Gibbs Phenomenon Research Articles

Wrinkled and wrinkle-free membranes

The extreme out-of-plane flexibility makes membranes vulnerable of wrinkling, which limits the applications with stringent-accuracy demands and high-frequency requirements. In this study, a theoretical analysis is proposed to predict the minimum of min-principal stress min(σmin), representing the wrinkling capability of the stretched membrane with an arbitrary aspect ratio and random curved edges, while the curved edges are represented by a Fourier series function. The Marguerre function is adopted to calculate the stress solutions and the sigma-approximation approach is employed to eliminate the Gibbs phenomenon. The determination of numbers of truncated terms of Marguerre function and the normalization strategy are helpful to solve the ill-posed problem. The accuracy and generality of our analytical solutions are verified by the finite element results. Moreover, the wrinkling criterion indicates that the entire membrane is taut and wrinkle-free if min(σmin) is positive. With maximizing the membrane area as the objective function, the positive min(σmin) as constraint, and the Fourier coefficients tuning edge profiles as design variables, the structural design approach gives the optimal configurations, whose wrinkle-free performance is verified by the finite element analyses and physical experiments.

Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation

This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equivalent least-squares formulation in the set of neural network functions with the ReLU activation function. The method is capable of approximating the discontinuous interface of the underlying problem automatically through the free hyper-planes of the ReLU neural network and, hence, outperforms mesh-based numerical methods in terms of the number of degrees of freedom. Numerical results of some benchmark test problems show that the method can not only approximate the solution with the least number of parameters, but also avoid the common Gibbs phenomena along the discontinuous interface. Moreover, a three-layer ReLU neural network is necessary and sufficient in order to well approximate a discontinuous solution with an interface in R2 that is not a straight line.

Oversampling Errors in Multimodal Medical Imaging Are Due to the Gibbs Effect

To analyze multimodal three-dimensional medical images, interpolation is required for resampling which—unavoidably—introduces an interpolation error. In this work we describe the interpolation method used for imaging and neuroimaging and we characterize the Gibbs effect occurring when using such methods. In the experimental section we consider three segmented three-dimensional images resampled with three different neuroimaging software tools for comparing undersampling and oversampling strategies and to identify where the oversampling error lies. The experimental results indicate that undersampling to the lowest image size is advantageous in terms of mean value per segment errors and that the oversampling error is larger where the gradient is steeper, showing a Gibbs effect.

Fault detection for rotating machinery using translation-invariant higher-density wavelet packet sliding window block thresholding

The weak incipient fault feature information of mechanical equipment is often submerged in strong background noise, which increases the difficulty of feature extraction. Herein, a novel method called the translation-invariant higher-density wavelet packet sliding window block thresholding is proposed. A higher-density wavelet transform processes one scale function and two wavelet functions, which has an advantage over single wavelet transform in matching various signal feature information. The translation-invariant higher-density wavelet packet transform effectively eliminates the Gibbs phenomenon, and the fine division of frequency bands breaks through the limitations of the higher-density wavelet transform to obtain more comprehensive feature information. At the same time, to accommodate the periodicity of the impact feature signal and the correlation between the decomposition coefficients, a sliding window block thresholding method is proposed. The sliding window is set to independently retain the weak fault features and block thresholding is applied to select the optimal block length and threshold by minimizing the Stein unbiased risk estimator. The results of the simulation and experiment indicate that the proposed method can more effectively reduce the interference of background noise and impurity frequencies, and successfully extract the characteristic frequencies of incipient fault diagnosis.

