soil structure interaction analysis methods - state of art ... · pdf filesoil structure...

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 2, No 1, 2011

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

Received on July 2011 published on November 2011 176

Soil structure interaction analysis methods - State of art-Review Siddharth G. Shah

1, Solanki C.H.

2, Desai J.A.

3

1, Research Scholar

2 Associate Professor,

SV National Institute of Technology Surat, Gujarat, India.

1 Asst. Prof, 3 Director & Professor

C G Patel Institute of Technology, Uka Tarsadia University, Bardoli, Gujarat, India.

[email protected]

doi:10.6088/ijcser.00202010101

ABSTRACT

Soil flexibility has to be considered in the analysis of massive structures to avoid failure

and ensure safe service. Post failure analysis of massive structures realized the

importance of SSI-soil structure interaction. In the literature as many as half dozen

methods are available but researches and designers are not clear about the history &

development in this field. Current paper attempts to review the stat of art about soil

structure interaction analysis methods. The review reveals that for simple analysis direct

methods-Global procedures are physible while for non linear analysis substructure

method is effective and simple to apply.

Keywords: Soil structure interaction, global procedure, local procedure, thin layer

method, boundary layer method, finite element method, transmitting boundary, infinite

element, absorbing boundary.

1. Introduction

Soil conditions have a great deal to do with damage to structures during earthquakes.

Foundation motions deviate from free-field motions for two principal reasons: (1) the

imposition of stiff foundation systems on (or in) a geologic medium experiencing

nonuniform shaking will result in foundation motions being reduced relative to those in

the free-field and (2) inertial forces developed in the structure will cause base shear and

moment, which in turn will induce relative foundation/free-field motions due to the

foundation compliance. These phenomena are commonly termed Soil-Structure

Interaction (SSI). The general SSI problem is subdivided into kinematic SSI, which is

concerned with first factor identified above, and inertial SSI, which is concerned with the

second factor. Depending mainly on the relative stiffness of the soil and structure, SSI

can have an impact on the response of the structure.

Over the last four decades, various methods have been proposed for the solution of wave

equations in unbounded domains. This paper summarizes briefly the existing literature

with particular emphasis on the dynamic soil-structure interaction. In general, these

approaches fall into two broad categories: global and local procedures. Global procedures

in Section 2 are divided into sub-sections presenting the boundary element method, thin

layer method, exact non-reflecting boundary conditions and the scaled boundary finite-

Soil structure interaction analysis methods - State of art-Review

Siddharth G. Shah et.al.,

International Journal of Civil and Structural Engineering

Volume 2 Issue 1 2011

177

element method. Local procedures in Section 3 are grouped as transmitting boundary

conditions, infinite elements and absorbing layers. An investigation on the capacity of

popular commercial finite-element packages for modeling unbounded domains is

presented in Section 4.

2. Global procedures

Global procedures have been proposed for the dynamic analysis of unbounded domains

from 1970s. They are constructed through integral operators with respect to space or time

leading to globality in space or time. This is consistent with the physical nature of wave

propagation. the global procedures are generally rigorous. Due to their high accuracy,

they can be placed immediately beyond the structure-media interface leading to a

reduction of the number of degrees of freedom in the bounded domain and thus total

computational time. However, because of the spatially and temporally global formulation

the computational effort in global procedures increases with the size of the problem.

Generally speaking, extension to unbounded domains of general anisotropic and non-

homogeneous media with arbitrary geometry increases the complexity of the global

procedures.

2.1 Boundary Element Method

The well known boundary element method based on boundary integral equations presents

an attractive computational framework especially for problems involving singularity and

unbounded domains. The basic idea of this method is to formulate the equation of motion

of the unbounded domain in the form of an integral equation instead of a differential

equation. Finally, this integral equation is solved numerically. The method has been

applied in various areas of engineering and science. A detailed literature on the

formulation of the method and its applications in different fields is addressed in the book

by Brebbia et al. (1984). A full literature review on the method is beyond the scope of

this paper.

Firstly, essential features of the method are summarized. Secondly, a brief review of the

recent researches for the dynamic soil-structure interaction analysis with the particular

emphasis on anisotropic and non-homogeneous soils is presented. Finally, new

developments for improving the efficiency of the method are summarized.

In the boundary element method (Beskos, 1987, 1997; Hall and Oliveto, 2003) (Figure

2.1) rather than throughout the interior domain (Ω), as its name suggests, only the

boundary (Γ) is discretized resulting in a reduction of the spatial discretization by one.

For instance, for a three-dimensional problem only a two-dimensional surface has to be

addressed. Surface and line elements are used to discretize the structure-soil interface in

three- and two-dimensional problems, respectively. Over each element the functions

under consideration, for example displacement in elastodynamics, can vary in much the

same manner as in finite elements. They are interpolated in terms of the nodal values by

the so-called shape functions in a local coordinate system. The weighting functions are

chosen as fundamental solutions, or Green’s functions, satisfying rigorously the radiation

condition at infinity and the governing equations. Generally speaking, the exact solution




Volume 2 Issue 1 2011 178

of the governing equation subjected to a concentrated unit load in an infinite domain

represents the fundamental solution.

Figure 2.1: Spatial discretization in boundary element method for an unbounded domain

The boundary element method is applicable to problems for which the fundamental

solutions can be calculated. Determining the fundamental solutions for general

anisotropic media is very complicated and sometimes even impossible. Applying the

fundamental solutions and the method of weighted residual to the governing equations

followed by integration by parts, yields the boundary integral equation. For example, a

typical boundary integral equation in the frequency domain is in the form

=

+

Where,

and are the corresponding displacements and tractions respectively,

describing the fundamental solutions.

B j (ω) is the body load.

U j (ω) and Pj (ω) are the boundary displacement and tractions respectively.

As can be observed in Eq. (2.1), only the contribution of body load B j (ω) is represented

by an integration over the domain Ω. Two other terms are expressed by the integration

over the boundary Γ. The boundary integral equation (Eq. (2.1)) is discretized into

boundary elements. Generally, the integrals involving fundamental solutions are

performed by using numerical integration schemes. The fundamental solutions are

generally the response to a concentrated force. They are singular. Special techniques have

to be applied to compute their integrals. Having calculated the integrations and

discretized Eq. (2.1) for different nodes, a global system of linear algebraic equations is





derived. After imposing the prescribed boundary conditions, the system of linear

algebraic equations can be solved by direct or iterative solvers for unknown nodal values.

The coefficient matrices are fully populated and non-symmetric. This leads to a high

storage requirement and computational costs. Boundary integral equation can also be

formulated in the time domain.

=

+

Convolution integrals are encountered. Evaluation of these convolution integrals leads to

a large amount of computational costs. The fundamental solutions for full-plane or full-

space are derived without satisfying any boundary condition. Using these fundamental

solutions, one has to discretize the free surfaces (half-planes or half-spaces) (Figure 2.2a).

In the boundary element method, the interface between two different materials when non-

homogeneity is encountered, has to be discretized (Figure 2.2a) too. Obviously, the

infinite extent of these boundaries requires a special treatment in any numerical scheme.

The most straightforward approach consists in restricting the discretization to a finite part

of the boundary, thereby truncating the boundary integrals leading to modeling errors. To

reduce the computational efforts, in addition to fundamental solutions for full-space or

full-plane in three- and two-dimensional problems, fundamental solutions for half-space

or half-plane may also be determined. These fundamental solutions satisfy the boundary

conditions on the free surfaces of half-plane or half-space. Doing so, the free-surfaces

need not to be discretized reducing considerably the amount of numerical work involved

in the solution of the problem (Figure 2.2b). An infinite boundary element has also been

developed by Zhang et al. (1989, 1991) for dynamic problems of three-dimensional half-

space by making use of the transformation technique from an infinite domain to a finite

domain. A decay-type shape function based on the asymptotic behavior of fundamental

solutions for three-dimensional dynamic problems was used in the element. The coupled

finite-element/boundary element method has been used in wave propagation problems as

well. In this case, the bounded domain is modeled by finite elements and the unbounded

domain by boundary element method.

Figure 2.2: (a) boundary discretization in the boundary element method when

fundamental solutions for full-plane are used and the discretization of interface between

different materials when non-homogeneity is encountered; (b) boundary discretization

when fundamental solutions for half-plane are used.





The earliest works on applying boundary element method to soil-structure interaction

problems address the homogeneous isotropic unbounded domains. Luco and Westmann

(1971) studied the dynamic response of a rigid circular footing on the surface of an

isotropic half-space using the mixed boundary value problem. Wong and Luco (1976)

computed the dynamic compliance of a surface rigid mass-less foundation of arbitrary

shape on an elastic half-space by dividing the soil-foundation interface into rectangular

elements. They used the integration of Lamb’s point load solution to obtain the relation

between the tractions over an element and the displacements on the soil surface. It was

actually a boundary element method with a half-space fundamental solution. Dominguez

(1978) applied for the first time the boundary element method to dynamic soil-structure

interaction problems. Using a frequency-domain formulation he obtained the dynamic

stiffness of rectangular foundations resting on, or embedded in, a viscoelastic half-space.

