ON A MULTI-WAVE ELASTODYNAMICAL QUADRILATERAL INFINITE ELEMENT FOR SOIL STRUCTURE INTERACTION O MULTIVALNOM ELASTODINAMIČKOM KVADRILATERALNOM BESKONAČNOM ELEMENTU ZA OBUHVATANJE INTERAKCIJE KONSTRUKCIJE I TLA

ON A MULTI-WAVE ELASTODYNAMICAL QUADRILATERAL INFINITE ELEMENT FOR SOIL STRUCTURE INTERACTION Konstantin S. KAZAKOV In this paper, a multi-wave elastodynamical quadrilateral infinite element is proposed. This kind of element is appropriate for multi-wave soil-structure interaction problem. The formulation is based on the standard steps which are the same as in the Finite element method after mapping the infinite to finite domain of the element. It is shown that in the case of only one wave function used in the formulation, only one frequency, the proposed multiwave elastodynamical infinite element is reduced as a special case to single-wave elastodynamical infinite element. In addition, the mapping and the Lagrangian isoparametric shape functions for a 2D axisymmetric four node multi-wave elastodynamical quadrilateral infinite element and for a 2D axisymmetric eight node multi-wave elastodynamical quadrilateral infinite element are given. The basic aspects of the C1 and the Cn continuity along the finite/infinite element (artificial boundary) line in two-dimensional substructure SoilStructure Interaction problems are discussed. In this class of models such a line marks artificial boundary between the near and the far field of the model.


INTRODUCTION
This section is devoted to review the historical background of infinite elements from the original works to the latest contribution. Exterior domain scattering problems appear naturally in many engineering fields such as electrodynamics, magnetic problems, fluid flow, thermal analyses and so on. Wave propagation in an elastic infinite media and scattering of waves on bodies in a fluid which extends infinitely are of particular interest. When numerical methods are used the main difficulty in such problems arises in unbounded domain that has to be discretizied. Many suggestions and ideas for the treatment of the exterior domain have been presented and discussed in a number of research papers over the period of three decades. The exterior (infinite) domain cannot be completely discretizied with standard finite elements, and a lot of efforts have been spent in the development of new infinite element techniques.
One possible approach is to just truncate the computational domain at some distance away and to impose "appropriate" boundary conditions. Such boundary is called "artificial" boundary. In this case socalled viscous, absorbing or transmitting boundary conditions can also be used. It is evident that the computational efficiency depends than on the localization of the "artificial" boundary and the type of the boundary conditions. In a lot of problems such techniques provide acceptable results. In Soil-Structure Interaction problems such techniques are known as Substructural approach.

INFINITE ELEMENT METHOD WORKS
Infinite element method was introduced about three decades ago in the original work of Bettless. Then the method was developed and refined in many works. The first works were the works of Pissanetzky on the magnetostatics and Kim on the magnetic field problems. The original Bettless formulation is based on and derived for the Laplace problems. This formulation is very similar to the finite element formulation except for the element domain. In this formulation infinite element domain extends toward infinity in one direction and the corresponding shape functions being non polynomial but integrable over the element. Subsequently, Infinite elements are directly applicable in the Finite element method. Similar to the Finite element method, the order of the approximation and the choice of shape functions directly relate to the accuracy of an infinite element.
The mapped infinite elements were developed by Bettess and Zienkiewicz. These elements allow using polynomial shape functions and the used technique generates shape functions which are consistent with the form of the solution of the exterior domain. A mathematically precise variational formulation of infinite elements has only recently been discussed.

PRACTICAL CLASSIFICATION OF INFINITE ELEMENTS
From practical point of view infinite element can be classified into five classes: • classical infinite elements, • decay infinite elements, • mapped infinite elements, • elastodynamical infinite elements and • Wave envelope infinite elements.
The origin of the idea and the development of every one of the above classes are difficult to be dated. The first class infinite elements are based on the original so called "classical" formulation of the infinite elements. Decay functions from different mathematical types are used in the decay infinite element formulation. The mapped infinite elements are developed by using mapping functions. These functions map the infinite domain of the element into a finite. By this approach the obtained infinite element is similar to the classical finite element. The latest researches of infinite elements are devoted to the development of the elestodynamical infinite elements and wave envelope infinite elements. The last two classes can be treated as a special combination of the mapped and decay infinite elements.

MULTI-WAVE ELASTODYNAMICAL QUADRI-LATERAL FOUR NODE INFINITE ELEMENT
The displacement field in the elastodynamical infinite element can be described in the standard form of the shape functions based on wave propagation functions [7] as: , n is the number of nodes for the element and m is the number of wave functions included in the formulation of the infinite element. For horizontal wave propagation basic shape functions can be expressed as: are horizontal wave functions. By taking into account only the real parts of the wave functions, the equations of the wave propagation can be written as: where s c , p c are the velocities of the S-waves and Pwaves respectively. Expanding these functions in a Fourier-like series for all wave functions included in the formulation of the infinite element, the shape functions can be written as: where ϖ is the lowest frequency and q ϖ ω = . The coefficients q A can be written as: Using introduced in [11] hypotheses for the shape functions, equation (4)  Then united shape function can be written as Now equation (1) can be expressed as It can be easily shown that in the case of only one wave function used in the computational model, only one frequency, the proposed multi-wave elastodynamical infinite element is reduced to single-wave elastodynamical infinite element. It can be treated as a special case. In this case

TWO DIMENSIONAL MAPPED INFINITE ELEMENT
Mapping functions and the Lagrangian isoparametric shape functions are given for a 2D axisymmetric four node quadrilateral mapping infinite element and for a 2D axisymmetric eight node quadrilateral mapping infinite element can be written as

Mass and stiffness matrices
The stiffness and mass matrices can be given in a standard of the Finite element method style as

CONTINUITY THROUGH FINITE AND INFINITE ELEMENTS
The continuity through finite and infinite elements can be enforced in exactly the same way as between two finite elements because they have same degrees of freedom and approximation polynomial degrees. A sketch of the boundary between finite and infinite elements is given in Fig.1   For the functions written as (4)

CONCLUSION
This paper deals with the mapping functions and the Lagrangian isoparametric shape functions for a 2D axisymmetric four node quadrilateral mapping infinite element and for a 2D axisymmetric eight node quadrilateral mapping infinite element. In addition, the basic aspects of the C1 and the Cn continuity along finite/infinite element line in two-dimensional substructure Soil-Structure Interaction problems are analyzed and discussed. In this class of models such a line marks artificial boundary between the near and the far field of the model. Finally, some important remarks about the C1 and the Cn continuity are reviewed when the author proposed using wave functions in Soil-Structure Interaction models.