A semi-analytical time integration method for linear second order systems based on impulse-preserving Heaviside series and the Generalized Integrating Factor

A semi-analytical time integration method is proposed for linear second order systems with classical normal modes. The homogeneous response is exact, through the Generalized Integrating Factor, and the load is represented by polynomials modulated by Heaviside steps (Heaviside Series, HS). In each step, the slope is the chord of the load and the intercept preserves its integral, so the impulse is exact for any step. When the load integrals are known, a moment expansion shows fourth order convergence at the time nodes, twice that of nodal interpolation, at the same cost. A line through the two Gauss points of each step is also fourth order for smooth loads; with a kink or jump inside a step neither is, but HS converges one order faster. For finite element models, the method is applied in a truncated modal basis with quasi-static correction, including the non-symmetric displacement-pressure vibroacoustic form. In a vibroacoustic example, the time integration error in a given modal basis is about 400 times smaller than with nodal interpolation. Mesh refinement up to about $10^6$ degrees of freedom shows that the cost reduction relative to direct integration comes from mode superposition, while HS avoids refining the time step in modal coordinates.