M.Sc Thesis
M.Sc Student Gur Ido Solution of large-scale aeroacoustics problems Using enriched finite elements Department of Aerospace Engineering Professor Dan Givoli

Abstract

The measurement and calculation of aircraft noise in the far field is an important subject in environmental engineering, since such noise has an acute effect on the community. A typical Large-Scale aero-acoustic problem consists of a moving source (airplane) that is passing over a changing topography. In some location there might be a human listener that hears the source. Until recent years, most of the work that was done in this context has been empirical in nature. Mathematically, the problem may be posed, in the frequency domain, as that of determining the Sound Pressure Level (SPL) distribution near the ground due to a point source of a given acoustic spectrum. The process of calculating the SPL distribution involves the repeated solution, for many different wave numbers, of the Helmholtz equation  in the upper half space domain with the given impedance boundary condition imposed on the ground. All these solutions are then combined to yield the SPL distribution.

For ground with a zero or infinite impedance the Helmholtz problem is easily solved analytically using the method of images. For finite impedance (even with a flat ground) the problem becomes much more complicated for analytic treatment.

Direct numerical treatment of the Helmholtz problem using the finite element method is difficult for two main reasons. First, the high frequency range is difficult to treat and requires special techniques due to the high resolution needed and the associated dispersion (pollution) error. Second, one has to cope with the unboundedness of the domain using techniques like Absorbing Boundary Conditions or Perfectly Matched Layers.

In this work two types of solution methods were developed for the solution of the Helmholtz equation above a ground with finite impedance. The first method is an extension of the mirror image method that will be called fictitious sources method and the second method is an extension of the finite element method (FEM) is such a way that the information about the sound waves (i.e. frequency and direction) is implemented inside the basic shape functions of the scheme. This method is called generalized finite element method (GFEM) or partition of unity (PUM) according to the specific use.