Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics

Abstract The subject of this paper is the stability, convergence and growth and decay of energy of the average acceleration method applied to a class of linear and nonlinear elastic problems encountered in structural dynamics. A discrete energy identity is obtained and the cause of the spurious growth and decay of energy, noted in nonlinear problems, is exhibited. A notion of stability in energy is defined which guarantees that for small time steps energy is asymptotically conserved and for large time steps amplification of higher modes does not occur. By way of the energy identity, the average acceleration method is proved to be stable in that sense. Furthermore, convergence is proved and the rate-of-convergence is shown to be second-order, with respect to time step, for both displacements and velocities.