From the early 1970s the boundary integral methods have been used for the dynamic

analysis of soil-structure interaction problems in anisotropic soils. A brief literature

review for the frequency-domain analysis is included here. Freedman and Keer (1972)

studied the response of a body resting on the surface of a transversely isotropic half-plane

and presented exact solutions in terms of dual integral equations. Wang and Rajapakse

(1991) studied the dynamic response of rigid strip foundations embedded in orthotropic

elastic soils. Rajapakse and Wang (1993) presented a boundary integral solution for the

dynamic response of three-dimensional problems in a transversely isotropic elastic half-

space. They applied the Fourier expansion with respect to the circumferential coordinate

and Hankel integral transforms with respect to the radial coordinates to derive general

solutions for equations of equilibrium expressed in terms of displacements. These general

solutions are used to derive the explicit solutions for Green’s functions corresponding to

a set of time-harmonic circular ring loads acting inside a half-space. Wang and

Achenbach (1995) obtained Green’s functions in an anisotropic medium by using the

Radon transform. This transformation reduces a three- or two-dimensional partial

differential equation to one-dimensional differential equations of the same kind. Having

solved the one-dimensional problem, the 3D and 2D solutions follow from the inverse

Radon transform. Dravinski and Zhang (2000) utilized the Radon transform to develop an

efficient algorithm for evaluation of time-harmonic Green’s functions for an orthotropic

full-space. Wang and Rajapakse (2000) used the boundary element method to investigate

the dynamic response of rigid mass-less cylindrical and hemispherical foundations

embedded in transversely isotropic elastic soils. Ahmad et al. (2001) analyzed the time-

harmonic two-dimensional elastodynamic problems of anisotropic media. They used the

full- and half-space Green’s functions developed by Rajapakse and Wang (1991), higher

order isoparametric curvilinear boundary elements and the self-adapting numerical

integration technique to deal with problems of orthotropic and non-orthotropic solids

with arbitrary geometries. They illustrated the effect of soil anisotropy on the compliance

of a rigid strip foundation resting on a two-layered media through an extensive

parametric study. Dravinski and Niu (2002) developed three-dimensional time-harmonic

Green’s functions based on Radon transform for the most general anisotropic material,

i.e., for a triclinic material. They used a symbolic computation system to efficiently

evaluate the finite integrals of the Green’s function. One- and two-dimensional Gauss-

Legendre quadratures were used. Niu and Dravinski (2003) used the time-harmonic

Green’s functions proposed in Dravinski and Niu (2002) to study the scattering of elastic





waves in 3D general anisotropic media (triclinic). They formulated the scattering

problems for scatterer in form of a cavity only. Denda et al. (2003) investigated two-

dimensional time harmonic response of solids of general anisotropy. They split the

fundamental solution obtained by Radon transform into static singular and dynamic

regular parts. The boundary integrals for the static singular part are evaluated analytically

and those for the dynamic regular part numerically over a unit circle. They also applied

the developed boundary element method to an eigen-value analysis. Arias and Achenbach

(2004) proposed a simple approach to correct the error introduced by the truncation of the

infinite boundary in the boundary element method modeling of elastodynamic wave

propagation in semi-infinite domains when Green’s functions of a full-space are used.

They exploited the asymptotic behavior of the solution and invoked the reciprocity

theorem of elastodynamics to adequately correct the boundary element method results.

Chen and Dravinski (2007) derived the displacement and stress Green’s functions for a

general anisotropic 2D half-space with embedded harmonic line load. Both displacement

and stress fields were expressed in terms of double Fourier integrals. The integrals were

evaluated by using contour integrations and a composite Gauss-Legendre quadrature over

a finite domain. Only a limited number of boundary element method studies have taken

into account the non-homogeneity of soils in a dynamic soil-structure interaction analysis.

Most of them are for cases when the foundation is located at the surface of the unbounded

soil. A summary of the existing literature is presented here. Guzina and Pak (1998)

studied the vertical vibration of a rigid circular footing resting on an elastic half-space

with a linear wave velocity profile in the vertical direction. Using a displacement-

potential representation and integral transforms, the problem is formulated as a set of dual

integral equations which are reducible to a Fredholm’s integral equation. Pak and Guzina

(1999) derived a regularized format of the time-harmonic direct boundary element

formulations for the three-dimensional elastodynamics for general anisotropic materials.

As in the conventional direct boundary element methods the evaluation of the Cauchy

principal values leads to mathematical and numerical complexities, they derived the

regularized boundary integral equation based on the decomposition of Green’s functions

into their singular and regular parts. This alternative form of the conventional boundary

integral equation involves weakly singular integrals only. They extended their

formulations to the general soil-structure interaction problems in the semi-infinite

domains including inhomogeneous and anisotropic media. Vrettos (1999) investigated the

vertical and rocking response of rigid rectangular foundations resting on a linear elastic

non-homogeneous half-space utilizing semi-analytical solutions for mixed boundary-

value integrations. Using the boundary integral method, Muravskii (2001) studied the

dynamic response of surface footings on a heterogeneous isotropic and transversely

isotropic elastic half-space. The variation of shear modulus with depth as linear functions

and as exponential functions to a finite or infinite depth are addressed.

Most of the studies for the dynamic analysis of soil-structure interaction problems in non-

homogeneous anisotropic soils using the boundary element method have been carried out

in the frequency domain. Here a summary of works done in the time domain is presented.

Wang and Achenbach (1992, 1993) developed two- and three-dimensional Green’s

functions for general anisotropic media based on Radon transform. Rajapakse and Gross

(1995) investigated the time-domain response of an orthotropic elastic half-plane

containing a cavity of arbitrary shape using the Laplace transformation. Wang et





al.(1996) presented a two-dimensional transient boundary element method using the

Green’s functions presented by Wang and Achenbach (1993) to solve elastodynamic

boundary initial-value problems in solids of general anisotropy. Richter and Schmid

(1999) derived the transient Green’s functions of the elastodynamic half-plane for cases

where source and observation points are situated beneath the traction-free surface. The

derivations are based on Laplace transform methods and the Cagniard-de Hoop inversion.

In the conventional boundary element approaches, the storage requirements and

computational time will tend to grow according to the square of the problem size. This is

because of the dense and non-symmetric nature of the boundary element matrix. To

improve the efficiency in terms of computational time and memory requirement, several

fast algorithms have been proposed in the past 20 years. Fast multi-pole method was first

introduced by Rokhlin (1983). It was developed for potential problems based on iterative

algorithms. It has been shown that this approach is efficient for the integral equations of

second kind. For the integral equations of first kind, appropriate matrix conditioning

techniques are necessary to result in similar computational costs because of the

unboundedness of the condition number of the matrix. Alpert et al. (1993) proposed

wavelet-based methods. They transformed the coefficient matrix to wavelet-like

coordinates leading to a sparse matrix. Grigoriev and Dargush (2004b) proposed a fast

multi-level boundary element method for the two-dimensional solution of Laplace

equation with mixed boundary condition. They used double-noded corners to facilitate

the implementation of a patch-by patch boundary element. They employed a bi-conjugate

gradient method as an iterative solver and also utilized the multi-grid method to

accelerate the convergence rate of the proposed iterative solver. They later extended the

fast multi-level boundary element into Helmholtz (Grigoriev and Dargush, 2004a) and

Stokes problems (Grigoriev and Dargush,2005). Wang et al. (2005) recently applied the

fast multi-level boundary element into the transient diffusion. In addition to fast

algorithms, the symmetric Galerkin boundary element methods have been proposed. The

symmetry property of the coefficient matrices permits the employment of efficient

symmetric solvers. The symmetric Galerkin boundary element methods are beneficial

from the point of view of coupling with finite elements as well. The main difficulty

associated with the symmetric Galerkin boundary element methods is the existence of

hyper singular integrals leading to more complex fundamental solutions. A fairly

extensive recent review on symmetric Galerkin boundary element methods can be found

in Bonnet et al. (1998). Perez-Gavilan and Aliabadi (2001) presented a symmetric

Galerkin boundary element formulation for the frequency-domain solution of two-

dimensional viscoelastic problems. Kallivokas et al. (2005) introduced a symmetric

Galerkin boundary element method for the solution of interior problems based on an

energy-based variational framework.

In summary, in the boundary element method only the boundary is discretized reducing

the spatial discretization by one. The fundamental solutions satisfy rigorously the

governing equations and the radiation condition at infinity. Although the boundary

element method in last four decades has gained the recognition as the logical tool for

dealing with unbounded domain problems, difficulties are encountered in its application

to many practical engineering problems owing to its reliance on the fundamental solution.

