Premium
Performance and numerical behavior of the second‐order scheme of precise time‐step integration for transient dynamic analysis
Author(s) -
Ma Hang,
Yin Feng,
Qin QingHua
Publication year - 2007
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20221
Subject(s) - discretization , heaviside step function , numerical integration , numerical analysis , order of accuracy , mathematics , newmark beta method , temporal discretization , dissipative system , spurious relationship , dynamic problem , finite element method , ordinary differential equation , algorithm , mathematical optimization , differential equation , mathematical analysis , numerical stability , statistics , physics , quantum mechanics , thermodynamics
Spurious high‐frequency responses resulting from spatial discretization in time‐step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time‐step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second‐order scheme of the precise integration method (PIM). Taking the Newmark‐β method as a reference, the performance and numerical behavior of the second‐order PIM for elasto‐dynamic impact‐response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock‐excited rod with rectangular, half‐triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine‐like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007