Approximation Theory Research Articles

Vibration analysis of size dependent micro FML cylindrical shell reinforced by CNTs based on modified couple stress theory

In this manuscript, the sequel of agglomeration on the vibration of fiber metal laminated (FML) cylindrical shell in the micro phase using developed couple stress theory (MCST). Hamilton’s principle has been carried out for deriving the non-classical equations of motion of size-dependent thin micro cylindrical shell on the basis of Love’s first approximation theory. Mori-Tanaka and extended rule of mixture are utilized to estimate the mechanical attributes of carbon nanotubes (CNTs) and equivalent fiber, respectively. These four phases CNTs/fiber/polymer/metal laminated (CNTFPML) micro cylindrical shell is analyzed applying beam modal function model for several boundary limitations. Then, an investigation is performed to study the impacts of differing input parameters namely material length scale parameter, agglomeration, the distributions of agglomerated CNTs, the mass fraction of equivalent fiber and the volume fraction of CNTs on the frequency response of micro agglomerated CNTFPML cylindrical shell. The main output illustrated that the growth of frequencies is directly dependent to the increase of material length scale parameter for this agglomerated CNTFPML cylindrical shell so that through increasing the values of agglomeration parameters η and μ and material length scale parameter l altogether, the frequencies of this cylindrical shell grow.

Analytic theory for the tangential YORP produced by the asteroid regolith

Context. The tangential YORP effect is a radiation pressure torque produced by asymmetric thermal emission by structures on the asteroid surface. Previous works considered these structures to be boulders of different shapes lying on the surface of the asteroid. Aims. We study the tangential YORP produced by the rough interface of the asteroid's regolith. Methods. We created an approximate analytic theory of heat conduction on a slightly non-flat sinusoidal surface. We analyzed the published data on the small-scale shape of the asteroid (162173) Ryugu and estimated its tangential YORP due to the surface roughness. Results. We derive an analytic formula that expresses the tangential YORP of a sinusoidal surface in terms of its geometric and thermal properties. The tangential YORP is highest at the thermal parameter on the order of unity and for shape irregularities on the order of the thermal wavelength. Application of this equation to Ryugu predicts a tangential YORP that is 5-70 times greater than its normal YORP effect. Conclusions. The contribution of the small-scale regolith roughness to the YORP effect of the asteroid can be comparable to the normal YORP and the tangential YORP produced by boulders. The same theory can describe the roughness of the asteroid boulders, thus adding a new term to the previously considered the tangential YORP created by boulders.

Roughness correction method for the measurement of attenuation coefficient using contact transducers

The surface quality of a material significantly affects the accurate measurement of the ultrasonic attenuation coefficient. In this research, a roughness correction method for experimental calculation of the attenuation coefficient using contact transducers is proposed. Firstly, the losses due to the scattering from a rough interface are analysed. According to the mechanism of ultrasonic waves reflected from a thin layer between two solid media, the frequency-dependent reflection coefficient of the coupling interface involving a roughness parameter is derived based on the phase-screen approximation theory. Then, a compensation model is established to correct the error caused by the surface roughness. 304 stainless steel and 45 steel specimens with different levels of surface roughness are prepared and ultrasonic measurements are implemented using both the through-transmission and pulse-echo methods. The experimental results show that the proposed correction method can effectively eliminate the losses caused by surface roughness and improve the measurement accuracy of the attenuation coefficient.

An Efficient MMW 3-D Imaging Algorithm for Near-Field MIMO-SAR With Nonuniform Transmitting Array

In this letter, an efficient millimeter-wave (MMW) frequency-domain imaging algorithm is presented for 3-D object reconstruction of the near-field multiple-input–multiple-output synthetic aperture radar (MIMO-SAR) with nonuniform transmitting array. For each single-input–multiple-output (SIMO) case of the MIMO-SAR setup, the precise spectral echo expression is derived by spherical-wave decomposition and fast Fourier transform (FFT) operations. To obtain the linear phase in range direction, the multiorder approximation theory is employed to expand the coupling phases, and thus the target subimage can be reconstructed by utilizing phase compensation, central wavenumber approximation, and inverse FFT. Finally, the fully focused 3-D image for MIMO setup will be realized through accumulating all the SIMO subimages. Compared with the existing representative algorithms, our method avoids complicated interpolation and achieves high-quality fast imaging with only matrix multiplication and a small amount of iteration operations. The experimental results validate the effectiveness of the proposed algorithm.

Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients

The paper discusses the analytical stability and numerical stability of differential equations with piecewise constant arguments with matrix coefficients. Firstly, the Runge-Kutta method is applied to the equation and the recurrence relationship of the numerical solution of the equation is obtained. Secondly, it is proved that the Runge-Kutta method can preserve the convergence order. Thirdly, the stability conditions of the numerical solution under different matrix coefficients are given by Pad$\acute{e}$ approximation and order star theory. Finally, the conclusions are verified by several numerical experiments.

Smart and sustainable city logistics: Design, consolidation, and regulation

Growing e-commerce delivery increases traffic congestion and demands for smart last-mile city distribution systems. In this context, municipalities and logistics service providers aim towards sustainable and profitable city logistics, which are often conflicting goals. Whether to use single-echelon or two-echelon systems with micro-depots and city freighters as well as quantifying the benefits of shipment consolidation between independent providers under different system designs are unanswered questions in academia and practice.We provide a general methodological framework based on continuous approximation theory to study the design of city logistics networks with multiple logistics service providers that choose transportation strategies, each defined by a network structure and specific transportation modes. We provide analytical results on the comparison of different transportation strategies. By providing thresholds on the benefit of consolidated systems, we analytically quantify shipment consolidation effects in city centers. We then embed this continuous approximation into a mixed-integer non-linear program and study parameter sensitivities for a real-world case study.Our results show that higher customer density and longer distances between depots and distribution areas promote 2-tier transportation strategies. Consolidation between logistics service providers allows for emission reductions independent of the transportation strategy and correlate with cost reductions up to 50% on the lower echelon. Outsourcing promotes 2-tier last-mile deliveries with city freighters and in a city with heterogeneous districts, it is beneficial to use a hybrid network that includes both single- and two-echelon strategies using multiple transportation modes for different demand classes.

Isotropic pressure-induced electronic band structure of BaTiO3, SrTiO3 and CaTiO3 with its impact on structural and optical properties: ab-initio calculation

ABSTRACT The present theoretical study offers a comprehensive investigation under the application of systematic external static isotropic pressure, phase stability, electronic-band structure and their impact on the optical characteristics of BaTiO3, SrTiO3 and CaTiO3. All the computational works were performed by Cambridge Serial Total Energy Package, using generalised gradient approximation and density functional theory, Perdue-Burke-Ernzerhof, along with the ultra-soft pseudo-potential for the frozen core. According to structural properties, there is no phase transformation and the structure of all compounds remains cubic. The band structure is a perfect tool to observe the nature of a material, such as an insulator, semiconductor or metal. When the pressure is applied, the band gap of BaTiO3, SrTiO3 and CaTiO3 increases and retains their indirect nature. The total/partial/elemental density of states has been estimated for the full description of the band structure. To verify the authenticity of BaTiO3, SrTiO3 and CaTiO3, various optical parameters have been reported such as the real dielectric functions, imaginary dielectric functions, absorption, refractive index, extinction coefficient, energy loss function and reflectivity. All discrepancies in structural parameters, band structure and optical properties occurred due to applied pressure, and BaTiO3, SrTiO3, and CaTiO3 are connected .

Study on Numerical Simulation and Electromagnetic Scattering of Time-Invariant Freak Waves

The occurrence probability of freak waves is related to the sea wave spectrum. In this paper, different wave spectrums are used to simulate time-invariant three-dimensional freak waves. Freak waves that meet the international standards are generated at fixed time and location by adjusting the energy of the wavelets. We studied the occurrence probability of freak waves under the conditions of different wave spectrums, different wind speeds, and different modulation ratios and optimized the calculation speed of the model. Simulation data show that the difference in the shape of the wave spectrum affects the probability of freak waves occurrence. The model conforms to the Benjamin–Feir index (BFI), and the ratio of wave steepness to spectrum bandwidth is the key. In this paper, the Kirchhoff approximation theory is used to study the electromagnetic scattering (EM scattering) properties of freak waves on the large scale. We ideally calculate the Normalized Radar Cross-Section (NRCS) from the sea surface with freak waves, under different wind speeds and different grazing angles. The NRCS of freak waves is extremely low, and the increase of wind speed and the decrease of the grazing angle will make the detection of freak waves more difficult. The possibility of detection of freak waves is higher at high grazing angles (low incidence angles). The numerical simulations provide reference for engineering.

