A finite element time discretization (FETD) is utilized to develop a new direct integration method. In the case of linear problems, the derived recurrence formula is implicit and self-starting. There are no undetermined parameters which are often used to control stability and accuracy in the traditional direct time integration methods. The pseudo-force method by Stricklin et al. is adopted in the formulation of the nonlinear problems considered. Defining a relative change and using a quadratic extrapolation, subiterations are suggested for each time step.
The properties of convergence, stability and consistency are examined analytically based on the spectral approach for the linear problem case. The recurrence formula is characterized by two amplification matrices when a quadratic interpolation is used. The spectral radii for the two amplification matrices exhibit a better behavior compared with other existing methods. It is shown to be unconditionally stable and second-order accurate. The accuracy is discussed with two measures; algorithmic damping ratio and relative period error. The results show that the present method exhibits a better accuracy especially when the time step parameter $\alpha$ is within 0.23 approximately. Most of the manipulation for this study has been done symbolically using Mathematica.
Two numerical examples of simple discrete systems are prepared to demonstrate the validity and capability of the proposed method. A quadratic interpolation is used for the approximation of the state-variable, the weighting-function and the pseudo-force vector. The test with a single degree-of-freedom linear model has supported the analysis of convergence and errors. Even with very large steps compared with the period, they follow the analytical solutions fairly well as predicted. It has been shown that the proposed method gives consistently better response history than the two well-known methods of Newmark and Zienkiewicz et al. even with larger time steps. A two degree-of-freedom Hertzian impact model is solved to show its performance for a typical nonlinear problem. It is concluded that the proposed method works very well with nice properties as expected.
The applicability of FETD to structural dynamic problems is next illustrated. The degenerated shell finite elements proposed by Ahmad et. al. and consistent mass matrix strategy are used for FE modeling of curved-shell structures. The transient dynamic response of shell structures is obtained by using the present method and compared with the one by the Adams predictor-corrector method. With a spherical glass cap under a concentrated or distributed loading conditions, the applicability and versatility of the method are shown numerically.
A practical problem of the dynamic analysis of low-velocity impact between an elastic spherical-shell structure and a sphere is considered. The discretized nonlinear impact equations are derived based on the degenerated shell FE formulation and the Hertzian contact theory. The time histories of contact force and other field quantities are calculated simultaneously. From the results, the proposed algorithm based on the pseudo-force approach is shown to work well. However, the choice of the initial estimate of the pseudo-force has significant influence on the number of subiterations, and need be studied further.
