动力问题的非时间步参数时间有限元法及弹性波传播数值模拟的格子法
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摘要
振动问题和波动问题是动力学研究领域的两大方面。由于结构形式和介质
的复杂性，对于这两个方面的问题所从事的解析研究取得的进展不是很迅速，
大部分问题还是要依靠数值方法来解决。合理、有效、计算精度高、耗时少，
且易于编制通用程序的数值计算方法的研究一直是动力学研究领域的热点。
本文基于弹性动力学问题中位移型Gurtin变分原理的思想，利用拉普拉斯
变换，将具有初值—边值的动力学问题变换成拉普拉斯空间上等价的边值问
题；并类似于原真实空间中解决边值问题的最小势能原理，在拉普拉斯空间构
造了等价边值问题的泛函。然后，将此拉普拉斯空间上的泛函恢复到原真实空
间；给出了适合于构造时间有限元方法的只含单重卷积的简单泛函表达式。这
种表达形式的简单泛函为时间有限元的构造创建了一个“平台”，由此可开展
时间有限元方法的一系列研究工作。
本文针对由Gurtin变分原理建立的时间有限元方法容易产生数值计算不
稳定这一问题，提出了非时间步参数的概念。在对时间域进行离散时，针对时
间域离散所选取的各种插值函数形式并采用一个或几个非时间步参数来配合，
能够有效地控制数值计算的稳定性，从而解决了基于Gurtin变分原理构造出的
时间有限元方法的稳定性不易实现这一问题。这种思想方法非常适用于结构动
力分析问题中的各种无条件稳定计算格式的构造。
本文利用给出的简单泛函表达式，并结合在时间域离散时加入非时间步参
数的思想，对一些时间域插值函数形式的时间有限元方法进行了研究。给出了
一些解决初值—边值混合动力学问题的时间有限元的方法。获得了一些有意义
的结果。
本文提出了一种对弹性波传播进行数值模拟的新的全离散方法—格子法。
在半无限介质中，该方法可以精细模拟任意复杂形状的自由表面和任意复杂的
内部介质界面，灵活地处理自由表面问题，自由表面条件自然引入，同时又不
需要特殊处理自由表面稳定性条件。该方法解决了其它差分方法在弹性波传播
数值模拟中较难处理的任意复杂形状的自由表面和内部介质界面问题。在空间
离散方面与有限元法类似，可以按照连续介质的形状及内部分界面任意剖分网
格，具有网格剖分的灵活性，这是一般规则网格差分方法不可比拟的，但是格
子法的计算量比有限元方法的计算量要少。格子法是一种显式计算方法，对计
算机内存需求的少，计算效率高，能够对更大范围的介质区域中的弹性波传播
进行数值模拟，这些方面比弹性波传播的有限元法要有优势。格子法是将节点
的运动微分方程进行积分处理，给出了被研究节点的运动方程的积分形式，该
方法不同于一般的用于弹性波传播数值模拟的有限差分法，一般的有限差分方
法是基于节点的动力学的运动微分方程。
在格子法思想的基础上，本文对三角形格子法进行了详尽的研究。对其计
os
一
算方法、稳定性和数值计算的频散特性做了全面的研究，同时给出了数值算例，
结果表明这一方法程序实现简单，计算精度高、耗时少，所需计算机内存较少，
且具有频散小，稳定性好，可适用于高泊松比材料的特点。该方法的计算耗时
和对计算机内存的需求与二阶规则网格交错差分法相当，但三角形格子法能精
细刻划和灵活处理自由表面和内部介质界面问题。通过对自由表面问题的具体
研究，得知格子法的自由表面边界多J牛是自然引入的，并且自由表面稳定性条
件是自动满足的，无须另外处理，这是格子法的一个显著特点一即可精细模拟
任意复杂形状的自由表面问题，又不需要特殊处理自由表面稳定性条件。
三角形格子法适用于高泊松比介质，但不太适用于液体介质，为了克服这
一不足，同时为了进一步研究计算效率高于原三角形格子法的方法，本文提出
了一种三角形和四边形的混合格子法。本文主耍是针对横观各向同性介质中弹
性波传播数值模拟进行了研究。在该方法中，对四边形格子内的应力作了均匀
性假定，只需要由四边形格子中心处场变量的空间导数计算每个格子中心处的
应力，在本构关系中并不含对空间变量的导数，因此，该方法适用于任意各向
异性介质模型惰况。这种混合格子法为复杂各向异性介质中弹性被传括数值模
拟提供了一种有力的工具。混合格子法中的四边形格子适用于模拟液体介质中
的弹性波传播，因此，混合格子法可用于模拟固－液混合体中的弹性波传播问
题；同时四边形格子的选用使得混合格子法能进一步减少计算耗时和对计算机
内存的需求。混合格子法同样可以精细刻划和灵活地处理自由表面问题，并且
自然满足复杂几何边界的自由表面稳定性条件。
本文最后基于格子法的思想，将格子法的研究工作推广到研究具有任意自
由表面形状的三维非均匀介质空间模型。文中给出了空间四面体和空间平行六
面体格子法以及由这两种格子法组合成的三维混合格子法，对空间四面体格子
法和空间平行六面体格子法的计算方法、稳定性条件以及相速度频散都做了详
尽的研究。采用三维混合格子?
