor Numerical Methods in Fluids,1983,3(5):507-526.
[51] Chuhan Z, Chongbin Z. Coupling method of finite and infinite elements for strip foundationwave problems. Earthquake Engineering&Structural Dynamics,1987,15(7):839-851.
[52] Zhao C, Valliappan S. A dynamic infinite element for three-dimensional infinite-domainwave problems. International Journal for Numerical Methods in Engineering,1993,36(15):2567-2580.
[53] Burnett D S. A three-dimensional acoustic infinite element based on a prolate spheroidalmultipole expansion. The Journal of the Acoustical Society of America,1994,96(5):2798-2816.
[54] Chadwick E, Bettess P, Laghrouche O. Diffraction of short waves modelled using newmapped wave envelope finite and infinite elements. International Journal for NumericalMethods in Engineering,1999,45(3):335-354.
[55] Israeli M, Orszag S A. Approximation of radiation boundary conditions. Journal ofComputational Physics,1981,41(1):115-135.
[56] Berenger J-P. A perfectly matched layer for the absorption of electromagnetic waves. Journalof Computational Physics,1994,114(2):185-200.
[57] Chew W C, Weedon W H. A3D perfectly matched medium from modified maxwell'sequations with stretched coordinates. Microwave and Optical Technology Letters,1994,7(13):599-604.
[58] Sacks Z S, Kingsland D M, Lee R, et al. A perfectly matched anisotropic absorber for use asan absorbing boundary condition. Antennas and Propagation, IEEE Transactions on,1995,43(12):1460-1463.
[59] Li Z, Cangellaris A C. GT-PML: generalized theory of perfectly matched layers and itsapplication to the reflectionless truncation of finite-difference time-domain grids. MicrowaveTheory and Techniques, IEEE Transactions on,1996,44(12):2555-2563.
[60] Sullivan D M. An unsplit step3-D PML for use with the FDTD method. Microwave andGuided Wave Letters, IEEE,1997,7(7):184-186.
[61] Hu F Q. On absorbing boundary conditons for linearized Euler equations by a perfectlymatched layer. Journal Name: Journal of Computational Physics; Journal Volume:129;Journal Issue:1; Other Information: PBD: Nov1996,1996:Medium: X; Size: pp.201-219.
[62] Chew W C, Liu Q H. Perfectly matched layers for elastodynamics: a new absorbing boundarycondition. Journal of Computational Acoustics,1996,04(04):341-359.
[63] Hastings F D, Schneider J B, Broschat S L. Application of the perfectly matched layer (PML)absorbing boundary condition to elastic wave propagation. The Journal of the AcousticalSociety of America,1996,100(5):3061-3069.
[64] Collino F, Tsogka C. Application of the perfectly matched absorbing layer model to the linearelastodynamic problem in anisotropic heterogeneous media. Geophysics,2001,66(1):294-307.
[65] Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for thesecond-order seismic wave equation. Geophysical Journal International,2003,154(1):146-153.
[66] Basu U, Chopra A K. Perfectly matched layers for time-harmonic elastodynamics ofunbounded domains: theory and finite-element implementation. Computer Methods inApplied Mechanics and Engineering,2003,192(11–12):1337-1375.
[67] Basu U, Chopra A K. Perfectly matched layers for transient elastodynamics of unboundeddomains. International Journal for Numerical Methods in Engineering,2004,59(8):1039-1074.
[68] Kausel E. Local Transmitting Boundaries. Journal of Engineering Mechanics,1988,114(6):1011-1027.
[69] Lysmer J, Kuhlemeyer R. Finite dynamic model for infinite media. Journal of the EngineeringMechanics Division,1969,98:859-877.
[70] Deeks A J, Randolph M F. Axisymmetric Time‐Domain Transmitting Boundaries. Journalof Engineering Mechanics,1994,120(1):25-42.
[71]刘晶波,吕彦东.结构-地基动力相互作用问题分析的一种直接方法.土木工程学报,1998,(03):55-64.
[72] Liu J, Du Y, Du X, et al.3D viscous-spring artificial boundary in time domain. EarthquakeEngineering and Engineering Vibration,2006,5(1):93-102.
[73]杜修力,赵密,王进廷.近场波动模拟的人工应力边界条件.力学学报,2006,(01):49-56.
[74] Engquist B, Majda A. Absorbing boundary conditions for numerical simulation of waves.Proceedings of the National Academy of Sciences,1977,74(5):1765-1766.
[75] Engquist B, Majda A. Radiation boundary conditions for acoustic and elastic wavecalculations. Communications on Pure and Applied Mathematics,1979,32(3):313-357.
[76] Bayliss A, Turkel E. Radiation boundary conditions for wave-like equations.Communications on Pure and Applied Mathematics,1980,33(6):707-725.
[77] Higdon R L. Absorbing boundary conditions for difference approximations to themulti-dimensional wave equation. Math. Comput.,1986,47(176):437-459.
[78] Higdon R L. Radiation boundary conditions for elastic wave propagation. SIAM Journal onNumerical Analysis,1990,27(4):831-869.
[79] Higdon R L. Radiation boundary conditions for dispersive waves. SIAM Journal onNumerical Analysis,1994,31(1):64-100.
[80] Liao Z P, Wong H L. A transmitting boundary for the numerical simulation of elastic wavepropagation. International Journal of Soil Dynamics and Earthquake Engineering,1984,3(4):174-183.
[81] Liao Z-P. Extrapolation non-reflecting boundary conditions. Wave Motion,1996,24(2):117-138.
[82] Givoli D. High-order local non-reflecting boundary conditions: a review. Wave Motion,2004,39(4):319-326.
[83] Collino F. High order absorbing boundary conditions for wave propagation models: straightline boundary and corner cases. Proceedings of the Second International Conference onMathematical and Numerical Aspects of Wave Propagation. Delaware: SIAM;1993. p161-171.
