Wave finite element method for computing the dynamic response of periodic structures with transition zones and subjected to moving loads: Application to railways tracks with damaged or reinforced zones

The dynamics of transition zones linking two semi-infinite periodic structures is the subject of numerous researches, particularly in the railway track domain. For periodic structures, the Wave Finite Element (WFE) method is a numerical method helping the computation of the dynamics of these structures by reducing the consideration of the spatial domain to a few periods consisting of domains whose properties differ from those of the two right and left semi-infinite periodic structures. The WFE method firstly consists in reducing the degrees of freedom (DoFs) of one spatial period (substructure) to those of the borders of this substructure. Then, using the Floquet's theorem, these DoFs are computed by the mean of a wave analysis. This article presents new developments in the Wave Finite Element (WFE) method to compute the mechanical response of transition zones linking two semi-infinite periodic structures, with the aim of making complex computations affordable and of reducing the computation time. The WFE method is applied on each periodic structure to write the response of the boundaries of the central zone in terms of left-going and right-going waves. Some amplitudes of these waves can be directly computed from the external load. To get the unknown wave amplitudes, the wave equations are combined with the dynamic equilibrium equation of the central zone. Thus, this method reduces the computation of the dynamics of a structure containing a transition zone linking two semi-infinite periodic zones to a wave problem at the boundaries of the transition zone coupled to a FEM modelling of the transition zone. In this paper, writing the problem only in terms of wave amplitude allows a much better conditioning of the linear system giving the solution to the problem compared to classical methods combining wave amplitudes and usual degrees of freedom at the mesh nodes. Special developments are made to account for moving loads on the whole infinite structure. The case of moving loads is particularly considered because of its applications to railways. For simple geometries, numerical studies show a strong agreement between results obtained with this method and other experimental and analytical results. More complex examples are given for railways tracks with damaged or reinforced zones. The calculation of stresses and damage criteria in components of the track for healthy, damaged and repaired railway tracks shows the interest in repair.