Vibration and wave motion are the two research fields of the elastodynamics. The
development in the research of analytical solutions is not very rapid because of the
complexity of the structural patterns and the media, so moat of the problems have to be
investigated by numerical methods. Research of numerical simulation methods that are
reasonable, effective, high accuracy, low computational cost and ease to program is the
highlight field of the elastodynaniics.
Based on Gurtin variational principle of displacement model and making use of
Laplace transform, the initial value problems of elastodynamics can be change into the
~equivalent?boundary value problems in the Laplace space. Similar to the principle of
minimum potential energy used for the conventional boundary value problems, A kind
of functional is constructed in the Laplace space. Then, a simple expression of
functional with single convolution integral is given by returning functional in the
Laplace space back into the real space. Such a simple functional establishes a
損latform?for the constructions of time finite element methods, so a series of research
of time finite element methods could be developed.
A concept of non-time-step-parameter is put forward in dealing with the stability
problem that often arises in the numerical calculation for the initial value problem of
dynamics. One or several non-time-step-parameter can be used in the interpolation
function for the discretization in time domain, so the stability in the numerical
calculation could be controlled effectively. This idea can also be applied to the
construction of other unconditional stable algorithms for the structural analysis of
dynamic response.
Based on the simple expression of functional and combining the idea of
non-time-step-parameter used in the interpolation function for the discretization in time
domain, some time finite element methods are studied by using some type of
interpolation function in time domain. Some time finite clement methods that could be
U?
used to solve the initial-boundary value problems of dynamics are put forward and
obtain some significant conclusions.
A new fully numerical modeling algorithm for elastic wave propagation in
heterogeneous media is presented. The scheme is called grid method. The method can
accurately model the surface topography and inner curved interfaces by using an
unstructured mesh and it is flexible in dealing with the free-surface problem for the
semi-infinite media; The complex geometrical free-surface boundary conditions are
satisfied naturally and they are stable. Grid method can solve the surface topography
and inner curved interface problems that the other finite difference methods give rise
difficulti s in incorporating surface topography and curved interfaces by using regular
grids for the numerical modeling of elastic wave propagation. Similar to the finite
element method in discretization of numerical mesh, the grid method is very flexible in
incorporating surface topdgraphy and curved interfaces. The conventional regular grid
finite-difference methods are not comparable to the grid method in such an aspect. On
the other hand, the grid method need not requirements to calculate and store the global
