Implementation of the finite element method in the fourier-transformed domain and coupling with analytical solutions

The dynamic soil-structure interaction has to be modeled in an appropriate way in order to provide a realistic assessment of arising vibrations. Modeling of the complete emission- transmission- immission system by means of the emissiontransmission causes a high computational effort. In general appropriate modifications are necessary in order to describe the infinite extension of the soil where artificial reflections at the boundaries have to be avoided. An efficient approach to model the complete system is to describe only the irregular geometries of the emission and immission systems with finite elements and to model the transmission system of the soil by using analytical solutions based on Integral Transform Methods. These solutions are obtained for infinite systems using a Fourier-Transformation from the original domain to the wavenumber-frequency domain. In order to couple both methods an adapted Finite Element Method which can be used in the wavenumber-frequency domain has to be applied. In the scope of this paper the following systems are coupled: a halfspace with a cylindrical cavity and a Finite Element mesh which is placed inside the cavity. Applying a transformation along the longitudinal coordinate of the tunnel, the system is 2.5 dimensional which means that the formulation of two-dimensional Finite Elements contains the complete three-dimensional information of the system after an inverse transformation. For each wavenumber-frequency combination, the computed stiffness matrix for the Finite Element mesh is coupled with the respective wavenumber-frequency stiffness of the analytical solution describing the halfspace with cylindrical cavity using the substructure technique.