An Improved Fourier-Series-Based IGBT Model by Mitigating the Effect of Gibbs Phenomenon at Turn on

The Fourier-series-based insulated-gate bipolar transistor (IGBT) model has been very effective in simulating its switching behaviors and meanwhile, reflecting carrier dynamics. However, in turn-on process, such a model has difficulty in distinguishing the boundary between undepleted area and depletion layer of the drift region due to Gibbs phenomenon and therefore, cannot accurately describe the carrier behavior of carrier storage region (CSR). In this study, the Fourier-series-based model is modified to determine the more precise depletion layer edge for turn on. This is achieved by utilizing the actual characteristics of the carrier concentration gradient at the edge of CSR in turn-on process. Such a modified model is implemented in MATLAB/Simulink and it can better reflect the depletion layer and simulate the distribution of both the carrier concentration and electric field during turn-on process. It corrects the unreasonable voltage tail caused by the error of drift region voltage calculation in previous models. The results of the proposed model under both the resistive load and inductive load are in good agreement with TCAD. Such a modified model can more accurately simulate both carrier dynamics and switching behaviors in the turn-on process.

Clarence J. Gibbs Effect and the “Creutzfeldt-Jakob Disease” Eponym

In 2014, American neurologist and Nobel laureate Stanley Prusiner reported that microbiologist Clarence Joseph Gibbs at the US NIH had intentionally, systematically, and mischievously used the eponym Creutzfeldt-Jakob disease, rather than Jakob-Creutzfeldt disease, because of the correspondence with Gibbs' own initials, to imply "Clarence Joseph disease." The present study examines temporal trends in the use of "Creutzfeldt-Jakob" and "Jakob-Creutzfeldt" in scientific articles and monographs from 1946 to 2019 to assess whether there was a "Clarence J. Gibbs effect" that influenced the general use of a specific eponym by the scientific community. During Gibbs' period of publication on Creutzfeldt-Jakob disease (CJD), there was an abrupt, dramatic, and steady increase in use of the CJD eponym while use of the Jakob-Creutzfeldt disease (JCD) eponym remained at a low level. In the period after Gibbs ceased to publish, there was a corresponding marked fall-off in use of the CJD eponym. Surviving collaborators thought Gibbs may have been joking, but in 1991 Gibbs had admitted what Prusiner reported. Regardless of motive, Gibbs strongly influenced the preferred eponym for this human prion disease by (1) publishing a seminal and highly referenced initial article in a high-profile journal; (2) sustained output of further important studies published in high-quality journals over more than 30 years; (3) professional affiliation with an esteemed national laboratory where he worked with a large number of high-profile colleagues; and (4) extensive collaborations with a large number of colleagues, who published multiple further articles using the eponym Gibbs preferred.

A high order flux reconstruction interface capturing method with a phase field preconditioning procedure

This paper presents a simple and highly accurate method for capturing sharp interfaces moving in divergence-free velocity fields using the high-order Flux Reconstruction approach on unstructured grids. A well-known limitation of high-order methods is their susceptibility to the Gibbs phenomenon; the appearance of spurious oscillations in the vicinity of discontinuities and steep gradients makes it difficult to accurately resolve shocks or sharp interfaces. In order to address this issue in the context of sharp interface capturing, a novel, preconditioned and localized phase-field method is developed in this work. The numerical accuracy of interface normal vectors is improved by utilizing a preconditioning procedure based on the level set method with localized artificial viscosity stabilization. The developed method was implemented in the framework of the multi-platform Flux Reconstruction open-source code PyFR [1]. Kinematic numerical tests in 2D and 3D conducted on different mesh types showed that the preconditioning procedure significantly improves accuracy with little added computational effort. The results demonstrate the conservativeness of the proposed method and its ability to capture highly distorted interfaces with superior accuracy when compared to conventional and high-order VOF and level set methods. Dynamic interface capturing validation tests were also carried out by coupling the proposed method to the entropically damped artificial compressibility variant of the incompressible Navier–Stokes equations. The high accuracy and locality of the proposed method offer a promising route to carrying out massively-parallel, high accuracy simulations of multi-phase, incompressible phenomena.

Semi-algebraic Approximation Using Christoffel–Darboux Kernel

We provide a new method to approximate a (possibly discontinuous) function using Christoffel–Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to nonlinear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with nonsmoothness implicitly so that a single scheme can be used to treat smooth or nonsmooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular, we observe empirically that our method does not suffer from the Gibbs phenomenon when approximating discontinuous functions.