For instance, when the material is general anisotropic, the complexity of the fundamental





solution increases dramatically. Furthermore, in the conventional boundary element

methods the coefficient matrices are densely populated and unsymmetric. It leads to

expensive computations. Fictitious frequencies embedded in singular integral equations

are encountered in the frequency-domain analysis (Brebbia et al., 1984) too.

2.2 Thin layer method

The rigorous thin layer method, also known as consistent boundary or hyperelement, was

first developed by Lysmer and Waas (1972) for the dynamic analysis of unbounded

domains subjected to anti-plane loads. This method discretizes the boundary in the

vertical direction only consistent with that used for finite elements (Figure 2.3). The

displacement functions in the horizontal direction are expressed analytically to satisfy the

radiation condition at infinity. This semi-analytical technique is well suited to model

horizontal layers Layered medium.

Figure 2.3: Spatial discretization in thin layer method

Absorbing boundaries with material properties varying in the vertical direction. The

boundary conditions on the free surfaces and on the interfaces between adjacent layers

are rigorously satisfied with the same computational effort as for a homogeneous

horizontal layer. Thin layer method can be easily implemented in a finite-element

analysis. Applying the semi-discretization to the governing equations of motion and

enforcing the relevant boundary conditions, result in a complex-valued quadratic eigen-

value problem in the frequency domain. After solving the complex eigen-value problem,

the wave numbers and the corresponding mode shapes of the waves in the medium are

determined yielding the dynamic-stiffness matrix. The Fourier transformation can be

applied to the displacement functions obtained in the frequency domain to compute the

response in the time domain. It is obtained step by step by: (1) formulating the equations

of motion in the frequency-wave-number domain; (2) solving a complex-valued quadratic

eigen-value problem in the wave-numbers; (3) integrating analytically the displacement

functions over wave-numbers; (4) integrating numerically the displacement functions

over frequencies by means of the Fast Fourier transform (Kausel, 1992). In the following

the literature for the thin layer method is briefly reviewed.

Waas (1972) extended the method to in-plane motion of layered soils. Kausel et al.

(1975) and Kausel and Roesset (1977) generalized the method to axisymmetric problems.





Kausel and Peek (1982) applied the method to obtain Green’s functions for point forces

acting within (or on) a layered medium. These Green’s functions were later extended by

Seale and Kausel (1989) for the modeling of layered media over elastic half-spaces. The

thin layer method was extended to poroelastic unbounded domains by Bougacha et al.

(1993). Kausel (1992) extended the thin layer method to the time-domain formulation.

The extension is restricted to a class of anisotropic materials for which the required linear

eigen-value problem involves only real, narrowly banded symmetric matrices. In this

approach the time-domain formulation is obtained by: (1) expressing the governing

equations in the frequency-wave-number domain; (2) solving a linear real-valued eigen-

value problem in the frequency variable; (3) performing an analytical integration of the

displacement functions over frequencies; and finally (4) performing a numerical

transformation of the displacement functions over wave-numbers. This formulation

avoids using the complex variables, involves only a real, linear eigen-value problem and

allows obtaining the Green’s functions directly in the time domain. This strategy is

particularly appealing for the problems when the response is required at only a few

receivers, when the dynamic loads do not vary sharply in space, when the loads are

applied impulsively, and when the system has little or no damping (Kausel, 1992). Kausel

(1999) presented Green’s functions for the solution of various types of point sources

acting within (or on) horizontally layered media: point forces, force dipoles, blast loads,

seismic double couples with no net resultant and moment dipoles. He modeled the full-

space in the examples with a homogeneous finite layer of unit depth (horizontal

layer) to which paraxial boundaries were added to simulate the infinite medium. Park and

Tassoulas (2002) extended the thin layer method into inclined boundaries with the so-

called zigzag shape for both anti-plane motion and plane-strain problems. In the thin

layer method, since it is necessary to consider a rigid rock under the layered media, the

formulation ignores any possible vertical radiation. The assumption of rigid bases and

horizontally layered media are not always close to reality. There are cases in practical

geotechnical problems where the soil deposit is not very rigid or the soil geometry is far

from being horizontally layered.

2.3 The Scaled boundary finite-element method

The scaled boundary finite-element method, a fundamental-solution-less boundary

element method, is an attractive alternative to the numerical schemes in computational

mechanics. It not only combines some important advantages of the finite-element and

boundary element methods but also has its own salient features. This method, which is

semi-analytical, is based on the finite-element technology so that it does not require

fundamental solutions. The radiation condition at infinity is satisfied rigorously. Like the

boundary element method only the boundary is discretized reducing the spatial

discretization by one and leading to the increase of computational efficiency. Problems

involving stress singularities and discontinuities can be modeled accurately. Anisotropic

media can be handled without additional computational efforts.

In the scaled boundary finite-element method, a so-called scaling center O is chosen in a

zone from which the total boundary, other than the straight surfaces passing through the

scaling center must be visible (Figure 2.4a). Only the boundary S directly visible from the





scaling center O is discretized as shown in Figure 2.4a. One-dimensional line element is

used (Figure 2.4b). The straight surfaces passing through the scaling center (side faces)

and the interfaces between different materials are not discretized (Figure 2.4a). The

geometry of an element on the boundary is interpolated using the shape functions

formulated in the local coordinate η in the same way as in the finite-element method. The

geometry of the Side faces

Figure 2.4: (a) Spatial discretization for an unbounded domain in scaled boundary finite-

element method; (b) three-node line element on boundary domain V is described by

scaling the boundary with the dimensionless radial coordinate ξ pointing from the scaling

center to a point on the boundary (Figure 2.4a).

The radial and circumferential coordinates ξ and η form the scaled boundary coordinates.

Along the radial lines passing through the scaling center and a node on the boundary, the

nodal displacement functions are introduced. The shape functions are employed in the

circumferential direction to interpolate the displacement functions piece wisely. Having

expressed the governing differential equations in the scaled boundary coordinates,

Galerkin’s weighted residual method or the principle of virtual work is applied in the

circumferential direction transforming the governing partial differential equations to the

ordinary differential equations with the radial coordinate ξ as an independent variable.

This set of Euler-Cauchy ordinary differential equations is called the scaled boundary

finite-element equation in displacement. The coefficient matrices of this scaled boundary

finite-element equation are calculated and assembled in the same way as the static-

stiffness and mass matrices in the finite-element method. For static analysis, the scaled

boundary finite-element equation can be transformed into a system of first-order ordinary

differential equations. An eigen value problem can be used to solve this system of first-

order ordinary differential equations. Thus, the displacement and stress fields are

described by semi-analytical solutions permitting the boundary condition at infinity to be

satisfied rigorously. Obtaining the nodal forces on the boundary, introducing the

definition of the dynamic-stiffness matrix of an unbounded domain on the boundary and

making use of the scaled boundary finite-element equation in displacement, the scaled

boundary finite-element equation in dynamic stiffness with the frequency as the

independent variable is derived. It is a system of non-linear first-order ordinary





differential equations to be solved numerically for the dynamic-stiffness matrix. The

radiation condition at infinity is satisfied using an asymptotic expansion of the dynamic-

stiffness matrix for high frequency. Applying the inverse Fourier transformation to the

scaled boundary finite-element equation in dynamic stiffness leads to the scaled boundary

finite-element equation in time domain involving the convolution integrals. The time-

discretization method is used to solve the scaled boundary finite-element equation in the

time domain.

A literature review on the evolution of the method from the first inspiration until March

2006 is presented by Vu (2006) in her PhD thesis. A brief summary with emphasis on the

application to the dynamic soil-structure interaction analysis is provided here.

3. Local procedures

To avoid the expensive computational problems associated with the global procedures,

various local procedures have been proposed over the last four decades. As was explained

in Section 1.4.2 local procedures are approximate. In them the response at a specified

location and time depends to the response at its immediate neighbors (spatially local) and

at a few previous times (temporally local). Most of them are constructed based on the

theory of wave propagation and using differential operators to enforce outgoing plane

waves (Wolf and Song, 1996). They can be constructed by enforcing artificial damping

to a finite layer to absorb the outgoing waves too. On the surface, the local procedures

appear to be vastly different from each other from the mathematical formulation,

conceptual basis and implementation point of views. However, many of them are

mathematically related and comparable in energy-absorbing performance (Kausel, 1988).

Generally speaking, local procedures are algorithmically simple so that most of them can

be easily implemented into the finite-difference or the finite-element methods. The

approximations in the local procedures lead to spurious reflections from the artificial

boundary. In order to obtain results of an acceptable level of accuracy a local procedure

has to be applied at a so-called artificial boundary sufficiently far away from the

structure-media interface. It increases the number of degrees of freedom in the

computational domain. Most of the local procedures have been developed for scalar wave

equations in unbounded domains of simple geometry and material property. However, a

reliable local procedure for modeling vector wave equations in unbounded domains of

arbitrary geometry and material property does not exist at present.

