Dynamic Analysis Of Soil-Structure Interaction Problems Considering Infinite Boundaries

: One of the limitations of the usage of the finite element method in dynamic soil-structure interaction arises when it is used for the modelling of an infinite domain if nothing is done to prevent from artificial reflections at the mesh boundary; errors are introduced into the results. To handle reflections, different artificial boundaries have been proposed. The aim of such boundaries is to make them behave as nearly as possible as if the mesh extends to infinity. In this paper they are known as transmitting, absorbing or silent boundaries. A brief description to two different approaches of absorbing boundaries is made, first by using infinite elements and the second by using viscous boundaries method. For this purpose the computer program named “MIXDYN” is modified in this study to “Mod-MIXDYN” by adding mapped infinite element model to the finite elements models of the program to be used for dynamic analysis of soil-structure interaction problems. A new derivation of the mapped functions is made in this study for the cases when the infinite direction is extended to the left horizontally (at negative ξ direction) and down vertically (at negative η direction). Two verification problems are solved to compare the results of the modified program with the results of other software, namely ANSYS and OpenSees representing other types of elements (dashpot elements) modelling boundaries as viscous boundary. It was found that the transmitting boundary absorbs most of the incident energy. The distinct reflections observed in the "fixed boundaries" case disappear in the "transmitted boundaries" case. This is true for both cases of using viscous boundaries or mapped infinite elements. The viscous boundaries are more effective in absorbing the waves resulting from dynamic loads than mapped infinite elements. This is clear when comparing the results of both types with those of transient infinite elements .


Introduction:
Two important characteristics that distinguish the dynamic soilstructure interaction system from other general dynamic structural systems are the unbounded nature and the nonlinearity of the soil medium.
Generally, when establishing numerical dynamic soilstructure interaction models, the following problems should be taken into account [Zuo, 2002] However, due to the complexity of dynamic soil-structure interaction, numerical modelling of this phenomenon still remains a challenge.
Various kinds of analytical formulations and computer programs have been developed to solve the complex problem.There still exist many difficulties to cover in one model all the problems listed above.Current models usually stress one or several of these problems [Zuo, 2002] .

Infinite Elements
It is convenient to classify infinite elements as of static or dynamic type, as the methods needed for the two types are quite different.For static infinite elements, mapped and decay function type which can be used for some dynamic problems will be discussed.

• Static Infinite Elements
The succeeding infinite element formulations have followed two main lines of development.These have been [Bettess, 1977[Bettess, , 1992]:a.Mapping of the element from finite to infinite domain.b.Using decay functions in conjunction with the ordinary finite element shape function.

• Mapped Infinite Elements
Many of the infinite elements proposed have used the idea of mapping, or can be cast in that form.
Beer and Meek also used a standard Gauss-Legendre numerical integration.They found that a simple 2 x 2 integration was beneficial, as a higher order integration tended to make the infinite domain elements too stiff [Beer and Meek, 1981].
Curnier (1983) characterized the two methods (decay function and mapping) as "descent shape function" and "ascent shape function", respectively.It was shown that the two methods can be made equivalent, depending upon the choice of shape function.
Pissanetzky (1984) used a similar approach of Beer and Meek (1981) but he carried out the integration in the infinite domain, and so had to modify the Gauss-Legendre abscissae and weights.
Okabe (1983) gave various possible shape functions for infinite domains, based on what he calls "The generalized Lagrange family for the cube".
The form in which Zienkiewicz mapping was originally given was simplified and systematized by Marques and Owen (1984), who worked out and tabulated the mapping function for large range of commonly used infinite elements, [Bettess, 1992].