[84] Hagstrom T, Hariharan S I. A formulation of asymptotic and exact boundary conditions usinglocal operators. Applied Numerical Mathematics,1998,27(4):403-416.
[85] Thompson L L, Huan R. Finite element formulation of exact non-reflecting boundaryconditions for the time-dependent wave equation. International Journal for NumericalMethods in Engineering,1999,45(11):1607-1630.
[86] Huan R, Thompson L L. Accurate radiation boundary conditions for the time-dependent waveequation on unbounded domains. International Journal for Numerical Methods in Engineering,2000,47(9):1569-1603.
[87] Thompson L L, Huan R. Implementation of exact non-reflecting boundary conditions in thefinite element method for the time-dependent wave equation. Computer Methods in AppliedMechanics and Engineering,2000,187(1–2):137-159.
[88] Thompson L L, Huan R, He D. Accurate radiation boundary conditions for thetwo-dimensional wave equation on unbounded domains. Computer Methods in AppliedMechanics and Engineering,2001,191(3–5):311-351.
[89] Guddati M N, Tassoulas J L. Continued-fraction absorbing boundary conditions for the waveequation. Journal of Computational Acoustics,2000,08(01):139-156.
[90] Givoli D, Neta B. High-order non-reflecting boundary scheme for time-dependent waves.Journal of Computational Physics,2003,186(1):24-46.
[91] Givoli D, Neta B. High-order non-reflecting boundary conditions for dispersive waves. WaveMotion,2003,37(3):257-271.
[92] Givoli D, Neta B. High-order nonreflecting boundary conditions for the dispersive shallowwater equations. Journal of Computational and Applied Mathematics,2003,158(1):49-60.
[93] Givoli D, Neta B, Patlashenko I. Finite element analysis of time-dependent semi-infinitewave-guides with high-order boundary treatment. International Journal for NumericalMethods in Engineering,2003,58(13):1955-1983.
[94] Hagstrom T, Warburton T. A new auxiliary variable formulation of high-order local radiationboundary conditions: corner compatibility conditions and extensions to first-order systems.Wave Motion,2004,39(4):327-338.
[95] Givoli D, Hagstrom T, Patlashenko I. Finite element formulation with high-order absorbingboundary conditions for time-dependent waves. Computer Methods in Applied Mechanicsand Engineering,2006,195(29–32):3666-3690.
[96] Hagstrom T, De Castro M L, Givoli D, et al. Local high-order absorbing boundary conditionsfor time-dependent waves in guides. Journal of Computational Acoustics,2007,15(01):1-22.
[97] Hagstrom T, Mar-Or A, Givoli D. High-order local absorbing conditions for the waveequation: Extensions and improvements. Journal of Computational Physics,2008,227(6):3322-3357.
[98] Bazyar M H, Song C. A continued-fraction-based high-order transmitting boundary for wavepropagation in unbounded domains of arbitrary geometry. International Journal for NumericalMethods in Engineering,2008,74(2):209-237.
[99] Birk C, Prempramote S, Song C. An improved continued-fraction-based high-ordertransmitting boundary for time-domain analyses in unbounded domains. International Journalfor Numerical Methods in Engineering,2012,89(3):269-298.
[100] Prempramote S, Song C, Tin-Loi F, et al. High-order doubly asymptotic open boundaries forscalar wave equation. International Journal for Numerical Methods in Engineering,2009,79(3):340-374.
[101] Geers T. Singly and doubly asymptotic computational boundaries. In: TL G, editor.Proceedings of the IUTAM Symposium on Computational Methods for Unbounded Domains.Dordrecht: Kluwer Academic Publishers;1998. p135-141.
[102] Wang X, Jin F, Prempramote S, et al. Time-domain analysis of gravity dam-reservoirinteraction using high-order doubly asymptotic open boundary. Computers&Structures,2011,89(7-8):668-680.
[103] Mazzoni S, McKenna F, Scott M H, et al. OpenSees command language manual. PacificEarthquake Engineering Research (PEER) Center,2006.
[104] Mckenna F T. Object-oriented finite element programming: frameworks for analysis,algorithms and parallel computing[Doctor]. Berkeley: University of California1997.
[105] Dasgupta G. A finite element formulation for unbounded homogeneous continua. Journal ofApplied Mechanics,1982,49(1):136-140.
[106] Wolf J P, Song C. Dynamic-stiffness matrix in time domain of unbounded medium byinfinitesimal finite element cell method. Earthquake Engineering&Structural Dynamics,1994,23(11):1181-1198.
[107] Song C, Wolf J. Consistent infinitesimal finite-element–cell method: out-of-plane motion.Journal of Engineering Mechanics,1995,121(5):613-619.
[108] Wolf J, Song C. Consistent infinitesimal finite element cell method: Three-dimensional scalarwave equation. Journal of Applied Mechanics,1996,63(3):650-654.
[109] Wolf J P, Song C. Consistent infinitesimal finite-element cell method: in-plane motion.Computer Methods in Applied Mechanics and Engineering,1995,123(1-4):355-370.
[110] Song C, Wolf J P. Consistent infinitesimal finite-element cell method: Three-dimensionalvector wave equation. International Journal for Numerical Methods in Engineering,1996,39(13):2189-2208.
[111] Wolf J P, Song C. Static stiffness of unbounded soil by finite-element method. Journal ofgeotechnical engineering,1996,122(4):267-273.
[112] Song C, Wolf J P. Consistent infinitesimal finite-element cell method for diffusion equationin unbounded medium. Computer Methods in Applied Mechanics and Engineering,1996,132(3-4):319-334.
[113] Deeks A J, Wolf J P. A virtual work derivation of the scaled boundary finite-element methodfor elastostatics. Computational Mechanics,2002,28(6):489-504.