stiffness matrix like the finite element methods and it is a kind of explicit scheme, so
the memory requirements and computational cost for the algorithm are much less than
that of finite e
的复杂性，对于这两个方面的问题所从事的解析研究取得的进展不是很迅速，
大部分问题还是要依靠数值方法来解决。合理、有效、计算精度高、耗时少，
且易于编制通用程序的数值计算方法的研究一直是动力学研究领域的热点。
本文基于弹性动力学问题中位移型Gurtin变分原理的思想，利用拉普拉斯
变换，将具有初值—边值的动力学问题变换成拉普拉斯空间上等价的边值问
题；并类似于原真实空间中解决边值问题的最小势能原理，在拉普拉斯空间构
造了等价边值问题的泛函。然后，将此拉普拉斯空间上的泛函恢复到原真实空
间；给出了适合于构造时间有限元方法的只含单重卷积的简单泛函表达式。这
种表达形式的简单泛函为时间有限元的构造创建了一个“平台”，由此可开展
时间有限元方法的一系列研究工作。
本文针对由Gurtin变分原理建立的时间有限元方法容易产生数值计算不
稳定这一问题，提出了非时间步参数的概念。在对时间域进行离散时，针对时
间域离散所选取的各种插值函数形式并采用一个或几个非时间步参数来配合，
能够有效地控制数值计算的稳定性，从而解决了基于Gurtin变分原理构造出的
时间有限元方法的稳定性不易实现这一问题。这种思想方法非常适用于结构动
力分析问题中的各种无条件稳定计算格式的构造。
本文利用给出的简单泛函表达式，并结合在时间域离散时加入非时间步参
数的思想，对一些时间域插值函数形式的时间有限元方法进行了研究。给出了
一些解决初值—边值混合动力学问题的时间有限元的方法。获得了一些有意义
的结果。
本文提出了一种对弹性波传播进行数值模拟的新的全离散方法—格子法。
在半无限介质中，该方法可以精细模拟任意复杂形状的自由表面和任意复杂的
内部介质界面，灵活地处理自由表面问题，自由表面条件自然引入，同时又不
需要特殊处理自由表面稳定性条件。该方法解决了其它差分方法在弹性波传播
数值模拟中较难处理的任意复杂形状的自由表面和内部介质界面问题。在空间
离散方面与有限元法类似，可以按照连续介质的形状及内部分界面任意剖分网
格，具有网格剖分的灵活性，这是一般规则网格差分方法不可比拟的，但是格
子法的计算量比有限元方法的计算量要少。格子法是一种显式计算方法，对计
算机内存需求的少，计算效率高，能够对更大范围的介质区域中的弹性波传播
进行数值模拟，这些方面比弹性波传播的有限元法要有优势。格子法是将节点
的运动微分方程进行积分处理，给出了被研究节点的运动方程的积分形式，该
方法不同于一般的用于弹性波传播数值模拟的有限差分法，一般的有限差分方
法是基于节点的动力学的运动微分方程。
在格子法思想的基础上，本文对三角形格子法进行了详尽的研究。对其计
os
一
算方法、稳定性和数值计算的频散特性做了全面的研究，同时给出了数值算例，
结果表明这一方法程序实现简单，计算精度高、耗时少，所需计算机内存较少，
且具有频散小，稳定性好，可适用于高泊松比材料的特点。该方法的计算耗时
和对计算机内存的需求与二阶规则网格交错差分法相当，但三角形格子法能精
细刻划和灵活处理自由表面和内部介质界面问题。通过对自由表面问题的具体
研究，得知格子法的自由表面边界多J牛是自然引入的，并且自由表面稳定性条
件是自动满足的，无须另外处理，这是格子法的一个显著特点一即可精细模拟
任意复杂形状的自由表面问题，又不需要特殊处理自由表面稳定性条件。
三角形格子法适用于高泊松比介质，但不太适用于液体介质，为了克服这
一不足，同时为了进一步研究计算效率高于原三角形格子法的方法，本文提出
了一种三角形和四边形的混合格子法。本文主耍是针对横观各向同性介质中弹
性波传播数值模拟进行了研究。在该方法中，对四边形格子内的应力作了均匀
性假定，只需要由四边形格子中心处场变量的空间导数计算每个格子中心处的
应力，在本构关系中并不含对空间变量的导数，因此，该方法适用于任意各向
异性介质模型惰况。这种混合格子法为复杂各向异性介质中弹性被传括数值模
拟提供了一种有力的工具。混合格子法中的四边形格子适用于模拟液体介质中
的弹性波传播，因此，混合格子法可用于模拟固－液混合体中的弹性波传播问
题；同时四边形格子的选用使得混合格子法能进一步减少计算耗时和对计算机
内存的需求。混合格子法同样可以精细刻划和灵活地处理自由表面问题，并且
自然满足复杂几何边界的自由表面稳定性条件。
本文最后基于格子法的思想，将格子法的研究工作推广到研究具有任意自
由表面形状的三维非均匀介质空间模型。文中给出了空间四面体和空间平行六
面体格子法以及由这两种格子法组合成的三维混合格子法，对空间四面体格子
法和空间平行六面体格子法的计算方法、稳定性条件以及相速度频散都做了详
尽的研究。采用三维混合格子?
Vibration and wave motion are the two research fields of the elastodynamics. The
development in the research of analytical solutions is not very rapid because of the
complexity of the structural patterns and the media, so moat of the problems have to be
investigated by numerical methods. Research of numerical simulation methods that are
reasonable, effective, high accuracy, low computational cost and ease to program is the
highlight field of the elastodynaniics.
Based on Gurtin variational principle of displacement model and making use of
Laplace transform, the initial value problems of elastodynamics can be change into the
~equivalent?boundary value problems in the Laplace space. Similar to the principle of
minimum potential energy used for the conventional boundary value problems, A kind
of functional is constructed in the Laplace space. Then, a simple expression of
functional with single convolution integral is given by returning functional in the
Laplace space back into the real space. Such a simple functional establishes a
損latform?for the constructions of time finite element methods, so a series of research
of time finite element methods could be developed.
