The project extends isotropic computation of vibrations in the soil to anisotropic material. Up to now mainly deterministic models have been used in practice.
The current project introduces a stochastic model in order to quantify the randomness of the solution that results from the large randomness of soil parameters. A random process of the shear modulus either into the depth or in the horizontal directions is assumed. The random process is separated into discrete random variables and functions in space by the Karhunen Loeve expansion that is a continuous version of the principal component analysis (PCA). The discrete variables are the basis of the chaos polynomial transformation. An iterative procedure is used for inverting the matrices.
The classical procedure to deal with a stratified isotopic medium is the Fourier transformation of the Helmhotz potentials. These potentials are initally unkown for anisotropic medium. A Fourier transformation of all coordinates in space and time of the differential equations of the displacements together with the displacement-stress relation gives a matrix for the displacements and a matrix relation between displacements and stresses. The generalized eigenvalues of the first matrix give a relationship between the vertical wavenumber and the horizontal wavenumbers and angular frequency. The eigenvectors determine the direction of the related wave. For a half space the upgoing wave types are deleted to fulfill Sommerfeld's radiation condition. The remaining wavenumbers and vectors are used to define a homogenous solution for the unloaded layers of the stratified soil based on Dirac delta functions. A back transformation with respect to the vertical axis converts the Dirac functions into complex exponential functions.
The stresses and displacements are coupled at the interfaces. This leads to a matrix equation of the displacements at the interfaces, that have to be solved.
If Dirac delta impacts are used as a load, Green's functions of the displacements and stresses can be derived numerically. These functions are used in a boundary element method in 2D to simulate tunnels and trenches. The boundary element method takes place in the numerically back transformed space about the horizontal axis y. The horizontal axis x is still in the transformed domain that decouples the BEM problem for every wavenumber kx as long as the geometry of the boundaries does not change with respect to the coordinate x. A back transformation about the axis x is the last step of the procedure. The method is also named 2.5D.
The current application involves primary and secondary emissions of sound and vibrations at railway tracks. The usage of models as described, is highly needed, as the construction of new railway lines faces increasing public and community opposition throughout Europe. Current Application: (secondary) emissions of sound and vibrations at railway tracks.