Legendre Orthogonal Polynomials Research Articles

Chained-Function Filter Synthesis Based on the Legendre Polynomials

A new kind of filter called Legendre chained-function (LCF) filter with characteristic function given by the product of low-degree Legendre orthogonal polynomials (seed functions) is studied in this paper. LCF filter magnitude response exhibits unequal ripple level in the passband compared to Chebyshev chained-function filter with identical edge ripple factor at the passband edge. A proper combination of seed functions, i.e., a product of them, is used to control the maximum ripple in the passband, which affects the return loss, selectivity and a group delay characteristics of a filter. By selecting one of the possible combinations of seed functions (thus obtaining various degrees of freedom in filter design), filters with improved performances compared to traditional approximation techniques can be obtained. The degrees of freedom increase if the degree of filter increases. Compared to existing Chebyshev chained-function (CCF) filters, whose performances are also presented, and Butterworth function filters as a special case of both LCF and CCF filters, the new family of LCF filters has many advantages. A table summarizing properties of CCF and LCF filters is given for design purpose.

Research on the Statistical Characteristics of Crosstalk in Naval Ships Wiring Harness Based on Polynomial Chaos Expansion Method

Abstract Crosstalk in wiring harness has been studied extensively for its importance in the naval ships electromagnetic compatibility field. An effective and high-efficiency method is proposed in this paper for analyzing Statistical Characteristics of crosstalk in wiring harness with random variation of position based on Polynomial Chaos Expansion (PCE). A typical 14-cable wiring harness was simulated as the object of research. Distance among interfering cable, affected cable and GND is synthesized and analyzed in both frequency domain and time domain. The model of naval ships wiring harness distribution parameter was established by utilizing Legendre orthogonal polynomials as basis functions along with prediction model of statistical characters. Detailed mean value, mean square error, probability density function and reasonable varying range of crosstalk in naval ships wiring harness are described in both time domain and frequency domain. Numerical experiment proves that the method proposed in this paper, not only has good consistency with the MC method can be applied in the naval ships EMC research field to provide theoretical support for guaranteeing safety, but also has better time-efficiency than the MC method. Therefore, the Polynomial Chaos Expansion method.

A New Analytical Model to Study the Ionospheric Effects on VHF/UHF Wideband SAR Imaging

With a view to detecting foliage-obscured/ground-obscured targets on a global scale, low frequency (i.e., very high frequency (VHF)/ultrahigh frequency (UHF) band) and wide bandwidth is a trend in future spaceborne synthetic aperture radar (SAR) system design. However, due to the dispersion of ionosphere, VHF/UHF wide bandwidth SAR signals inevitably experience adverse effects. In contrast to narrow bandwidth SAR at VHF/UHF, quadratic and cubic ionospheric phase errors will introduce noticeable effects on future wide-bandwidth SAR systems. Traditional evaluation models based on Taylor series expansion may become inaccurate when an extremely wide bandwidth is considered. With a focus on this this issue, first, the shortcoming of Taylor series expansion of ionospheric phase errors is briefly discussed in this paper. Then, a new analytical model based on Legendre orthogonal polynomials is developed, which is expected to be widely applicable, especially for low-frequency and wide-bandwidth SAR systems. Finally, compared with previous models based on Taylor series expansion, numerical simulations and evaluations show the superiority of the new model.

Thermo-mechanical post-buckling analysis of variable angle tow composite plate assemblies

