A simplified variant of the time finite element methods based on the shape functions of an axial finite bar

In the field of structural dynamics, the structural responses in the time domain are of major concern. There already exist many methods proposed previously including widely used direct time integration methods such as ones in the β-Newmark family, Houbolt’s method, and Runge-Kutta method. The time finite element methods (TFEM) that followed the well-posed variational statement for structural dynamics are found to bring about a superior accuracy even with large time steps (element sizes), when compared with the results from methods mentioned above. Some high-order time finite elements were derived with the procedure analogous to the conventional finite element methods. In the formulation of these time finite elements, the shape functions are like the ones for a (spatial) 2-order finite beam. In this article, a simplified variant for the TFEM is proposed where the shape functions similar to the ones for a (spatial) axial bar are used. The accuracy in the obtained results of some numerical examples is found to be comparable with the accuracy in the previous results.

Exact Solutions to the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

In this paper, a trial function method is employed to find exact solutions to the nonlinear Schrödinger equations with high-order time-dependent coefficients. This system might be used to describe the propagation of ultrashort optical pulses in nonlinear optical fibers, with self-steepening and self-frequency shift effects. The new general solutions are found for the general case a 0 ≠ 0 including the Jacobi elliptic function solutions, solitary wave solutions, and rational function solutions which are presented in comparison with the previous ones obtained by Triki and Wazwaz, who only studied the special case a 0 = 0 .

Fourier Spectral High-Order Time-Stepping Method for Numerical Simulation of the Multi-Dimensional Allen–Cahn Equations

This paper introduces the Fourier spectral method combined with the strongly stable exponential time difference method as an attractive and easy-to-implement alternative for the integration of the multi-dimensional Allen–Cahn equation with no-flux boundary conditions. The main advantages of the proposed method are that it utilizes the discrete fast Fourier transform, which ensures efficiency, allows an extension to two and three spatial dimensions in a similar fashion as one-dimensional problems, and deals with various boundary conditions. Several numerical experiments are carried out on multi-dimensional Allen–Cahn equations including a two-dimensional Allen–Cahn equation with a radially symmetric circular interface initial condition to demonstrate the fourth-order temporal accuracy and stability of the method. The numerical results show that the proposed method is fourth-order accurate in the time direction and is able to satisfy the discrete energy law.

Time-Periodic and High-Order Time-Invariant Linearized Models of Rotorcraft: A Survey

The objective of this paper is to summarize the relevant published research studies on the extraction of linear time-periodic (LTP) systems and their higher order linear time-invariant (LTI) reformulations from rotorcraft physics-based models and on the identification of LTP systems from rotorcraft experimental data. The paper begins with an introductory overview of LTP system theory. Next, the relevant methods for the extraction of LTP and high-order LTI systems from physics-based models are presented. The paper continues with an overview of LTP model identification methods, followed by a discussion on the application of these methods toward the identification of the rotor dynamics alone and the coupled rigid-body/rotor dynamics. Final remarks summarize the overall findings of the study and identify areas for future work including, but not limited to, the context of the Future Vertical Lift (FVL) program.