Response Analysis of the Three-Degree-of-Freedom Vibroimpact System with an Uncertain Parameter.

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point.

CM-CPPA: Chaotic Map-Based Conditional Privacy-Preserving Authentication Scheme in 5G-Enabled Vehicular Networks.

The security and privacy concerns in vehicular communication are often faced with schemes depending on either elliptic curve (EC) or bilinear pair (BP) cryptographies. However, the operations used by BP and EC are time-consuming and more complicated. None of the previous studies fittingly tackled the efficient performance of signing messages and verifying signatures. Therefore, a chaotic map-based conditional privacy-preserving authentication (CM-CPPA) scheme is proposed to provide communication security in 5G-enabled vehicular networks in this paper. The proposed CM-CPPA scheme employs a Chebyshev polynomial mapping operation and a hash function based on a chaotic map to sign and verify messages. Furthermore, by using the AVISPA simulator for security analysis, the results of the proposed CM-CPPA scheme are good and safe against general attacks. Since EC and BP operations do not employ the proposed CM-CPPA scheme, their performance evaluation in terms of overhead such as computation and communication outperforms other most recent related schemes. Ultimately, the proposed CM-CPPA scheme decreases the overhead of computation of verifying the signatures and signing the messages by 24.2% and 62.52%, respectively. Whilst, the proposed CM-CPPA scheme decreases the overhead of communication of the format tuple by 57.69%.

A Provably Secure IBE Transformation Model for PKC Using Conformable Chebyshev Chaotic Maps under Human-Centered IoT Environments.

The place of public key cryptography (PKC) in guaranteeing the security of wireless networks under human-centered IoT environments cannot be overemphasized. PKC uses the idea of paired keys that are mathematically dependent but independent in practice. In PKC, each communicating party needs the public key and the authorized digital certificate of the other party to achieve encryption and decryption. In this circumstance, a directory is required to store the public keys of the participating parties. However, the design of such a directory can be cost-prohibitive and time-consuming. Recently, identity-based encryption (IBE) schemes have been introduced to address the vast limitations of PKC schemes. In a typical IBE system, a third-party server can distribute the public credentials to all parties involved in the system. Thus, the private key can be harvested from the arbitrary public key. As a result, the sender could use the public key of the receiver to encrypt the message, and the receiver could use the extracted private key to decrypt the message. In order to improve systems security, new IBE schemes are solely desired. However, the complexity and cost of designing an entirely new IBE technique remain. In order to address this problem, this paper presents a provably secure IBE transformation model for PKC using conformable Chebyshev chaotic maps under the human-centered IoT environment. In particular, we offer a robust and secure IBE transformation model and provide extensive performance analysis and security proofs of the model. Finally, we demonstrate the superiority of the proposed IBE transformation model over the existing IBE schemes. Overall, results indicate that the proposed scheme posed excellent security capabilities compared to the preliminary IBE-based schemes.

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics.

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.

Chebyshev apodized fiber Bragg gratings.

In this paper, a new apodized fiber Bragg grating (FBG) structure, the Chebyshev apodization, is proposed. The Chebyshev polynomial distribution has been widely used for the optimal design of antennas and filters, but it has not been used for designing FBGs. Unlike the function of traditional Gaussian-apodized FBGs, the Chebyshev polynomial is a discrete function. We demonstrate a new methodology for designing Chebyshev-apodized FBGs: the grating region is divided by discrete n sections with uniform gratings, while the index change is used to express the Chebyshev polynomial. We analyze the Chebyshev-apodized FBGs by using coupled mode theory and the piecewise-uniform approach. The reflection spectrum and the dispersion of Chebyshev-apodized FBGs are calculated and compared with those of Gaussian FBGs. Moreover, a sidelobe suppression level (SSL), a parameter of the Chebyshev polynomial, along with the maximum ac-index change of FBGs are analyzed. Assume that the grating length is 20mm, SSL is 100 dB, the section number is 40, and the maximum ac-index change is 2 × 10-4. The reflection spectrum of Chebyshev apodized FBGs shows flattened sidelobes with an absolute SSL of -95.9 dB (corresponding to SSL=100 dB). The simulation results reveal that at the same full width at half maximum, the Chebyshev FBGs have lower sidelobe suppression than the Gaussian FBGs, but their dispersion is similar. We demonstrate the potential of using Chebyshev-apodized FBGs in optical filters, dispersion compensators, and sensors; Chebyshev apodization can be applied in the design of periodic dielectric waveguides.

