Orthogonal Function Expansion Research Articles

Using Refined Theory to Studied Elastic Wave Scattering and Dynamic Stress Concentrations in Plates with Two Cutouts

In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied. Applying the orthogonal function expansion method, the problem to be solved can be reduced into the solution of a set of infinite algebraic equations. According to free boundary conditions, numerical results of dynamic moment concentration factors in thick plates with two circular cutouts analyze that: there will be more complex interaction changes between two-cutout situation than single cutout situation. In the case of low frequency or high frequency and thin plate, the hole-spacing in the absence of coupling interactions was larger or smaller. The numerical results and method can be used to analyze the dynamics and strength of plate-like structures.

Research on the Seismic Strain Field Before Earthquake Above <i>M</i>s7, CHINESE Mainland

In the paper, seismic strain field is considered as a variable of seismic activity. The strain field is calculated before the earthquake above <i>M</i>s7 in mainland China since 1980, and the abnormal change of time factor is extracted before the earthquake, by means of the natural orthogonal function expansion method. The results show that the time factor before the large earthquake will change suddenly, jump or jump down on stationary background. At least 3 typical fields will appear abnormal changes in the first 4 ones of seismic strain field before the earthquake, having the characteristics of multi-component display anomalies. The earliest time of abnormal occurrence is about 3 years before the earthquake. Moreover, there are some short-term time factor anomalies before some earthquakes. Through case studies the results of case studies show that the time factor anomaly extracted by strain field is more unique than seismic energy field in analyzing the anomaly of seismic activity. The difference between strain field and energy field is also discussed.

Matrix Formulation of the Cauer Ladder Network Method for Efficient Eddy-Current Analysis

Advanced power control often requires detailed eddy-current analyses of electric machines handling thin skin depth due to high frequency switching. The Cauer circuit [1], [2] is a very efficient and exact representation of the eddy-current field in magnetic sheets and cylinders for wide frequency range, where magnetic and electric fields are optimally expanded by orthogonal polynomials. Recently, the Cauer circuit representation is extended to describe general eddy-current fields powered by the finite element method (FEM). This method is called Cauer ladder network (CLN) method [3], [4] and still retains clear physical meaning based on the orthogonal function expansion. The generality of CLN method has a similarity to model order reduction (MOR) methods [5], [6]. The relation between the CLN method and other MOR methods has not been clear yet because of the physics-based derivation of CLN method. This article derives a matrix formulation of CLN method for clear understanding of mathematical aspects of the CLN method.

Kinetic damping in the spectra of the spherical impedance probe

The impedance probe is a measurement device to measure plasma parameters, such as electron density. It consists of one electrode connected to a network analyzer via a coaxial cable and is immersed into a plasma. A bias potential superposed with an alternating potential is applied to the electrode and the response of the plasma is measured. Its dynamical interaction with the plasma in an electrostatic, kinetic description can be modeled in an abstract notation based on functional analytic methods. These methods provide the opportunity to derive a general solution, which is given as the response function of the probe–plasma system. It is defined by the matrix elements of the resolvent of an appropriate dynamical operator. Based on the general solution, a residual damping for vanishing pressure can be predicted and can only be explained by kinetic effects. In this paper, an explicit response function of the spherical impedance probe is derived. Therefore, the resolvent is determined by its algebraic representation based on an expansion in orthogonal basis functions. This allows one to compute an approximated response function and its corresponding spectra. These spectra show additional damping due to kinetic effects and are in good agreement with former kinetically determined spectra.

Analysis of Wind Field Characteristics of Satallite Remote Sensing Image Based on EOF Change

By using empirical orthogonal function (EOF) and statistical analysis methods expansion, the paper analyzed the daily and monthly data of the ASCAT satellite in the Taiwan Strait in 2016 to explore the seasonal variation characteristics and spatial pattern of the sea wind field in the Taiwan Strait. The results showed the sea surface wind field of 2016 in the Taiwan Strait had obvious seasonal change with a trend of decreasing after increasing firstly throughout the year of 2016, the wind speed decreased from January to June, and increased from June to December through the whole year. The EOF expansion of wind speed showed that the maximum center of outliers deviation existed in the narrow straits of Taiwan Strait and distributed along the coastline from the first modal space pattern, revealed the influences of Taiwan island topography were obvious. Furthermore, the winter-summer monsoon oscillation was reflected from the time curve of the year of 2016, and the typhoon influence signal could be well corresponded in details, shown as strong amplitude in the time curve. The spatial pattern of second modal presented in zonal distribution, characterized by reverse structure from south to north. The EOF expansion of wind direction showed the spatial pattern distributions of the first and second modals were zonal, and the maximum center of outliers deviation existed in the north and south sides of Taiwan island affected by the island terrain, in particular, the typhoon influences on the direction of the wind were obvious from the time curves.