The increasing use of composite materials for lightweight structural applications and the extended tailoring capabilities offered by variable stiffness laminates requires rapid and robust analysis tools that adequately describe the mechanical behaviour of such structures. In this work, a Rayleigh–Ritz solution for generally restrained multilayered stiffened variable angle tow plates in the post-buckling regime is presented. The plate model is based on first-order shear deformation theory and accounts for geometrical nonlinearity through von Kármán’s assumptions. General symmetric and unsymmetric stacking sequences are considered and Legendre orthogonal polynomials are employed to approximate the unknown displacement field. Stiffened variable angle tow plates are modeled as an assembly of plate-like elements and penalty techniques are used to enforce the displacements continuity of the assembled multidomain structure and also to apply the kinematical boundary conditions. The developed postbuckling analysis is sufficiently versatile to model a wide range of configurations and load cases for multi-component, variable angle tow, composite structures, and provide the same accuracy level as finite element analysis. The proposed solution is validated by comparison with literature and finite elements analysis and original results are presented for the thermo-mechanical post-buckling solution of multilayered stiffened variable angle tow plates. The effectiveness of the developed analysis tool for both stiffened plates and a tapered stiffened wing box is shown, with a reduced number of unknowns and simplified data preparation compared to finite element analysis.

Modelling lactation curves of dairy goats by fitting random regression models using Legendre polynomials or B-splines

A total of 17 356 test-day milk yield (TDMY) records from 642 first lactations of Alpine goats were used to model variations in lactation curve using random regression models (RRM). Orthogonal Legendre polynomials and B-splines were evaluated to obtain adequate and parsimonious models for the estimation of genetic parameters. The analysis was performed using a single-trait RRM, including the additive genetic, permanent environmental, and residual effects. We estimated the mean trend of milk yield, and the additive genetic and permanent environmental covariance functions through random regression using different orders of orthogonal Legendre polynomial (three to six) and B-spline functions (linear, quadratic, and cubic, with three to six knots). This study further evaluated different number of classes of residual variances. The covariance components and the genetic parameters were estimated using the restricted maximum likelihood method. Heritability estimates presented similar trends for both functions. T...

Development of an Aeroelastic Formulation for Deformable Airfoils Using Orthogonal Polynomials

In this paper, an aeroelastic formulation is developed to analyze aeroelastic behavior of flexible airfoils with arbitrary camber deformations. The camberwise bending deformation of flexible airfoils, described by using the orthogonal Legendre polynomials, is considered in addition to traditional rigid-body plunging and pitching motions. The complete set of aeroelastic equations of motion is derived by following Hamilton’s principle, where a two-dimensional finite-state unsteady aerodynamic theory is applied to calculate the aerodynamic loads of flexible airfoils with both rigid-body motions and arbitrary camber deformations. The finite-state aerodynamic theory is also modified to involve magnitudes of the Legendre polynomials in the aerodynamic load equations. The aeroelastic equations, featuring rigid-body motions and camber deformation magnitudes as independent degrees of freedom, may facilitate the analysis of camber effects on aeroelastic characteristics of flexible airfoils. Numerical studies of this paper validate the developed aerodynamic and aeroelastic formulations by comparison with other published research and computational results. Finally, the impacts of camber flexibility on static and dynamic aeroelastic characteristics of flexible airfoils are explored.

Analysis of symmetrical flatness actuator efficiencies for UCM cold rolling mill by 3D elastic–plastic FEM

As the basis for a flatness control system, flatness actuator efficiency describes an actuator’s control ability, but it is difficult to obtain an actuator efficiency factor accurately through rolling tests because of the complicated subsidiary facilities of the mill. This paper proposes a novel simulation approach that is applied to obtain the actuator efficiency factors in terms of work roll bending, intermediate roll bending, and intermediate roll shifting for a six-high Universal Crown Control mill (UCM mill). A three-dimensional (3D) finite element model of the mill was developed to simulate the dynamic strip rolling process. The validation of rolling experiments shows that this model has enough precision, where the relative error of strip thickness between simulated values and actual values is less than 1.0%. The effects of the actuators on strip thickness profile, crown, edge drop, and elongation difference of longitudinal fibers were investigated. In the case of different actuator parameters, the curves of actuator efficiency factors were obtained and quantitatively descripted by truncated Legendre orthogonal polynomials. The mechanism of flatness control was studied based on an analysis of the actuator’s influence rule on the elastic deflection of rolls and 3D distribution of rolling pressure. The results indicate that the curves of actuator efficiency factors have a symmetrical upside-down v-shaped distribution and contain the quadratic and quartic flatness components. The actuator efficiency factors of intermediate roll shifting have a nonignorable variation with the change of actuator parameters. This study is the first attempt to obtain actuator efficiency factors for UCM mill using an elastic–plastic finite element method.

