Discrete equivalent time integration methods for transient analysis

It was recently shown in a series of papers that the frequency response of temporally discretized finite element equations and, consequently, the achievable accuracy cannot be manipulated independently in different frequency ranges. In addition, there exist limitations on the achievable accuracy of a time integration method no matter what the order of accuracy of the method is. Motivated by this fact, a family of time integration methods is derived in the time domain based on the principle that the exact solution of the semidiscrete equation of a system and the solution of the time integration method match at discrete time steps. It is necessary to pursue an exact match at discrete time steps, i.e. discrete equivalence, since the solutions of semidiscrete equations are obtained only at the time steps. Two time integration methods, that are exact at the time steps, are obtained based on the impulse and ramp response invariance principles. Numerical examples are presented to show the advantage of the proposed methods and to compare the performance of them with the performance of some popular methods.