Zienkiewicz Mapped Infinite Elements:
There is no doubt that the Zienkiewicz approach (1983) leads to a clarification and simplification of this class of method.The mapping functions and derivatives are given in Table (1) for twodimensional quadratic serendipity mapped infinite element shown in Figure ( 1), [Bettess, 1992].In this table, M refers to the mapping functions while ξ and η are the local coordinates.
A precisely analogous procedure to derive these mapping functions is described in detail by Dawood (2006).
Using the same procedure, mapping functions of the infinite elements in two cases will be derived ; the first when the infinite element extends to infinity in the negative ξ direction and in the second case, the infinite element extends to infinity in the negative η direction as shown in Figure (2).The mapping functions and their derivatives for these two cases are derived here and shown in Tables (2) and (3  Figure ( 5) shows the finite element mesh of this problem.The mesh, material properties and analysis information are listed in Tables (4) and ( 5).
Figures ( 6) and ( 7) present the time history of the vertical displacement at top node as predicted by the program (ANSYS) considering fixed and viscous boundaries, respectively.It is assumed that 15.24 cm thick concrete lining is added on the surface of the open-mined space so that the inside dimensions of the opening remain the same as in the unsupported case.
The material properties of the half-space are shown in Table (6).
Under the effect of this loading condition, the plane strain problem is solved by the finite element method with the coupling of finite and infinite elements.
The finite element mesh is shown in Figure ( 15), while the finite element mesh including infinite elements is shown in Figure (16).
In Figure ( 17), the finite element mesh with viscous boundaries is drawn.Table (

Conclusions:
A dynamic finite-element analysis is carried out for soil-structure interaction problems considering transmitting boundaries.Two types of boundaries are considered: viscous boundaries and mapped infinite elements.The results are compared for three cases; the first one using finite elements only, the second using 5-node mapped infinite elements and the third one using viscous boundaries.The computer program named "MIXDYN" (Owen and Hinton, 1980) is modified in this study to "Mod-MIXDYN" by adding 5-node coding of infinite element presented by Selvadurai and Karpurapu (1988).A new derivation of the shape and mapped functions is made in this study for the cases when the infinite direction is extended to the left and down.The following conclusions are drawn: 1) The transmitting boundary absorbs most of the incident energy.The distinct reflections observed in the "fixed boundaries" case disappear in the "transmitted boundaries" case.This is true for both cases 2): Serendipity eight-node two dimensional infinite element extending to infinity representing case a.The mapping functions and derivatives for element extending to negative ξ direction, as derived in this study.

Verification
Problem No. (1) A research at the University of Washington using the program (OpenSees) (which is a finite element tool developed by Berkley University), zero-length dashpot elements with viscous components normal and tangent to a given boundary are used to simulate the transmitting condition [u.washington.eduWebsite].The dashpot coefficients are determined in terms of the material properties of the semi-infinite domain, as shown in Figure (2).As a verification problem, the results of the above mentioned research on that website are used in this study to assess this problem.A simple 1-Dimensional case is analyzed using the program (ANSYS) in addition to the program (OpenSees).The 1-D condition is enforced constraining both sides of the model to move the same amount.The analysis is performed using fixed boundary condition at the bottom.The model details are shown in Figure (3) and the loading function is drawn in Figure (4).

Figures ( 8
Figures (8)  and (9) present the time history of the vertical displacement at mid-node for the same conditions.Figures(10) to (13)  show the time history of the vertical displacement at top and mid-nodes as predicted by the program (OpenSees).A comparison of recorded displacements at the top and middle nodes shows that the transmitting boundary absorbs most of the incident energy.The distinct reflections observed in the "fixed" case disappear in the "transmitted" case.A comparison between Figures (6) to (9) and Figures (10) to (13) show that the results of the program (ANSYS) adopting fixed boundary with dashpot elements are in good agreement with those of the program OpenSees which adopts transmitting viscous boundaries, using zero-length dashpot elements.

Fig
Fig. (11): Vertical displacements at top node as predicted by the program (OpenSees) for viscous boundary (VB).

Fig
Fig. (19): Displacement versus time at points A, B and C considering viscous boundary (VB) as predicted by (ANSYS).

Fig. ( 20
Fig. (20): Displacement versus time at points A, B and C considering fixed boundaries as predicted by (Mod-MIXDYN).

Fig
Fig. (21): Displacement versus time at points A, B and C considering fixed boundaries as predicted by (ANSYS).