Frequency‐domain analysis of time‐integration methods for semidiscrete finite element equations—part II: Hyperbolic and parabolic–hyperbolic problems

Abstract

Time-integration methods for semidiscrete ÿnite element equations of hyperbolic and parabolichyperbolic types are analysed in the frequency domain. The discrete-time transfer functions of six popular methods are derived, and subsequently the forced response characteristics of single modes are studied in the frequency domain. Three characteristic numbers are derived which eliminate the parameter dependence of the frequency responses. Frequency responses and L 2 -norms of the phase and magnitude errors are calculated, and comparisons are given of the methods. As shown; the frequency-domain analysis explains all time-domain properties of the methods, and gives more insight into the subject.

Real‐time hybrid testing using the unconditionally stable explicit CR integration algorithm

“…Equations (3a) and (3b) indicate that the CR integration algorithm is explicit for both the displacement and velocity, making it well suited for application to real-time testing. Ramirez [20], Mugan and Hulbert [21,22] showed that integration algorithms for linear elastic structures can be expressed in the form of discrete transfer functions. A discrete transfer function is used in control engineering to represent the relationship between the output and the input of a discrete linear time-invariant system.…”

Discrete equivalent time integration methods for transient analysis

“…see References [13] and [14]), and there is a tradeo in the frequency domain that has to be satisÿed by time integration methods [20]. Some of the existing higher-order accurate time integration methods are derived based on approximations to the exact solutions of semidiscrete equations that require the forcing terms and boundary conditions to be known; however, these functions may not be known exactly and it is common practice that values of forcing terms and boundary conditions are available only at discrete time steps.…”

“…One of the constraints in the frequency domain is that temporally discretized equation of a system cannot emulate the associated semidiscrete equation of the system beyond the Nyquist frequency = t, where t is the time step; furthermore, signiÿcant phase and magnitude errors occur as the frequency of the forcing terms approaches to the Nyquist frequency = t, e.g. see References [13] and [14]. In brief, it is impossible to resolve the response of a structure in response to inputs containing frequencies beyond the Nyquist frequency = t no matter what the order of accuracy of the time integration method is.…”

“…In Reference [12], frequency response characteristics of the Newmark family of algorithms are studied. In References [13] and [14], some characteristic (non-dimensional) numbers are proposed to eliminate the parameter dependence of frequency response of time integration methods. Magnitude and phase error characteristics of the following methods are studied in the frequency domain: the generalized trapezoidal method [4], one-parameter ( ) predictor-corrector method [4], and one-parameter ( ) iterative predictorcorrector method [4] are studied in Reference [13]; on the other hand, the Newmark method [4], the Houbolt method [4], the Single Step Houbolt (SSH) method [15], the optimal collocation schemes [16], the Hilber-Hughes-Taylor-(HHT-) method [17] and the implicit generalized-method [18] are studied in Reference [14].…”

“…It is shown in References [13] and [14] that temporally discretized semidiscrete ÿnite element equations can be cast into the transfer function form whose behaviour can be determined by its steady-state response to sinusoidal inputs, i.e. the Bode plots [19].…”