Number Of Fourier Terms Research Articles

The research on TBATS and ELM models for prediction of human brucellosis cases in mainland China: a time series study

BackgroundHuman brucellosis is a serious public health concern in China. The objective of this study is to develop a suitable model for forecasting human brucellosis cases in mainland China.MethodsData on monthly human brucellosis cases from January 2012 to December 2021 in 31 provinces and municipalities in mainland China were obtained from the National Health Commission of the People’s Republic of China website. The TBATS and ELM models were constructed. The MAE, MSE, MAPE, and RMSE were calculated to evaluate the prediction performance of the two models.ResultsThe optimal TBATS model was TBATS (1, {0,0}, -, {< 12,4 >}) and the lowest AIC value was 1854.703. In the optimal TBATS model, {0,0} represents the ARIMA (0,0) model, {< 12,4 >} are the parameters of the seasonal periods and the corresponding number of Fourier terms, respectively, and the parameters of the Box-Cox transformation ω are 1. The optimal ELM model hidden layer number was 33 and the R-squared value was 0.89. The ELM model provided lower values of MAE, MSE, MAPE, and RMSE for both the fitting and forecasting performance.ConclusionsThe results suggest that the forecasting performance of ELM model outperforms the TBATS model in predicting human brucellosis between January 2012 and December 2021 in mainland China. Forecasts of the ELM model can help provide early warnings and more effective prevention and control measures for human brucellosis in mainland China.

A validity of phenomenological models of turbulence in experimental studies

Estimates of the velocity of the chaotic medium motion using information of "narrow" spectral intervals of stationary random signals generated by the motion of such media on the sensors located at two points along the direction of medium motion are considered. The question about the physical meaning of the minimum spectral interval required for such measurements with a given accuracy is posed and discussed based on the main relations given in the article and formulated using the theory of homogeneous isotropic turbulence. It is shown, that within suggested relations the process of filtration may be described using a turbulent term and so the results of the filtration could be seen as a virtual turbulent dissipation. The latter allowed semiquantitatively describing the results of the numeric calculations presented in the article and forming the ways of the further development of the approach suggested which after appropriate updating might be called a "full filtered correlation analysis" (FFCA). In particular, a method of turbulent dissipation rate measurement by means of two (or more) filters with different bandpasses applied to signals in two points of cross-correlation measurements and an approach to the physically clear description of the optimal number of Fourier-terms in the representation of the signal passed the filter are outlined as well as the approach to a turbulence spectral shape evaluation. Apart from that, with the relations given in the article, the well-known fact that Doppler measurements are not applicable to the remote sensing with use of wideband signals as an information carrier as well as the reason why such measurements are applicable in the case of signals with narrow spectra receive its clear physical explanation. The latter allows forming the definition of a monochromatic signal as it is from the point of view of a cross-correlation analysis.

Reconstructing free-energy landscapes for nonequilibrium periodic potentials.

We present a method for reconstructing the free-energy landscape of overdamped Brownian motion on a tilted periodic potential. Our approach exploits the periodicity of the system by using the k-space form of the Smoluchowski equation and we employ an iterative approach to determine the nonequilibrium tilt. We reconstruct landscapes for a number of example potentials to show the applicability of the method to both deep and shallow wells and near-to- and far-from-equilibrium regimes. The method converges logarithmically with the number of Fourier terms in the potential.

On the Operator Splitting and Integral Equation Preconditioned Deferred Correction Methods for the “Good” Boussinesq Equation

We study two numerical methods for the “Good” Boussinesq (GB) equation. Both methods are designed to solve the spatial-temporal pseudo-spectral collocation formulations of the GB equation using the deferred correction methods in one time marching step, where the Fourier series based pseudo-spectral formulation is applied in the spatial direction. The main idea is to iteratively apply a low order method to solve an error equation and refine the provisional solutions until they converge to the high order pseudo-spectral solutions both in space and time. In the first method, an operator splitting approach is introduced as the low order preconditioner in the deferred correction procedure for the temporal Gauss collocation formulation of the original GB equation. The method shows good numerical properties when the deferred correction procedure is convergent and the accuracy requirement is achievable. However, due to the stiffness of the linear differential operators, the Krylov deferred correction (KDC) method has to be applied in order to make the iterations converge. And also, due to the spectral differentiation operator involved, the condition number of the algorithm scales as $$O(N^4)$$ , where N is the number of Fourier terms in the spatial direction. To improve the numerical stability and efficiency, an integral equation approach is applied to “precondition” the GB equation in the second proposed numerical method, by inverting the linear terms of the GB equation analytically. As the nonlinear term of the GB equation is non-stiff, the simple forward Euler’s method preconditioned spectral deferred correction (SDC) iterations converge more efficiently than existing Jacobian-Free Newton–Krylov (JFNK) based KDC implementations, and the condition number of the new formulation is O(1), which leads to a machine precision accuracy at each discrete time step.

Solution procedure of residue harmonic balance method and its applications

This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems. In this solution procedure, no small parameter is assumed. The harmonic residue of balance equation is separated in two parts at each step. The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement. The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again. Three kinds of different differential equations involving general, fractional and delay ordinary differential systems are given as numerical examples respectively. Highly accurate limited cycle frequency and amplitude are captured. The results match well with the exact solutions or numerical solutions for a wide range of control parameters. Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems. Moreover, the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.