Efficient circuit representation of eddy-current fields

PurposeThis paper aims to discuss the relationship between the continued fraction form of the analytical solution in the frequency domain, the orthogonal function expansion and their circuit realization to derive an efficient representation of the eddy-current field in the conducting sheet and wire/cylinder. Effective frequency ranges of representations are analytically derived.Design/methodology/approachThe Cauer circuit representation is derived from the continued fraction form of analytical solution and from the orthogonal polynomial expansion. Simple circuit calculations give the upper frequency bounds where the truncated circuit and orthogonal expansion are applicable.FindingsThe Cauer circuit representation and the orthogonal polynomial expansions for the magnetic sheet in the E-mode and for the wire in the axial H-mode are derived. The upper frequency bound for the Cauer circuit is roughly proportional to N4 with N inductive elements, whereas the frequency bound for the finite element eddy-current analysis with uniform N elements is roughly proportional to N2.Practical implicationsThe Cauer circuit representation is expected to provide an efficient homogenization method because it requires only several elements to describe the eddy-current field over a wide frequency range.Originality/valueThe applicable frequency ranges are analytically derived depending on the conductor geometry and on the truncation types.

Rubber Composition–Properties Relationships during Tire Numerical Simulation and Design Optimization

ABSTRACT Numerical simulation based on finite element analysis is now widely used during the design optimization of tires, thereby drastically reducing the time investment in the design process and improving tire performance because it is obtained from the optimized solution. Rubber material models that are used in numerical calculations of stress–strain distributions are nonlinear and may include several parameters. The relations of these parameters with rubber formulations are usually unknown, so the designer has no information on whether the optimal set of parameters is reachable by the rubber technological possibilities. The aim of this work was to develop such relations. The most common approach to derive the equation of the state of rubber is based on the expansion of the strain energy in a series of invariants of the strain tensor. Here, we show that this approach has several drawbacks, one of which is problems that arise when trying to build on its basis the quantitative relations between the rubber composition and its properties. An alternative is to use a series expansion in orthogonal functions, thereby ensuring the linear independence of the coefficients of elasticity in evaluation of the experimental data and the possibility of constructing continuous maps of “the composition to the property.” In the case of orthogonal Legendre polynomials, the technique for constructing such maps is considered, and a set of empirical functions is proposed to adequately describe the dependence of the parameters of nonlinear elastic properties of general-purpose rubbers on the content of the main ingredients. The calculated sets of parameters were used in numerical tire simulations including static loading, footprint analysis, braking/acceleration, and cornering and also in design optimization procedures.

Influence of kinetic effects on the spectrum of a parallel electrode probe

Active plasma resonance spectroscopy (APRS) denotes a class of diagnostic techniques which utilize the natural ability of plasmas to resonate on or near the electron plasma frequency . One particular class of APRS can be described in an abstract notation based on functional analytic methods in electrostatic approximation. These methods allow for a general solution of the kinetic model in arbitrary geometry. This solution is given as the response function of the probe-plasma system and is defined by the resolvent of an appropriate dynamical operator. The general response predicts an additional damping due to kinetic effects. This manuscript provides the derivation of an explicit response function of the kinetic APRS model in a simple geometry. Therefore, the resolvent is determined by its matrix representation based on an expansion in orthogonal basis functions. This allows to compute an approximated response function. The resulting spectra show clearly a stronger damping due to kinetic effects.

Dynamic Stress Concentrations in Infinite Plates with a Circular Cutout Based on Refined Theory

In this paper, based on the refined dynamic equations of plate bending vibration, the wave function expansion method is employed to investigate the elastic wave scattering and dynamic stress concentrations in infinite plates with a circular cutout. By applying the orthogonal function expansion method, the problem is reduced into solving a set of infinite algebraic equations. Then, by using the series truncation method, the above infinite algebraic equations reduce to the finite algebraic equations. As examples, the numerical results of dynamic moment concentration factor (DMCF) in infinite plates with one circular cutout under free boundary conditions subjected to an incident flexural wave are presented, and then the influences of the parameters including incident wavenumber, thickness of plates and frequency on the dynamic moment distributions are also analysed. The analyses indicate that the wave type of scattering waves around the circular cutout converts into the propagating wave from the attenuating wave when the cutoff frequency calculated by this study is 0.861 and the DMCFs vary remarkably nearby the cutoff frequency. This phenomenon is more prominent in thick plates.

