Higher-order Nonlinear Functions Research Articles

Nonlinear eigen modes in optical media supported by cubic and quintic nonlinearities with Parity-Time symmetric hyperbolic complex potential

This work explored optical waves in different nonlinear media, specifically in self-focusing cubic–quintic media, and investigated how these waves evolve and maintain their characteristics under the influence of various factors such as nonlinear interactions, diffraction, and Parity-Time (PT) symmetric complex field with hyperbolic real and imaginary distributions. We explored how the range of the complex field parameter over which PT-symmetry collapse occurs is modified by the order of the nonlinear function, modulation strength that makes up the potential, the separation between peaks, and the width of the PT-symmetric field. A comparison of the soliton solutions with Kerr and quintic nonlinearities clarifies that the span of sustainable PT-symmetry in the self-focusing medium expands with the higher order nonlinear function. We discussed the propagation-related nonlinear evolutions of unstable and robust PT solitons. The conditions necessary for sustained soliton in PT-symmetric and broken PT-symmetric domains were identified by the study of the linear stability of solitons.

An improved nonlinear constitutive model for the magneto-elastic coupling behavior of ferromagnetic materials

Based on the micro-domain theory, the macro-thermodynamic relationship, and the effective field theory, an improved nonlinear constitutive model is established for ferromagnetic materials. The constitutive model includes new expressions of some nonlinear functions and high-order nonlinear functions and has simple expressions for some ferromagnetic materials. Compared with the existing models, the nonlinear model established in this paper has a great improvement in the prediction accuracy of the magneto-elastic coupling behavior. In other words, it can accurately predict the complicated nonlinear behaviors of magnetostriction and magnetization under the magnetic field and applied stress. Furthermore, it can correctly describe magnetostriction with high tensile stress and low magnetic fields, the complicated magnetostriction behavior with stress and middle magnetic fields, the complicated magnetization behavior with stress and high magnetic fields, and the maximum point of magnetostriction according to applied stress for ferromagnetic materials. A detailed analysis of the complicated nonlinear behaviors for ferromagnetic materials demonstrates the high effectiveness of the presented model.

Reliability analysis of tunnels using a metamodeling technique based on augmented radial basis functions

Metamodeling techniques have been developed and used for years in engineering reliability analysis involving expensive response simulations. In practical tunnel engineering problems where finite element (FE) simulations are required, the limited state/performance functions are in general implicit and nonlinear, and it is difficult to apply traditional gradient-based or sampling-based reliability methods, especially for large-scale problems. There is a need to develop accurate and efficient metamodels for practical tunnel engineering applications. In this paper, a metamodeling technique for reliability analysis of tunnels was studied based on augmented radial basis functions (RBFs). With a relatively small size of samples, the RBFs were used to create accurate approximate functions for different types of responses including linear and higher-order nonlinear functions. With the RBF-based metamodel constructed to express a limit state/performance function, Monte Carlo simulations (MCS) were applied to evaluate failure probability. The failure probability and reliability index obtained using the RBF-based metamodeling method were found to have good accuracy with a reasonable number of sample points. The reliability analyses of two existing tunnel examples showed that the augmented RBF metamodeling approach was efficient and effective for tunnel engineering problems.

Approximate modeling for high order non-linear functions using small sample sets

Having high learning accuracies, neural networks are widely applied for solving function approximation problems. Nevertheless, it is difficult to train a neural network to recognize a non-linear function using a small training sample set. Because the errors between real values and estimated values are significant and almost impossible to figure out using insufficient samples. This study develops an algorithm combining segmentation technique and artificial samples to overcome this situation. The strategy is to shorten the model range to minimize the total estimation error. Another supportive strategy known as artificial samples generation is also employed to fill information gaps. At the end of this research, the results of the computational examples indicate that learning accuracy can be significantly improved using the proposed method involving a very small data set.