Accuracy of a composite implicit time integration scheme for structural dynamics

Abstract

Summary A comprehensive study of the two sub‐steps composite implicit time integration scheme for the structural dynamics is presented in this paper. A framework is proposed for the convergence accuracy analysis of the generalized composite scheme. The local truncation errors of the acceleration, velocity, and displacement are evaluated in a rigorous procedure. The presented and proved accuracy condition enables the displacement, velocity, and acceleration achieving second‐order accuracy simultaneously, which avoids the drawback that the acceleration accuracy may not reach second order. The different influences of numerical frequencies and time step on the accuracy of displacement, velocity, and acceleration are clarified. The numerical dissipation and dispersion and the initial magnitude errors are investigated physically, which measure the errors from the algorithmic amplification matrix's eigenvalues and eigenvectors, respectively. The load and physically undamped/damped cases are naturally accounted. An optimal algorithm‐Bathe composite method (Bathe and Baig, 2005; Bathe, 2007; Bathe and Noh, 2012) is revealed with unconditional stability, no overshooting in displacement, velocity, and acceleration, and excellent performance compared with many other algorithms. The proposed framework also can be used for accuracy analysis and design of other multi‐sub‐steps composite schemes and single‐step methods under physical damping and/or loading. Copyright © 2016 John Wiley & Sons, Ltd.

A‐stable two‐step time integration methods with controllable numerical dissipation for structural dynamics

“…As the eigenvalues are conjugate pairs, only the results of λ 1 and λ 3 are given. The exact results referring to the work of Zhang et al are also compared, where some high‐frequency results are truncated to show a clear feature. It should be mentioned that the method only in the limit cases ( Ω → 0, Ω → ∞ or ρ ∞ → 1.0) shows multiple eigenvalues ( λ 01 = λ 02 , λ 03 = λ 04 , λ ∞1 = λ ∞2 , λ ∞3 = λ ∞4 , or λ 3 = λ 4 = −1 with ρ ∞ = 1.0), whereas in a general frequency regime, the method gives distinct eigenvalues, which renders Equation or Equation satisfied.…”

“…For the physically undamped case, we use two ways to measure the spectral radius, with one based on all eigenvalues in Equation and another one just based on the principal roots as ρ P = | λ 1 | = | λ 2 |. The exact results are also given by referring to the work of Zhang et al…”

“…For the physically damped case, we only discuss the results of the OALTS method with some typical ρ ∞ . For some results of other methods, one can refer to the work of Zhang et al All algorithms show ρ P well approximating to the exact spectral radius in low frequencies with a dissipation characteristic. However, they are not consistent with the exact spectral radius in high frequencies.…”

“…Here, we do not measure the algorithmic errors by displacement individually but consider velocity and acceleration together. As we have known, in the HHT‐α method, the TPOS (G‐α) method, and most of the GSSSS algorithms, although their displacement and velocity are second‐order accurate, their accelerations are only first‐order accurate …”

“…Implicit algorithms tend to be numerically stable, permit a large time step, and are most effective for structural dynamics problems where the response is controlled by a relatively small number of low‐frequency modes . A number of implicit integration algorithms have been proposed (see review articles by Subbaraj and Dokainish, Tamma et al, and Fung as well as the references therein); representative members of these algorithms are, among others, the Newmark method, the Wilson‐ θ method, the HHT‐ α method, the WBZ‐ α method, the three‐parameters optimal schemes (TPOS) by Shao et al or, equivalently, the generalized‐ α (G‐ α ) method, the generalized single‐step single‐solve (GSSSS) method, the η method, and the composite implicit time integration schemes . It is important to note that many typical algorithms can be unified under the umbrella of the well‐known GSSSS framework .…”