Dynamic Stress Intensity Factors for an Interfacial Crack near a Non-Circular Cavity in Piezoelectric Bi-Materials

Abstract A theoretical analysis is followed to calculate the dynamic stress intensity factors (DSIFs) due to existence of an interfacial crack near the right edge of a non-circular cavity, in transversely isotropic piezoelectric bi-materials. The model is subjected to dynamic incident anti-plane shearing (SH-wave). Green’s functions are constructed Based on complex variable and conformal mapping methods. DSIFs at the crack inner and outer tips are obtained by conjunction and cracks-deviation techniques. The boundary value problems are solved by applying the orthogonal function expansion technique. Based on FORTRAN language program, some numerical calculations with an elliptic cavity are provided for different elliptic axial length ratios, different wave numbers and different piezoelectric parameters. For calibration, a comparison is accomplished between the present model and similar model with a crack emerging from a circular cavity edge. Calculating results showed the influences on DSIFs and how affected the efficiency of piezoelectric devices and materials.

A scalable theoretical mean-field model for the electron component of an ultracold neutral plasma

The electron component of an ultracold neutral plasma (UCP) is modelled based on a scalable method using a self-consistently determined mean-field approximation. Representative sampling of discrete electrons within the UCP is used to project the electron spatial distribution onto an expansion of orthogonal basis functions. A collision operator acting on the sample electrons is employed in order to drive the distribution toward thermal equilibrium. These equilibrium distributions can be determined for non-zero electron temperatures even in the presence of spherical symmetry-breaking applied electric fields. This is useful for predicting key macroscopic UCP parameters, such as the depth of the electrons’ confining potential given other plasma parameters. Dynamics such as electron oscillations in UCPs with non-uniform density distributions can also be treated by this model.

Methods and applications for modeling of continuous-wave ultrasound fields

Techniques are presented for efficient and accurate computation of three-dimensional, continuous-wave radiated fields from ultrasound transducers of canonical geometries. Fields of flat circular pistons are computed using exact series expressions based on orthogonal function expansions of the Rayleigh integral. Fields of concave circular pistons are computed using analogous series expansions for an integral of effective source contributions over the radiating surface; this approach accurately approximates radiated fields of many focused ultrasound transducers of interest. Fields of flat and focused rectangular transducers are computed using closed-form expressions based on a modified Fresnel approximation; accurate fields are obtained at distances large compared to the radiating aperture dimensions, so that the near field for large transducers can be accurately simulated by subdividing the radiating aperture. Efficient methods for field computation facilitate design optimization for therapy and imaging de...

Roth’s orthogonal function method in discrepancy theory and some new connections

In this survey we give a comprehensive, but gentle introduction to the circle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth (Mathematika 1:73–79, 1954) and based on multiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most important questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, probability (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate.

Global Wavenumber Spectrum with Corrections for Altimeter High-Frequency Noise

AbstractThe altimetry wavenumber spectra of sea surface height (SSH) provide a unique dataset for testing of geostrophic turbulence. While SSH spectral slopes of k−11/3 and k−5 are expected from theories and numerical simulations, the altimetry spectra from the original unfiltered and instrument noise–corrected data often are too shallow, falling between k−2 and k−3. In this study, the possibility that the flattened spectral slopes are partly due to contamination by unresolved high-frequency (&lt;10 days) motions is tested. A spatiotemporal filter based on empirical orthogonal function expansion (EOF) is used to remove the temporally incoherent signals. The resulting spectral slopes are much steeper than in the previous studies. Over 70% of the revised global spectral estimates, excluding the tropics, are above k−3. Moreover, in high energy regions like the Gulf Stream and Kuroshio, the spectral slopes are about k−5, which is consistent with the classical quasigeostrophic (QG) turbulence. The spectral slopes are validated with the eddy kinetic energy (EKE) spectra from shipboard acoustic Doppler current profiler (ADCP) measurements in the high and low energy regions.

Dynamic anti-plane analysis for symmetrically radial cracks near a non-circular cavity in piezoelectric bi-materials

Dynamic incident anti-plane shearing (SH-wave) analysis is considered to calculate the dynamic stress intensity factors (DSIFs) in transversely isotropic piezoelectric bi-materials, which contain symmetrically radial cracks, near the edges of a non-circular cavity. Green’s functions are established based on complex variable and conformal mapping methods. Conjunction and crack-deviation techniques are used to evaluate DSIFs at the cracks’ tips. Boundary conditions are solved by applying the orthogonal function expansion technique. Models with elliptic cavities are studied numerically based on FORTRAN language program. A comparison is accomplished to adjust program validity. Results clarified the influences on DSIFs under proper conditions.

