Gibbs phenomena for Lq-best approximation in finite element spaces

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretisation methods in non-standard function spaces, such as L q -type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of the numerical observations. In particular, we investigate the Gibbs phenomena for L q -best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

Photovoltaic Power Quality Analysis Based on the Modulation Broadband Mode Decomposition Algorithm

The Broadband Mode Decomposition (BMD) method was previously proposed to solve the Gibbs phenomenon that occurs during photovoltaic signal decomposition; its main idea is to build a dictionary which contains signal features, and to search in the dictionary to solve the problem. However, BMD has some shortcomings; especially if the relative bandwidth of the decomposed signal is not small enough, it may treat a square wave signal as several narrowband signals, resulting in a deviation in the decomposition effect. In order to solve the problem of relative bandwidth, the original signal is multiplied by a high-frequency, single-frequency signal, and the wideband signal is processed as an approximate wideband signal. This is the modulation broadband mode decomposition algorithm (MBMD) proposed in this article. In order to further identify and classify the disturbances in the photovoltaic direct current (DC) signal, the experiment uses composite multi-scale fuzzy entropy (CMFE) to calculate the components after MBMD decomposition, and then uses the calculated value in combination with the back propagation (BP) neural network algorithm. Simulation and experimental signals verify that the method can effectively extract the characteristics of the square wave component in the DC signal, and can successfully identify various disturbance signals in the photovoltaic DC signal.

The Generalised Plug-in Algorithm for the Diffeomorphism Kernel Estimate

The optimal value of the smoothing parameter of the Kernel estimator can be obtained by the well known Plug-in algorithm. The optimality is realised in the sense of Mean Integrated Square Error (MISE). In this paper, we propose to generalise this algorithm to the case of the difficult problem of the estimation of a distribution which has a bounded support. The proposed algorithm consists in searching the optimal smoothing parameter by iterations from the expression of MISE of the kernel-diffeomorphism estimator. By some simulations applied to some distribution having a support bounded and semi bounded, we show that the support of the pdf estimator respects the one of the theoretical distribution. We also prove that the proposed method minimizes the Gibbs phenomenon.

101 MRI Brain Solutions, 1st Edition: A Book Review

This book is written by a group of Neuroradiologists from an Indian university and published by Jaypee Brothers Medical Publishers (P) Ltd see (Figure 1). The first chapter discusses the physical principles of MRI and the 2nd chapter illustrates brain anatomy. From the 3rd to 11th chapter, all chapters discuss different malformations and disorders that affect the brain. Chapters 12th and 13th are glossary and acronyms respectively. The only colored pictures are in the plate and the rest of the book illustrations has no colors. Each chapter is written by a Neuroradiologists of the 14 authors of the book (3 editors and 11 contributors). The used method in 3rd to 11th chapter is a presentation of a case as an example of the brain malformation or disorder on an MRI scan. The book uses Radioland’s style in presenting the cases. Each case will have a case presentation and history, MRI scan pictures, MRI findings, comments and explanations, opinions, and clinical discussion. This book is (101), so it is for beginners and it does not cover all MRI neuroradiology topics since it has 101 cases only. The book has too many English language mistakes and a significant lack of proofreading. Some of the MRI images have poor image quality. The book is printed on high-quality papers. In some chapters of the book, there are some CT scans to show the difference between MRI and CT in the ability to show some of the abnormalities. The medical illustration is not high-quality illustration; for example; the Percheron artery was illustrated in the book poorly. Nobody should expect to see professional illustrations like Netter’s style in illustrations in this book. In the book, they rarely use arrows to indicate the location of the pathology. The book does not mention some of the famous neuroradiology signs. In the subdural hematoma case, they present a case of acute on chronic hematoma as subdural hematoma case which would be better to show acute in one case, chronic in one case, and acute on chronic on a third case because this book is for beginners. Some anatomical illustrations are out of the chapter which discuss the anatomy of the brain like the illustration of the cerebral venous system on page 137, figure 28. From page 139 to 144, the book presents three anatomical variations in the brain’s vasculatures (carotid cavernous fistula, persistent occipital sinus, and fetal posterior cerebellar artery) which would be better to be in a separate chapter about anatomical variations. Sometimes, the book will mention a famous neuroradiology sign without providing any MRI image as a demonstration. As well, the book will describe some findings which is invisible on MRI images and the authors will write “not seen in these images”. If the finding is not demonstrated for the readers, the authors should not mention it in the findings. Instead, they can include this undemonstrated finding in the comments and explanation section of the case. This book will mention too many things (radiological signs, findings, associated conditions, etc.) without showing figures for what they are writing about like in the astrocytoma case they mention astrocytoma’s types without differentiating them on figures. Plus, the cases are not mentioned in an organized fashion. For example, the book will speak about a tumor that affects a specific region (let’s call it region number 1) then they will write about another tumor in a different region (let’s call it region number 2) then they jump back to the first region and write about another tumor that affects region number 1. The book needs to be organized to write; for example; the tumors in cerebellopontine angle together which will help the reader to collect and compare all the information about the tumors that affect this region then move to another region. The book will provide a low image quality with too many artifacts; for example; the diffusion image in the hypoglycemia case. The artifacts chapter discusses only three common types of artifacts which are the Gibbs phenomenon, zipper artifact, and susceptibility artifact. The chapter can be improved by organizing the artifacts like; patient-related, signal processing related, and machinery (hardware) related artifacts. The artifacts’ chapter is placed in between clinical cases which should be removed to the beginning or at the end of the book. The last chapter collected different cases under miscellaneous.

Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

Development of a Harmonic Balance Method-Based Numerical Strategy for Blade-Tip/Casing Interactions: Application to NASA Rotor 37

In this study, a frequency-domain approach based on the harmonic balance method coupled to a predictor-corrector continuation algorithm is implemented for the qualitative analysis of blade-tip/casing contacts in aircraft engines. Unilateral contact and dry friction are taken into account through a regularized penalty law. To enhance the robustness of the methodology, particular attention is paid to the mitigation of the Gibbs phenomenon. To this end, the employed Alternating Frequency/Time scheme features a Lanczos σ-approximation so that spurious oscillations of the computed nonlinear contact forces become negligible. This approach is applied in combination with a model reduction technique on an industrial compressor blade: NASA rotor 37. In order to assess the influence of both the contact law regularization and the Lanczos σ-approximation, obtained results are thoroughly compared to an existing time integration-based numerical strategy relying on a Lagrange multiplier-based approach for contact treatment and that was previously confronted to experimental results. Presented results underline the very good agreement between the proposed methodology and the reference time integration numerical strategy. The proposed developments thus complement existing results on blade-tip/casing contact adding a much needed qualitative understanding of the interaction and an accurate assessment of the contact stiffening phenomenon.

Development of a Harmonic Balance Method-Based Numerical Strategy for Blade-Tip/Casing Interactions: Application to NASA Rotor 37

In this study, a frequency-domain approach based on the harmonic balance method coupled to a predictor-corrector continuation algorithm is implemented for the qualitative analysis of blade-tip/casing contacts in aircraft engines. Unilateral contact and dry friction are taken into account through a regularized penalty law. To enhance the robustness of the methodology, particular attention is paid to the mitigation of the Gibbs phenomenon. To this end, the employed Alternating Frequency/Time scheme features a Lanczos σ-approximation so that spurious oscillations of the computed nonlinear contact forces become negligible. This approach is applied in combination with a model reduction technique on an industrial compressor blade: NASA rotor 37. In order to assess the influence of both the contact law regularization and the Lanczos σ-approximation, obtained results are thoroughly compared to an existing time integration-based numerical strategy relying on a Lagrange multiplier-based approach for contact treatment and that was previously confronted to experimental results. Presented results underline the very good agreement between the proposed methodology and the reference time integration numerical strategy. The proposed developments thus complement existing results on blade-tip/casing contact adding a much needed qualitative understanding of the interaction and an accurate assessment of the contact stiffening phenomenon.

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.