The trapezoidal rule (constant acceleration rule) is unconditionally stable for linear systems, it is second order accurate and it has no numerical dissipation so it preserves the energy of the systems oscillations. It has the lowest amplitude decay (AD) and period elongation (PE) of the linear multistep time integration methods. However ringing (spurious oscillations) may occur for stiff problems or sudden changes in loads or time steps.

Here, we compare the procedures to perform a simple dynamic analysis of a linear finite element model in the ABAQUS and ADINA codes with the trapezoidal rule.

The model consists of 5 single-degree-of-freedom (SDOF) oscillators, each modelled as one-node springs with spring rates so that the unity mass oscillators have selected frequencies of 10⁰ = 1, 10⁰·⁵ ≈ 3.16 , 10¹ = 10 , 10¹·⁵ ≈ 31.6, 10² = 100 Hz. The transient problems are started with initial velocities equal to each oscillators natural frequency ωᵢ, which gives each oscillator unity magnitude in the resultant oscillations.

The dynamic direct integration procedures are used in both codes, using 2000 time steps with 0.005 s fixed increments and a total simulation time 10 s. This leads to 20 time steps per period for the 10 Hz oscillator, so that it will be well resolved and high accuracy is expected. The higher frequency oscillators at 31 and 100 Hz will not be well resolved, which the analysis will show in respect of amplitude decay and period elongation.

In ABAQUS the *DYNAMIC DIRECT INTEGRATION method is inherently nonlinear. When the model is linear as in this case, the convergence checks are done anyway and the model converges in the first iteration in each time step. It is therefore much more cost-effective to use the *MODAL DYNAMIC procedure for linear models in ABAQUS. For nonlinear models numerical dissipation is often wanted to stabilize the problem. The implicit time integration methods are unconditionally stable, meaning that the are stable regardless of selected time step, but only for linear problems. Therefore, numerical dissipation is a wanted feature for nonlinear problems. 

The HHT-α method in is a one-parameter method based on the original two-parameter Newmark method. The Newmark parameters relates as γ = 0.5-α and β = 1/4 (1-α)². Letting α = 0 yields γ = 0.5 and β = 1/4, which then corresponds to the trapezoidal rule. 

In ABAQUS there are default values of α. Transient fidelity α = -0.005, Moderate dissipation α =-0.41421, and Quasi static α =-0.333?. The latter can be used because it can be advantageous to solve quasi static problems with a dynamic method so that the inertia and damping stabilizes the model and ease convergence. For truly dynamic events the numerical dissipation should be kept as low as possible.

In ABAQUS the *MODAL DYNAMIC method is inherently linear, and no numerical dissipation is needed for numerical stability. The trapezoidal rule is used for *MODAL DYNAMIC.

In ADINA the DYNAMIC-DIRECT-INTEGRATION method is automatically linear or nonlinear. When the solver recognizes that the model is linear, the next time step solution can be calculated as an implicit algebraic equation directly which is very computationally efficient.  The Newmark scheme should be selected with default parameters δ = 0.5 and α = 0.25, which corresponds to the trapezoidal rule.  

in ADINA when the problem is nonlinear, the DYNAMIC-DIRECT-INTEGRATION response is calculated with iterations. The Newmark scheme can be used with selected parameters to introduce numerical dissipation for stability, just like the HHT-α method, or the composite Bathe method can be selected. The Bathe method displays high accuracy of the resolved low frequency oscillations and effective high numerical dissipation of the high frequency non-resolved (spurious) oscillations.

in ADINA the Bathe scheme offers accuracy and stability for both linear and nonlinear problems. With the ρ∞-Bathe method, the one-parameter ɣ can be set in the 0 - 1 interval to control the high frequency spectral radius. Using ɣ = 0 yields the original Bathe method, with maximum high-frequency numerical dissipation. Using ɣ = 1 yields no numerical dissipation, and the method turns into a 2-substep method with the trapezoidal rule in both. The Bathe scheme is attractive in linear and non-linear dynamics as well as wave propagation analyses, however with a higher computational cost per time step since it is divided into two substeps.

In ADINA the MODAL-TRANSIENT method can be used for linear problems also, which then uses the trapezoidal rule for time integration. Here, we have run the problem also using the MODAL-TRANSIENT method in ADINA.

The results show that:

1. The results are identical for the ABAQUS *DYNAMIC DIRECT INTEGRATION, and the ADINA DYNAMIC-DIRECT-INTEGRATION and MODAL-TRANSIENT simulations using the trapezoidal rule time integration method. As expected.

2. The solution times differs, the ABAQUS *DYNAMIC DIRECT INTEGRATION took 34 seconds, the ADINA DYNAMIC-DIRECT-INTEGRATION 1.7 seconds and MODAL-TRANSIENT 1.6 seconds. As expected.

3. All oscillators show no amplitude decay (AD), although the high frequency oscillators are not well resolved. As expected for the trapezoidal rule.

4.  The low frequency oscillators show little period elongation, however the 31.6 Hz show 29.4 Hz and the 100 Hz merely 63.9 Hz response in the FFT plot due to period elongation (PE). As expected for the trapezoidal rule.

References:

N. M. Newmark, A Method of Computation for Structural Dynamics, 1959

K. J. Bathe, E. L. Wilson, Stability and Accuracy Analysis of Direct Integration Methods, 1973

H. M. Hilber, T. J. R, Hughes, Collocation, Dissipation and Overshoot for Time Integration Schemes in Structural Dynamics, 1978

K-J. Bathe, M. M. I. Baig, On a Composite Implicit Time Integration Procedure for Nonlinear Dynamics, 2005

K-J. Bathe, Conserving Energy and Momentum in Nonlinear Dynamics: A simple Implicit Time Integration Scheme, 2007

K-J. Bathe, G. Noh, Insight into an Implicit Time Integration Scheme for Structural Dynamics, 2012

G. Noh, K-J. Bathe, The Bathe Time Integration Method with Controllable Spectral Radius, The ρ∞-Bathe Method, 2019