[114] Song C, Wolf J P. The scaled boundary finite-element method: analytical solution infrequency domain. Computer Methods in Applied Mechanics and Engineering,1998,164(1-2):249-264.
[115] Song C, Wolf J P. Body loads in scaled boundary finite-element method. Computer Methodsin Applied Mechanics and Engineering,1999,180(1-2):117-135.
[116] Song C. A matrix function solution for the scaled boundary finite-element equation in statics.Computer Methods in Applied Mechanics and Engineering,2004,193(23-26):2325-2356.
[117] Song C. Weighted block-orthogonal base functions for static analysis of unbounded domains.Proceedings of the6th World Congress on Computational Mechanics. Beijing;2004. p615-620.
[118] Song C. A super-element for crack analysis in the time domain. International Journal forNumerical Methods in Engineering,2004,61(8):1332-1357.
[119] Song C, Wolf J P. Semi-analytical representation of stress singularities as occurring in cracksin anisotropic multi-materials with the scaled boundary finite-element method. Computers&Structures,2002,80(2):183-197.
[120] Song C, Vrcelj Z. Evaluation of dynamic stress intensity factors and T-stress using the scaledboundary finite-element method. Engineering Fracture Mechanics,2008,75(8):1960-1980.
[121] Yang Z. Fully automatic modelling of mixed-mode crack propagation using scaled boundaryfinite element method. Engineering Fracture Mechanics,2006,73(12):1711-1731.
[122] Ooi E T, Yang Z J. A hybrid finite element-scaled boundary finite element method for crackpropagation modelling. Computer Methods in Applied Mechanics and Engineering,2010,199(17–20):1178-1192.
[123] Ooi E T, Song C, Tin-Loi F, et al. Polygon scaled boundary finite elements for crackpropagation modelling. International Journal for Numerical Methods in Engineering,2012,91(3):319-342.
[124] Doherty J P, Deeks A J. Scaled boundary finite-element analysis of a non-homogeneouselastic half-space. International Journal for Numerical Methods in Engineering,2003,57(7):955-973.
[125] Hassanen M, El-Hamalawi A. Two-dimensional development of the dynamic coupledconsolidation scaled boundary finite-element method for fully saturated soils. Soil Dynamicsand Earthquake Engineering,2007,27(2):153-165.
[126] Deeks A J, Cheng L. Potential flow around obstacles using the scaled boundary finite-elementmethod. International Journal for Numerical Methods in Fluids,2003,41(7):721-741.
[127] Song C, Wolf J P. The scaled boundary finite element method—alias consistent infinitesimalfinite element cell method—for diffusion. International Journal for Numerical Methods inEngineering,1999,45(10):1403-1431.
[128] Wolf J P, Song C. Finite-Element Modelling of Unbounded Media. New York: Wiley;1996.
[129] Song C. Dynamic analysis of unbounded domains by a reduced set of base functions.Computer Methods in Applied Mechanics and Engineering,2006,195(33-36):4075-4094.
[130] Song C, Bazyar M H. A boundary condition in Padé series for frequency-domain solution ofwave propagation in unbounded domains. International Journal for Numerical Methods inEngineering,2007,69(11):2330-2358.
[131] Zhang X, Wegner J L, Haddow J B. Three-dimensional dynamic soil–structure interactionanalysis in the time domain. Earthquake Engineering&Structural Dynamics,1999,28(12):1501-1524.
[132] Wegner J L, Yao M M, Zhang X. Dynamic wave–soil–structure interaction analysis in thetime domain. Computers&Structures,2005,83(27):2206-2214.
[133] Yan J, Zhang C, Jin F. A coupling procedure of FE and SBFE for soil–structure interaction inthe time domain. International Journal for Numerical Methods in Engineering,2004,59(11):1453-1471.
[134] Lehmann L. An effective finite element approach for soil-structure analysis in thetime-domain. Structural Engineering and Mechanics,2005,21(4):437-450.
[135] Radmanovi B, Katz C. A high performance scaled boundary finite element method. IOPConference Series: Materials Science and Engineering,2010,10(1):012214.
[136]杜建国.基于SBFEM的大坝—库水—地基动力相互作用分析[博士学位论文].大连:大连理工大学2007.
[137] Wolf J P, Song C. The scaled boundary finite-element method-a primer: derivations.Computers&Structures,2000,78(1-3):191-210.
[138] Westergaard H M. Water pressure on dams during earthquakes. Transactions of the AmericanSociety of Civil Engineering,1933,98:418-433.
[139]杜修力,王进廷.动水压力及其对坝体地震反应影响的研究进展.水利学报,2001,(07):13-21.
[140] Zienkiewicz O C, Bettess P. Fluid-structure dynamic interaction and wave forces. Anintroduction to numerical treatment. International Journal for Numerical Methods inEngineering,1978,13(1):1-16.
[141] Kü ükarslan S, Coskun S B, TaskIn B. Transient analysis of dam-reservoir interactionincluding the reservoir bottom effects. Journal of Fluids and Structures,2005,20(8):1073-1084.
[142] Soares Jr D, von Estorff O, Mansur W J. Efficient non-linear solid–fluid interaction analysisby an iterative BEM/FEM coupling. International Journal for Numerical Methods inEngineering,2005,64(11):1416-1431.
[143] Seghir A, Tahakourt A, Bonnet G. Coupling FEM and symmetric BEM for dynamicinteraction of dam-reservoir systems. Engineering Analysis with Boundary Elements,2009,33(10):1201-1210.
[144] Lin G, Du J, Hu Z. Dynamic dam-reservoir interaction analysis including effect of reservoirboundary absorption. Science in China Series E: Technological Sciences,2007,50:1-10.
[145] Lin G, Wang Y, Hu Z. An efficient approach for frequency-domain and time-domainhydrodynamic analysis of dam–reservoir systems. Earthquake Engineering&StructuralDynamics,2012,41(13):1725-1749.