Dynamic stress concentrations in thick plates with two holes based on refined theory

Based on complex variables and conformal mapping, the elastic wave scattering and dynamic stress concentrations in the plates with two holes are studied by the refined dynamic equation of plate bending. The problem to be solved is changed to a set of infinite algebraic equations by an orthogonal function expansion method. As examples, under free boundary conditions, the numerical results of the dynamic moment concentration factors in the plates with two circular holes are computed. The results indicate that the parameters such as the incident wave number, the thickness of plates, and the spacing between holes have great effects on the dynamic stress distributions. The results are accurate because the refined equation is derived without any engineering hypothese.

Multivariate Regression Reconstruction and Its Sampling Error for the Quasi-Global Annual Precipitation from 1900 to 2011

Abstract This paper provides a multivariate regression method to estimate the sampling errors of the annual quasi-global (75°S–75°N) precipitation reconstructed by an empirical orthogonal function (EOF) expansion. The Global Precipitation Climatology Project (GPCP) precipitation data from 1979 to 2008 are used to calculate the EOFs. The Global Historical Climatology Network (GHCN) gridded data (1900–2011) are used to calculate the regression coefficients for reconstructions. The sampling errors of the reconstruction are analyzed in detail for different EOF modes. The reconstructed time series of the global-average annual precipitation shows a 0.024 mm day−1 (100 yr)−1 trend, which is very close to the trend derived from the mean of 25 models of phase 5 of the Coupled Model Intercomparison Project. Reconstruction examples of 1983 El Niño precipitation and 1917 La Niña precipitation demonstrate that the El Niño and La Niña precipitation patterns are well reflected in the first two EOFs. Although the validation in the GPCP period shows remarkable skill at predicting oceanic precipitation from land stations, the error pattern analysis through comparison between reconstruction and GHCN suggests the critical importance of improving oceanic measurement of precipitation.

Cosmic string detection in radio surveys

We consider the detection of cosmic strings using observations of the anisotropy of the Cosmic Microwave Background. Several methods for detecting cosmic strings are analyzed, using a threshold filter and expansion in orthogonal Haar functions. Computer simulation provides estimates of the noise present in experiments aimed at detection of cosmic strings. Attempts to detect cosmic strings were carried out using the full-sky ILC map obtained as a result of the WMAP space mission. A list of cosmic string candidates has been compiled using the Haar function method.

An update global model of hmF2 from values estimated from ionosonde and COSMIC/FORMOSAT-3 radio occultation

In this paper, we present our recent work on developing an updated global model of the ionospheric F2 peak height hmF2 parameter by combining data from the Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC/FORMOSAT-3) radio occultation (RO) measurements and from the extended global ionosonde stations. In particular, 10 Chinese ionosonde stations’ data are newly introduced into this study. The modeling technique used is based on a two-layer empirical orthogonal function (EOF) expansion. Global distributions of hmF2 maps calculated using the newly constructed global model and the one provided by the International Reference Ionosphere model (IRI-ITU-R) are compared with the global distributions of hmF2 obtained by the COSMIC RO measurements and quantitative statistical analysis of the differences between the model results and those of the COSMIC RO measurements is made for the low (2008) and high (2012) solar activity years. The obtained average root-mean-square differences (RMSEs) for our model are 27.7km (11.1%) and 31.0km (9.8%), respectively for the years 2008 and 2012, whereas those for the IRI-ITU-R model are 39.9km (16.9%) and 35.0km (11.6%), respectively. Comparison of the results calculated both by our model and the IRI-ITU-R model with the digisonde observation is also made. The comparisons show that the newly constructed global hmF2 model can reproduce reasonably well the observations and perform better than IRI-ITU-R model.

A spectral approach for fuzzy uncertainty propagation in finite element analysis

Uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge. This paper proposes a Polynomial Chaos type spectral approach based on orthogonal function expansion. A fuzzy variable is represented as a set of interval variables via the membership function. The interval variables are further transformed into the standard interval [−1,1]. Smooth nonlinear functions of standard interval variables are projected in the basis of Legendre polynomials by exploiting its orthogonal properties over the interval [−1,1]. The coefficients associated with the basis functions are obtained by a Galerkin type of error minimisation. The method is first illustrated using scalar functions of multiple fuzzy variables. Later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables. The response of such systems can be expressed in the complete basis of multivariate Legendre polynomials. The coefficients, obtained by Galerkin type of error minimisation, can be calculated from the solution of an extended set of linear algebraic equations. An eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way. A numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods. Linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results.