A concept of non-time-step-parameter is put forward in dealing with the stability
problem that often arises in the numerical calculation for the initial value problem of
dynamics. One or several non-time-step-parameter can be used in the interpolation
function for the discretization in time domain, so the stability in the numerical
calculation could be controlled effectively. This idea can also be applied to the
construction of other unconditional stable algorithms for the structural analysis of
dynamic response.
Based on the simple expression of functional and combining the idea of
non-time-step-parameter used in the interpolation function for the discretization in time
domain, some time finite element methods are studied by using some type of
interpolation function in time domain. Some time finite clement methods that could be
U?
used to solve the initial-boundary value problems of dynamics are put forward and
obtain some significant conclusions.
A new fully numerical modeling algorithm for elastic wave propagation in
heterogeneous media is presented. The scheme is called grid method. The method can
accurately model the surface topography and inner curved interfaces by using an
unstructured mesh and it is flexible in dealing with the free-surface problem for the
semi-infinite media; The complex geometrical free-surface boundary conditions are
satisfied naturally and they are stable. Grid method can solve the surface topography
and inner curved interface problems that the other finite difference methods give rise
difficulti s in incorporating surface topography and curved interfaces by using regular
grids for the numerical modeling of elastic wave propagation. Similar to the finite
element method in discretization of numerical mesh, the grid method is very flexible in
incorporating surface topdgraphy and curved interfaces. The conventional regular grid
finite-difference methods are not comparable to the grid method in such an aspect. On
the other hand, the grid method need not requirements to calculate and store the global
stiffness matrix like the finite element methods and it is a kind of explicit scheme, so
the memory requirements and computational cost for the algorithm are much less than
that of finite e
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Boore D.M., Finite-difference methods for seismic wave propagation in heterogeneous materials, in Methods in computational physics, 11: B.A. Bolt, ed.,Academic Press, Inc., 1972
Bruch J.C. and Zyvoloski G., Transient two-dimensional heat conduction problems solved by the finite element method, Internal. J. Numer. Methods Engrg. 1974, 8, 481-494
Cella A., Lucchesi M. and Pasquinelli G., Space-time elements for the shock wave propagation problem, Internal. J. Numer. Methods Engrg. 1980, 15, 1475-1488
Chapman C.H., A new method for computing seismograms, Geophys. J. Roy. astr.Soc., 1978, 54, 481-518
Chapman C.H. and Drummond R., Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory, Bull. Seism. Soc. Am., 1982, 72: S277-S317
Chapman C.H., Ray theory and its extentions: WKBJ and Maslov seismograms, J.Geophy., 1985, 58, 27-43
Cerveny V., Molotkov I.A. and Psencik I., Ray Method in Seismology, Praha, Karlova Universita, 1977
Cerveny V. and Psencik I., Gaussian beams in two—dimensional elastic inhomoge-neous media. Geophys. J. Roy. Astr. Soc.,1983,72 : 417-433
Cerveny V., Ray synthetic seismograms for complex two dimensional and three dimensional structures, J. Geophy. 1985, 58, 2-26
Cerveny V., Popov M.M. and Psencik I., Gaussian beam synthetic seismograms, J. Geophy., 1985b, 58, 44-72
Chung K.S., The fourth-dimension concept in the finite element analysis of transient heat transfer problems, Internal. J. Numer.Methods Engrg. 1981, 17, 315-325
Cook Robert D., Concepts and Applications of Finite Element Analysis, Second Edition, John Wiley & Sons, 1981
Dablain M.A., The application of high-order differenceing to the scalar wave equation, Geophysics, 1986, 51, 54-66