[146] Li S M. Diagonalization procedure for scaled boundary finite element method in modelingsemi-infinite reservoir with uniform cross-section. International Journal for NumericalMethods in Engineering,2009,80(5):596-608.
[147]王翔.高阶双渐近透射边界及其在大坝-库水相互作用中的应用[博士学位论文].北京:清华大学2010.
[148] Chopra A K, Chakrabarti P. Earthquake analysis of concrete gravity dams includingdam-water-foundation rock interaction. Earthquake Engineering&Structural Dynamics,1981,9(4):363-383.
[149] Hall J, Chopra A. Dynamic analysis of arch dams including hydrodynamic effects. Journal ofEngineering Mechanics,1983,109(1):149-167.
[150] Fenves G, Chopra A K. Earthquake analysis of concrete gravity dams including reservoirbottom absorption and dam-water-foundation rock interaction. Earthquake Engineering&Structural Dynamics,1984,12(5):663-680.
[151] Tan H, Chopra A K. Earthquake analysis of arch dams including dam-water-foundation rockinteraction. Earthquake Engineering&Structural Dynamics,1995,24(11):1453-1474.
[152] Birk C, Song C. A continued-fraction approach for transient diffusion in unbounded medium.Computer Methods in Applied Mechanics and Engineering,2009,198(33–36):2576-2590.
[153]杜建国,林皋.比例边界有限元子结构法研究.世界地震工程,2007,(01):67-72.
[154] Prempramote S. Development of High-Order Doubly Asymptotic Open Boundaries For WavePropagation In Unbounded Domains By Extending The Scaled Boundary Finite ElementMethod[Doctor]. Sydney: University of New South Wales2011.
[2] Brebbia C A, Telles J C F, Wrobel L C. Boundary element techniques: theory andapplications in engineering. Berlin and New York, Springer-Verlag,1984,478p,1984.
[3] Beskos D E. Boundary element methods in dynamic analysis. Applied Mechanics Reviews,1987,40(1):1-23.
[4] Beskos D E. Boundary element methods in dynamic analysis: Part II (1986-1996). AppliedMechanics Reviews,1997,50(3):149.
[5] Tai G R C, Shaw R P. Helmholtz-equation eigenvalues and eigenmodes for arbitrary domains.The Journal of the Acoustical Society of America,1974,56(3):796-804.
[6] Karabalis D L, Beskos D E. Dynamic response of3-D rigid surface foundations by timedomain boundary element method. Earthquake Engineering&Structural Dynamics,1984,12(1):73-93.
[7] Abascal R, Domínguez J. Vibrations of footings on zoned viscoelastic soils. Journal ofEngineering Mechanics,1986,112(5):433-447.
[8] Spyrakos C C, Beskos D E. Dynamic response of rigid strip-foundations by a time-domainboundary element method. International Journal for Numerical Methods in Engineering,1986,23(8):1547-1565.
[9] Sánchez-Sesma F J, Esquivel J A. Ground motion on alluvial valleys under incident plane SHwaves. Bulletin of the Seismological Society of America,1979,69(4):1107-1120.
[10] Mossessian T K, Dravinski M. Scattering of elastic waves by three-dimensional surfacetopographies. Wave Motion,1989,11(6):579-592.
[11] Sánchez-Sesma F J, Ramos-Martínez J, Campillo M. An indirect boundary element methodapplied to simulate the seismic response of alluvial valleys for incident P, S and Rayleighwaves. Earthquake Engineering&Structural Dynamics,1993,22(4):279-295.
[12] Bonnet M, Maier G, Polizzotto C. Symmetric Galerkin boundary element method. AppliedMechanics Reviews,1998,51(11):669-704.
[13] Beer G, Watson J O. Infinite boundary elements. International Journal for Numerical Methodsin Engineering,1989,28(6):1233-1247.
[14] Chuhan Z, Chongmin S, Guanglun W, et al.3-D infinite boundary elements and simulation ofmonolithic dam foundations. Communications in Applied Numerical Methods,1989,5(6):389-400.
[15] Chuhan Z, Chongmin S, Pekau O A. Infinite boundary elements for dynamic problems of3-Dhalf space. International Journal for Numerical Methods in Engineering,1991,31(3):447-462.
[16] Song C, Wolf J P. The scaled boundary finite-element method--alias consistent infinitesimalfinite-element cell method--for elastodynamics. Computer Methods in Applied Mechanicsand Engineering,1997,147(3-4):329-355.
[17] Lysmer J. Lumped mass method for Rayleigh waves. Bulletin of the Seismological Society ofAmerica,1970,60(1):89-104.
[18] Lysmer J, Waas G. Shear waves in plane infinite structures. Journal of EngineeringMechanics,1972.
[19] Kausel E, Waas G, Roesset J M. Dynamic analysis of footings on layered media. Journal ofthe Engineering Mechanics Division,1975,101(5):679-693.
[20] Kausel E, Roesset J M. Semianalytic hyperelement for layered strata. Journal of theEngineering Mechanics Division,1977,103(4):569-588.
[21] Kausel E, Peek R. Dynamic loads in the interior of a layered stratum: an explicit solution.Bulletin of the Seismological Society of America,1982,72(5):1459-1481.
[22] Bougacha S, Tassoulas J, Ro sset J. Analysis of Foundations on Fluid‐Filled poroelasticstratum. Journal of Engineering Mechanics,1993,119(8):1632-1648.
[23] Kausel E. Thin‐layer method: Formulation in the time domain. International Journal forNumerical Methods in Engineering,1994,37(6):927-941.
[24] Park J, Kausel E. Numerical dispersion in the thin-layer method. Computers&Structures,2004,82(7–8):607-625.
[25] Kausel E, Park J. Response of Layered Half-Space Obtained Directly in the Time Domain,Part II: SV-P and Three-Dimensional Sources. Bulletin of the Seismological Society ofAmerica,2006,96(5):1810-1826.