Vibration Analysis of Rotating Combined Thin-Walled Shells With Multiple Conical Segments

Abstract In this paper, vibration analysis of rotating combined thin-walled shells with multiple conical segments has been carried out. Considering the centrifugal force, Coriolis force, and initial hoop tension due to rotation, the elastic strain energy and kinetic energy of a single rotating conical shell are expressed based on Love’s first approximation theory. The artificial springs are introduced to simulate the connections of adjacent conical shells and the boundaries of the rotating combined thin-walled shells. Taking characteristic orthogonal polynomial series as the admissible functions, the Rayleigh–Ritz method is employed to derive the frequency equations of the combined shell and corresponding vibration characteristics are then obtained. Given that the cylindrical shell and annular plates can be regarded as conical shells with semi-vertex angles of 0 deg and 90 deg, respectively, the solution given is also available for the vibration analysis of rotating combined thin-walled shells comprised of any segments of cylindrical, conical shell, and annular plate. As examples of rotating combined thin-walled shells with two and five segments, vibrations of rotating conical–conical joined shell and cylindrical–conical–cylindrical–conical–cylindrical joined shell are investigated in the paper. Traveling wave frequencies and corresponding mode shapes are shown, and the effects of rotating speed, circumferential wave number, spring stiffness, and semi-vertex angles on the vibration behavior are given in detail.

Experimentally and Theoretically Revealing the Co-Doping Effects of Ce-Sr in Gd2Zr2O7.

To safely dispose radioactive waste (including, e.g., thorium and radiostrontium), Ce4+ and Sr2+ were chosen as simulated surrogates of α and β waste and were introduced into the Gd3+ site in Gd2Zr2O7 to maintain the average cationic radius and to compensate for charge. A series of Gd2-xSrx/2Cex/2Zr2O7 (0.00 ≤ x ≤ 0.25) compounds were examined by experimental and theoretical calculations to investigate the co-doping effects of α and β waste in a Gd2Zr2O7-based matrix. The effects of Ce4+ and Sr2+ content on the phase, unit cell parameters, active modes, mechanical property, and microstructure were studied systematically. Moreover, the limit of incorporation of Ce4+ and Sr2+ in Gd2Zr2O7 pyrochlore and the lattice parameters were also calculated through virtual crystal approximation theory, and the results were found to well agree with experimental results.

Frequency domain resilient consensus of multi-agent systems under IMP-based and non IMP-based attacks

This paper presents a frequency domain resilient consensus problem of time-delayed multi-agent systems (MASs) under attacks. An internal model principle-based attack, a kind of network attack, is firstly analyzed and modeled in frequency domain. Then, a Quasi-H∞ robust controller is designed for resilient consensus of the MAS with the help of an internal model controller. This scheme is different from the existing works in the sense that the attacker may inject an attacking signal on the node of an agent instead of a communication link. It is also novel because it can eliminate the impacts of attack without excluding compromised nodes and restricting the number of compromised nodes. The proposed control scheme is applied to a linear delayed MAS with single-input single-output agents and then transformed into multi-input and multi-output agents. An innovative approach is employed for the tuning of the proposed controller to track the performance of the MAS and its robustness. Some necessary and sufficient conditions are extracted using Padé approximation and graph theory to minimize the impacts of time delay and attacks simultaneously. The results are extended for resilient consensus of the MAS under non-internal model principle-based attack. The simulation results are presented to demonstrate the validity of the proposed control strategy.

Simple fluid with broken time-reversal invariance.

We characterize a system of hard spheres with a simple collision rule that breaks time-reversal symmetry but conserves energy. The collisions lead to an achiral, isotropic, and homogeneous stationary state whose properties are determined in simulations and compared to an approximate theory originally developed for elastic hard spheres. In the nonequilibrium fluid state, velocities are correlated, a phenomenon known from other nonequilibrium stationary states. The correlations are long-ranged decaying like 1/r^{D} in D dimensions. Such correlations are expected on general grounds far from equilibrium and had previously been observed in driven or nonstationary systems.

