ABSTRACT
In the finite element method, we usually seek an approximate solution of a system of differential equations over a domain Ω with boundary ∂Ω:
L
C
( )
( )
u f
u f v
=
= ∂ ⎧
⎨
⎩
in
over
Ω
Ω (3.1)
Here, L and denote differential operators characterizing the system; fv and fs are the associated loading terms. The unknown variable u is usually approximated in the form of a weighted sum:
u ui i
φ ,
(3.2)
The set of functions { },φt i are called trial functions. They must satisfy all or part of the boundary conditions. Vector { }ui contains N unknowns to be determined. The direct substitution of Equation 3.2 into Equation 3.1 leads to an error or residual, given by
R L( ) ( )u u fv= − (3.3)
This residual is only equal to zero for the exact solution of Equation 3.1. A classic way to solve for the unknown vector { }ui is to set the integral
over domain Ω of the residual, weighted by N arbitrary but integrable functions ψw i, , to zero (Weighted residuals method):
ψ φw i t i iu dV, ,( ) , 0 Ω
(3.4)
This system of N equations can be solved directly for the unknown vector { }.ui In this case, the trial functions need to satisfy all boundary conditions. It can also be transformed using an integral formulation to eliminate part of these boundary conditions. The accuracy of the approximation depends on the selection of the trial functions, the weight functions, and the number of terms in the approximation. This chapter summarizes, briefly, the main integral formulations used to solve the classical equations of structural acoustics and vibration. A thorough discussion can be found in several textbooks (e.g., Finlayson 1972; Lanczos 1986; Reddy 1993; Géradin and Rixen 1997). The different methods differ in the choice of test functions and the used integral formulation (=Rayleigh-Ritz, Petrov-Galerkin, Galerkin, least squares, collocations,…). These methods are illustrated through an example in Appendix 3A. In this chapter, we will however focus on describing a general method of construction of the integral formulation, called weak formulation. Then, we will discuss the link between this approach and the variational formulation of the problem. Finally, we will recall the fundamental energy-based principles commonly used in engineering for the direct construction of the weak integral formulation.