Orthogonal fast spherical Bessel transform on uniform grid

We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on a uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast Fourier transform and the orthogonal transform founded on the derivatives of the discrete Legendre orthogonal polynomials. The method utility is illustrated by its implementation for the problem of a two-atomic molecule in a time-dependent external field simulating the one utilized in the attosecond streaking technique.

Effect of dam weight and pregnancy nutrition on average lactation performance of ewe offspring over 5 years

The foetal mammary gland is sensitive to maternal weight and nutrition during gestation, which could affect offspring milk production. It has previously been shown that ewes born to dams offered maintenance nutrition during pregnancy (day 21 to 140 of gestation) produced greater milk, lactose and CP yields in their first lactation when compared with ewes born to dams offered ad libitum nutrition. In addition, ewes born to heavier dams produced greater milk and lactose yields when compared with ewes born to lighter dams. The objective of this study was to analyse and compare the 5-year lactation performance of the previously mentioned ewes, born to heavy or light dams that were offered maintenance or ad libitum pregnancy nutrition. Ewes were milked once per week, for the first 6 weeks of their lactation, for 5 years. Using milk yield and composition data, accumulated yields were calculated over a 42-day period for each year for milk, milk fat, CP, true protein, casein and lactose using a Legendre orthogonal polynomial model. Over the 5-year period, ewes born to heavy dams produced greater average milk (P=0.04), lactose (P=0.01) and CP (P=0.04) yields than offspring born to light dams. In contrast, over the 5-year period dam nutrition during pregnancy did not affect average (P>0.05) offspring milk yields or composition, but did increase milk and lactose accumulated yield (P=0.03 and 0.01, respectively) in the first lactation. These results indicate that maternal gestational nutrition appears to only affect the first lactational performance of ewe offspring. Neither dam nutrition nor size affected grand-offspring live weight gain to, or live weight at weaning (P>0.05). Combined these data indicate that under the conditions of the present study, manipulating dam weight or nutrition in pregnancy can have some effects of offspring lactational performance, however, these effects are not large enough to alter grand-offspring growth to weaning. Therefore, such manipulations are not a viable management tool for farmers to influence lamb growth to weaning.

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.

Spectral element method for band-structure calculations of 3D phononic crystals

The spectral element method (SEM) is a special kind of high-order finite element method (FEM) which combines the flexibility of a finite element method with the accuracy of a spectral method. In contrast to the traditional FEM, the SEM exhibits advantages in the high-order accuracy as the error decreases exponentially with the increase of interpolation degree by employing the Gauss–Lobatto–Legendre (GLL) polynomials as basis functions. In this study, the spectral element method is developed for the first time for the determination of band structures of 3D isotropic/anisotropic phononic crystals (PCs). Based on the Bloch theorem, we present a novel, intuitive discretization formulation for Navier equation in the SEM scheme for periodic media. By virtue of using the orthogonal Legendre polynomials, the generalized eigenvalue problem is converted to a regular one in our SEM implementation to improve the efficiency. Besides, according to the specific geometry structure, 8-node and 27-node hexahedral elements as well as an analytic mesh have been used to accurately capture curved PC models in our SEM scheme. To verify its accuracy and efficiency, this study analyses the phononic-crystal plates with square and triangular lattice arrangements, and the 3D cubic phononic crystals consisting of simple cubic (SC), bulk central cubic (BCC) and faced central cubic (FCC) lattices with isotropic or anisotropic scatters. All the numerical results considered demonstrate that SEM is superior to the conventional FEM and can be an efficient alternative method for accurate determination of band structures of 3D phononic crystals.

Kernel polynomial representation for imaginary-time Green’s functions in continuous-time quantum Monte Carlo impurity solver**Project supported by the National Natural Science Foundation of China (Grant No. 11504340).

