Sensitivity Analysis of Electromagnetic Scattering from Dielectric Targets with Polynomial Chaos Expansion and Method of Moments

In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.

Opinion consensus incorporating higher-order interactions in individual-collective networks

In the current information society, the dissemination mechanisms and evolution laws of individual or collective opinions and their behaviors are the research hot topics in the field of opinion dynamics. First, in this paper, a two-layer network consisting of an individual-opinion layer and a collective-opinion layer is constructed, and a dissemination model of opinions incorporating higher-order interactions(i.e. OIHOI dissemination model) is proposed. Furthermore, the dynamic equations of opinion dissemination for both individuals and groups are presented. Using Lyapunov's first method,two equilibrium points, including the negative consensus point and positive consensus point, and the dynamic equations obtained for opinion dissemination, are analyzed theoretically. In addition, for individual opinions and collective opinions,some conditions for reaching negative consensus and positive consensus as well as the theoretical expression for the dissemination threshold are put forward. Numerical simulations are carried to verify the feasibility and effectiveness of the proposed theoretical results, as well as the influence of the intra-structure, inter-connections, and higher-order interactions on the dissemination and evolution of individual opinions. The main results are as follows.(i) When the intra-structure of the collective-opinion layer meets certain characteristics, then a negative or positive consensus is easier to reach for individuals.(ii) Both negative consensus and positive consensus perform best in mixed type of inter-connections in the two-layer network.(iii) Higher-order interactions can quickly eliminate differences in individual opinions, thereby enabling individuals to reach consensus faster.

Time–frequency co-movement and risk connectedness among cryptocurrencies:new evidence from the higher-order moments before and during the COVID-19 pandemic

Analyzing comovements and connectedness is critical for providing significant implications for crypto-portfolio risk management.However,most existing research focuses on the lower-order moment nexus(i.e.the return and volatility interactions).For the first time,this study investigates the higher-order moment comovements and risk connectedness among cryptocurrencies before and during the COVID-19 pandemic in both the time and frequency domains.We combine the realized moment measures and wavelet coherence,and the newly proposed time-varying parameter vector autoregression-based frequency connectedness approach(Chatziantoniou et al.in Integration and risk transmission in the market for crude oil a time-varying parameter frequency connectedness approach.Technical report,University of Pretoria,Department of Economics,2021)using intraday high-frequency data.The empirical results demonstrate that the comovement of realized volatility between BTC and other cryp-tocurrencies is stronger than that of the realized skewness,realized kurtosis,and signed jump variation.The comovements among cryptocurrencies are both time-dependent and frequency-dependent.Besides the volatility spillovers,the risk spillovers of high-order moments and jumps are also significant,although their magnitudes vary with moments,making them moment-dependent as well and are lower than volatility connectedness.Frequency connectedness demonstrates that the risk connectedness is mainly transmitted in the short term(1–7 days).Furthermore,the total dynamic connectedness of all realized moments is time-varying and has been significantly affected by the outbreak of the COVID-19 pandemic.Several practical implications are drawn for crypto investors,portfolio managers,regulators,and policymakers in optimizing their investment and risk management tactics.

Series expansion feasibility of singular integral in method of moments

When calculating electromagnetic scattering using method of moments （MoM）, integral of the singular term has a significant influence on the results. This paper transforms the singular surface integral to the contour integral. The integrand is expanded to Taylor series and the integral results in a closed form. The cut-off error is analyzed to show that the series converges fast and only about 2 terms can agree wel with the accurate result. The comparison of the perfect electric conductive （PEC） sphere＇s bi-static radar cross section （RCS） using MoM and the accurate method validates the feasibility in manipulating the singularity. The error due to the facet size and the cut-off terms of the series are analyzed in examples.

Measurement of remanent magnetic moment using a torsion pendulum with single frequency modulation method

In Tian Qin spaceborne gravitational-wave detectors, the stringent requirements on the magnetic cleanliness of the test masses demand the high resolution ground-based characterization measurement of their magnetic properties. Here we present a single frequency modulation method based on a torsion pendulum to measure the remanent magnetic moment mr of 1.1 kg dummy copper test mass, and the measurement result is(6.45 ± 0.04(stat) ± 0.07(syst)) × 10^(-8)A · m^(2). The measurement precision of the mr is about 0.9 n A · m^(2), well below the present measurement requirement of Tian Qin. The method is particularly useful for measuring extremely low magnetic properties of the materials for use in the construction of space-borne gravitational wave detection and other precision scientific apparatus.

CONVERGENCE PROPERTIES OF HIGHER-ORDER MOMENT AND CUMULANT ESTIMATES

The sample estimates of higher-order statistics are studied. Under certain conditions, the almost sure convergence of the third- and fourth-order moment and cumulant estimates of stationary processes is established. The rate of almost sure convergence is obtained for the sample estimates of third- and fourth-order moment and cumulant. Additionally, it is shown that the third- and fourth-order moment and cumulant estimates are asymptotic normal.

