Retraction notice to “A hybrid Finite element-Meshfree method based on partition of unity for transient wave propagation problems in homogeneous and inhomogeneous media” Applied Mathematical Modelling 85, September 2020, 192-209

In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. The edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; then, the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced. To effectively solve the transient wave propagation problems, the Bathe time integration scheme is employed to perform the involved time integration. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; then, the temporal numerical dispersion error in the calculated solutions can also be notably controlled. A number of supporting numerical examples are considered to examine the capabilities of the present approach. The numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM.

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Numerical Analysis of Transient Wave Propagation in Nonlinear One-Dimensional Waveguides by Using the Spectral Finite Element Method

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4028015 ◽

2015 ◽

Vol 10 (5) ◽

Cited By ~ 1

Author(s):

Yu Liu ◽

Andrew J. Dick

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Wave Propagation ◽

Geometric Nonlinearity ◽

Material Nonlinearity ◽

Transient Wave ◽

One Dimensional ◽

Transient Wave Propagation ◽

1D Model ◽

Element Method

In this paper, transient wave propagation in nonlinear one-dimensional (1D) waveguides is studied. A complete nonlinear (CN) 1D model accounting for both axial and transverse displacements is developed and geometric and material nonlinearities are separately modeled. The alternating frequency-time finite element method (AFT-FEM) is implemented for this complete 1D model. Numerical simulations are conducted and the response behaviors for axial and transverse motions are analyzed. Comparison of the responses for the geometrically nonlinear (GN) model with a corresponding linear model supports predictions made from the previous analytical studies that the geometric nonlinearity has limited influence on the response of transient transverse waves in the intermediate strain regime. On the contrary, strong nonlinear behavior appears in the response for the materially nonlinear (MN) models. Depending on the local nonlinear property of the material in the intermediate strain regime, the amplitude of the response can be significantly influenced and additional dispersion can be introduced into the response. An exploration of the interaction between the geometric nonlinearity and the material nonlinearity for a rod model in a large strain regime is also conducted and the responses are analyzed by using time-frequency analysis. The competing effect of the geometric nonlinearity and the material nonlinearity can result in a pseudolinear response in a strong nonlinear system for a given range of impact loading.

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