Efficient computation of wave propagation along axisymmetric pipes under non-axisymmetric loading

This paper presents an efficient formulation of the problem of wave propagation along the length of axisymmetric pipes under non-axisymmetric loading such as leaks or new cracks so that wave characteristics in a pipe can be identified without the excessive computational time associated with most current 3D modeling techniques. The axisymmetric geometry of the pipe is simplified by reducing the problem to 2D while the non-axisymmetric loading is represented by the summation of Fourier series. Since the pipe stiffness matrix as conventionally formulated represents the greatest single computational load, the strain–displacement matrix is partitioned in such a way that numerical integration components are decoupled from θ (the angular parameter) and n (the number of Fourier terms). A single numerical integration of the strain–displacement matrix is performed and utilized for all the iterations of Fourier terms to represent the non-axisymmetric load. The numerical formulation is conducted using spectral elements, which also reduce computational time since these elements yield a diagonal mass matrix. The computational efficiency of the developed method is compared with conventional finite element tools.

Residue harmonic balance solution procedure to nonlinear delay differential systems

This paper develops the residue harmonic balance solution procedure to predict the bifurcated periodic solutions of some autonomous delay differential systems at and after Hopf bifurcation. In this solution procedure, the zeroth-order solution employs just one Fourier term. The unbalanced residues due to Fourier truncation are considered by solving linear equation iteratively to improve the accuracy. The number of Fourier terms is increased automatically. The well-known sunflower equation and van der Pol equation with unit delay are given as numerical examples. Their solutions are verified for a wide range of system parameters. Comparison with those available shows that the residue harmonic balance method is effective to solve the autonomous delay differential equations. Moreover, the present method works not only in determining the amplitude but also the frequency at bifurcation.

Interlaminar stresses in composite laminates: Thermoelastic deformation

An approximate elasticity solution for prediction of the displacement, stress and strain fields within the m-layer, symmetric and balanced angle-ply composite laminate of finite-width and subjected to uniform axial extension was developed earlier [4]. In the present paper, the authors have extended that solution to treat thermal stresses and deformations induced by a uniform change in laminate temperature. The results have revealed not only the complex fields within the laminate, but also inter-relationships between the lamina axial and shearing coefficients of thermal expansion and the effective laminate coefficients of thermal expansion. Further, the solution is shown to recover laminated plate theory predictions for thermally induced fields at interior regions of the laminate, thereby confirming the boundary layer nature of the interlaminar phenomena for the thermoelastic case. Finally, the results exhibit the anticipated response in congruence with the mechanical solution of Ref. [4] and the thermoelastic results satisfy the conditions of self-equilibration necessary for the finite-width laminate subjected to free thermal deformation. Integration of the stress σ x over the laminate cross-section in the y– z plane is shown to converge to zero as the number of Fourier terms is increased. While the exact solution for mechanical loading is known to exhibit singular behavior, non-convergence of the interlaminar shearing strain is also seen to occur at the intersection of the free edge and planes between lamina of + θ and − θ orientation under thermal loading. The analytical results show excellent agreement with the finite-element predictions for the same boundary-value problem.

Advantages and drawbacks of applying periodic time-variant modal analysis to spur gear dynamics

A simplified torsional model with a reduced number of degrees-of-freedom is used in order to investigate the potential of the technique. A time-dependent gear mesh stiffness function is introduced and expanded in a Fourier series. The necessary number of Fourier terms is determined in order to ensure sufficient accuracy of the results. The method of time-variant modal analysis is applied, and the changes in the fundamental and the parametric resonance frequencies as a function of the rotational speed of the gears, are found. By obtaining the stationary and parametric parts of the time-dependent modes shapes, the importance of the time-varying component relative to the stationary component is investigated and quantified. The method used for calculation and subsequent sorting of the left and right eigenvectors based on a first order Taylor expansion is explained. The advantages and drawbacks of applying the methodology to wind turbine gearboxes are addressed and elucidated.

A study on the application of Fourier series in IMRT treatment planning

In intensity-modulated radiotherapy, a set of x-ray fluence profiles is iteratively adjusted until a desired absorbed dose distribution is obtained. The purpose of this article is to present a method that allows the optimization of fluence profiles based on the Fourier series decomposition of an initial approximation to the profile. The method has the advantage that a new fluence profile can be obtained in a precise and controlled way with the tuning of only two parameters, namely the phase of the sine and cosine terms of one of the Fourier components, in contrast to the point-by-point tuning of the profile. Also, because the method uses analytical functions, the resultant profiles do not exhibit numerical artifacts. A test case consisting of a mathematical phantom with a target wrapped around a critical structure is discussed to illustrate the algorithm. It is shown that the degree of conformality of the absorbed dose distribution can be tailored by varying the number of Fourier terms made available to the optimization algorithm. For the test case discussed here, it is shown that the number of Fourier terms to be modified depends on the number of radiation beams incident on the target but it is in general in the order of 10 terms.

Quantum lithography with classical light: Generation of arbitrary patterns

Based on our previous scheme [Hemmer et al., Phys. Rev. Lett. 96, 163603 (2006)] we show the procedures to obtain any one- or two-dimensional pattern by multiple exposures. A key modification to the resonance condition is made to achieve subwavelength pattern resolution. The level separation of the substrate does not pose an upper limit to the frequency summation. So the fundamental frequency can be high and the number of Fourier terms can be large, making our scheme very suitable for subwavelength arbitrary patterns.

STATIC ANALYSIS OF DOUBLE-PLANE CABLE-STAYED BRIDGES.

A new method for the torsional analysis of cable-stayed bridges is presented. The method is based on the potential energy of the stayed deck, and the solution is obtained through an iterative procedure. The iterative procedure is reasonably convergent, and the use of only a small number of Fourier terms is sufficient. The method is simple and quick, and it is also suitable for use on both small electronic computers and pocket calculators. The computer program prepared by the author solves any double-plane cable-stayed bridge literally in seconds. Numerical examples on two-and three-span bridges are worked out using the new method. If the bridge is loaded symmetrically about its longitudinal centreline, the problem reduces to the case of a single-plane cable-stayed bridge.(a)