Abstract
This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.
Highlights
The step-by-step time integration method is the most commonly used numerical method for transient analysis of nonlinear dynamics, and can in general be classified into an explicit and implicit scheme
To solve the linear initial value problems (IVPs) of ordinary differential equations (ODEs) in Equation (5), a time-domain finite element method (FEM) based on the Galerkin weak form [25,34,42] was used in the present paper
The present algorithm automatically degrades into the so-called ‘GW (Galerkin Weak form) method’ which has been already proposed [34] for linear elastodynamic problems, and turns out to be an unconditionally stable time integration method with the spectral radii ρ(A) = 1, similar to quadratic and cubic elements
Summary
The step-by-step time integration method is the most commonly used numerical method for transient analysis of nonlinear dynamics, and can in general be classified into an explicit and implicit scheme. The stability of an integration algorithm can be estimated by spectral analysis, whereas in nonlinear dynamics spectral analysis is only one of the requirements to remain stable, and the conservation or decay of total energy within a time step interval should be the very sufficient condition for effective solutions [3]. From this point of view, the time integration method can be classified into three categories: One of numerical dissipation, one of enforced conservation of energy and one of algorithmic conservation of energy. Of the paper, these contents will be presented one by one
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