Finite Element Method

Abstract

Finite element method (FEM) is widely used in various engineering fields to solve problems with too many complexities to be dealt with by certain conventional approaches. In 1943, Courant proposed the theoretical basis of the method, and, in 1956, Turner et al. published both fundamental theory and application of FEM, namely “Stiffness deflection analysis”, to solve structural problems [1, 2]. A decade and several years latter, FEM has been come to be applied to solve various acoustic problems [3, 4, 5]. Compared with other numerical techniques, FEM is advantageous in its broad range of applicability. However, FEM requires discretization of the domain, which results in huge amount of degrees-of-freedom especially when a three-dimensional domain is analyzed. Nevertheless, the matrices constructed in a standard FEM procedure have rather simple mathematical structures with sparseness. The simplicity makes their computation more efficient especially when they are processed on a parallel/vector processors. Generally speaking, the iterative methods are suitable for solving such a sparse matrix equation efficiently with far less memory space on a computer. In this chapter, fundamentals, improvements, and applications of FEM on acoustic problems are explored.