Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 140

In cases where the geometry of the acoustical space is complicated and where the lumped‐element approach cannot be used, then it is necessary to use numerical approaches. In the late 1960s, with the advent of powerful computers, the acoustical finite element method (FEM) became feasible. In this approach, the fluid volume is divided into a number of small fluid elements (usually rectangular or triangular), and the equations of motion are solved for the elements, ensuring that the sound pressure and volume velocity are continuous at the node points where the elements are joined. The FEM has been widely used to study the acoustical performance of elements in automobile mufflers and cabins (See Chapter 10 of this book.).

The boundary element method (BEM) was developed a little later than the FEM. In the BEM approach the elements are described on the boundary surface only, which reduces the computational dimension of the problem by one. This correspondingly produces a smaller system of equations than the FEM. BEM involves the use of a surface mesh rather than a volume mesh. BEM, in general, produces a smaller set of equations that grows more slowly with frequency, and the resulting matrix is full; whereas the FEM matrix is sparse (with elements near and on the diagonal). Thus computations with FEM are generally less time‐consuming than with BEM. For sound propagation problems involving the radiation of sound to infinity, the BEM is more suitable because the radiation condition at infinity can be easily satisfied with the BEM, unlike with the FEM. However, the FEM is better suited than the BEM for the determination of the natural frequencies and mode shapes of cavities (See Chapter 10 of this book.).

Recently, FEM and BEM commercial software has become widely available. The FEM and BEM are described in Refs. [35, 36] and in chapters 12 and 13 in the Handbook of Acoustics [1].