Keywords: wave propagation, numerical modelling, perfectly matched layers, tunnel-ground interaction.

The simulation and study of problems related to vibrations induced by traffic in tunnels is a difficult and complex task. The semi-infinitive character of the domain surrounding the tunnel associated with the three-dimensional characteristics of the problem, as well as to the movement of the train, are some of the major causes of that complexity. However, during recent years some approaches have been proposed for dealing with this kind of problems, generally taking advantage of the periodicity or invariability of the cross-section of the tunnel-ground system. Aiming to achieve a high computational performance, semi-analytical models were proposed, namely by Hussein and Hunt [1] and by Muller [2]. However the semi-analytical approaches are shown to be very limited in terms of adaptability to the geometries usually found in practical applications. In order to overturn those limitations, Yang and Hung [3] suggested the use of two and a half dimensional finite elements for the simulation of the whole domain, i.e., the tunnel and the surrounding soil, being the artificial boundaries treated using infinite elements in order to minimize the spurious reflections of waves. However, it is well known that the efficiency of these types of procedure for the treatment of the artificial boundaries is limited and dependent on the wavenumber of the propagating waves. Alternatively, Gupta et al. [4] presented a periodic model based on the coupling of the finite element and boundary element methods. Despite the high accuracy achieved with that model, it leads to a high cost in terms of numerical computing.

An efficient and accurate solution can also be achieved using two and a half dimensional techniques in the context of the finite element method (FEM), and adopting perfectly matched layers (PMLs) for the treatment of the boundaries arising from the truncation of the finite element mesh. This is the approach followed in the present paper. Since the use of the FEM-PML approach is not very usual, the equations of the two and a half dimensional PML are derived, highlighting the compatibility with the behaviour of the FEM. After that brief presentation of the model, a range of validation examples are depicted, showing the accuracy of the model.

Finally, an application of the method to the study of vibration induced by traffic in tunnels is presented, being the solution compared with the theoretical one obtained by resource to the model PiP previously presented by Hussein and Hunt [1].

From the studies here presented it is found that the two and half dimensional FEM-PML approaches is a reliable method for the study of vibrations induced by traffic. The use of two and half dimensional finite elements for the discretization of the cross-section is very suitable for dealing with complex geometries, the two and half dimensional PML approach providing a guarantee of accuracy for the local treatment of the artificial boundaries.

1

M. Hussein, H. Hunt, "A numerical model for calculating vibration from a railway tunnel embedded in a full-space", Journal of Sound and Vibration, 305(3), 401-431, 2007. doi:10.1016/j.jsv.2007.03.068

2

K. Muller, H. Grundmann, S. Lenz, "Nonlinear interaction between a moving vehicle and a plate elastically mounted on a tunnel", Journal of Sound and Vibration, 310, 558-586, 2008. doi:10.1016/j.jsv.2007.10.042

3

Y. Yang, H. Hung, "Soil Vibrations Caused by Underground Moving Trains", Journal of Geotechnical and Geoenvironmental Engineering, 134(11), 1633-1644, 2008. doi:10.1061/(ASCE)1090-0241(2008)134:11(1633)

4

S. Gupta, et al., "A comparison of two numerical models for the prediction of vibrations from underground railway traffic", Soil Dynamics and Earthquake Engineering, 27(7), 608-624, 2007. doi:10.1016/j.soildyn.2006.12.007

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