Rotary Flexible Joint Control Using Adaptive Fuzzy Sliding Mode Scheme.

An adaptive fuzzy sliding control (AFSMC) approach is adopted in this paper to address the problem of angular position control and vibration suppression of rotary flexible joint systems. AFSMC consists of fuzzy-based singleton control action and switching control law. By adjusting fuzzy parameters with the self-learning ability to discard system nonlinearities and uncertainties, singleton control based on fuzzy approximation theory can estimate the perfect control law of feedback linearization. To enhance robustness, an additional switching control law is incorporated to reduce the approximation error between the derived singleton control action and the perfect control law of feedback linearization. AFSMC's closed-loop stability will be demonstrated via sliding surface and Lyapunov function analysis of error function. In order to demonstrate the effectiveness of the AFSMC in tracking performance as well as its ability to respond to model uncertainties and external perturbations, simulations are carried out using Simulink and Matlab in order to demonstrate how well it adapts to these situations. Based on these results, it can be concluded that the AFSMC performs satisfactorily in terms of tracking.

On <img src=image/13427444_01.gif>-Ideal Statistically Convergent of Double Sequences in n-Normed Spaces over Non-Archimedean Fields

The main aim of this work is to investigate some important properties of statistical convergence sequence in non-Archimedean fields. Statistical convergence has been discussed in various fields of mathematics namely approximation theory, measure theory, probability theory, trigonometric series, number theory, etc. The concept of summability over valued fields is a significant area of mathematics that has many applications in analytic continuation, quantum mechanics, probability theory, Fourier analysis, approximation theory, and fixed point theory. The theory of statistical convergence plays a notable space in the summability theory and functional analysis. The purpose of this work is to provide certain characterizations of <img src=image/13427444_01.gif> ideal statistical convergence of sequence and <img src=image/13427444_01.gif> ideal statistical Cauchy sequence in n-normed spaces and the establishment of relevant results in non-Archimedean fields. The <img src=image/13427444_01.gif> ideal statistical convergence of sequence and <img src=image/13427444_01.gif> ideal statistically Cauchy sequence are defined. A few related theorems are proved in field <img src=image/13427444_02.gif>. The results of this work are extended to establish statistical convergence of double sequences in n-normed space and some new results have been proved. In this work, the main concept is <img src=image/13427444_01.gif> ideal statistical convergence of double sequences in n-normed space over a complete, non-trivially valued, non-Archimedean field. Throughout this article, <img src=image/13427444_02.gif> is a complete, non-trivially valued, non-Archimedean field.

Joint Optimization of Trajectory, Task Offloading, and CPU Control in UAV-Assisted Wireless Powered Fog Computing Networks

This paper studies a rotary-wing unmanned aerial vehicle (UAV)-assisted wireless powered fog computing network, where a UAV serves as both a mobile wireless energy source and a mobile fog server to provide charging and computing services to a group of ground sensors. The UAV first broadcasts radio frequency (RF) signals to charge the sensors, and then the sensors use the harvested energy to complete their tasks by local computing and fog computing, where the partial offloading policy is employed. In order to achieve green communication system design, an optimization problem is formulated to minimize the UAV’s energy consumption by jointly optimizing the UAV’s trajectory, the task offloading allocation and computing resource allocation, where the piecewise nonlinear energy harvesting (EH) model, the charging requirements of sensors and velocity constraints of UAV are taken into account together. To solve the non-convex problem, an iterative method is designed based on the successive convex approximation (SCA) theory. Simulation results show that our presented scheme converges within a few iterations, by using which the UAV’s energy consumption can be greatly reduced compared to benchmark schemes. It is also observed that the harvested energy by the linear EH model have obvious bias compared to that obtained by the nonlinear EH model. And in the case of nonlinear EH model, as circuit saturation threshold increases, the UAV moves closer to the sensors such that their tasks can be completed, which may result in a higher energy consumption of the UAV. Besides, the velocity variation of UAV is relatively small by introducing the velocity constraint, which avoids drastic steering of the UAV in our considered scenario.