Fried I., Finite element analysis of time-dependent phenomena, AIAA J. 1969, 7, 1170-1173
Fried I., Numerical Solution of Differential Equations, Academic Press, New York, 1979
Ghaboussi J. and Wilson E.L., Eng. Mech. Div. ASCE., 1972, 98, 947-963
Graves R.W., Simulating Seismic Wave Propagation in 3D Elastic Media Using Staggered-Grid Finite Differences, Bull. Seism. Soci. Am., 1996, 86(4), 1091-1106
Gurtin M.E., Variational Principles for Linear Elastodynamics, Archive for Rational Mechanics and Analysis, 1964, 16, 34～50
Gurtin M.E., Encyclopedia of physics, Via/2, Mechanics of solids, 1972, Ⅱ,223～232
Haskell N.A., The dispersion of surface waves in multilayered media, Bull. Seism.Soc. Am., 1953, 43, 17-34
Hughes T.J.R. and Hulbert G.M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comp Methods Appl. Mech. Eng., 1988, 66, 339-363
Hulbert G. M. and Hughes T.J.R., Space-time finite element methods for secondorder hyperbolic equations, Comp Methods Appl. Mech. Eng., 1990, 84, 327-348
Hulbert G. M., Time finite element methods for structural dynamics, Int. J. Numer.Methods Eng. 1992, 33, 307-331
Kelly K.R., Ward R.W., Treitel S. and Alford R.M., Synthetic seismograms: a finite difference approach, Geophysics, 1976, 41, 2-27
Kennett B.L.N. and Kerry N.J., Seismic waves in a stratified half-space, Geophys J., 1979, 57, 557-583
Kosloff D. and Baysal E., Forward modeling by a Fourier method, Geophysics, 1982, 47, 1402-1312
Kosloff D., Reshel M. and Lowenthal D., Elastic wave calculations by the Fourier method, Bull. Seism. Soc. Am., 1984, 74, 875-891
Kosloff D., Kessler D., Filho A.Q., Tessmer E., Behle A. and Strahilevitz R., Solution of the equations of the dynamic elasticity by a Chebyshev spectral method, Geophysics, 1990, 55(4), 734-748
Kummer B and Behle A., Second-order finite-difference modeling of SH-wave propagation in laterally inhomogeneous media, Bull. Seism. Soc. Am., 1982, 72, 793-808
Levander A.R., Fourth-order finite-difference P-SV seismograms, Geophysics, 1988, 53, 1425～1436
Lewis D.L., Lund J. and Bowers K.L., The space-time Sine-Galerkin method for parabolic problems, Internal. J. Numer. Methods Engrg. 1987, 24, 1629-1644
Liu D.K., Gai B.Z. and Tao G.Y., Applications of the Method of Complex Functions to Dynamic Stress Concentrations, Wave Motion, 1982, 4, 293-304
Liu D.K. and Han F., Scattering of plane SH-wave by a cylindrical canyon of arbitrary shape in anisotropic media, ACTA Mechenica Sinica, 1990, 6(3)
Liu D.K. and Han F., Scattering of plane SH-wave on cylindrical canyon of arbitrary shape, Soil Dynamics and Earthquake Engineering, 1991, 10(5), 249-255
Lysmer J., and Drake L.A., A finite element method for seismology, in Methods in Computational Physics, Vol. 11, ed. Bolt B.A., Academic Press, New York, 1972,181-216
Magnier S.-A., Mora P.and Tarantola A., Finite differences on minimal grids, Geophysics, 1994, 59, 1435-1443
Moczo P., Finite-difference technique for SH waves in 2-D media using irregular grids: application to the seismic response problem, Geophys. J.Int., 1989, 99, 321-329
Morse P.M. and Feshbach H., Methods of theoretical physics, 1953
Oden J.T., A general theory of finite elements Ⅰ. Topological Considerations, Internal. J.Numer. Methods in Engrg., 1969, 1,205-221
Oden J.T., A general theory of finite elements Ⅱ. Applications, Internal. J. Numer. Methods in Engrg., 1969, 1,247-259
Pao Y.H. and Mow C.C., Diffraction of Elastic Waves and Dynamic Stress Concerntrations, Crane and Russak Company Inc., New York, 1973
Pekeris C.L., The Seismic Surface Pulse, Proc. Nat. Acad. Sic., 1955, 41, 469-480
Peng J.S., R.W. Lewis and J.Y. Zhang, A Semi-analytical Approach for Solving Forced Vibration Problems Based on A Convolution-type Variational Principle, Computers & Structures, 1996, 59(1), 167-177
Peng J.S., J.Y. Zhang and J. Liu, Formulation of a Semi-Analytical Approach Based on Gurtin Variational Principle for Dynamic Response of General Thin Plates, Applied Mathematics and Mechanics, 1997, 18(11), 1059-1063