Parameter Interval Uncertainty Analysis of Internal Resonance of Rotating Porous Shaft-Disk-Blade Assemblies Reinforced by Graphene Nanoplatelets.

This paper conducts a parameter interval uncertainty analysis of the internal resonance of a rotating porous shaft-disk-blade assembly reinforced by graphene nanoplatelets (GPLs). The nanocomposite rotating assembly is considered to be composed of a porous metal matrix and graphene nanoplatelet (GPL) reinforcement material. Effective material properties are obtained by using the rule of mixture and the Halpin-Tsai micromechanical model. The modeling and internal resonance analysis of a rotating shaft-disk-blade assembly are carried out based on the finite element method. Moreover, based on the Chebyshev polynomial approximation method, the parameter interval uncertainty analysis of the rotating assembly is conducted. The effects of the uncertainties of the GPL length-to-width ratio, porosity coefficient and GPL length-to-thickness ratio are investigated in detail. The present analysis procedure can give an interval estimation of the vibration behavior of porous shaft-disk-blade rotors reinforced with graphene nanoplatelets (GPLs).

Testing fractional unit roots with non-linear smooth break approximations using Fourier functions.

In this paper, we present a testing procedure for fractional orders of integration in the context of non-linear terms approximated by Fourier functions. The test statistic has an asymptotic standard normal distribution and several Monte Carlo experiments conducted in the paper show that it performs well in finite samples. Various applications using real life time series, such as US unemployment rates, US GNP and Purchasing Power Parity (PPP) of G7 countries are presented at the end of the paper.

Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering.

This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We define spectral filters via the LB operator on a graph and explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters. We then update the LB operator for pooling in the LB-CNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and Open Access Series of Imaging Studies (OASIS) to demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of two datasets, we showed that the LB-CNN slightly improves classification accuracy compared to the spectral graph-CNN. The three polynomials had a similar computational cost and showed comparable classification accuracy in the LB-CNN or spectral graph-CNN. The LB-CNN trained via the ADNI dataset can achieve reasonable classification accuracy for the OASIS dataset. Our findings suggest that even though the shapes of the three polynomials are different, deep learning architecture allows us to learn spectral filters such that the classification performance is not dependent on the type of the polynomials or the operators (graph Laplacian and LB operator).

Solution of the Fokker-Planck Equation by Cross Approximation Method in the Tensor Train Format.

We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.

Deep learning for computational structural optimization.

We investigate a novel computational approach to computational structural optimization based on deep learning. After employing algorithms to solve the stiffness formulation of structures, we used their improvement to optimize the structural computation. A standard illustration of 10 bar-truss was revisited to illustrate the mechanism of neural networks and deep learning. Several benchmark problems of 2D and 3D truss structures were used to verify the reliability of the present approach, and its extension to other engineering structures is straightforward. To enhance computational efficiency, a constant sum technique was proposed to generate data for the input of multi-similar variables. Both displacement and stress enforcements were the constraints of the optimized problem. The optimization data for cross sections with the objective function of total weight were then employed in the context of deep learning. The stochastic gradient descent (SGD) with Nesterov's accelerated gradient (NAG), root mean square propagation (RMSProp) and adaptive moment estimation (Adam) optimizers were compared in terms of convergence. In addition, this paper devised Chebyshev polynomials for a new approach to activation functions in single-layer neural networks. As expected, its convergence was quicker than the popular learning functions, especially in a short training with a small number of epochs for tested problems. Finally, a split data technique for linear regression was proposed to deal with some sensitive data.

Buckling Behavior of Nanobeams Placed in Electromagnetic Field Using Shifted Chebyshev Polynomials-Based Rayleigh-Ritz Method.

In the present investigation, the buckling behavior of Euler-Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen's nonlocal theory. Critical buckling load for all the classical boundary conditions such as "Pined-Pined (P-P), Clamped-Pined (C-P), Clamped-Clamped (C-C), and Clamped-Free (C-F)" are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the "Pined-Pined (P-P)" boundary condition using Navier's technique, and the numerical results obtained for the "Pined-Pined (P-P)" boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.