Improvement of Discrete-Mechanics-Type Time Integration Schemes by Utilizing Balance Relations in Integral form Together with Picard-Type Iterations

Abstract

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature.

Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study

“…We now present a discussion of the computational method published in [11] and how to apply it in the present context. Our method is a time-stepping procedure; in the following, we consider time intervals of equal length T. The normal form of the balance equation in Equation ( 11) is brought into its integral form by integrating over the finite interval (time step) τ ∈ [0, T], the local time in the time interval being defined as τ = t − (n − 1)T, where n = 1, 2, .…”

“…The necessary analytic operations, such as Taylor-series representations and integration, can be easily performed by means of symbolic computation. Since we approximate the relations of balance in their integrated (global balance) form, and not directly in their differential one and since we start the iteration using the well-established explicit fourth-order Runge-Kutta scheme, convergence is generally fast; for large free non-linear vibrations of the pendulum, see the detailed comparative study in [11]. The necessary analytic operations of the procedure, such as Taylor-series representations and integration, can be easily performed by means of symbolic computation.…”

“…For the development of dynaimc structural models in linear non-linear earthquake engineering, as well as for suitable time-integration methods and non-linear control formulations, the reader is moreover referred exemplarily to [5][6][7][8][9][10]. In the following, a novel efficient time-integration scheme recently derived by the present authors, see [11], is applied for simulating non-linear forced and free vibrations of mechanical structures, and its advantages against standard numerical time-integration methods, such as NDSolve, available in Wolfram Mathematica [12], are exemplarily demonstrated. Our technique consists in an application of an extended Picard iteration to the time-integrated (global balance) form of the equations of motion, see [13] for the original Picard iteration and [14] for the structural equations of motion in integral form, i.e., for the global relations of balance of momentum.…”

“…A main feature is that, thanks to the power of symbolic methods, our technique can be formulated so that the Picard iteration and the necessary symbolic operations need to be performed only once, before the time-stepping procedure starts. In our previous investigation [11], computational advantages have been demonstrated for large free vibrations of a hanging rigid pendulum, for which exact solutions do exist, and which is used in the literature as a benchmark example for comparison with numerical time integration methods. It was shown in [11] that a large number of free non-linear vibration cycles can be accurately computed, where the simulation remains in the close vicinity of the exact free phase-plane trajectory of the pendulum, due to the algorithmic satisfaction of the relations of global balance.…”

“…In our previous investigation [11], computational advantages have been demonstrated for large free vibrations of a hanging rigid pendulum, for which exact solutions do exist, and which is used in the literature as a benchmark example for comparison with numerical time integration methods. It was shown in [11] that a large number of free non-linear vibration cycles can be accurately computed, where the simulation remains in the close vicinity of the exact free phase-plane trajectory of the pendulum, due to the algorithmic satisfaction of the relations of global balance. Excellent accuracy with respect to the exact solution and competing numerical schemes was observed, also for large observation periods.…”