Keywords: halfspace dynamics, tunnel dynamics, integral transform technique.

In the dynamic calculation of tunnels subjected to time depending moving loads one has to consider the tunnel structure itself with its finite dimensions in the lateral and its infinite extension in the longitudinal direction. For the soil, a system which extends to infinity in both horizontal directions and in the positive z-direction, an excavation for the tunnel has to be taken into account. This situation requires a description which allows to represent the "local" effects as well as the effects of the infinity of the tunnel in the longitudinal and of the soil additionally in the lateral and vertical direction. Correspondingly it is advantageous to apply a FEM/BEM or a FEM/ITM coupling [1,2,3]. In the paper the tunnel and a portion of the surrounding soil shall be modelized by Finite Elements in a Fourier transformed wavenumber/frequency domain (regarding the longitudinal direction and the time) in order to take account of the respective infinity. The soil as a whole will be described in a domain transformed additionally in regard of the second horizontal direction, the y-direction. (The chosen approach leads to a favourable description particularly in view of the possibility of a coupling to moving vehicles.)

Only a very small FEM mesh (in the Fourier transformed domain) has to be considered, if it is coupled at its outer boundary to a stiffness matrix in the same transformed domain representing the infinite halfspace exterior to the excavation for the FEM mesh.

The stiffness matrix is derived by the aid of ITM starting with the halfspace without any excavation: A sufficiently large set of properly selected shape functions for dynamic loadings is applied along a preselected internal closed surface surrounding the tunnel. This surface is narrower than the surface of the later FEM excavation. The dynamic reaction of the full halfspace (without excavation) subjected to these fictitious loadings which are -Dirac distributed in the directions normal to the internal surface can be found in analytical expressions. The respective stresses and displacements are evaluated (with an inverse transform regarding y) for a second surface which is identical with the later FEM boundary. (The two different surfaces - the first one for the application of the -loadings and the second one for the evaluation of the respective stresses and displacements - have to be considered in order to observe requirements of an error minimization.) In order to arrive at the dynamic stiffness matrix which represents the excavated halfspace, the evaluated stresses and displacements have to be brought into a corresponding form. This can be achieved by means of a residuum minimization procedure along this second boundary taking into account the local shape functions of the FEM approach.

After the stiffness matrix is applied at the boundary, the interaction problem can be solved by a 2D FEM calculation in the transformed () domain. The evaluation of the final result for the original domain requires an inverse Fourier transform. This task is the crucial point insofar as the computational effort is concerned. In earlier papers [4] and also in current research [5] remarkable progress was made in the reduction of the effort by the introduction of an additional (Wavelet) transform technique.

The application shall be illustrated for a simple situation to show the possibilities and the efficiency of the dynamic FEM/ITM coupling.

1

Beskos, D.E., "Boundary element methods in dynamic analysis", Part II, Applied Mechanics Reviews 50 (3), 149-197, 1997. doi:10.1115/1.3101695

2

Grundmann, H., "Dynamic interaction of structures with the subsoil", EURODYN '99, Frýba & Náprstek (eds.), Balkema, Rotterdam, 31-41, 1999.

3

Zirwas, G., "Ein hybrides Verfahren zur Behandlung der Bauwerk-Bodenwechselwirkung mit analytischen Integraltransformationen und numerischen Ansätzen", PhD Thesis, Technische Universität München, 1996.

4

Lieb, M., "Adaptive numerische Fouriertransformation in der Bodendynamik unter Verwendung einer Waveletzerlegung", PhD Thesis, Technische Universität München, 1997.

5

Grundmann, H., Lenz, S., "Nonlinear interaction between a moving SDOF system and a Timoshenko beam/halfspace support", Archive of Applied Mechanics 72 (11-12), 830-842, 2003.

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