3.1 Transmitting boundary conditions

Transmitting boundary conditions have been introduced since the late 1960s. Most of

them are based on the mathematical representation of plane wave propagation to

eliminate the incident waves at special angle of incident. Lysmer and Kuhlemeyer (1969)

proposed the first transmitting boundary for elastodynamics often referred to as the

classical viscous boundary condition. It absorbs plane waves propagating perpendicularly

to the artificial boundary. For two-dimensional cases the viscous boundary condition is

formulated as ∂u





σ = a ρ Cp

τ = b ρ Cs

where,

σ and τ are the normal and shear stresses on the boundary, respectively;

ρ is the mass density;

Cp and Cs are the longitudinal and shear wave velocities, respectively and

u and v are the normal and tangential displacements on the boundary, respectively.

Dimensionless parameters a and b are chosen to minimize the reflected energy for an

incident plane wave impinging at a given angle of incidence. Lysmer and Kuhlemeyer

(1969) found that the choice of a = b = 1 leads to good absorptions. The viscous

boundary condition can easily be implemented in finite-element codes for both

frequency-domain and transient analyses. It is algorithmically simple, geometrically

universal and frequency independent. As dashpots have no static stiffness, the viscous

boundary condition is not able to model a static problem as the limiting case of a dynamic

problem at low frequency. White et al. (1977) proposed the unified boundary condition

which is a viscous boundary condition applicable to anisotropic media with a certain

choice of the parameters a and b. By discretizing the domain using the finite elements,

and then determining the linear relationship between stresses and velocities on the

boundary, they obtained the parameters a and b. They presented formulations for both

plane-strain and Axisymmetric conditions. Akiyoshi (1978) presented a viscous boundary

for shear waves called compatible viscous boundary. It is actually a correction to the

viscous boundary to account for the discretization scheme used for the domain. This

approach has the disadvantage of involving a convolution integral in its formulation. It

loses the local character of the boundary condition. Smith (1974) proposed the

superposition boundary condition to solve both the scalar and elastic wave propagation

problems. The superposition boundary averages the solutions from two sets of boundary

conditions corresponding to symmetry and anti-symmetry, which eliminates the reflected

waves for a single boundary. The formulation is independent of both frequency and angle

of incidence. This boundary condition is not able to eliminate multiple reflections. The

superposition boundary condition was later modified to overcome multiple reflections by

introducing two overlapping narrow boundary neighborhoods in which the reflected

waves are canceled as they occur (Cundall et al., 1978; Kunar and Marti, 1981).

Underwood and Geers (1981) introduced the doubly-asymptotic boundary condition for

dynamic soil-structure interaction. In this boundary, dashpots and coupled static springs

are used which are asymptotically exact at high and low frequencies for plane waves

propagating perpendicularly to the boundary, respectively. They used the boundary

element method to determine the static-stiffness matrix for the medium leading to fully

coupled and non-symmetric coefficients. The doubly-asymptotic boundary results in

errors for modeling the intermediate frequencies. The approach is temporally local but

spatially global. The low accuracy is the most important concern associated with the





above mentioned transmitting boundary conditions named as Sommerfeld-like

transmitting boundary conditions (Givoli, 2004). Since late 1970s, high-order

transmitting boundary conditions have been proposed for modeling unbounded domains.

They have the potential of leading to accurate results with increasing orders of

approximation. At the same time, they are computationally efficient owing to the local

formulation. Lindman (1975) proposed a very general set of transmitting boundary

conditions using projection operators leading to a high degree of effectiveness for the

time-dependent scalar wave equations. He determined coefficients for the high-order

boundary conditions to minimize reflections over a broad range of angle of incidence for

both traveling and evanescent waves. Engquist and Majda (1977) and Clayton and

Engquist (1977) proposed the so-called paraxial boundary conditions, which are closely

related to the boundary conditions proposed by Lindman (1975), using rational

approximations (Padé) to pseudo-differential equations for scalar and elastic wave

equations. This paraxial boundary constructs a differential equation which favors

outgoing waves by splitting the differential operator of the wave equation for plane

waves. It was the first time that the method of rational approximation appeared as a tool

to derive local absorbing boundary conditions. The paraxial boundary condition is exact

for plane waves propagating perpendicularly to the boundary and any wave with oblique

incidence will necessarily cause some reflections. The paraxial boundary condition leads

to a formulation with high-order derivatives. The first and second-order paraxial

boundary conditions for scalar wave equations in rectangular coordinates are formulated

as

u = 0

u = 0

The first-order boundary condition (Eq. (2.8)) is identical to the viscous boundary

condition. They showed that for any given angle of incidence other than right angle, the

second-order boundary condition (Eq. (2.9)) generates less reflections than the first-order

one (Eq. (2.8)). The Engquist-Majda boundary conditions are easily implemented in a

finite-difference scheme. There are instability problems associated with higher order

boundaries and for inclined body waves (Wolf, 1988). Bayliss and Turkel (1980)

proposed a sequence of transmitting boundary conditions based on an asymptotic

expansion of an exact solution at large distances for time-dependent wave-like equations

in polar coordinate systems. They constructed a set of special local differential relations

that identically eliminates a prescribed number of leading terms in the corresponding

series. For three-dimensional scalar wave equation in spherical coordinates, the mth

Bayliss and Turkel boundary condition is formulated as





The high-order formulations lead to higher derivatives. The first order Bayliss and Turkel

boundary coincides with the first-order Engquist and Majda boundary. Bayliss et al.

(1982) extended the approach to Helmholtz and Laplace equations and implemented

them into finite elements. Utilizing the idea of factorizing the high-order operators,

Higdon (1986) constructed multi-directional boundary conditions for the scalar and

elastic wave equations at preselected angles of incidence. He demonstrated that all

transmitting boundary conditions based on a rational approximation of a pseudo-

differential operator can be cast in this form so that they are special cases of his multi-

directional boundary condition. Higdon’s boundary condition is expressed as

where,

α is the angle of incidence.

The multi-directional boundary condition which is exact for plane waves propagating at

preselected angle of incidence, can be regarded as the general case of the viscous and

paraxial boundaries. The higher the number of orders, the higher the accuracy is. In his

numerical studies, Higdon found instabilities for third-orders of these boundary

conditions. Same as in the paraxial boundary, number of orders higher than two is not

practical. Higdon later extended this idea to stratified media (Higdon, 1992) and

dispersive systems (Higdon, 1994). Liao and Wong (1984) proposed the extrapolation

boundary condition in elastodynamics which calculates the displacements at the artificial

boundary by extrapolating the data in the interior nodes at earlier times requiring an

estimate of the propagation velocity. Using higher order extrapolation or smaller distance

between the data points, can lead to more accurate results. For frequencies less than the

cut-off frequency, increasing the order of extrapolation does not improve accuracy.

Extrapolation boundary condition is more convenient to be implemented into finite

elements. Later Liao (1996) generalized the original extrapolation boundary condition by

coupling a space extrapolation to the space-time extrapolation and by introducing

multiple artificial wave speeds. Explicit time-integration schemes can only be used to

perform a time-domain analysis. The so-called doubly asymptotic multi directional

transmitting boundary (Wolf and Song, 1995b) combines the advantages of the doubly-

asymptotic and multi-directional boundary conditions. Wolf and Song (1995b) used the

interaction forces rather than displacements in the doubly asymptotic operators and

discretized the boundary condition with a forward difference formula. The static or low-

frequency behavior is modeled using a doubly asymptotic boundary implemented

implicitly. This boundary condition is temporally local but spatially global due to fully

coupled static-stiffness matrix in doubly asymptotic formulation. They used an

approximate banded static-stiffness matrix to construct a spatially local boundary

condition. Kellezi (2000) developed the so called linear cone and cone boundary

conditions for 2D and 3D transient models, respectively, which can be considered as

doubly asymptotic ones and a generalization of the viscous boundary. In constructing

these boundary conditions, it is assumed that the body waves in plane-strain analysis

should propagate radially outward along a cylindrical and hemispherical wave front for





two- and three-dimensional cases, respectively. In the original high-order transmitting

boundary conditions the order of derivatives in the formulations increases with the order

of the transmitting boundary so that beyond the second order, the implementation in a

finite-element computer program becomes complex and instability may occur. Their

second-order terms are still widely used. (Givoli, 2004) considers these original ones as

low-order transmitting boundary conditions. Introducing auxiliary variables to high-order

transmitting boundary conditions to eliminate higher-order derivatives has recently

attracted interest of many researchers. Most of them have been developed for scalar wave

equations with simple geometry and isotropic and homogeneous materials. For certain

types of waves and geometries of unbounded domains, a two-and three-dimensional

problem can be decomposed into a series of one-dimensional problems. The method of

separation of variables is applicable to the Helmholtz equation in cylindrical and

spherical unbounded domains. Starting from one-dimensional wave equations,

approximate procedures can be constructed. They lead to a system of linear first-order

ordinary differential equations in the time domain. These high-order transmitting

boundary conditions are increasingly recognized as a potential technique for modeling

unbounded domains. They have the potential of leading to accurate results with

increasing orders of approximation. The order of boundary condition can be easily

increased to a desired level. They are local in time but non-local in space. They not only

can produce numerical results of high accuracy but also can be straightforwardly

implemented into finite elements. However, the application of current methods for

constructing high-order transmitting boundary conditions is usually limited to unbounded

domains of simple geometry. The extension to elastic wave propagation in unbounded

domains of arbitrary geometry and material property is not a straightforward task. Here

the literature of the existing high-order transmitting boundary conditions is briefly

reviewed. Grote and Keller (1995a) developed high-order local boundary conditions for

three-dimensional time-dependent scalar wave equations based on spherical harmonic

transformations. These boundary conditions are constructed for three-dimensional

problems as they are based on special properties of the spherical harmonics. Grote and

Keller (1996) implemented the boundary condition developed in Grote and Keller

(1995a) within finite difference and finite-element methods and proved the uniqueness of

the solution and discussed the stability issues associated with their boundary condition.

