Numerical approximation of scalar fields propagation in lossy media. Application to acoustic problems

This doctoral thesis deals with the use of the Finite-Difference Time-Domain (FDTD) method for solving problems of acoustic propagation and scattering. Since this method has been originally developed in the framework of electromagnetism as a stable and reliable numerical scheme for the solution of Maxwell’s equations and since little progress had been conducted towards the direction of applying this background in acoustic wave propagation, it is evident that a considerable need for an analogous technique in acoustics exists. This is the objective of the present work and the whole document is organized as follows. The first chapter constitutes an introductory note, which refers to the most important attempts made in the respective scientific domain as well as to the motivation for the selection of the specific research theme. There is also an analytical description of the content of every individual chapter. In the second chapter, the mechanism of the production and propagation of acoustic waves is presented in order to indicate the basic differences and similarities between electromagnetic and acoustic phenomena. The concept of the finite difference approximation is sufficiently described and finally the first FDTD scheme is properly built for the one-dimensional acoustic problem. This is the simplest case and provides the occasion for a first-hand evaluation of the method’s competence. The third chapter contains the development of the complete, three-dimensional, FDTD scheme in acoustics. In parallel, the crucial topic of the proper termination of the computational domain is discussed, and the technique of the perfectly matched layer (PML), the most effective boundary condition, is implemented in acoustics after the appropriate modifications. Taking into account all the individual characteristics of the acoustic propagation and scattering that could cause instabilities and other problems, we conclude to a stable and effective numerical scheme. The selection of specific problems, for which the exact analytical solutions are possible to be obtained, enables us to proceed to the comparison of the respective results and thus to the verification of the reliability of the suggested scheme. However, the aforementioned absorption technique of the PML does not work in cases where lossy media are concerned. This is of vital importance in acoustics as in most case losses due to thermal conduction and viscosity effects must be taken into account. The solution to this problem constitutes the content of the fourth chapter. Based on the idea of the stretched coordinates we introduce a generalized PML method that performs significant absorption of traveling acoustic waves for both lossless and lossy media. Numerical results support the suggested PML theory as well as reveal the proper modifications, which lead to the achievement of the optimum absorbing boundary condition. In the last chapter the proposed methodology is extended to more complicated problems, such as guided sound waves in ducts and rooms. The general characteristics of sound propagation in ducted domains are studied and the analytical solution in the simplest case of the rectangular duct with rigid walls is derived. The application of the developed FDTD scheme permits us to verify its effectiveness and to move further in cases with bent ducts or pipes of varying cross-section and Y junctions with flexible or rigid walls