Finite Element Method for Nonlinear Wave Propagation

1985 ◽  
Vol 54 (2) ◽  
pp. 544-554 ◽  
Author(s):  
Ichiro Kawakami ◽  
Masamitsu Aizawa ◽  
Katsumi Harada ◽  
Hiroyuki Saito
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Propagation ◽  
Element Method

Modeling and Analysis of Nonlinear Wave Propagation in One-Dimensional Phononic Structures

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
M. Liu ◽  
W. D. Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Band Structure ◽  
Band Gap ◽  
Nonlinear Wave ◽  
Elastic Waves ◽  
Propagation Characteristics ◽  
Nonlinear Wave Propagation ◽  
Weakly Nonlinear

Different from elastic waves in linear periodic structures, those in phononic crystals (PCs) with nonlinear properties can exhibit more interesting phenomena. Linear dispersion relations cannot accurately predict band-gap variations under finite-amplitude wave motions; creating nonlinear PCs remains challenging and few examples have been studied. Recent studies in the literature mainly focus on discrete chain-like systems; most studies only consider weakly nonlinear regimes and cannot accurately obtain some relations between wave propagation characteristics and general nonlinearities. This paper presents propagation characteristics of longitudinal elastic waves in a thin rod and coupled longitudinal and transverse waves in an Euler–Bernoulli beam using their exact Green–Lagrange strain relations. We derive band structure relations for a periodic rod and beam and predict their nonlinear wave propagation characteristics using the B-spline wavelet on the interval (BSWI) finite element method. Influences of nonlinearities on wave propagation characteristics are discussed. Numerical examples show that the proposed method is more effective for nonlinear static and band structure problems than the traditional finite element method and illustrate that nonlinearities can cause band-gap width and location changes, which is similar to results reported in the literature for discrete systems. The proposed methodology is not restricted to weakly nonlinear systems and can be used to accurately predict wave propagation characteristics of nonlinear structures. This study can provide good support for engineering applications, such as sound and vibration control using tunable band gaps of nonlinear PCs.

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

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Yu Liu ◽  
Andrew J. Dick
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Time Domain ◽  
Analytical Formula ◽  
Nonlinear Behavior ◽  
High Fidelity ◽  
Nonlinear Wave Propagation ◽  
Time Frequency ◽  
Element Method

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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