K-Functionals and Extreme Problems of the Approximation Theory for Classes of Analytic Functions in a Circle. I

K-Functionals and Extreme Problems of the Approximation Theory for Classes of Analytic Functions in a Circle. I

Optimal Control for Unmanned Systems With One-Way Broadcast Communication

Unmanned systems (USs) including unmanned aerial vehicles, unmanned underwater vehicles, and unmanned ground vehicles have great application prospects in military and civil fields, among which the process of finding feasible and optimal paths for the agents in USs is a kernel problem. Traditional path finding algorithms are hard to adequately obtain optimal paths in real-time under fast time-varying and poor communication environments. We propose an online optimal control algorithm for USs based on a one-way broadcast communication mode under the assumption of a poor communication environment, mobile targets, radars (or sonar), and missiles (or torpedoes). With the principle of receding horizon control, optimal (or suboptimal) paths are then generated by the approximation theory of neural networks and gradient optimization techniques, with low computation requirements. Also, we give a convergence analysis for our algorithm, and show that each agent can reach its target in finite time under some conditions on agents, targets and radar-missiles. Moreover, simulations demonstrate that the agents in USs can generate optimal (or suboptimal) paths in real time using our algorithm while effectively avoiding collision with other agents or detection by enemy radars.

Virtual element method on polyhedral meshes for bi-harmonic eigenvalues problems

In this paper we study a C 1 Virtual Element Method (VEM) on polyhedral meshes for bi-harmonic eigenvalue problems in three dimensions. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained by using the approximation theory of compact self-adjoint operators. Finally, a set of tests will numerically prove the theoretical result of the proposed scheme.

Optimal Resource Allocation of 5G Machine-Type Communications for Situation Awareness in Active Distribution Networks

Advanced metering infrastructure targets the real-time collection of massive multisource data with smart meters (SMs), forming a key component of active distribution networks (ADNs). To achieve comprehensive situation awareness for ADNs, the use of high-speed and reliable two-way communications, such as 5G communications, is the basis. However, it is challenging to optimally allocate wireless resources for massive data, especially when emergency events occur randomly. This study discusses an energy-efficient and optimal wireless resource allocation method employing 5G uplink-based machine-type communications for achieving ADN situation awareness. We propose a 5G-based framework that can reasonably reserve resource blocks for emergency data and model the resource allocation scheme as a problem of maximizing energy efficiency (EE). Then, by using the sample average approximation theory, we transform the stochastic constraint and summarize the scheme as a mixed-integer linear programming problem. Finally, we construct an iterative algorithm based on the Lagrange dual algorithm for searching the global optimal allocation scheme. The results of simulations and experiments illustrate that even if emergency events occur randomly, our algorithm can optimally support the cooperative transmission of all sampled data while providing the best EE for large-scale SM communications.

WARPd: A Linearly Convergent First-Order Primal-Dual Algorithm for Inverse Problems with Approximate Sharpness Conditions

Sharpness conditions directly control the recovery performance of restart schemes for first-order optimization methods without the need for restrictive assumptions such as strong convexity. However, they are challenging to apply in the presence of noise or approximate model classes (e.g., approximate sparsity). We provide a first-order method: weighted, accelerated, and restarted primal-dual (WARPd), based on primal-dual iterations and a novel restart-reweight scheme. Under a generic approximate sharpness condition, WARPd achieves stable linear convergence to the desired vector. Many problems of interest fit into this framework. For example, we analyze sparse recovery in compressed sensing, low-rank matrix recovery, matrix completion, TV regularization, minimization of $\|Bx\|_{l^1}$ under constraints ($l^1$-analysis problems for general $B$), and mixed regularization problems. We show how several quantities controlling recovery performance also provide explicit approximate sharpness constants. Numerical experiments show that WARPd compares favorably with specialized state-of-the-art methods and is ideally suited for solving large-scale problems. We also present a noise-blind variant based on a square-root LASSO decoder. Finally, we show how to unroll WARPd as neural networks. This approximation theory result provides lower bounds for stable and accurate neural networks for inverse problems and sheds light on architecture choices. Code and a gallery of examples are available online as a MATLAB package.