They later extended the high-order boundary condition to three-dimensional

elastodynamics (Grote and Keller, 2000) and Maxwell equations (Grote and Keller,

1998). Thompson and Huan (2000) rederived the high-order boundary conditions

presented in Grote and Keller (1995a, 1996) to improve the scaling of the related first-

order system of equations. They (Thompson and Huan, 1999, 2000) implemented the

local high-order boundary condition into the standard finite-element method with several

alternative implicit and explicit time integrators. Most recently, Grote (2006) derived a

spatially local formulation for the boundary condition proposed in Grote and Keller

(1998) for three-dimensional time-dependent Maxwell equations. It only involves first

time derivatives and second tangential derivatives of the electromagnetic field and is of

certain auxiliary functions. This new boundary condition does not need any vector

spherical harmonics or inner products with them. It leads to a somewhat easier and

cheaper implementation. Hagstrom and Hariharan (1998) derived local high-order

boundary conditions for two- and three-dimensional time-dependent scalar and Maxwell





wave equations based on Bayliss and Turkel (1980) operators representing outgoing

solution of the equations. Two-dimensional boundary conditions are asymptotic while the

three-dimensional ones are exact. Huan and Thompson (2000) rederived the sequence of

boundary conditions formulated in Hagstrom and Hariharan (1998) based on the

hierarchy of local boundary operators used by Bayliss and Turkel (1980) and a recursion

relation for the expansion coefficients appearing in a radial asymptotic expansion. They

later in Thompson et al. (2001) extended this idea into two-dimensional scalar wave

equations on a circle. Zhao and Liu (2002) rederived the local high-order boundary

conditions proposed in Hagstrom and Hariharan (1998) for two-dimensional scalar

problems using the operator splitting method. They later in Zhao and Liu (2003) used the

splitting operator method to derive a high-order local boundary condition for a two-

dimensional infinite horizontal layer. Ruge et al. (2001) used a mixed-variable

formulation to transfer the solution of an unbounded domain approximated by a Padé

series in the frequency domain, into a high-order local boundary condition used for the

solution directly in the time domain. The mixed-variable formulation possesses a

continued-fraction form. The technique is applicable to scalar and vector wave equations,

arbitrary geometries and anisotropic and non-homogeneous materials. However, the

dynamic-stiffness matrix has to be specified at discrete frequencies using the global

procedures, for instance boundary element method or the scaled boundary finite-element

method, resulting in a high computational cost. Krenk (2002) derived a high-order

transmitting boundary condition from a rational function approximation of the plane

wave representation for scalar waves. This boundary condition is a modified form of that

proposed by Lindman (1975). It is suitable for the finite-element formulation. Guddati

and Tassoulas (2000) developed a high-order local boundary condition based on

recursive continued fraction of the dispersion relation for scalar wave equations in

Cartesian coordinates. This continued-fraction boundary was limited to straight

computational boundaries. Most recently, Guddati and Lim (2006) extended the

continued-fraction absorbing boundary condition to polygonal computational domains. In

their new derivations, the infinite domain is replaced by a computationally tractable

finite-element mesh to absorb the propagating waves. The infinite domain is recursively

split to an infinite number of finite element layers. Length of each layer is frequency

dependent and determined by choosing reference phase velocities. They found a link

between the continued fraction and the complex coordinate stretching of perfectly

matched layers. Zahid and Guddati (2006) modified the continued-fraction absorbing

boundary condition with adding padding elements to model dispersive waves. Special

techniques were used to discretize the resulting system of evolution equations which are

different form the conventional second-order systems in dynamic problems.

Givoli and Patlashenko (2002) proposed a frequency-domain high-order local boundary

condition for two-dimensional Helmholtz equation based on the localization of the non-

local DtN map in the frequency domain. The boundary geometry is of 2D cylindrical and

2D wave-guide configurations. Givoli and Neta (2003) developed high-order boundary

conditions based on Higdon (1986) boundary condition for both dispersive and non-

dispersive linear time-dependent scalar waves. It was limited to a geometry with a single

plane boundary (wave guide). In his review paper, Givoli (2004) summarizes the key

aspects of different high-order local boundary conditions developed by that time.

Hagstrom and Warburton (2004) generalized Givoli and Neta (2003) boundary condition





to a full-space configuration and enhanced its stability by deriving corner compatibility

conditions for the auxiliary variable equations. van Joolen et al. (2005) extended the

high-order boundary condition proposed in Givoli and Neta (2003) in unbounded

domains with a rectangular boundary. Kechroud et al. (2005) proposed a high-order

Padé-type non-reflecting boundary condition for two-dimensional acoustic scattering

problems. The boundary condition can be applied on any convex fictitious boundary.

Givoli et al. (2006) incorporated the boundary condition developed in Hagstrom and

Warburton (2004) in a finite-element scheme and compared its performance with the

boundary condition developed in Givoli and Neta (2003).

3.2 Infinite Elements

In the standard finite-element method, it is only possible to discretize a finite part of the

domain. To overcome the difficulty in satisfying the radiation condition at the infinity for

modeling unbounded domains, infinite elements (Bettess, 1992; Astley, 2000) have been

developed. The infinite elements use decay functions representing the wave propagation

toward infinity as the shape functions of the displacements. The decay rate and the phase

velocity must be specified. The form of the shape functions in the unbounded direction is

usually derived from exact solutions of associated one-dimensional wave propagation

problems. Infinite elements can capture only a few propagation modes so that they are

generally not accurate even if the finite-element mesh is very fine. The sequence of

decaying functions may be expressed as reciprocal (1/rn), exponential (1/enr ) or

logarithmic (1/ln r)n types, in the radial coordinate r. The formulation of infinite elements

is spatially local. In this section a summary of the literature for this method is provided.

Bettess and Zienkiewicz (1977) proposed the original unconjugated infinite elements

using the exponential decay functions for radial distance. These infinite elements led to

symmetric discrete problems but were unable to simulate the correct asymptotic decay.

Astley (1983) introduced the so-called wave envelope elements using the complex

conjugate weighting functions in a Petrov-Galerkin scheme, along with mapping for the

exterior problems in acoustics. These infinite elements represent the correct asymptotic

behavior. This technique led to remarkable simplifications in the problem due to

cancellation of the oscillatory terms within integrands but results in local non-symmetric

matrices which destroy the symmetric structure of the semi-discretization of the interior

domains. Zienkiewicz et al. (1983) developed the mapped infinite elements. This

approach incorporates an amplitude decay which is asymptotically correct. The original

infinite elements use cylindrical or spherical boundaries to truncate the exterior domains.

For slender bodies like submarines, truncating the exterior domains using a spherical

boundary leads to a large computational bounded domains to be processed. To tackle this

problem for long and thin bodies, Burnett (1994) developed a novel unconjugated

spheroidal infinite elements in terms of prolate spheroidal coordinates for Helmholtz

equations. In addition to choose of spheroidal coordinate system, separate shape

functions in the radial and transverse directions are used. It leads to analytical radial

integrals. Later Burnett and Holford (1998a,b) extended this approach to ellipsoidal and

oblate acoustic infinite elements. The convergence of the ellipsoidal multiple expansions

is proved. As three axes of an ellipsoid can be chosen independently, an ellipsoid

boundary leads to a small number of meshes in the computational domains. It results in a

greater computational speed than those derived for the spheroidal elements. Shirron and





Babuska (1998) formulated a conjugated spheroidal infinite element based on the Burnett

formulation and compared various local boundary conditions with infinite elements. They

mentioned that the rate of convergence for the conjugated infinite elements was slower

than the unconjugated ones. Once the domain tends to infinity, the unconjugated infinite

elements diverge from the exact solutions whereas the conjugated ones converge. The

instability of the unconjugated infinite elements is due to the bilinear form on which they

are based.

