Strongly cumulative second harmonic generation in a plate with quadratic nonlinearity: Finite element simulation

2013 ◽  
Author(s):  
Yang Liu ◽  
Cliff J. Lissenden ◽  
J. L. Rose
Keyword(s):  
Finite Element ◽  
Second Harmonic Generation ◽  
Finite Element Simulation ◽  
Harmonic Generation ◽  
Second Harmonic ◽  
Quadratic Nonlinearity ◽  
Element Simulation

scholarly journals Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

2021 ◽  
Vol 11 (1) ◽  
pp. 417-431
Author(s):  
Jing Yang ◽  
Ting Zhou
Keyword(s):  
Second Harmonic Generation ◽  
Small Parameter ◽  
Harmonic Generation ◽  
Reduction Method ◽  
Second Harmonic ◽  
Positive Function ◽  
Quadratic Nonlinearity ◽  
Single Peak ◽  
Degenerate Critical Point ◽  
Schrödinger System

Abstract We are concerned with the following Schrödinger system with coupled quadratic nonlinearity − ε 2 Δ v + P ( x ) v = μ v w , x ∈ R N , − ε 2 Δ w + Q ( x ) w = μ 2 v 2 + γ w 2 , x ∈ R N , v > 0 , w > 0 , v , w ∈ H 1 R N , $$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$ which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x 0 is a non-degenerate critical point of Δ(P + Q) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution (vε , wε ) concentrating at x 0 for ε small enough.

Time-Domain Spectral Finite Element Method for Modeling Second Harmonic Generation of Guided Waves Induced by Material, Geometric and Contact Nonlinearities in Beams

2020 ◽  
Vol 20 (10) ◽  
pp. 2042005
Author(s):  
Shuai He ◽  
Ching-Tai Ng ◽  
Carman Yeung
Keyword(s):  
Finite Element ◽  
Second Harmonic Generation ◽  
Damage Detection ◽  
Harmonic Generation ◽  
Time Domain ◽  
Second Harmonic ◽  
Breathing Crack ◽  
Geometric Nonlinearities ◽  
Contact Nonlinearity ◽  
Spectral Finite Element

This study proposes a time-domain spectral finite element (SFE) method for simulating the second harmonic generation (SHG) of nonlinear guided wave due to material, geometric and contact nonlinearities in beams. The time-domain SFE method is developed based on the Mindlin–Hermann rod and Timoshenko beam theory. The material and geometric nonlinearities are modeled by adapting the constitutive relation between stress and strain using a second-order approximation. The contact nonlinearity induced by breathing crack is simulated by bilinear crack mechanism. The material and geometric nonlinearities of the SFE model are validated analytically and the contact nonlinearity is verified numerically using three-dimensional (3D) finite element (FE) simulation. There is good agreement between the analytical, numerical and SFE results, demonstrating the accuracy of the proposed method. Numerical case studies are conducted to investigate the influence of number of cycles and amplitude of the excitation signal on the SHG and its performance in damage detection. The results show that the amplitude of the SHG increases with the numbers of cycles and amplitude of the excitation signal. The amplitudes of the SHG due to material and geometric nonlinearities are also compared with the contact nonlinearity when a breathing crack exists in the beam. It shows that the material and geometric nonlinearities have much less contribution to the SHG than the contact nonlinearity. In addition, the SHG can accurately determine the crack location without using the reference data. Overall, the findings of this study help further advance the use of SHG for damage detection.

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