INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2010

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2010 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN 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. siddharth.shah@utu.ac.in 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 Received on July 2011 published on November

2 Soil structure interaction analysis methods State of art Review Siddharth G. Shah et.al., 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 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 nonhomogeneous 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 177

3 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 178

4 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 fullspace 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 nonhomogeneity 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 twodimensional 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 threedimensional 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. 179

5 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 halfspace. 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 timeharmonic 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 180

6 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 twodimensional 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 displacementpotential 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 boundaryvalue 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 nonhomogeneous anisotropic soils using the boundary element method have been carried out 181

7 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 twodimensional 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 182

8 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 eigenvalue 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 183

9 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 eigenvalue 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 fullspace 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 socalled 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 184

10 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 finiteelement 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 staticstiffness 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 firstorder ordinary differential equations. Thus, the displacement and stress fields are 185

11 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 dynamicstiffness 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 timediscretization 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 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) 186

12 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 C p τ = b ρ C s where, σ and τ are the normal and shear stresses on the boundary, respectively; ρ is the mass density; C p and C s 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 187

13 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 188

14 series. For three dimensional scalar wave equation in spherical coordinates, the m th 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 pseudodifferential operator can be cast in this form so that they are special cases of his multidirectional 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 doublyasymptotic 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 lowfrequency behavior is modeled using a doubly asymptotic boundary implemented implicitly. This boundary condition is temporally local but spatially global due to fully 189

15 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 firstorder 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 190

16 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 twodimensional 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 191

17 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 nondispersive 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 192

18 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 timedomain 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 193

19 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. 194