This article develops a new time integration family for second-order dynamic equations. A combination of the trapezoidal rule and higher-order Newton backward extrapolation functions are utilized in the formulation. Five members of the suggested family are extensively studied in this article. Most members of the presented time integration family are new. The stability and accuracy of the proposed time integration schemes are investigated by solving some benchmark problems. Numerical results are checked and compared with well-known strategies. The findings of the article show the efficiency, accuracy and robustness of the suggested techniques.

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Plastic films are largely produced using the film blowing process. Branched polymers such as LDPE are commonly used in this process to produce a variety of plastic films. In this study, recently developed eXtended Pom-Pom (XPP) model, which focuses on describing rheological behavior of branched polymers, is employed to analyze the non-isothermal flow in the film blowing process. Furthermore, the Nakamura equation for crystallization kinetics is employed to consider crystallization effects and improve prediction of the state of the stress in the film. Results of the numerical analysis show good agreement with the experimental values reported by Tas (1994) for the bubble shape and film velocity of the LDPE films. Predictions for the strain rates in machine and hoop directions which are critical for the estimation of the stress are also compared with the reported data and showed reasonably good agreement.

This paper develops the optimal time-step ratio and the critical damping for nonlinear structural analysis, when using dynamic relaxation method (DRM). The convergence rate of the DRM depends on the value of the time step and the critical damping factor. In the present study, the effect of the time-step ratio is separated from the time-step value. Additionally, the effect of the eigenvalue on the error function is investigated. Using the eigenvalue definition, the critical damping is updated in each dynamic relaxation (DR) step to decrease the required computations. Moreover, the optimal time-step ratio is calculated. It is also shown that the value of the constant time step has no effect on the rate of convergence. The proposed procedure provides a simple and accelerated DRM, which can be used in the linear and nonlinear analyses of structures. The efficiency of the new method was verified by its application to a wide range of typical structures. The results show that the suggested scheme accelerates the convergence rate of the DR process.