Generalized A-GPSFS and modeling of vibrating elements with internal complexities

Mechanical components containing internal complexities in free vibration mode are analyzed through generalized bases made of global piecewise smooth functions. Such bases introduced in 2002, as GPSFs, were made adaptive in 2015 to reduce the computational efforts (A-GPSFs) while, a recent study in 2018 used supplementary analytical conditions along with A-GPSFs to study free vibrations of thin-walled beams and plates. In this paper, generalized bases of A-GPSFs are introduced (i) in a self-contained manner, id est, without needing any supplementary analytical conditions and even (ii) extending the possibility of modeling mechanical components containing internal jumps. It is shown how such generalized A-GPSFs can be used to model continuous functions which have different degrees of smoothness; even discontinuous functions (i.e. containing jumps) can be modeled and always without involving significant approximations or letting rise to any Gibbs phenomenon. Thus, a free vibration analysis of mechanical components including internal complexities of engineering interest becomes part of a simple unified procedure based on a global approximation where unknown eigenvalues and eigenfunctions can be singled out only through generalized bases of A-GPSFs; the results of previous formulations are then obtained as particular cases. The efficiency and ability of the proposed models result from the comparison between calculated eigen-parameters and those of other models presented in the open literature.

Arbitrary depth slice imaging based on cone beam x-rays multi-angle sampling

Arbitrary internal depth slice imaging is an effective way to realize rapid industrial scanning and reconstruction method in x-rays imaging. This paper proposes a novel arbitrary internal depth slice imaging (AIDSI) method for cone beam x-rays imaging. AIDSI builds the relationships between voxels of depth slices and pixels of sampling images under the cone beam x-ray source, which serves to refocus sampling data to specified depth slices. To remove artifacts of direct refocusing, a 3-point weighted practical ramp filtering method is used for sparse sampling data. The side lobes of the filter converge fast, artifacts caused by Gibbs effects are effectively reduced compared with common Shepp-Logan and Ram-Lak filters. Additionally, a weighting function is designed to reduce artifacts caused by structures of uninteresting layers. The feasibility and effectiveness of AIDSI is evaluated through simulation and real experiments. Results show that the proposed method has the ability to directly reconstruct arbitrary depth slice images based on random sparse sampling data, which is expected to be used for rapid detection of a large number of samples in industry.

A fifth-order nonlinear spectral difference scheme for hyperbolic conservation laws

We develop in this paper a fifth-order nonlinear spectral difference method for solving hyperbolic conservation laws, whose solutions often admit discontinuities. To avoid instability caused by the Gibbs phenomenon arising from interpolation across discontinuities, a fifth-order nonlinear interpolation scheme is proposed within a single cell, keeping the compactness of the original linear spectral difference method. Some numerical results are also presented to demonstrate the accuracy and effectiveness of the proposed method.

The harmonic balance method with arc-length continuation in blade-tip/casing contact problems

This article presents a Harmonic Balance Method-based numerical strategy to provide a qualitative numerical characterization of a compressor blade’s dynamics when structural contacts occur with the surrounding casing. The mitigation of the Gibbs phenomenon follows a two-pronged approach: (1) a regularization of the unilateral contact law and (2) the adjustment of the Fourier coefficients by means of a Lanczos filtering technique. In order to validate the proposed approach, it is first applied to an academic cantilever rod undergoing unilateral contact constraints. An in-depth comparative analysis of the obtained results—with an emphasis on displacements, contact forces and velocities—with respect to a time integration-based reference numerical strategy underlines the relevance and accuracy of the proposed methodology. The latter is then applied to the vibration analysis of an industrial compressor blade: NASA rotor 37. For a given contact configuration, obtained results are thoroughly compared to those obtained with a previously published time integration-based numerical strategy with a distinct contact treatment algorithm. Particular attention is paid to demonstrate the accuracy of the methodology for the prediction of displacements, contact forces, velocities as well as stress fields within the blade. Notably, it is evidenced that the proposed methodology, contrary to the reference time integration-based numerical strategy, is able to capture the exact location of the blade’s nonlinear resonance.