Inspired by the recently proposed Legendre orthogonal polynomial representation for imaginary-time Green’s functions G(τ), we develop an alternate and superior representation for G(τ) and implement it in the hybridization expansion continuous-time quantum Monte Carlo impurity solver. This representation is based on the kernel polynomial method, which introduces some integral kernel functions to filter the numerical fluctuations caused by the explicit truncations of polynomial expansion series and can improve the computational precision significantly. As an illustration of the new representation, we re-examine the imaginary-time Green’s functions of the single-band Hubbard model in the framework of dynamical mean-field theory. The calculated results suggest that with carefully chosen integral kernel functions, whether the system is metallic or insulating, the Gibbs oscillations found in the previous Legendre orthogonal polynomial representation have been vastly suppressed and remarkable corrections to the measured Green’s functions have been obtained.

A Flexible Functional Form Approach to Mortality Modeling

There has been a rapid growth in life expectancy during the past twenty years, resulting in increased pressure on personal and public finances. The increasing amount of attention paid on longevity risk and funding for old has created the needs for precise mortality models and accurate future mortality forecasts. Orthogonal polynomials have been widely used in technical fields but there are very few applications in mortality modeling. In this paper we adopt a flexible functional form approach using two-dimensional Legendre orthogonal polynomials to fit and forecast mortality rates. Unlike the existing mortality models in the literature, the model we propose does not impose any restrictions on the age, time or cohort structure of the data and thus allows for different model designs for different countries' mortality experience. We conduct an empirical study using male mortality data from a range of developed countries and compare our model with well known mortality models in the literature. The fitting results show that the proposed model provides comparable fitting but with a much smaller number of parameters. We also show that the proposed model works well and produces clean residual plots without the incorporation cohort effect. Moreover, based on the 5-year-ahead forecasting results, it can be concluded that our model improves the overall accuracy of the future mortality forecast.

A Flexible Functional Form Approach To Mortality Modeling: Do We Need Additional Cohort Dummies?

The increasing amount of attention paid to longevity risk and funding for old age has created the need for precise mortality models and accurate future mortality forecasts. Orthogonal polynomials have been widely used in technical fields and there have also been applications in mortality modeling. In this paper we adopt a flexible functional form approach using two‐dimensional Legendre orthogonal polynomials to fit and forecast mortality rates. Unlike some of the existing mortality models in the literature, the model we propose does not impose any restrictions on the age, time or cohort structure of the data and thus allows for different model designs for different countries' mortality experience. We conduct an empirical study using male mortality data from a range of developed countries and explore the possibility of using age–time effects to capture cohort effects in the underlying mortality data. It is found that, for some countries, cohort dummies still need to be incorporated into the model. Moreover, when comparing the proposed model with well‐known mortality models in the literature, we find that our model provides comparable fitting but with a much smaller number of parameters. Based on 5‐year‐ahead mortality forecasts, it can be concluded that the proposed model improves the overall accuracy of the future mortality projection. Copyright © 2016 John Wiley & Sons, Ltd.

Investigation of the coupled Lamb waves propagation in viscoelastic and anisotropic multilayer composites by Legendre polynomial method

In this paper, the attenuation of guided waves in damped multilayered arbitrary anisotropic composite laminates is investigated using a Legendre orthogonal polynomial approach. The laminate is made of the graphite–epoxy composite material with the layering sequence [0/ϕ/0]. The validity of the polynomial approach is illustrated through a comparison with exact solution obtained from the Stiffness matrix method and a comparison with earlier known Lamb wave solution of pure elastic multilayered plate. The dispersion and attenuation of coupled Lamb waves with different layering sequences are investigated. The viscous effect on dispersion curves is shown. The combination of frequencies and modes with minimum attenuation in high and low frequencies are determined. These frequencies and modes are of great importance for NDT applications since they play a critical role in the selection of the optimal inspection frequencies.

Random Regression Models Using Legendre Polynomials to Estimate Genetic Parameters for Test-day Milk Protein Yields in Iranian Holstein Dairy Cattle.