Peters D. A. and Izadpanah A. P., hp-version finite elements for the space-time domain, Comput. Mech. 1988, 3, 73-88
Rice J.M., and Sadd M.H., Propagation and Scattering of SH Waves in Semi-Infinite Domains Using a Time-Dependent Boundary Element Method, ASME. J. Appl. Mech., 1984, 51, 641-645
Riff R. and Baruch M, Time Finite Element Discretization of Hamilton's Law of Varying Action, AIAA Journal, 1984, 22(9), 1310-1318
Seybert A.F., Soenarko B., Rizzo F.J., et al, Application of BIE Method to Sound Radiation Problems Using an Isoparametric Element, the ASME Winter Annual Meeting, Phoenix, USA, 1982
Shaw P.P., Boundary Integral Equation Methods Applied to Wave Problems In "Development in BEM I"eds Banerjee P.K. and Butterfield R., Appl. Sci., London, 1979, 121-153
Simkins T.E., Finite Elements for Initial Value Problems in Dynamics, AIAA Journal, Vol. 1981, 19(10), 1357-1362
Smith W.D., The application of finite—element analysis to body wave Propaga-tion problems. Geophys. J. Roy. Astr. Soc., 1975,42, 747～768
Tessmer E., Kosloff D. and Behle A., Elastic wave propagation simulation in the presence of surface topography, Geophys.J.Int., 1992, 108, 621-632
Tessmer E. and Kosloff D., 3-D elastic modeling with surface topography by a Chebyshev spectral method, Geophysics, 1994, 59, 464—473
Timoshenko s., Vibration problems in engineerings, 3rd ed., 1955
Trifunac M.D., Scattering of plane SH wave by a semi-cylindrical canyon, Int. J. Earthquake Engineering and Structure Dynamics, 1973, 1,267-281
Virieux J., SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics, 1984, 49, 1933-1957
Virieux J., P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics, 1986, 51, 889-901
Warburton G.B., The Dynamical Behaviour of Structure, Second Edition,Oxford, Pergamon Press, 1976
Wilson E.L. and Nickellr E., Nuc. Eng. Des., 1969, 4, 276-286
Yu J.R. and Hsu T.R., Analysis of heat conduction in solid by space-time finite element method, Internal. J. Numer. Methods in Engrg, 1985, 21(12), 2001-2012
Zhang Jianfeng, Quadrangle-grid velocity-stress finite-difference method for elastic-wave-propagation simulation. Geophys. J.Int., 1997, 131, 127-134
Zhang Jianfeng and Li Youming, Numerical simulation of elastic wave propagation in inhomogeneous media. Wave Motion, 1997,25: 109-125
Zienkiewicz O.C., The Finite Element Method, Third Edition, London, McGRAW-HILL Book Company(UK) Limited, 1977, 569-570
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Alford R.M., Kelly K.R. and Boore D.M., Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics, 1974, 39, 834-842
Alterman Z. and Karal F.C., Propagation of elastic waves in layered media by finite difference methods, Bull. Seism. Soc. Am., 1968, 58, 367-398
Alterman Z. and Loewenthal D., Computer generated seismograms, in Methods in Computational Physics, Vol.12, Academic Press, New York, 1972
Argyris J. H. and Scharpf D. W., Finite elements in Time and Space, Nucl. Engrg.Des., 1969, 10, 456-464
Atluri S., AIAA Journal, 1973, 11, 1028-1031
Bailey C.D., The Method of Ritz Applied to the equation of Hamilton, Comp Methods Appl. Mech. Eng, 1976, 7, P235
Bailey C.D., On a More Precise Statement of Hamilton's Principie, Foundations of Physics, 1981, 11(3,4), 279-296
Bailey C.D., Hamilton's Law or Hamilton's Principle: A Response to Ulvi Yurtsever, Foundations of Physics, 1983, 13(5), 539-544
Bajer C., Triangular and tetrahedral space-time finite elements in vibrational analysis, Internal. J. Numer. Methods Engrg. 1986, 23, 2031-2048
Baruch M. and Riff R., Hamilton's principle. Hamilton's Law-6" correct formulations, AIAA J. 1982, 20, 687-692
Bathe K.J., Wilson E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, New Jersey, 1976
Bayliss A., Jordan K.E., Lemesurier B.J. and Turkel E., A fourth-order accurate finite-difference scheme for the computation of elastic wave, Bull. Seism. Soc. Am.,1986, 76, 1115-1132
Boore D.M., Finite-difference methods for seismic wave propagation in heterogeneous materials, in Methods in computational physics, 11: B.A. Bolt, ed.,Academic Press, Inc., 1972