Transmission of a 112-Gbit/s 16-QAM over a 1440-km SSMF with parallel Kramers-Kronig receivers enabled by an overlap approach and bandwidth compensation.

We investigate the parallelized performance of the conventional Kramers-Kronig (KK) and without the digital up-sampling KK (WDU-KK) receivers in a 112-Gbit/s 16-ary quadrature amplitude modulation (16-QAM) system over a 1440-km standard single-mode fiber (SSMF). A joint overlap approach and bandwidth compensation filter (OLA-BC) architecture is presented to mitigate the edge effect caused by the Hilbert transform and the Gibbs phenomenon induced by the FIR filter, respectively. Moreover, the computational complexity of the OLA-BC based parallelized KK/WDU-KK receiver is also discussed. Parallelized KK/WDU-KK receivers based on the presented OLA-BC architecture can effectively mitigate the edge effect and the Gibbs phenomenon together with more than two orders of magnitude improvement in terms of bit-error-ratio (BER) compared with parallelized KK/WDU-KK receivers without OLA-BC receivers in back-to-back (B2B) case. Finally, we successfully transmit the 16-QAM signals over 960-km SSMF with a BER lower than 7% hard-decision forward error correction (HD-FEC) threshold (3.8 × 10-3) and 1440-km SSMF with a BER lower than 20% soft-decision FEC (SD-FEC) threshold (2 × 10-2).

Mathematical Modeling of Coal Dust Screening by Means of Sieve Analysis and Coal Dust Combustion Based on New Methods of Piece-Linear Function Approximation

This study focuses on the development of new methodological approaches to dust-preparation and burning of separated particles, including through the use of polyfractional ensembles. Coal dust screening by means of sieve analysis is described in standard methods. However, in order to further use the results obtained during mathematical modeling of particle motion in fuel-air mixture and exothermal reactions of oxidation while burning in a torch, it must be possible to differentiate and integrate continuous functions. The methodology is based on the continuity of particle motion in a mixture with air in the calculation of aerodynamic and heat-mass exchange processes in a torch. The paper employs new scientific approaches to transforming and normalizing a continuously differentiable function described by the Gauss curve. We propose to combine mathematical modeling of such functions with methods of approximation of piece-linear functions developed by Professor S. V. Aliukov. The implementation of such methods helps reduce calculation errors of particle size and deviations thereof from average equivalent diameter and to avoid the Gibbs effect while differentiating. The paper contains analytical calculations based on the proposed method and experimental data. Quantitative and qualitative results of comparing analytical and experimental data are also presented. We provide recommendations on the further use and extension of the range of the results obtained in a computer simulation of fuel production and burning processes in a torch.

A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.

On a class of splines free of Gibbs phenomenon

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.

Displacement Estimation of Self-Sensing Magnetic Bearings Based on Biorthogonal Spline Wavelet

By collecting the ripple current in the control coil, the self-sensing magnetic bearings can obtain rotor displacement information without a displacement sensor, which has the advantages of low cost and high integration. According to the problems of incomplete and inaccurate displacement information extraction caused by the traditional displacement estimation method using Fourier method to analyze the ripple current with non-stationary characteristics to estimate the rotor displacement with the Gibbs effect. Therefore, this paper proposes a rotor displacement estimation algorithm based on multiresolution filter bank biorthogonal spline wavelet for a magnetic bearing motor based on two-level switching power amplifier. This algorithm utilizes the biorthogonal spline wavelet with the characteristics of generalized linear phase and tight support, which can accurately demodulate the ripple current in the coil and extract the displacement information in the ripple current. Therefore, it is able to overcome the Gibbs effect in the traditional Fourier analysis estimation algorithm, to reduce the influence of ripple on rotor displacement estimation, and to improve the accuracy and stability of displacement estimation. In order to verify the correctness and effectiveness of the proposed algorithm, this paper establishes a simulation model of magnetic bearing in MATLAB Simulink and designs the proposed displacement estimator. Simulation results demonstrate that the proposed algorithm has higher accuracy and better stability than traditional displacement estimation algorithms. Experimental results show that the maximum deviation rate of the displacement estimation method proposed in this paper does not exceed 1% when the radial magnetic bearing gap is 0.5 mm, which can effectively estimate the rotor displacement.