Astley (1998a) formulated conjugated infinite elements with geometric weight factor 1/r2

and based on Burnett’s spheroidal trial solutions (Burnett, 1994). Astley (2000)

summarized the developments of the infinite elements during 1980s and 1990s and

discussed their merits and drawbacks. Comparing different convergence studies, he

mentions that the unconjugated formulations lead to the most accurate results in the near

field but is limited by ill-conditioning as the radial orders increase. The conjugated

elements, on the other hand, are stable and less accurate in the near fiel dbut converge to

the exact solution in the far field. Astley and Coyette (2001) studied the performance of

the prolate spheroidal infinite elements. They investigated the convergence of these

infinite elements for conjugated and unconjugated test functions. They analyzed

examples for high and low frequencies and also low- and high-aspect ratios. Most of the

developments for the infinite elements have been performed for time harmonic analyses.

They tend to not performing well in a transient analysis since the wave shapes are not

constant and changing with time. Only low radial orders (up to order three) have been

implemented. Astley (1996) extended the conjugated infinite elements into a time-

domain analysis of acoustic problems by taking an inverse Fourier transform of the

discrete infinite element equation in the frequency domain. For conjugated elements, the

coefficient matrices can be written in terms of frequency-independent mass, stiffness and

damping matrices. However, for unconjugated elements this is not the case so that it leads

to convolution integrals. Later Astley (1998b) implemented the transient formulation in

spheroidal coordinate systems. He analyzed several examples to investigate the required

element orders, effect of temporal step size and the performance of the iterative solutions.

Astley and Hamilton (2006) studied the stability of infinite elements for transient acoustic

problems. Numerically mapped elements and analytically formulated spheroidal elements

were considered. In both cases the form of the interface between the finite and infinite

element meshes was shown to be critical in determining whether the transient equations

yield stable solutions.

In summary, the infinite elements use decay displacement shape functions to represent

the wave propagation toward infinity. The decay rate and phase velocity must be

specified. It overcomes the problem of satisfying the radiation condition at the infinity in

the standard finite-element method. However, instability and ill-conditioning problems

are associated with high-order elements. The low-order ones suffer from low accuracy.

The infinite elements do not perform well in a transient analysis.

3.3 Absorbing layers

The absorbing layers are alternatives to local boundary conditions discussed in the

previous sections. They are constructed by replacing the unbounded domain by an





absorbing layer of finite thickness with the properties that appreciably reduces the wave

reflection into computational domain. Israeli and Orszag (1981) developed the sponge

layers in which artificial material damping is introduced to absorb waves propagating

through the layer. In the damping solvent extraction method (Song and Wolf, 1994), the

effect of the artificial damping on the outgoing waves is extracted leading to higher

accuracy.

Figure 2.5: Perfectly matched layer technique

The perfectly matched layers are the most developed absorbing layers. The theory and

concept of perfectly matched layers (PMLs), was initially introduced by Berenger (1994)

for electromagnetic waves (Maxwell equation). In this technique as shown in Figure 2.5,

the computational domain is surrounded by a finite-thickness layer in which artificial

attenuation of wave propagation in a pre-selected direction pointing to infinity is

introduced to the governing equations for the unbounded domain. As the wave

propagation in the other independent directions is not attenuated, the transmission of

plane waves from the domain of interest to the surrounding absorbing layer is improved.

Propagating waves of all non-tangential angles-of-incidence and of all non-zero

frequencies can be absorbed (Basu and Chopra, 2003). The perfectly matched medium

may be equivalently interpreted as an non-homogeneous viscoelastic medium as material

damping is introduced through the governing equations. Computational domain perfectly

matched layers in the pioneering paper; Berenger (1994) presented a finite-difference

time-domain formulation (FDTD) for 2D electromagnetic split fields.

Later Berenger (1996) extended the perfectly matched layers to three-dimensional

electromagnetic waves. Beranger’s computational domains are Cartesian rectangle and

parallelepiped in two and three dimensions, respectively. It has been demonstrated that

the Beranger’s original split-field formulation is weakly ill-posed (Tsynkov, 1998).

Abarbanel and Gottlieb (1998) used the non-split field components to obtain well-posed

formulations. Perfectly matched layers have been extended to various linear wave

equations: the Helmholtz equation (Qi and Geers, 1998; Harari et al., 2000), the

linearized Euler equations (Hu, 1998) and the wave equation for poroelastic media (Zeng





et al., 2001). A literature review of perfectly matched layers applied to fields other than

elastodynamics is not covered in this paper.

4. Conclusions

This paper has described in brief, different methods for dynamic analysis of unbounded

domains. Various global and local procedures with particular emphasis on those

developed for elastodynamics are covered. Their mathematical backgrounds, potentials

and limitations are stated. In spite of existence of different techniques for treating

unbounded domains, a reliable and efficient technique applicable to vector wave

equations in unbounded domains of arbitrary geometry and material property does not

exist at present.

5. References

1. Abarbanel, S. and Gottlieb, D. (1998), “On the construction and analysis of

absorbing layers in CEM”. Applied Numerical Mathematics, 27, pp 331–340.

2. Achenbach, J. D. (1984), “Wave propagation in elastic solids. Elsevier

Science”. ADINA (2004). ADINA User Interface Primer. Version 8.2,

ADINA R & D, Inc. (http://www.adina.com).

3. Ahmad, S., Leyte, F., and Rajapakse, R. K. N. D. (2001), “BEM analysis of

two-dimensional elastodynamics problems of anisotropic solids”. Journal of

Engineering Mechanics, ASCE, 127, pp 149–156.

4. Akiyoshi, T. (1978), “Compatible viscous boundary for discrete models”.

Journal of Engineering Mechanics, ASCE, 104, pp 1253–1265.

5. Alpert, B., Beylkin, G., Coifman, R., and Rokhlin, V. (1993), “Wavelet-like

bases for the fast solution of second-kind integral equations”. SIAM Journal

on scientific and statistical computing, 14, pp 159–184.

6. Alpert, B., Greengard, L., and Hagstrom, T. (2002), “Non-reflecting boundary

conditions for the time-dependent wave equation”. Journal of Computational

Physics, 180, pp 270–296.

7. ANSYS (2003), Multyphysics. Version 8.0, Canonsburg, P.A: ANSYS, Inc.

(http://www.ansys.com).

8. Arias, I. and Achenbach, J. D. (2004), “Rayleigh wave correction for the BEM

analysis of two-dimensional elastodynamic problems in a half-space”.

International Journal for Numerical Methods in Engineering, 60, pp 2131–

2146.

9. Astley, R. J. (2000), “Infinite elements for wave problems: a review of current

formulations and an assessment of accuracy”. International Journal for

Numerical Methods in Engineering, 49, pp 951–976.





10. Bartels, R. H. and Stewart, J. L. (1972), “Solution of the matrix equation AX

+ X B = C.” Communications of the ACM, 15: pp 820–826.

11. Basu, U. and Chopra, A. K. (2003), “Perfectly matched layers for time-

harmonic elastodynamics of unbounded domains: theory and finite-element

implementation”. Computational Methods in Applied Mechanics and

Engineering, 192, pp 1337–11375.

12. Bayliss, A. and Turkel, E. (1980), “Radiation boundary conditions for wave-

like equations”. Communications on Pure and Applied Mathematics, 33, pp

707–725.

13. Bazyar, M. H. and Song, C. (2006a), “Time-harmonic response of non-

homogeneous elastic unbounded domains using the scaled boundary finite-

element method. Earthquake Engineering and Structural Dynamics, 35, pp

357–383.

14. Becache, E., Fauqueux, S., and Joly, P. (2003), “Stability of perfectly matched

layers, group velocities and anisotropic waves”. Journal of Computational

Physics, 188, pp 399–433.

15. Bender, C. M. and Orszag, S. A. (1978), “Advanced Mathematical Methods

for Scientists and Engineers”. McGraw-Hill, Singapore.

16. Berenger, J. P. (1994), “A perfectly matched layer for the absorption of

electromagnetic waves. Journal of Computational Physics, 114, pp 185–200.

17. Beskos, D. E. (1997), “Boundary element methods in dynamic analysis: Part

II (1986-1996). Applied Mechanics Reviews, ASME, 50, pp 149–197.

18. Bettess, P. (1992).”Infinite Elements”. Penshaw Press, Sunderland, UK.

19. Bougacha, S., Tassoulas, J. L., and Roesset, J. M. (1993), “Analysis of

foundations on fluid-filled poroelastic stratum”. Journal of Engineering

Mechanics, ASCE, 119, pp 1632–1648.

20. Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C. (1984), “Boundary Element

Techniques”. Springer-Verlag.

21. Burnett, D. S. (1994), “A three-dimensional acoustic infinite element based on

a prolate spheroidal multipole expansion. Journal of Acoustic Society of

America, 96, pp 2798–2816.

