Finite Difference and Finite Element Methods

Abstract

Applications of the finite difference and finite element techniques to vibroacoustic problems are presented. The basic ideas and the mathematical descriptions are outlined for both of the methods and examples are given to demonstrate the potential of such numerical techniques. The finite difference method is illustrated by studying the resonant frequencies and forced response of a cavity closed by an elastic plate. The Helmholtz and Kirchhoff plate equations are the starting points for the discretization. It is also demonstrated how the Richardson extrapolation method can be used to minimize the errors in the numerical calculations. For the finite element method, the idea is first illustrated by solving a simple acoustic duct problem. It is further shown how models can be made for vibrating plates based on a thick plate theory, and for wave propagation in elastic solids. The coupling of plates and acoustic fields is again illustrated by calculations of resonant frequencies for different plate cavity geometries. Another example considered is the excitation of a cylinder on the ocean floor. As the acoustic damping by porous materials is of importance in noise control, it is shown how a finite element model can be made for a porous elastic material (Biot theory). Use of the model is illustrated by a study of sound transmission through a wall made up by a porous materia! sandwiched between two elastic plates.