Bruch J.C. and Zyvoloski G., Transient two-dimensional heat conduction problems solved by the finite element method, Internal. J. Numer. Methods Engrg. 1974, 8, 481-494
Cella A., Lucchesi M. and Pasquinelli G., Space-time elements for the shock wave propagation problem, Internal. J. Numer. Methods Engrg. 1980, 15, 1475-1488
Chapman C.H., A new method for computing seismograms, Geophys. J. Roy. astr.Soc., 1978, 54, 481-518
Chapman C.H. and Drummond R., Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory, Bull. Seism. Soc. Am., 1982, 72: S277-S317
Chapman C.H., Ray theory and its extentions: WKBJ and Maslov seismograms, J.Geophy., 1985, 58, 27-43
Cerveny V., Molotkov I.A. and Psencik I., Ray Method in Seismology, Praha, Karlova Universita, 1977
Cerveny V. and Psencik I., Gaussian beams in two—dimensional elastic inhomoge-neous media. Geophys. J. Roy. Astr. Soc.,1983,72 : 417-433
Cerveny V., Ray synthetic seismograms for complex two dimensional and three dimensional structures, J. Geophy. 1985, 58, 2-26
Cerveny V., Popov M.M. and Psencik I., Gaussian beam synthetic seismograms, J. Geophy., 1985b, 58, 44-72
Chung K.S., The fourth-dimension concept in the finite element analysis of transient heat transfer problems, Internal. J. Numer.Methods Engrg. 1981, 17, 315-325
Cook Robert D., Concepts and Applications of Finite Element Analysis, Second Edition, John Wiley & Sons, 1981
Dablain M.A., The application of high-order differenceing to the scalar wave equation, Geophysics, 1986, 51, 54-66
Fried I., Finite element analysis of time-dependent phenomena, AIAA J. 1969, 7, 1170-1173
Fried I., Numerical Solution of Differential Equations, Academic Press, New York, 1979
Ghaboussi J. and Wilson E.L., Eng. Mech. Div. ASCE., 1972, 98, 947-963
Graves R.W., Simulating Seismic Wave Propagation in 3D Elastic Media Using Staggered-Grid Finite Differences, Bull. Seism. Soci. Am., 1996, 86(4), 1091-1106
Gurtin M.E., Variational Principles for Linear Elastodynamics, Archive for Rational Mechanics and Analysis, 1964, 16, 34～50
Gurtin M.E., Encyclopedia of physics, Via/2, Mechanics of solids, 1972, Ⅱ,223～232
Haskell N.A., The dispersion of surface waves in multilayered media, Bull. Seism.Soc. Am., 1953, 43, 17-34
Hughes T.J.R. and Hulbert G.M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comp Methods Appl. Mech. Eng., 1988, 66, 339-363
Hulbert G. M. and Hughes T.J.R., Space-time finite element methods for secondorder hyperbolic equations, Comp Methods Appl. Mech. Eng., 1990, 84, 327-348
Hulbert G. M., Time finite element methods for structural dynamics, Int. J. Numer.Methods Eng. 1992, 33, 307-331
Kelly K.R., Ward R.W., Treitel S. and Alford R.M., Synthetic seismograms: a finite difference approach, Geophysics, 1976, 41, 2-27
Kennett B.L.N. and Kerry N.J., Seismic waves in a stratified half-space, Geophys J., 1979, 57, 557-583
Kosloff D. and Baysal E., Forward modeling by a Fourier method, Geophysics, 1982, 47, 1402-1312
Kosloff D., Reshel M. and Lowenthal D., Elastic wave calculations by the Fourier method, Bull. Seism. Soc. Am., 1984, 74, 875-891
Kosloff D., Kessler D., Filho A.Q., Tessmer E., Behle A. and Strahilevitz R., Solution of the equations of the dynamic elasticity by a Chebyshev spectral method, Geophysics, 1990, 55(4), 734-748
Kummer B and Behle A., Second-order finite-difference modeling of SH-wave propagation in laterally inhomogeneous media, Bull. Seism. Soc. Am., 1982, 72, 793-808
Levander A.R., Fourth-order finite-difference P-SV seismograms, Geophysics, 1988, 53, 1425～1436
Lewis D.L., Lund J. and Bowers K.L., The space-time Sine-Galerkin method for parabolic problems, Internal. J. Numer. Methods Engrg. 1987, 24, 1629-1644
Liu D.K., Gai B.Z. and Tao G.Y., Applications of the Method of Complex Functions to Dynamic Stress Concentrations, Wave Motion, 1982, 4, 293-304
Liu D.K. and Han F., Scattering of plane SH-wave by a cylindrical canyon of arbitrary shape in anisotropic media, ACTA Mechenica Sinica, 1990, 6(3)
