Abstract
This paper presents a modified Precise Integration Method (PIM) for long-time duration dynamic analysis. The fundamental solution and loading operator matrices in the first time substep are numerically computed employing a known unconditionally stable method (referred to as original method in this paper). By using efficient recursive algorithms to evaluate these matrices in the first time-step, the same numerical results as those using the original method with small time-step are obtained. The proposed method avoids the need of matrix inversion and numerical quadrature formulae, while the particular solution obtained has the same accuracy as that of the homogeneous solution. Through setting a proper value of the time substep, satisfactory accuracy and numerical dissipation can be achieved.
Highlights
In structural dynamic analysis, direct time integration algorithms are widely used to solve the motion equation [1]
This paper aims to modify and improve the existing Precise Integration Method (PIM) for solving long-time duration dynamics problems
The technologies used to evaluate the particular solution are basically based on some approximations that are only suitable for a small time-step
Summary
Direct time integration algorithms are widely used to solve the motion equation [1]. An alternative way to evaluate the dynamic response is to use the Precise Integration Method (PIM-L) [5], where a numerical exponential matrix function is accurately computed to approximate the fundamental solution matrix and the applied loading is simulated by using piecewise linear functions. For linear dynamic homogeneous systems, the PIM-L can obtain precise numerical results approaching the exact solution. Lin et al [6] applied Fourier series expansions to simulate the applied loading and proposed the PIM-F method. Both the PIM-L and PIM-F require matrix inversion to obtain the particular solution. If the objective matrix is singular or nearly singular, its inverse matrix either does not exist or is hard to obtain steadily
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