[26] Park J, Kausel E. Response of Layered Half-Space Obtained Directly in the Time Domain,Part I: SH Sources. Bulletin of the Seismological Society of America,2006,96(5):1795-1809.
[27] Tsynkov S V. Numerical solution of problems on unbounded domains. A review. AppliedNumerical Mathematics,1998,27(4):465-532.
[28] Keller J B, Givoli D. Exact non-reflecting boundary conditions. Journal of ComputationalPhysics,1989,82(1):172-192.
[29] Givoli D, Keller J B. Non-reflecting boundary conditions for elastic waves. Wave Motion,1990,12(3):261-279.
[30] Givoli D. Non-reflecting boundary conditions. Journal of Computational Physics,1991,94(1):1-29.
[31] Givoli D, Keller J B. A finite element method for large domains. Computer Methods inApplied Mechanics and Engineering,1989,76(1):41-66.
[32] Grote M J, Keller J B. On Nonreflecting Boundary Conditions. Journal of ComputationalPhysics,1995,122(2):231-243.
[33] Harari I, Patlashenko I, Givoli D. Dirichlet-to-Neumann Maps for Unbounded Wave Guides.Journal of Computational Physics,1998,143(1):200-223.
[34] Givoli D, Cohen D. Nonreflecting Boundary Conditions Based on Kirchhoff-Type Formulae.Journal of Computational Physics,1995,117(1):102-113.
[35] Ting L, Miksis M J. Exact boundary conditions for scattering problems. The Journal of theAcoustical Society of America,1986,80(6):1825-1827.
[36] Alpert B, Greengard L, Hagstrom T. Nonreflecting Boundary Conditions for theTime-Dependent Wave Equation. Journal of Computational Physics,2002,180(1):270-296.
[37] Grote M, Keller J. Exact Nonreflecting Boundary Conditions for the Time Dependent WaveEquation. SIAM Journal on Applied Mathematics,1995,55(2):280-297.
[38] Grote M J, Keller J B. Nonreflecting Boundary Conditions for Maxwell's Equations. Journalof Computational Physics,1998,139(2):327-342.
[39] Keller J, Grote M. Exact Nonreflecting Boundary Condition For Elastic Waves. SIAMJournal on Applied Mathematics,2000,60(3):803-819.
[40] Grote M J. Local nonreflecting boundary condition for Maxwell’s equations. ComputerMethods in Applied Mechanics and Engineering,2006,195(29–32):3691-3708.
[41] Sofronov I L. Artificial boundary conditions of absolute transparency for two-andthree-dimensional external time-dependent scattering problems. European Journal of AppliedMathematics,1998,9(06):561-588.
[42] Geers T L, Sprague M A. A residual-potential boundary for time-dependent, infinite-domainproblems in computational acoustics. The Journal of the Acoustical Society of America,2010,127(2):675-682.
[43]赵崇斌,张楚汉,张光斗.用无穷元模拟半无限平面弹性地基.清华大学学报(自然科学版),1986,(03):51-64.
[44] Astley R J. Infinite elements for wave problems: a review of current formulations and anassessment of accuracy. International Journal for Numerical Methods in Engineering,2000,49(7):951-976.
[45] Bettess P. More on infinite elements. International Journal for Numerical Methods inEngineering,1980,15(11):1613-1626.
[46] Zienkiewicz O C, Bando K, Bettess P, et al. Mapped infinite elements for exterior waveproblems. International Journal for Numerical Methods in Engineering,1985,21(7):1229-1251.
[47] Zienkiewicz O C, Emson C, Bettess P. A novel boundary infinite element. InternationalJournal for Numerical Methods in Engineering,1983,19(3):393-404.
[48] Bettess P. Infinite elements. International Journal for Numerical Methods in Engineering,1977,11(1):53-64.
[49] Bettess P, Zienkiewicz O C. Diffraction and refraction of surface waves using finite andinfinite elements. International Journal for Numerical Methods in Engineering,1977,11(8):1271-1290.
[50] Astley R J. Wave envelope and infinite elements for acoustical radiation. International Journalfor Numerical Methods in Fluids,1983,3(5):507-526.
[51] Chuhan Z, Chongbin Z. Coupling method of finite and infinite elements for strip foundationwave problems. Earthquake Engineering&Structural Dynamics,1987,15(7):839-851.
[52] Zhao C, Valliappan S. A dynamic infinite element for three-dimensional infinite-domainwave problems. International Journal for Numerical Methods in Engineering,1993,36(15):2567-2580.
[53] Burnett D S. A three-dimensional acoustic infinite element based on a prolate spheroidalmultipole expansion. The Journal of the Acoustical Society of America,1994,96(5):2798-2816.
[54] Chadwick E, Bettess P, Laghrouche O. Diffraction of short waves modelled using newmapped wave envelope finite and infinite elements. International Journal for NumericalMethods in Engineering,1999,45(3):335-354.
[55] Israeli M, Orszag S A. Approximation of radiation boundary conditions. Journal ofComputational Physics,1981,41(1):115-135.
[56] Berenger J-P. A perfectly matched layer for the absorption of electromagnetic waves. Journalof Computational Physics,1994,114(2):185-200.
[57] Chew W C, Weedon W H. A3D perfectly matched medium from modified maxwell'sequations with stretched coordinates. Microwave and Optical Technology Letters,1994,7(13):599-604.
[58] Sacks Z S, Kingsland D M, Lee R, et al. A perfectly matched anisotropic absorber for use asan absorbing boundary condition. Antennas and Propagation, IEEE Transactions on,1995,43(12):1460-1463.
[59] Li Z, Cangellaris A C. GT-PML: generalized theory of perfectly matched layers and itsapplication to the reflectionless truncation of finite-difference time-domain grids. MicrowaveTheory and Techniques, IEEE Transactions on,1996,44(12):2555-2563.