SPECTRAL ANALYSIS AND FORECASTING OF THE 25TH SOLAR CYCLES

In the present work, we propose an extrapolation method, developed on the basis of spectral analysis, digital filtering, and the principle of demodulation of a complex signal, for predicting the beginning of cycle 25 of solar activity. The Wolf number and other measured characteristics of solar activity have a very complex spectral composition. The Sun, by the nature of its radiation, contributes a significant stochastic component to the observational data. The experimental data are known only up to the present, and the prediction is about bridging the gap in our data set. Mathematically, the prediction problem boils down to extrapolation of discontinuous functions, which leads to a Gibbs phenomenon that occurs at the point of discontinuity and makes prediction into the future impossible. To overcome this discontinuity, additional physical models describing a continuous process are most often used. This paper uses only the Wolf series of numbers from 1818 to 2020. The authors developed an original forecasting technique using Fourier series, digital filtering and representation of the complex process as modulated and subsequent demodulation. As a result of decomposing the complex signal by Fourier series into separate components, the spectral ranges characteristic of the Wolf number were singled out. Taylor's series was used for construction of prediction or extrapolation algorithms. The extraction of spectral ranges, characteristic for the investigated process, is carried out by means of sequential digital filtering methods and information compression in accordance with the cut-off frequency of the digital filter. For example, when selecting eleven-year cycles of solar activity, we have to compress the information by a factor of 160. With such a processing scheme, the forecasting starts with the ultralow-frequency component with a period of more than 11 years, successively moving to the ranges of higher frequencies. The use of spectral analysis and Chebyshev filtering showed the possibility to predict the low-frequency component for the full cycle period. The eleven-year component forecast obtained by the authors is in good agreement with the data of the Brussels Royal Center.

The Calculation Method of PV Direct Current Energy Based on Modulated Broadband Mode Decomposition and Compound Simpson Integral Algorithm

The application of photovoltaic power is becoming more and more extensive presently, when the existing time-frequency analysis methods are used to analyze photovoltaic mutation signals such as transient rise and transient fall, the cornered signal will produce Gibbs phenomenon and leads to errors. Thus, the Broadband Mode Decomposition (BMD) method is proposed. The main idea of BMD is to search in the associated dictionary that contains the wideband and narrowband signals. However, the BMD algorithm may treat the wideband signal as several narrowband components, when applied to a wideband signal interfered by strong noise, because the relative bandwidth is not sufficiently small. Therefore, a modulated broadband modal decomposition (MBMD) based on the modulation differential operator is proposed. By multiplying a high-frequency single-frequency signal, the relative bandwidth of the effective wideband signal is much smaller than 1, and the wideband signal is processed as an approximate wideband signal. Meanwhile, because the errors of the traditional dot product sum algorithm in DC energy measurement cannot be positive and negative offset, while the compound Simpson integral algorithm has small error and high calculation accuracy. Therefore, a photovoltaic DC power calculation method based on the MBMD and compound Simpson integration is proposed.

A Unified Meshfree Pseudospectral Method for Solving Both Classical and Fractional PDEs

In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. These important merits distinguish it from other numerical methods for fractional PDEs. Moreover, our method is simple and easy to handle complex geometry and local refinement, and its computer program implementation remains the same for any dimension $d \ge 1$. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving PDE problems. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the literature. Our studies suggest that to obtain good accuracy the shape parameter cannot be too small or too big, and the optimal shape parameter might depend on the RBF center points and the solution properties.