22. Chen, Z. and Dravinski, M. (2007), “Numerical evaluation of harmonic

Green’s functions for triclinic half-space with embedded sources-part I: A 2D

model”. International Journal for Numerical Methods in Engineering, 69, pp

347–366.





23. Chew, W. C. and Liu, Q. H. (1996). “Perfectly matched layers for

elastodynamics: a new absorbing boundary condition”. Journal of

Computational Acoustics, 4,341–359.

24. Chidszey, S. and Deeks, A. J. (2005), “Determination of coefficients of crack

tip asymptotic fields using the scaled boundary finite-element method”.

Engineering Fracture Mechanics, 72, pp 2019–2036.

25. Chopra, A. K. (1995), “Dynamics of Structures: Theory and Applications to

Earthquake Engineering”. Prentice-Hall, New Jersey.

26. Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. (2002). “Concepts

and Applications of Finite Element Analysis. John Wiley & Sons.

27. Dasgupta, G. (1982), “A finite element formulation for unbounded

homogeneous continua”. Journal of Applied Mechanics, ASME, 49, pp 136–

140.

28. Deeks, A. J. (2004), “Prescribed side-face displacements in the scaled

boundary finite-element method”. Computers & Structures, 82, pp 1153–1165.

29. Doherty, J. and Deeks, A. (2003), “Elastic response of circular footings

embedded in a non- homogeneous half-space”. Geotechnique, 53, pp 703–714.

30. Dominguez, J. (1978), “Dynamic stiffness of rectangular foundations”.

Research Report R78-20, Department of Civil Engineering, Massachusetts

Institute of Technology, Cambridge, Massachusetts.

31. Dravinski, M. and Niu, Y. (2002), “Three-dimensional time-harmonic Green’s

functions for a triclinic full-space using a symbolic computation system”.

International Journal for Numerical Methods in Engineering, 53, pp 455–472.

32. Dravinski, M. and Zhang, T. (2000), “Numerical evaluation of three-

dimensional time-harmonic Green’s functions for anisotropic full-space”.

Wave motion, 32, pp 141–151.

33. Ekevid, T., Lane, H., and Wiberg, N. E. (2006), “Adaptive solid wave

propagation influences of boundary conditions in high-speed train

applications”. Computational Methods in Applied Mechanics and Engineering,

195: pp 236–250.

34. Engquist, B. and Majda, A. (1977), “Absorbing boundary conditions for the

numerical simulation of waves”. Mathematics of Computation, 31, pp 629–

651.

35. Engquist, B. and Majda, A. (1979), “Radiation boundary conditions for

acoustic and elastic wave calculations”. Communications on Pure and Applied

Mathematics, 32, pp 314–358.





36. Fan, S. C., Li, S. M., and Yu, G. Y. (2005), “Dynamic fluid-structure

interaction analysis using boundary element method-finite element method”.

Journal of Applied Mechanics, ASME, 72, pp 591–598.

37. Freedman, L. M. and Keer, L. M. (1972), “Vibration motion of a body on an

orthotropic half-plane”. Journal of Applied Mechanics, ASME, 94, pp 1033–

1039.

38. Fried, I. and Malkus, D. S. (1975), “Finite element mass matrix lumping by

numerical integration with no convergence rate loss”. International Journal of

Solid Structures, 11, pp 461–466.

39. Gazetas, G. (1981), “Strip foundations on cross-anisotropic soil layer

subjected to static and dynamic loadings”. Geotechnique, 31, pp 161–179.

40. Gazetas, G. and Dobry, R. (1984), “Simple radiation damping model for piles

and footings”. 110, pp 937–956.

41. Geers, T. L. (1997), proceedings of the IUTAM Symposium on

Computational Methods for Unbounded Domains. Kluwer Academic

Publishers.

42. Genes, M. C. and Kocak, S. (2005), “Dynamic soil-structure interaction

analysis of layered unbounded media via a coupled finite element/boundary

element/scaled boundary finite element model”. International Journal for


43. Gibson, R. E. (1967), “Some results concerning displacements and stresses in

a non- homogeneous elastic half-space”. Geotechnique, 17, pp 58–67.

44. Gilat, A. (2004), MATLAB: an introduction with applications. Wiley,

Chichester.

45. Givoli, D. (1991), “Non-reflecting boundary conditions: a review”. Journal of

Computational Physics, 8, pp 1–29.

46. Grigoriev, M. M. and Dargush, G. F. (2004a), “A fast multi-level boundary

element method for the Helmholtz equation”. Computational Methods in

Applied Mechanics and Engineering, 193, pp 165–203.

47. Grigoriev, M. M. and Dargush, G. F. (2004b), “A fast multi-level boundary

element method for the two-dimensional steady heat diffusion”. Numerical

Heat Transfer, 46: pp 329–356.

48. Grote, M. (2006), “Local nonreflecting boundary condition for Maxwell’s

equations”. Computational Methods in Applied Mechanics and Engineering,





195, pp 3691–3708.

49. Grote, M. and Keller, J. (1995a), “Exact non-reflecting boundary conditions

for the time-dependent wave equation”. SIAM Journal of Applied

Mathematics, 55, pp 280–297.

50. Grote, M. and Keller, J. (1995b), “On non-reflecting boundary conditions”.

Journal of Computational Physics, 122, pp 231–243.

51. Guddati, M. and Lim, K. (2006), “Continued-fraction absorbing boundary

conditions for convex polygonal domains”. International Journal for


52. Guddati, M. and Tassoulas, J. L. (1998b), “An efficient numerical algorithm

for transient analysis of exterior scalar wave propagation in a homogeneous

layer”. Computational Methods in Applied Mechanics and Engineering, 167,

pp 261–273.

53. Gurtin, M. E. (1972), “The linear theory of elasticity”. Section 74, Mechanics

of Solids II,C. Truesdell, ed., in Encyclopedia of physics, Vol. VIa/2, Springer,

Berlin, pp 257-264.

54. Guzina, B. B. and Pak, R. Y. S. (1998), “Vertical vibration of a circular

footing on a linear-wave-velocity half-space”. Geotechnique, 48, pp 159–168.

55. Hagstrom, T. (2003), “New results on absorbing layers and radiation

boundary conditions”. M. Ainsworth, et al. (eds). Springer: New York.

56. Hall, W. S. and Oliveto, G. (2003), “Boundary Element Methods for Soil-

Structure Interaction”. Kluwer Academic Publishers.

57. Harari, I. and Albocher, U. (2006), “Studies of FE/PML for exterior problems

of time-harmonic elastic waves”. Computational Methods in Applied

Mechanics and Engineering, 195, pp 3854–3879.

58. Hasting, F. D., Schneider, J. B., and Broschat, S. L. (1996), “Application of

the perfectly matched layer (PML) absorbing boundary condition to elastic

wave propagation”. Journal of Acoustic Society of America, 100, pp 3061–

3069.

59. Higdon, R. L. (1986), “Absorbing boundary conditions for difference

approximations to the multi-dimensional wave equation”. Mathematics of

Computation, 176, pp 437–459.

60. Hu, F. Q. (1998), “On absorbing boundary conditions for linearized Euler

equations by a perfectly matched layer”. Journal of Computational Physics,

129, pp 201–219.





61. Huan, R. N. and Thompson, L. L. (2000), “Accurate radiation boundary

conditions for the time-dependent wave equation on unbounded domains”.


1603.

62. Hughes, T. J. R. (1987), “The Finite Element Method: Linear Static and

Dynamic Finite Element Analysis”. Prentice-Hall.

63. Israeli, M. and Orszag, S. (1981), “Approximation of radiation boundary

conditions”. Journal of Computational Physics, 41, pp 115–135.

64. Jafargandomi, A., Fatemi Aghda, S. M., Suzuki, S., and Nakamaru, T. (2004),

“Strong ground motions of the 2003 Bam Earthquake, southeast of Iran

(Mw=6.5)”. Bulletin of the Earthquake Research Institute, University of

Tokyo, 79, pp 47–57.

65. Kallivokas, L. F., Juneja, T., and Bielak, J. (2005), “A symmetric Galerkin

BEM variational framework for multi-domain interface problems”.

Computational Methods in Applied Mechanics and Engineering, 194, pp

3607–3636.

66. Kausel, E. (1988), “Local transmitting boundaries”. Journal of Engineering

Mechanics, ASCE, 114, pp 1011–1027.

67. Kausel, E. (1992), “Thin layer method: formulation in time domain”.

International Journal for Numerical Methods in Engineering, 37, pp 927–941.

68. Keller, J. B. and Givoli, D. (1989), “Exact non-reflecting boundary

conditions”. Journal of Computational Physics, 82, pp 172–192.

69. Kellezi, K. (2000), “Local transmitting boundaries for transient elastic

analysis”. Soil Dynamics and Earthquake, 19, pp 533–547.

70. Keys, R. G. (1985), “Absorbing boundary conditions for acoustic media”.

Geophysics, 50, pp 892–902.