[60] Sullivan D M. An unsplit step3-D PML for use with the FDTD method. Microwave andGuided Wave Letters, IEEE,1997,7(7):184-186.
[61] Hu F Q. On absorbing boundary conditons for linearized Euler equations by a perfectlymatched layer. Journal Name: Journal of Computational Physics; Journal Volume:129;Journal Issue:1; Other Information: PBD: Nov1996,1996:Medium: X; Size: pp.201-219.
[62] Chew W C, Liu Q H. Perfectly matched layers for elastodynamics: a new absorbing boundarycondition. Journal of Computational Acoustics,1996,04(04):341-359.
[63] Hastings F D, Schneider J B, Broschat S L. Application of the perfectly matched layer (PML)absorbing boundary condition to elastic wave propagation. The Journal of the AcousticalSociety of America,1996,100(5):3061-3069.
[64] Collino F, Tsogka C. Application of the perfectly matched absorbing layer model to the linearelastodynamic problem in anisotropic heterogeneous media. Geophysics,2001,66(1):294-307.
[65] Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for thesecond-order seismic wave equation. Geophysical Journal International,2003,154(1):146-153.
[66] Basu U, Chopra A K. Perfectly matched layers for time-harmonic elastodynamics ofunbounded domains: theory and finite-element implementation. Computer Methods inApplied Mechanics and Engineering,2003,192(11–12):1337-1375.
[67] Basu U, Chopra A K. Perfectly matched layers for transient elastodynamics of unboundeddomains. International Journal for Numerical Methods in Engineering,2004,59(8):1039-1074.
[68] Kausel E. Local Transmitting Boundaries. Journal of Engineering Mechanics,1988,114(6):1011-1027.
[69] Lysmer J, Kuhlemeyer R. Finite dynamic model for infinite media. Journal of the EngineeringMechanics Division,1969,98:859-877.
[70] Deeks A J, Randolph M F. Axisymmetric Time‐Domain Transmitting Boundaries. Journalof Engineering Mechanics,1994,120(1):25-42.
[71]刘晶波,吕彦东.结构-地基动力相互作用问题分析的一种直接方法.土木工程学报,1998,(03):55-64.
[72] Liu J, Du Y, Du X, et al.3D viscous-spring artificial boundary in time domain. EarthquakeEngineering and Engineering Vibration,2006,5(1):93-102.
[73]杜修力,赵密,王进廷.近场波动模拟的人工应力边界条件.力学学报,2006,(01):49-56.
[74] Engquist B, Majda A. Absorbing boundary conditions for numerical simulation of waves.Proceedings of the National Academy of Sciences,1977,74(5):1765-1766.
[75] Engquist B, Majda A. Radiation boundary conditions for acoustic and elastic wavecalculations. Communications on Pure and Applied Mathematics,1979,32(3):313-357.
[76] Bayliss A, Turkel E. Radiation boundary conditions for wave-like equations.Communications on Pure and Applied Mathematics,1980,33(6):707-725.
[77] Higdon R L. Absorbing boundary conditions for difference approximations to themulti-dimensional wave equation. Math. Comput.,1986,47(176):437-459.
[78] Higdon R L. Radiation boundary conditions for elastic wave propagation. SIAM Journal onNumerical Analysis,1990,27(4):831-869.
[79] Higdon R L. Radiation boundary conditions for dispersive waves. SIAM Journal onNumerical Analysis,1994,31(1):64-100.
[80] Liao Z P, Wong H L. A transmitting boundary for the numerical simulation of elastic wavepropagation. International Journal of Soil Dynamics and Earthquake Engineering,1984,3(4):174-183.
[81] Liao Z-P. Extrapolation non-reflecting boundary conditions. Wave Motion,1996,24(2):117-138.
[82] Givoli D. High-order local non-reflecting boundary conditions: a review. Wave Motion,2004,39(4):319-326.
[83] Collino F. High order absorbing boundary conditions for wave propagation models: straightline boundary and corner cases. Proceedings of the Second International Conference onMathematical and Numerical Aspects of Wave Propagation. Delaware: SIAM;1993. p161-171.
[84] Hagstrom T, Hariharan S I. A formulation of asymptotic and exact boundary conditions usinglocal operators. Applied Numerical Mathematics,1998,27(4):403-416.
[85] Thompson L L, Huan R. Finite element formulation of exact non-reflecting boundaryconditions for the time-dependent wave equation. International Journal for NumericalMethods in Engineering,1999,45(11):1607-1630.
[86] Huan R, Thompson L L. Accurate radiation boundary conditions for the time-dependent waveequation on unbounded domains. International Journal for Numerical Methods in Engineering,2000,47(9):1569-1603.
[87] Thompson L L, Huan R. Implementation of exact non-reflecting boundary conditions in thefinite element method for the time-dependent wave equation. Computer Methods in AppliedMechanics and Engineering,2000,187(1–2):137-159.
[88] Thompson L L, Huan R, He D. Accurate radiation boundary conditions for thetwo-dimensional wave equation on unbounded domains. Computer Methods in AppliedMechanics and Engineering,2001,191(3–5):311-351.
[89] Guddati M N, Tassoulas J L. Continued-fraction absorbing boundary conditions for the waveequation. Journal of Computational Acoustics,2000,08(01):139-156.
[90] Givoli D, Neta B. High-order non-reflecting boundary scheme for time-dependent waves.Journal of Computational Physics,2003,186(1):24-46.
[91] Givoli D, Neta B. High-order non-reflecting boundary conditions for dispersive waves. WaveMotion,2003,37(3):257-271.
[92] Givoli D, Neta B. High-order nonreflecting boundary conditions for the dispersive shallowwater equations. Journal of Computational and Applied Mathematics,2003,158(1):49-60.