Liu D.K. and Han F., Scattering of plane SH-wave on cylindrical canyon of arbitrary shape, Soil Dynamics and Earthquake Engineering, 1991, 10(5), 249-255
Lysmer J., and Drake L.A., A finite element method for seismology, in Methods in Computational Physics, Vol. 11, ed. Bolt B.A., Academic Press, New York, 1972,181-216
Magnier S.-A., Mora P.and Tarantola A., Finite differences on minimal grids, Geophysics, 1994, 59, 1435-1443
Moczo P., Finite-difference technique for SH waves in 2-D media using irregular grids: application to the seismic response problem, Geophys. J.Int., 1989, 99, 321-329
Morse P.M. and Feshbach H., Methods of theoretical physics, 1953
Oden J.T., A general theory of finite elements Ⅰ. Topological Considerations, Internal. J.Numer. Methods in Engrg., 1969, 1,205-221
Oden J.T., A general theory of finite elements Ⅱ. Applications, Internal. J. Numer. Methods in Engrg., 1969, 1,247-259
Pao Y.H. and Mow C.C., Diffraction of Elastic Waves and Dynamic Stress Concerntrations, Crane and Russak Company Inc., New York, 1973
Pekeris C.L., The Seismic Surface Pulse, Proc. Nat. Acad. Sic., 1955, 41, 469-480
Peng J.S., R.W. Lewis and J.Y. Zhang, A Semi-analytical Approach for Solving Forced Vibration Problems Based on A Convolution-type Variational Principle, Computers & Structures, 1996, 59(1), 167-177
Peng J.S., J.Y. Zhang and J. Liu, Formulation of a Semi-Analytical Approach Based on Gurtin Variational Principle for Dynamic Response of General Thin Plates, Applied Mathematics and Mechanics, 1997, 18(11), 1059-1063
Peters D. A. and Izadpanah A. P., hp-version finite elements for the space-time domain, Comput. Mech. 1988, 3, 73-88
Rice J.M., and Sadd M.H., Propagation and Scattering of SH Waves in Semi-Infinite Domains Using a Time-Dependent Boundary Element Method, ASME. J. Appl. Mech., 1984, 51, 641-645
Riff R. and Baruch M, Time Finite Element Discretization of Hamilton's Law of Varying Action, AIAA Journal, 1984, 22(9), 1310-1318
Seybert A.F., Soenarko B., Rizzo F.J., et al, Application of BIE Method to Sound Radiation Problems Using an Isoparametric Element, the ASME Winter Annual Meeting, Phoenix, USA, 1982
Shaw P.P., Boundary Integral Equation Methods Applied to Wave Problems In "Development in BEM I"eds Banerjee P.K. and Butterfield R., Appl. Sci., London, 1979, 121-153
Simkins T.E., Finite Elements for Initial Value Problems in Dynamics, AIAA Journal, Vol. 1981, 19(10), 1357-1362
Smith W.D., The application of finite—element analysis to body wave Propaga-tion problems. Geophys. J. Roy. Astr. Soc., 1975,42, 747～768
Tessmer E., Kosloff D. and Behle A., Elastic wave propagation simulation in the presence of surface topography, Geophys.J.Int., 1992, 108, 621-632
Tessmer E. and Kosloff D., 3-D elastic modeling with surface topography by a Chebyshev spectral method, Geophysics, 1994, 59, 464—473
Timoshenko s., Vibration problems in engineerings, 3rd ed., 1955
Trifunac M.D., Scattering of plane SH wave by a semi-cylindrical canyon, Int. J. Earthquake Engineering and Structure Dynamics, 1973, 1,267-281
Virieux J., SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics, 1984, 49, 1933-1957
Virieux J., P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics, 1986, 51, 889-901
Warburton G.B., The Dynamical Behaviour of Structure, Second Edition,Oxford, Pergamon Press, 1976
Wilson E.L. and Nickellr E., Nuc. Eng. Des., 1969, 4, 276-286
Yu J.R. and Hsu T.R., Analysis of heat conduction in solid by space-time finite element method, Internal. J. Numer. Methods in Engrg, 1985, 21(12), 2001-2012
Zhang Jianfeng, Quadrangle-grid velocity-stress finite-difference method for elastic-wave-propagation simulation. Geophys. J.Int., 1997, 131, 127-134
Zhang Jianfeng and Li Youming, Numerical simulation of elastic wave propagation in inhomogeneous media. Wave Motion, 1997,25: 109-125
Zienkiewicz O.C., The Finite Element Method, Third Edition, London, McGRAW-HILL Book Company(UK) Limited, 1977, 569-570