71. Krenk, S. (2002), “Unified formulation of radiation conditions for the wave

equations”. International Journal for Numerical Methods in Engineering, 53,

pp 275–295.

72. Kunar, R. R. and Marti, J. (1981), “A non-reflecting boundary for explicit

calculations. Computational methods for infinite domain media-structure

interaction”, AMD (Ed. A.J.Kalinowski), 46, pp 183–204.

73. Lehmann, L. (2005), “An effective finite element approach for soil-structure

analysis in the time-domain”. Structural Engineering and Mechanics, 21, pp

437–450.





74. Lehmann, L. and Ruberg, T. (2006), “Application of hierarchical matrices to

the simulation of wave propagation in fluids”. Communications in Numerical

Methods in Engineering, 22, pp 489–503.

75. Liao, Z. P. (1996), “Extrapolation non-reflecting boundary conditions”. Wave

motion, 24, pp 117–138.

76. Liao, Z. P. and Wong, H. L. (1984), “A transmitting boundary for the

numerical simulation of elastic wave propagation”. Soil Dynamics and

Earthquake Engineering, 3, pp 174–183.

77. Lin, H. and Tassoulas, J. L. (1986), “A hybrid method for three-dimensional

problems of dynamics of foundations”. Earthquake Engineering and Structural

Dynamics, 14, pp 61–74.

78. Lindemann, J. and Becker, W. (2002a), “Analysis of the free-edge effect in

composite laminates by the boundary finite-element method”. Mechanics of

Composite Materials, 36, pp 207–214.

79. Lindemann, J. and Becker, W. (2002b), “Free-edge stresses around holes in

laminates by the boundary finite-element method”. Mechanics of Composite

Materials, 38, pp 407–416.

80. Luco, J. E. (1982), “Linear soil-structure interaction: a review”. Earthquake

Ground Motion and Its Effects on structures, AMD (Ed. S.K. Datta), ASME,

53, pp 41–57.

81. Luco, J. E. and Westmann, R. A. (1971), “Dynamic response of circular

foundations”. Journal of Engineering Mechanics, ASCE, 97, pp 1381–1395.

82. Lysmer, J. (1970), “Lumped mass method for Rayleigh waves”. Bulletin of

the Seismological Society of America, 43, pp 17–34.

83. Lysmer, J. and Kuhlemeyer, R. L. (1969), “Finite dynamic model for infinite

media”. Journal of Engineering Mechanics, ASCE, 95, pp 859–877.

84. Muravskii, G. (2001), “Mechanics of non-homogeneous and anisotropic

foundations”. Springer, Berlin.

85. Nicolas-Vullierme, B. (1991), “A contribution to doubly asymptotic

approximation: an operator top-down approach”. Journal of Vibration and

Acoustics, ASME, 113, pp 409–415.

86. Park, S. and Tassoulas, R. S. (2002), “Time-harmonic analysis of wave

propagation in unbounded layered strata with zigzag boundaries”. Journal of

Engineering Mechanics, ASCE, 128, pp 359–368.

87. Paronesso, A. and Wolf, J. P. (1995), “Global lumped-parameter model with





physical representation for unbounded medium”. Earthquake Engineering and

Structural Dynamics, 24, pp 637–654.

88. Rajapakse, R. K. N. D. and Gross, D. (1995), “Transient response of an

orthotropic elastic medium with a cavity”. Wave motion, 21, pp 231–252.

89. Rajapakse, R. K. N. D. and Wang, Y. P. (1991), “Elastodynamic Green

functions of orthotropic half-plane”. Journal of Engineering Mechanics,

ASCE, 117, pp 588–603.

90. Rajapakse, R. K. N. D. and Wang, Y. P. (1993), “Green’s functions for

transversely isotropic elastic half space”. Journal of Engineering Mechanics,

ASCE, 119, pp 1725–1746.

91. Richter, C. and Schmid, G. (1999), “A Green function time-domain boundary

element method for the elastodynamics half-plane”. International Journal for


92. Rokhlin, V. (1983), “Rapid Solution of Integral Equations of Classical

Potential Theory”. Journal of Computational Physics, 60, pp 187–207.

93. Ruge, P., Trinks, C., and Witte, S. (2001), “Time-domain analysis of

unbounded media using mixed-variable”. Earthquake Engineering and

Structural Dynamics, 30, pp 899–925.

94. Seale, S. H. and Kausel, E. (1989), “Point loads in cross-anisotropic, layered

half spaces”. Journal of Engineering Mechanics, ASCE, 115, pp 509–524.

95. Shirron, J. J. and Babuska, I. (1998), “A comparison of approximate boundary

conditions and infinite element method for exterior Helmholtz problems”.

Computer Methods in Applied Mechanics and Engineering, 164, pp 121–139.

96. Sommerfeld, A. (1949), “Partial Differential Equations in Physics”, chapter

28. Academic Press, New York.

97. Song, C. (2004a). “A matrix function solution for the scaled boundary finite-

element equation in statics”. Computational Methods in Applied Mechanics

and Engineering, 193, pp 2325–2356.

98. Song, C. (2004b), “Weighted block-orthogonal base functions for static

analysis of unbounded domains”; Proceedings of the 6th World Congress on

Computational Mechanics, Beijing, China, 5-10 September; pp 615-620.

99. Song, C. and Wolf, J. P. (1994), “Dynamic stiffness of unbounded medium

based on damping-solvent extraction”. Earthquake Engineering and Structural

Dynamics, 23: pp 169–181.

100. Song, C. and Wolf, J. P. (2002), “Semi-analytical representation of stress





singularity as occurring in cracks in anisotropic multi-materials with the

scaled boundary finite-element method”. Computers & Structures, 80, pp

183–197.

101. Tan, H. and Chopra, A. (1996), “Dam-foundation rock interaction effects in

earthquake response of arch dams. Journal of Structural Engineering”, ASCE,

122, pp 528–538.

102. Thatcher, R. W. (1978), “On the finite element method for unbounded

regions”. SIAM Journal of the Institute of Mathematics and Its Applications,

15, pp 466–477.

103. Thompson, L. L. and Huan, R. N. (1999), “Finite-element formulation of

exact non-reflecting boundary condition for the time-dependent wave

equation”. International Journal for Numerical Methods in Engineering, 45, pp

1607–1630.

104. Ting, L. and Miksis, M. J. (1986), “Exact boundary conditions for scattering

problems”. Journal of Acoustic Society of America, 80: pp 1825–1827.

105. Tsynkov, S. V. (1998), “Numerical solution of problems on unbounded

domains: a review”. Applied Numerical Mathematics, 27, pp 465–532.

106. Underwood, P. and Geers, T. L. (1981), “Doubly asymptotic boundary-

element analysis of dynamic soil-structure interaction”. International Journal

of Solid Structures, 17, pp 687–697.

107. van Joolen, V., Neta, B., and Givoli, D. (2005), “High-order Higdon-like

boundary conditions for exterior transient wave problems”. International

Journal for Numerical Methods in Engineering, 63, pp 1041–1068.

108. Velez, A., Gazetas, G., and Krishnan, R. (1983), “Lateral dynamic response of

constrained head piles”. Journal of Geotechnical Engineering, ASCE, 199, pp

1063–1081.

109. Vrettos, C. H. (1999), “Vertical and rocking impedence for rigid rectangular

foundations on soils with bounded non-homogeneity”. Earthquake

Engineering and Structural Dynamics, 28, pp 1525–1540.

110. Wang, C. H., Grigoriev, M. M., and Dargush, G. F. (2005), “A fast multi-level

convolution boundary element method transient diffusion problems”.


1926.

111. Wang, C. Y. and Achenbach, J. D. (1995), “3-dimensional time-harmonic

elastodynamics Green’s functions for anisotropic solids”. Proceedings of the

Royal Society London Series A-Mathematical and Physical Sciences, 449, pp

441–458.





112. Wegner, J. L., Yao, M. M., and Zhang, X. (2005), “Dynamic wave-soil-

structure interaction analysis in the time domain”. Computers & Structures, 83,

pp 2206–2214.

113. Wegner, J. L. and Zhang, X. (2001), “Free-vibration analysis of a three-

dimensional soil-structure system”. Earthquake Engineering and Structural

Dynamics, 30, pp 43–57.

114. Wolf, J. P. (1985), “Dynamic Soil-Structure Interaction”. Prentice-Hall,

Englewood Cliffs.

115. Wolf, J. P. (1986), “A comparison of time-domain transmitting boundaries”.

Earthquake Engineering and Structural Dynamics, 14, pp 655–673.

116. Wolf, J. P. (1988), “Soil-Structure-Interaction Analysis in Time Domain”.

Prentice-Hall, Englewood Cliffs.

117. Wolf, J. P. (2003), “The Scaled Boundary Finite Element Method”. John

Wiley & Sons, Chichester, England.

soil structure interaction analysis methods - state of art ... · pdf filesoil structure...

Documents