A unified set of single-step asymptotic annihilation algorithms for structural dynamics

Abstract

When solving the equations of structural dynamics using direct time integration methods, algorithmic damping is useful in controlling spurious high-frequency oscillations. Ideally, an algorithm should possess asymptotic annihilation of the high-frequency response, i.e., spurious oscillations in the high frequencies are eliminated after one time step. In this paper, a new class of asymptotic annihilation algorithms for structural dynamics is presented that possesses little numerical dissipation in the low-frequency regime. The algorithms are based upon using finite elements in time. The displacement and velocity fields are interpolated independently using time-discontinuous functions. The equations of motion, displacement-velocity compatibility and the time continuity of displacement and velocity are satisfied weakly. Asymptotic annihilation is achieved by choosing the displacement and velocity fields to be of equal order. Algorithms of any desired order of temporal accuracy can be obtained by appropriate choice of the finite element interpolations in time. An analysis of the proposed class of algorithms is given proving the asymptotic annihilation property and the spectral equivalence of the algorithms to the upper diagonal of the Padé approximation table. Results from finite difference analyses are presented showing the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low frequency regime.