Finite Element Methods

Finite difference methods are well suited to problems defined in either one spatial dimension, or problems defined over a simple geometry in two dimensions (such as rectangular or circular). This covers all string models, 1D tube models, bars, and rectangular or circular percussion instruments. In general, finite differences methods rely on the use of regular grids, and, for finer modeling in irregular geometries, (perhaps geared towards musical instrument design rather than sound synthesis), are not always the best choice. There are certain strategies which may be employed: the use of form-fitting coordinates is one, and domain decomposition [41] is another. On the other hand, for extremely complex geometries, such approaches become unwieldy.

Finite element methods are designed to deal with such problems in irregular geometries; they are based on a distinct point of view; instead of approximating an underlying differential equation, pointwise, over a grid. The solution is approximated using a collection of so-called shape-functions, which may or may not lie in a regular arrangement. In addition, the numerical solution is often approached using integral and variational methods, rather than differential approximations. In the end, however, finite element methods for problems in dynamics operate much as finite difference methods do, through matrix recursions at a given (or perhaps variable) sample rate. Needless to say, the literature on finite element methods is vast; only the briefest introduction can be supplied here. The downside is that writing finite element code is considerably more involved than in the case of finite differences--generally, one must rely on software packages, which can be good or bad, depending on one's point of view.

In §, the alternate variational form of the model equation is introduced, followed in § by the definition of several simple shape functions of local support, in both one and two spatial dimensions. Strategies for grid generation are discussed, briefly, in §, and in § and §, the stiff string and linear plate models are reexamined from the point of finite element simulations. Finally, in §, some general trends in finite element modeling of musical instrument modeling are presented.

References for this chapter include: [10,79,59,169,152]