Tesi di LAUREA SPECIALISTICA Titolo Numerical simulation of a train traveling in a tunnel Data 2010-05-03 Autore/i Proverbio, Alessandro Relatore Formaggia, L. Relatore Peirò, J. Full text non disponibile Abstract Over the past 40 years trains have gradually gained importance. Nowdays they play a dierent role than in the past because they have passed from being a slow transportation to represent an alternative at the airplane. The fact that their speed has tripled introduces aerodynamic problems and they require to study the shape both in order to have higher performances than to have more comfortable trains. This work aims to analyze the problem of a train traveling in a tunnel. This is a very interesting scenary because of the critical stresses that structures have to deal. Understanding the aerodynamic phenomena allows to better design the structures and to increase passengers comfort. When a train enters in a tunnel, pressure waves propagate form its nose all along the tunnel and they are re ected by the exit portal. Also the tail generates waves that are re ected. All these phenomena and their interactions require to be studied with a numerical method. The physical phenomenon is 3D, turbulent, unsteady and viscous but it has been possible to reduce the model to a monodimensional hyperbolic system of non linear dierential equations written in a conservative form. In the present work we have used a discontinuous Galerkin (DG) approach, as explained in chapter 2. This method does not introduce numerical viscosity in the spatial discretization and this is very useful property in a wave propagation problem. The only numerical viscosity is due to the Runge{Kutta or Strong Stability Preserving Runge{Kutta methods adopted for the time stepping. The method presents unexpected oscillations both in time than in space over the tunnel length. They are very strong and require high polynomial orders of the DG basis or large numbers of mesh elements in order to be reduced. This necessity dramatically increases computational costs. Two type of errors could generate the oscillations: a numerical quadrature error or an error due to the projection of the functions on the polynomial basis. In chapter 3 has been presented a deep analysis of the oscillations that appears to be connected to the movement of the train along the mesh elements. In chapter 4 we have described the implementation of an implicit method. It does not have any stability constrain and gives us the possibility to x an arbitrary timestep. This fact shows that a timestep that moves the train of one element per time keeps the error constant. Because of the results in chapter 3, in chapter 5 we have adopted a new formulation, able to do not project the area and its derivative on the polynomial basis and consequently avoiding their projection error. The function representing the area reproduces the eect of the train movement in our problem. This fact allows us to study the only integration error as done in section 5.1.2. Next, in section 5.2, we have erased the integration error. In the end, in section 5.2, we have gone back to the original formulation