Implementation of second-order absorbing boundary conditions in frequency-domain computations

The finite element method is widely used in soil-structure interaction studies when dealing with problems involving complicated boundaries and inhomogeneities. Finite elements are employed in the near field (neighborhood of the source of excitation or region of interest). Because of the unboundedness of the soil region, appropriate conditions must be applied on the boundary of the near field in order to reproduce the effect of the far field (unbounded region). Since the objective is to allow absorption or transmission of the waves impinging on the boundary, the conditions have become known as absorbing or transmitting conditions. In one of the existing treatments, namely, the consistent transmitting boundary, the finite element method is used to obtain modes es of vibration in the far field. Then, any motion in the unbounded region can be synthesized as a linear combination of those modes which satisfy the radiation and boundedness conditions where only discretization with respect to the vertical direction is employed. Then, a dynamic stiffness matrix relating the nodal forces and nodal displacements at the boundary is obtained. This approach leads to a plane consistent transmitting boundary for a layered stratum in antiplane shear and plane strain and a cylindrical one for both axisymmetric and non-axisymmetric vibrations of a layered stratum. The procedure is rigorously applicable only to soil layers on a rigid base. To extend the applicability to layers over a homogeneous half space, paraxial approximations of the impedances of the halfspace have been derived and applied on the bottom of the soil layers. Although such impedances resulting from second-order paraxial approximations have been incorporated in the calculation of semidiscrete Green functions, their use with conventional finite elements has not been investigated. In this work, the implementation of these approximations within the framework of the finite element method is examined in two-dimensional as well as axisymmetric settings. The performance of the approximations is assessed by means of applications to problems of foundation dynamics.;...