Dynamic Direct Integration. Spectral Radius of HHT-α and 𝜌∞-Bathe methods 

In dynamic direct integration several time integration methods are used in practical finite element analyses. Among the implicit methods non-dissipative and dissipative methods can be chosen.

Implicit methods are often chosen because they are unconditionally stable, however that only holds for linear analyses. For non-linear analyses numerical dissipation is a wanted feature to attenuate high frequency oscillations and provide stability.

High frequency attenuation is like the treble control on your stereo

The HHT-α method is implemented in several commercial finite element programs (ABAQUS, ANSYS) and the Bathe method in others (ADINA, NX-NASTRAN, LS-DYNA). For example.

The HHT-α (1976) method 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.

In ABAQUS the predefined values α = -0.05 (transient fidelity) and α = -0.414214 (moderate dissipation) can be chosen, depending on the problem to be solved. Letting α = 0 yields γ = 0.5 and β = 1/4, which then corresponds to the trapezoidal rule. 

The 𝜌∞-Bathe (2019) method is a one-parameter composite method based on the trapezoidal rule for the first substep, and an adjustable method for the second substep. Letting 𝜌∞=0 renders the original Bathe method (2005). Letting 𝜌∞=1 the method becomes a two-substep trapezoidal rule.

In ADINA the parameter can be chosen 𝜌∞ = 0.9 or 𝜌∞ = 0.6 to achieve approximately the same dissipation as in ABAQUS. Letting 𝜌∞ = 0 provides maximum dissipation and hence stability, and accuracy in wave propagation problems.

The amplitude decay (AD) and period elongation (PE) of the methods will be shown in separate posts.

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, R. L. Taylor, Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics, 1976

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, 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