Abstract A simple unified set of displacement and velocity approximations is presented to express various two-stage composite time integration schemes. Based on the unified approximations, two novel sets of optimized parameters are newly proposed for the implicit composite schemes to enhance the capability of conserving total energy. Two special cases of the unified approximations are also considered to overcome some shortcomings of the existing schemes. Besides, the newly proposed unified set of approximations can include many of the existing composite time integration schemes. To be specific, both implicit and explicit composite schemes can be expressed by using the unified set of approximations. Thus, both implicit and explicit types of composite schemes can be selected from the unified set by simply changing algorithmic parameters. To demonstrate advantageous features of the proposed unified set of approximations, various numerical examples are solved, and results are analyzed.

Abstract In this article, third- and fourth-order accurate explicit time integration methods are developed for effective analyses of various linear and nonlinear dynamic problems stated by second-order ordinary differential equations in time. Two sets of the new methods are developed by employing the collocation approach in the time domain. To remedy some shortcomings of using the explicit Runge-Kutta methods for second-order ordinary differential equations in time, the new methods are designed to introduce small period and damping errors in the important low-frequency range. For linear cases, the explicitness of the new methods is not affected by the presence of non-diagonal damping matrix. For nonlinear cases, the new methods can handle velocity dependent problems explicitly without decreasing order of accuracy. The new methods do not have any undetermined algorithmic parameters. Improved numerical solutions are obtained when they are applied to various linear and nonlinear problems.

In this study, mixed finite element models of plate bending are developed to include other variables (e.g., the membrane forces and shear forces) in addition to the generalized displacements to investigate their effect on nonlinear response. Various finite element models are developed using the weighted-residual statements of suitable equations. The classical plate theory and the first-order shear deformation plate theory are used in this study and the von Karman nonlinear strains are accounted for. Each newly developed model is examined and compared with displacement finite element models to evaluate their performance. Numerical results show that the new mixed models developed herein show better accuracy than existing displacement based models.