[93] Givoli D, Neta B, Patlashenko I. Finite element analysis of time-dependent semi-infinitewave-guides with high-order boundary treatment. International Journal for NumericalMethods in Engineering,2003,58(13):1955-1983.
[94] Hagstrom T, Warburton T. A new auxiliary variable formulation of high-order local radiationboundary conditions: corner compatibility conditions and extensions to first-order systems.Wave Motion,2004,39(4):327-338.
[95] Givoli D, Hagstrom T, Patlashenko I. Finite element formulation with high-order absorbingboundary conditions for time-dependent waves. Computer Methods in Applied Mechanicsand Engineering,2006,195(29–32):3666-3690.
[96] Hagstrom T, De Castro M L, Givoli D, et al. Local high-order absorbing boundary conditionsfor time-dependent waves in guides. Journal of Computational Acoustics,2007,15(01):1-22.
[97] Hagstrom T, Mar-Or A, Givoli D. High-order local absorbing conditions for the waveequation: Extensions and improvements. Journal of Computational Physics,2008,227(6):3322-3357.
[98] Bazyar M H, Song C. A continued-fraction-based high-order transmitting boundary for wavepropagation in unbounded domains of arbitrary geometry. International Journal for NumericalMethods in Engineering,2008,74(2):209-237.
[99] Birk C, Prempramote S, Song C. An improved continued-fraction-based high-ordertransmitting boundary for time-domain analyses in unbounded domains. International Journalfor Numerical Methods in Engineering,2012,89(3):269-298.
[100] Prempramote S, Song C, Tin-Loi F, et al. High-order doubly asymptotic open boundaries forscalar wave equation. International Journal for Numerical Methods in Engineering,2009,79(3):340-374.
[101] Geers T. Singly and doubly asymptotic computational boundaries. In: TL G, editor.Proceedings of the IUTAM Symposium on Computational Methods for Unbounded Domains.Dordrecht: Kluwer Academic Publishers;1998. p135-141.
[102] Wang X, Jin F, Prempramote S, et al. Time-domain analysis of gravity dam-reservoirinteraction using high-order doubly asymptotic open boundary. Computers&Structures,2011,89(7-8):668-680.
[103] Mazzoni S, McKenna F, Scott M H, et al. OpenSees command language manual. PacificEarthquake Engineering Research (PEER) Center,2006.
[104] Mckenna F T. Object-oriented finite element programming: frameworks for analysis,algorithms and parallel computing[Doctor]. Berkeley: University of California1997.
[105] Dasgupta G. A finite element formulation for unbounded homogeneous continua. Journal ofApplied Mechanics,1982,49(1):136-140.
[106] Wolf J P, Song C. Dynamic-stiffness matrix in time domain of unbounded medium byinfinitesimal finite element cell method. Earthquake Engineering&Structural Dynamics,1994,23(11):1181-1198.
[107] Song C, Wolf J. Consistent infinitesimal finite-element–cell method: out-of-plane motion.Journal of Engineering Mechanics,1995,121(5):613-619.
[108] Wolf J, Song C. Consistent infinitesimal finite element cell method: Three-dimensional scalarwave equation. Journal of Applied Mechanics,1996,63(3):650-654.
[109] Wolf J P, Song C. Consistent infinitesimal finite-element cell method: in-plane motion.Computer Methods in Applied Mechanics and Engineering,1995,123(1-4):355-370.
[110] Song C, Wolf J P. Consistent infinitesimal finite-element cell method: Three-dimensionalvector wave equation. International Journal for Numerical Methods in Engineering,1996,39(13):2189-2208.
[111] Wolf J P, Song C. Static stiffness of unbounded soil by finite-element method. Journal ofgeotechnical engineering,1996,122(4):267-273.
[112] Song C, Wolf J P. Consistent infinitesimal finite-element cell method for diffusion equationin unbounded medium. Computer Methods in Applied Mechanics and Engineering,1996,132(3-4):319-334.
[113] Deeks A J, Wolf J P. A virtual work derivation of the scaled boundary finite-element methodfor elastostatics. Computational Mechanics,2002,28(6):489-504.
[114] Song C, Wolf J P. The scaled boundary finite-element method: analytical solution infrequency domain. Computer Methods in Applied Mechanics and Engineering,1998,164(1-2):249-264.
[115] Song C, Wolf J P. Body loads in scaled boundary finite-element method. Computer Methodsin Applied Mechanics and Engineering,1999,180(1-2):117-135.
[116] Song C. A matrix function solution for the scaled boundary finite-element equation in statics.Computer Methods in Applied Mechanics and Engineering,2004,193(23-26):2325-2356.
[117] Song C. Weighted block-orthogonal base functions for static analysis of unbounded domains.Proceedings of the6th World Congress on Computational Mechanics. Beijing;2004. p615-620.
[118] Song C. A super-element for crack analysis in the time domain. International Journal forNumerical Methods in Engineering,2004,61(8):1332-1357.
[119] Song C, Wolf J P. Semi-analytical representation of stress singularities as occurring in cracksin anisotropic multi-materials with the scaled boundary finite-element method. Computers&Structures,2002,80(2):183-197.
[120] Song C, Vrcelj Z. Evaluation of dynamic stress intensity factors and T-stress using the scaledboundary finite-element method. Engineering Fracture Mechanics,2008,75(8):1960-1980.
[121] Yang Z. Fully automatic modelling of mixed-mode crack propagation using scaled boundaryfinite element method. Engineering Fracture Mechanics,2006,73(12):1711-1731.
[122] Ooi E T, Yang Z J. A hybrid finite element-scaled boundary finite element method for crackpropagation modelling. Computer Methods in Applied Mechanics and Engineering,2010,199(17–20):1178-1192.
