Initial conditions in frequency-domain analysis: the FEM applied to the scalar wave equation

The present paper describes a procedure to consider initial conditions in the frequency-domain analysis of continuous media discretized by the FEM. The frequency-domain formulation presented here is based on a standard DFT procedure, the FFT algorithm being employed to transform from time to the frequency domain and vice versa. The standard Galerkin finite element method (displacement model) is used to replace the original differential governing equation by an integral equation amenable to numerical solution. The scalar wave equation (one- and two-dimensional) is used to illustrate the proposed approach. At the end of the paper, examples of wave propagation for a taut string, a one-dimensional rod and a membrane are presented to illustrate the robustness of the formulation presented here.