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization(RBMDO)of engineering structures,the saddlepoint approximation(SA)method can be utilized to enhance the accuracy and efficiency of reliability evaluation.However,the random variables involved in SA should be easy to handle.Additionally,the corresponding saddlepoint equation should not be complicated.Both of them limit the application of SA for engineering problems.The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments.However,the traditional moment matching method is not very accurate generally.In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization,a fourth-moment saddlepoint approximation(FMSA)method is introduced into RBMDO.In FMSA,the approximate cumulative generating functions are constructed based on the first four moments of the limit state function.The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function.Furthermore,the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment,which is based on the performance measure approach strategy.Two engineering examples are introduced to verify the effectiveness of proposed method.

Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow

把大旋涡模拟与照时刻方法相结合的一个计算模特儿被雇用在一口限制侵犯的喷气学习 nanoparticle 进化。调查粒子尺寸在短暂政体被限制，并且粒子碰撞核被使用流动匹配的理论获得。模拟被把它与试验性的结果作比较验证。协调结构扮演统治粒子数字紧张的数字结果表演，尺寸和多分散性分布，和它也导致粒子沉重的喷气被冲淡由周围。在侵犯的喷气流动的粒子动力学的进化是强烈，与雷纳兹有关，数字和 nozzle-to-plate 距离，和他们的关系被分析。

Chebyshev polynomial-based Ritz method for thermal buckling and free vibration behaviors of metal foam beams

This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.

Two-Stream Approximation to the Radiative Transfer Equation:A New Improvement and Comparative Accuracy with Existing Methods

Mathematical modeling of the interaction between solar radiation and the Earth's atmosphere is formalized by the radiative transfer equation(RTE), whose resolution calls for two-stream approximations among other methods. This paper proposes a new two-stream approximation of the RTE with the development of the phase function and the intensity into a third-order series of Legendre polynomials. This new approach, which adds one more term in the expression of the intensity and the phase function, allows in the conditions of a plane parallel atmosphere a new mathematical formulation of γparameters. It is then compared to the Eddington, Hemispheric Constant, Quadrature, Combined Delta Function and Modified Eddington, and second-order approximation methods with reference to the Discrete Ordinate(Disort) method(δ –128 streams), considered as the most precise. This work also determines the conversion function of the proposed New Method using the fundamental definition of two-stream approximation(F-TSA) developed in a previous work. Notably,New Method has generally better precision compared to the second-order approximation and Hemispheric Constant methods. Compared to the Quadrature and Eddington methods, New Method shows very good precision for wide domains of the zenith angle μ 0, but tends to deviate from the Disort method with the zenith angle, especially for high values of optical thickness. In spite of this divergence in reflectance for high values of optical thickness, very strong correlation with the Disort method(R ≈ 1) was obtained for most cases of optical thickness in this study. An analysis of the Legendre polynomial series for simple functions shows that the high precision is due to the fact that the approximated functions ameliorate the accuracy when the order of approximation increases, although it has been proven that there is a limit order depending on the function from which the precision is lost. This observation indicates that increasing the order of approximation of the phase function of the RTE leads to a better precision in flux calculations. However, this approach may be limited to a certain order that has not been studied in this paper.

A method of moments for calculating dynamic responses beyond linear response theory

A method of moments for calculating the dynamic response of periodically driven overdamped nonlinear stochastic systems in the general response sense is proposed, which is a modification of the method of moments confined within linear response theory. The calculating experience suggests that the proposed technique is simple and efficient in implementation, and the comparison with stochastic simulation shows that the first three orders of susceptibilities calculated by the proposed technique have high accuracy. The dependence of the spectral amplification parameters at the first three harmonics on the noise intensity is also investigated, and another observed phenomenon of stochastic resonance in the systems induced by the location of a single periodic orbit is disclosed and explained.

Monte Carlo Integration Technique for Method of Moments Solution of EFIE in Scattering Problems

An integration technique based on use of Monte Carlo Integration is proposed for Method of Moments solution of Electric Field Integral Equation. As an example numerical analysis is carried out for the solution of the integral equation for unknown current distribution on metallic plate structures. The entire domain polynomial basis functions are employed in the MOM formulation which leads to only small number of matrix elements thus saving significant computer time and storage. It is observed that the proposed method not only provides solution of the unknown current distribution on the surface of the metallic plates but is also capable of dealing with the problem of singularity efficiently.

Computation of Protection Zone of a Lightning Rod Using Method of Moments and Monte Carlo Integration Technique

An accurate determination of lightning protection zone is an important issue in the analysis and design of an appropri-ate lightning protection system. This paper presents a fast and accurate protection zone determination methodology for metallic lightning rod. The methodology is based on Quasi Monte Carlo Integration technique applied to Method of Moments (MoM) solution of Integral Equations. As an example, solution of the integral equation for unknown charge distribution on lightning rod is obtained. The electric field in the region surrounding the rod is then computed and the protection zone plotted accordingly. The effect of the thickness of the rod on the protection zone is also studied.

Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

Anti-plane deformations around arbitrary-shaped canyons on a wedge-shape half-space:moment method solutions

The wave propagation behavior in an elastic wedge-shaped medium with an arbitrary shaped cylindrical canyon at its vertex has been studied.Numerical computation of the wave displacement field is carried out on and near the canyon surfaces using weighted-residuals(moment method).The wave displacement fields are computed by the residual method for the cases of elliptic,circular,rounded-rectangular and flat-elliptic canyons,The analysis demonstrates that the resulting surface displacement depends,as in similar previous analyses,on several factors including,but not limited,to the angle of the wedge,the geometry of the vertex,the frequencies of the incident waves,the angles of incidence,and the material properties of the media.The analysis provides intriguing results that help to explain geophysical observations regarding the amplification of seismic energy as a function of site conditions.

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model(PBM). A typical population balance equation(PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods(QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function(NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments(QMOM), direct quadrature method of moments(DQMOM),extended quadrature method of moments(EQMOM), conditional quadrature method of moments(CQMOM),extended conditional quadrature method of moments(ECQMOM) and hyperbolic quadrature method of moments(Hy QMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics(CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

Rolling Force and Rolling Moment in Spline Cold Rolling Using Slip-line Field Method

Rolling force and rolling moment are prime process parameter of external spline cold rolling. However, the precise theoretical formulae of rolling force and rolling moment are still very fewer, and the determination of them depends on experience. In the present study, the mathematical models of rolling force and rolling moment are established based on stress field theory of slip-line. And the isotropic hardening is used to improve the yield criterion. Based on MATLAB program language environment, calculation program is developed according to mathematical models established. The rolling force and rolling moment could be predicted quickly via the calculation program, and then the reliability of the models is validated by FEM. Within the range of module of spline m=0.5-1.5 mm, pressure angle of reference circle α=30.0°-45.0°, and number of spline teeth Z=19-54, the rolling force and rolling moment in rolling process （finishing rolling is excluded） are researched by means of virtualizing orthogonal experiment design. The results of the present study indicate that： the influences of module and number of spline teeth on the maximum rolling force and rolling moment in the process are remarkable; in the case of pressure angle of reference circle is little, module of spline is great, and number of spline teeth is little, the peak value of rolling force in rolling process may appear in the midst of the process; the peak value of rolling moment in rolling process appears in the midst of the process, and then oscillator weaken to a stable value. The results of the present study may provide guidelines for the determination of power of the motor and the design of hydraulic system of special machine, and provide basis for the farther researches on the precise forming process of external spline cold rolling.

An analysis of the radar backscatter from oil-covered sea surfaces using moment method and Monte-Carlo simulation: preliminary results

An analysis of the radar backscattering from the ocean surface covered by oil spill is presented using a mi- crowave scattering model and Monte-Carlo simulation. In the analysis, a one-dimensional rough sea sur- face is numerically generated with an ocean waveheight spectrum for a given wind velocity. A two-layered medium is then generated by adding a thin oil layer on the simulated rough sea surface. The electric fields backscattered from the sea surface with two-layered medium are computed with the method of moments （MoM）, and the backscattering coefficients are statistically obtained with N independent samples for each oil-spilled surface using the Monte-Carlo technique for various conditions of surface roughness, oil-layer thickness, frequency, polarization and incidence angle. The numerical simulation results are compared with theoretical models for clean sea surfaces and SAR images of an off-spilled sea surface caused by the Hebei （Hebei province, China） Spirit oil tanker in 2007. Further, conditions for better oil spill extraction are sought by the numerical simulation on the effects of wind speed and oil-layer thickness at different inci- dence angles on the backscattering coefficients.

Nanoparticle coagulation in a planar jet via moment method

Large eddy simulations of nanoparticle coagulation in an incompressible planar jet were performed. The particle is described using a moment method to approximate the particle general dynarnics equations. The time-averaged results based on 3000 time steps for every case were obtained to explore the influence of the Schmidt number and the Damkohler number on the nanoparticle dynamics. The results show that the changes of Schmidt number have the influence on the number concentration of nanoparticles only when the particle diameter is less than 1 nm for the fixed gas parameters. The number concentration of particles for small particles decreases more rapidly along the flow direction, and the nanoparticles with larger Schmidt number have a narrower distribution along the transverse direction. The smaller nanoparticles Coagulate and disperse easily, grow rapidly hence show a stronger polydispersity. The smaller coagulation time scale can enhance the particle collision and coagulation. Frequented collision and coagulation bring a great increase in particle size. The larger the Damkohler number is, the higher the particle polydispersity is.