[123] Ooi E T, Song C, Tin-Loi F, et al. Polygon scaled boundary finite elements for crackpropagation modelling. International Journal for Numerical Methods in Engineering,2012,91(3):319-342.
[124] Doherty J P, Deeks A J. Scaled boundary finite-element analysis of a non-homogeneouselastic half-space. International Journal for Numerical Methods in Engineering,2003,57(7):955-973.
[125] Hassanen M, El-Hamalawi A. Two-dimensional development of the dynamic coupledconsolidation scaled boundary finite-element method for fully saturated soils. Soil Dynamicsand Earthquake Engineering,2007,27(2):153-165.
[126] Deeks A J, Cheng L. Potential flow around obstacles using the scaled boundary finite-elementmethod. International Journal for Numerical Methods in Fluids,2003,41(7):721-741.
[127] Song C, Wolf J P. The scaled boundary finite element method—alias consistent infinitesimalfinite element cell method—for diffusion. International Journal for Numerical Methods inEngineering,1999,45(10):1403-1431.
[128] Wolf J P, Song C. Finite-Element Modelling of Unbounded Media. New York: Wiley;1996.
[129] Song C. Dynamic analysis of unbounded domains by a reduced set of base functions.Computer Methods in Applied Mechanics and Engineering,2006,195(33-36):4075-4094.
[130] Song C, Bazyar M H. A boundary condition in Padé series for frequency-domain solution ofwave propagation in unbounded domains. International Journal for Numerical Methods inEngineering,2007,69(11):2330-2358.
[131] Zhang X, Wegner J L, Haddow J B. Three-dimensional dynamic soil–structure interactionanalysis in the time domain. Earthquake Engineering&Structural Dynamics,1999,28(12):1501-1524.
[132] Wegner J L, Yao M M, Zhang X. Dynamic wave–soil–structure interaction analysis in thetime domain. Computers&Structures,2005,83(27):2206-2214.
[133] Yan J, Zhang C, Jin F. A coupling procedure of FE and SBFE for soil–structure interaction inthe time domain. International Journal for Numerical Methods in Engineering,2004,59(11):1453-1471.
[134] Lehmann L. An effective finite element approach for soil-structure analysis in thetime-domain. Structural Engineering and Mechanics,2005,21(4):437-450.
[135] Radmanovi B, Katz C. A high performance scaled boundary finite element method. IOPConference Series: Materials Science and Engineering,2010,10(1):012214.
[136]杜建国.基于SBFEM的大坝—库水—地基动力相互作用分析[博士学位论文].大连:大连理工大学2007.
[137] Wolf J P, Song C. The scaled boundary finite-element method-a primer: derivations.Computers&Structures,2000,78(1-3):191-210.
[138] Westergaard H M. Water pressure on dams during earthquakes. Transactions of the AmericanSociety of Civil Engineering,1933,98:418-433.
[139]杜修力,王进廷.动水压力及其对坝体地震反应影响的研究进展.水利学报,2001,(07):13-21.
[140] Zienkiewicz O C, Bettess P. Fluid-structure dynamic interaction and wave forces. Anintroduction to numerical treatment. International Journal for Numerical Methods inEngineering,1978,13(1):1-16.
[141] Kü ükarslan S, Coskun S B, TaskIn B. Transient analysis of dam-reservoir interactionincluding the reservoir bottom effects. Journal of Fluids and Structures,2005,20(8):1073-1084.
[142] Soares Jr D, von Estorff O, Mansur W J. Efficient non-linear solid–fluid interaction analysisby an iterative BEM/FEM coupling. International Journal for Numerical Methods inEngineering,2005,64(11):1416-1431.
[143] Seghir A, Tahakourt A, Bonnet G. Coupling FEM and symmetric BEM for dynamicinteraction of dam-reservoir systems. Engineering Analysis with Boundary Elements,2009,33(10):1201-1210.
[144] Lin G, Du J, Hu Z. Dynamic dam-reservoir interaction analysis including effect of reservoirboundary absorption. Science in China Series E: Technological Sciences,2007,50:1-10.
[145] Lin G, Wang Y, Hu Z. An efficient approach for frequency-domain and time-domainhydrodynamic analysis of dam–reservoir systems. Earthquake Engineering&StructuralDynamics,2012,41(13):1725-1749.
[146] Li S M. Diagonalization procedure for scaled boundary finite element method in modelingsemi-infinite reservoir with uniform cross-section. International Journal for NumericalMethods in Engineering,2009,80(5):596-608.
[147]王翔.高阶双渐近透射边界及其在大坝-库水相互作用中的应用[博士学位论文].北京:清华大学2010.
[148] Chopra A K, Chakrabarti P. Earthquake analysis of concrete gravity dams includingdam-water-foundation rock interaction. Earthquake Engineering&Structural Dynamics,1981,9(4):363-383.
[149] Hall J, Chopra A. Dynamic analysis of arch dams including hydrodynamic effects. Journal ofEngineering Mechanics,1983,109(1):149-167.
[150] Fenves G, Chopra A K. Earthquake analysis of concrete gravity dams including reservoirbottom absorption and dam-water-foundation rock interaction. Earthquake Engineering&Structural Dynamics,1984,12(5):663-680.
[151] Tan H, Chopra A K. Earthquake analysis of arch dams including dam-water-foundation rockinteraction. Earthquake Engineering&Structural Dynamics,1995,24(11):1453-1474.
[152] Birk C, Song C. A continued-fraction approach for transient diffusion in unbounded medium.Computer Methods in Applied Mechanics and Engineering,2009,198(33–36):2576-2590.
[153]杜建国,林皋.比例边界有限元子结构法研究.世界地震工程,2007,(01):67-72.
[154] Prempramote S. Development of High-Order Doubly Asymptotic Open Boundaries For WavePropagation In Unbounded Domains By Extending The Scaled Boundary Finite ElementMethod[Doctor]. Sydney: University of New South Wales2011.
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