LAUSR

A spectral consistent starting procedure is proposed for the first‐order‐type A‐stable linear two‐step (LTS) time integration methods in structural dynamics analysis. The accuracy analysis for the LTS methods in structural dynamics is presented based on the first‐order model, which enables the algorithms in first‐order transient systems to be extended to structural dynamics under the umbrella of a consistent framework. It is indicated that the approximation of the loads plays an essential role in the equivalence between the LTS methods and some single‐step methods. An algorithmic equivalence feature, which is stricter than the spectral equivalence, is revealed between the two‐leg generalized‐α methods in first‐order systems and the A‐stable LTS methods. The measures of the error constant in the LTS methods and the ultimate spectral radius are extended to optimize a class of single‐step methods of a given convergence order. The spectral consistent starting procedure for the A‐stable LTS methods is developed by utilizing the algorithmic equivalence of the single‐step methods, which results in an optimal A‐stable LTS (OALTS) method possessing a consistent spectral radius with controllable numerical dissipation in the starting step and the steps thereafter. Comparing with the equivalent single‐step methods, the present OALTS method does not need the auxiliary variables, and its displacement, velocity and acceleration can achieve second‐order accuracy simultaneously. The performance of the present OALTS method is verified by numerical examples including physical damping, external loads, and/or nonlinearity.