Vibrations in soils and fluids with random properties

In connection with the construction of new buildings, vibrations in soils are awaking increasing interest, since as a result of the dense civilization in Europe traffic lines and industrial plants have to be planned close to residential zones. Due to tightened legal restraints, direct airborne noise has been reduced by a multitude of measures, thus making ground borne noise and secondary airborne noise highly significant. The goal of the project is to investigate the wave propagation in horizontally stratified half spaces and fluids. In the past, methods based on Fourier and Hankel transforms have proved useful for calculating the wave emission up to a large number of wavelengths. The project applicant himself has already dealt with deterministic as well as random models. The new approach proposed assumes a random distribution of the system parameters along the surface, in order to account for irregularities of top layers or at a barrier layer between water and the sea bottom. To this end, transformations with respect to time and the horizontal directions are applied, yielding ordinary differential equations, which are solvable analytically for deterministic cases. Moreover, for random cases or cases of varying properties with respect to depth, e.g. random shear modulus, approximate solutions based on finite elements in the transformed space have been derived. The advantage of transform methods is their orthogonality, which provides the opportunity to calculate the spectral bins separately for every frequency and wave number. As a final step, the back transformation from the wave number domain is necessary, in order to determine the frequency spectrum of the deflections of each desired position in the half space. Applying random properties varying along the horizontal direction straightforward neutralizes the orthogonality, and disturbs the advantage of the transform method. But, applying the polynomial chaos transform additionally and putting higher moments to the load side allows to establish an iterative procedure that preserves the orthogonality of the transform. Compared with the deterministic cases, the additional numerical effort is limited to a tolerable extent. The results of the inverse transforms are frequency spectra of the moments for any desired position in space, e.g. mean value and standard deviation. By using an approximation based on higher moments, the nonlinearity of the inversion of random system matrices is taken into account.

Strategies that deal with the reduction of traffic induced vibrations have become more and more important due to the construction of high speed railway lines in or near residential areas. One part of these strategies is to develop computer models that simulate the propagation of waves in soil layers. With such models it is possible to reduce planning costs, since simulation can detect places where special measurements and constructions may be necessary. Up to now mostly deterministic models, i.e. models with fixed material parameters, have been used, but since in general it is quite hard to determine these parameters exactly, it is reasonable to use a method which also has some stochastic components. These models certainly have the disadvantage that they use high amounts of computing power, which makes them hardly applicable for practical purposes. During this project, a stochastic model was developed that allows for stochastic material parameters, but only uses a reasonable amount of computer resources. With this model it is possible to calculate the propagation of waves that are induced by moving loads in soil layers with stochastic material parameters. The basic idea behind this implementation is to split the system into deterministic and stochastic components. Operations which need a lot of calculations are only applied to the deterministic part which is easier to handle. The stochastic part is integrated into the global system with the help of a special iteration scheme. The project combines ideas from different scientific areas, for example mechanics (Finite Element Methods), signal processing (Fourier- and wavelet-analysis) and numerical mathematics (development and implementation of algorithms), and therefore acts as an interesting opportunity for transdisciplinary research.