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Research Papers

Alternating Frequency–Time Finite Element Method: High-Fidelity Modeling of Nonlinear Wave Propagation in One-Dimensional Waveguides

[+] Author and Article Information
Yu Liu

Nonlinear Phenomena Laboratory,
Department of Mechanical Engineering,
Rice University,
Houston, TX 77005

Andrew J. Dick

Nonlinear Phenomena Laboratory,
Department of Mechanical Engineering,
Rice University,
Houston, TX 77005
e-mail: andrew.j.dick@rice.edu

1Corresponding author.

Manuscript received June 20, 2013; final manuscript received May 26, 2015; published online June 30, 2015. Assoc. Editor: Carmen M. Lilley.

J. Comput. Nonlinear Dynam 11(1), 011003 (Jan 01, 2016) (9 pages) Paper No: CND-13-1150; doi: 10.1115/1.4030746 History: Received June 20, 2013; Revised May 26, 2015; Online June 30, 2015

In this paper, a spectral finite element method (SFEM) based on the alternating frequency–time (AFT) framework is extended to study impact wave propagation in a rod structure with a general material nonlinearity. The novelty of combining AFT and SFEM successfully solves the computational issue of existing nonlinear versions of SFEM and creates a high-fidelity method to study impact response behavior. The validity and efficiency of the method are studied through comparison with the prediction of a qualitative analytical study and a time-domain finite element method (FEM). A new analytical approach is also proposed to derive an analytical formula for the wavenumber. By using the wavenumber equation and with the help of time–frequency analysis techniques, the physical meaning of the nonlinear behavior is studied. Through this combined effort with both analytical and numerical components, distortion of the wave shape and dispersive behavior have been identified in the nonlinear response. The advantages of AFT-FEM are (1) high-fidelity results can be obtained with fewer elements for high-frequency impact shock response conditions; (2) dispersion or dissipation is not erroneously introduced into the response as can occur with time-domain FEM; (3) the high-fidelity properties of SFEM enable it to provide a better interpretation of nonlinear behavior in the response; and (4) the AFT framework makes it more computationally efficient when compared to existing nonlinear versions of SFEM which often involve convolution operations.

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References

Figures

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Fig. 1

The flowchart for the AFT-FEM

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Fig. 2

A diagram of semi-infinite rod

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Fig. 3

The predicted linear response when the impact loading magnitude is F = 500 kN. The (a) response at the impacted location and (b) wave propagation.

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Fig. 4

The predicted nonlinear response at the impacted location when the nonlinear coefficient α2 = 20 and impact loading magnitudes F = 100 kN, 300 kN, and 500 kN. The (a) velocity and (b) nonlinear component of the velocity.

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Fig. 5

The predicted nonlinear response at the impacted location when the impact loading magnitude F = 500 kN and the nonlinear coefficient α2 = 1, 10, 20. The (a) velocity and (b) nonlinear component of the velocity.

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Fig. 6

Nonlinear component of the nonlinear wave propagation with the AFT-FEM when impact loading F = 500 kN and the nonlinear coefficient α2 = 20

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Fig. 7

Sketch showing the velocity profile at different locations with the method of multiple scales when α2 > 0. The nonlinearity causes the wave shape to distort through wave propagation.

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Fig. 8

Comparison between the AFT-FEM and the nonlinear time-domain FEM when the impact loading F = 500 kN and the nonlinear coefficient α2 = 20

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Fig. 9

Comparison between the AFT-FEM and the nonlinear time-domain FEM when the impact loading F = 500 kN, the nonlinear coefficient α2 = 20, and the impact duration Tp = 20 μs

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Fig. 10

Comparison of convergence between the AFT-FEM and the nonlinear time-domain FEM when the impact loading F = 500 kN and the nonlinear coefficient α2 = 10

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Fig. 11

Comparison between n = 3 and n = 2 when the impact loading F = 500 kN

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Fig. 12

Surface plot of the normalized group speed

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Fig. 13

Contour plot of the PSD obtained by STFT when the impact loading F = 500 kN and the nonlinear coefficient α2 = 20

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Fig. 14

Comparison of STFT results between nonlinear model and linear dispersive model for the reflected wave

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