The objective of this study was to estimate the genetic parameters of milk protein yields in Iranian Holstein dairy cattle. A total of 1,112,082 test-day milk protein yield records of 167,269 first lactation Holstein cows, calved from 1990 to 2010, were analyzed. Estimates of the variance components, heritability, and genetic correlations for milk protein yields were obtained using a random regression test-day model. Milking times, herd, age of recording, year, and month of recording were included as fixed effects in the model. Additive genetic and permanent environmental random effects for the lactation curve were taken into account by applying orthogonal Legendre polynomials of the fourth order in the model. The lowest and highest additive genetic variances were estimated at the beginning and end of lactation, respectively. Permanent environmental variance was higher at both extremes. Residual variance was lowest at the middle of the lactation and contrarily, heritability increased during this period. Maximum heritability was found during the 12th lactation stage (0.213±0.007). Genetic, permanent, and phenotypic correlations among test-days decreased as the interval between consecutive test-days increased. A relatively large data set was used in this study; therefore, the estimated (co)variance components for random regression coefficients could be used for national genetic evaluation of dairy cattle in Iran.

A robust nonparametric framework for reconstruction of stochastic differential equation models

In this paper, we employ a nonparametric framework to robustly estimate the functional forms of drift and diffusion terms from discrete stationary time series. The proposed method significantly improves the accuracy of the parameter estimation. In this framework, drift and diffusion coefficients are modeled through orthogonal Legendre polynomials. We employ the least squares regression approach along with the Euler–Maruyama approximation method to learn coefficients of stochastic model. Next, a numerical discrete construction of mean squared prediction error (MSPE) is established to calculate the order of Legendre polynomials in drift and diffusion terms. We show numerically that the new method is robust against the variation in sample size and sampling rate. The performance of our method in comparison with the kernel-based regression (KBR) method is demonstrated through simulation and real data. In case of real dataset, we test our method for discriminating healthy electroencephalogram (EEG) signals from epilepsy ones. We also demonstrate the efficiency of the method through prediction in the financial data. In both simulation and real data, our algorithm outperforms the KBR method.

Optimisation for stealth target detection based on stratospheric balloon‐borne netted radar system

This study deals with an improvement of radar cross section (RCS) of stealth target model such as F-117A with a higher aspect vision, and proposes a new detection technique based on stratospheric balloon-borne netted radar system. Normally, it is very difficult to statistically model a stealth target by conventional analytically probability density function (pdf) expressions. Thus, a novel non-parametric detection technique using time difference of arrival (TDOA) and physical optics (PO) approximation method is proposed. Finally, Legendre orthogonal polynomials (LOP) using to reconstruct the pdf of stealth RCS predicted by PO. The potential problem of proposed scheme is instability of the platform caused by sudden gusts of winds. The balloon flight control is studied in detail by considering external force of wind that is typical random process with Dryden turbulence spectrum. The proposed geometrical structure of radar system is composed of one transmitter and multiple receivers. To achieve high accuracy of locating a stealth target, the proposed scheme uses accuracy RCS measurement to compute Stealth target position. Simulations demonstrate that proposed scheme using one transmitter and multiple balloon-borne receivers give much higher location accuracy comparing to other geometrical systems because of increasing the PO - scattered field with higher aspect angles.

Computation of dispersion relations of functionally graded rectangular bars

The propagation characteristics of elastic wave in the functional graded rectangular bars are studied by the two dimensional Rayleigh–Ritz method. Legendre orthogonal polynomials are used as trial functions. The material properties are assumed to be graded along the width direction. This dispersion equation is obtained from the 3D variational formulation governing elastodynamics. The convergence and accuracy are demonstrated by comparisons with the available solution of the scaled boundary finite element method. Finally, we give some phase and group curves of various rectangle waveguides and analyze influences of the width to length ratio and volume fraction indexes on dispersive behavior for wave propagation.

Free vibration analysis of symmetrically laminated composite circular plates with curvilinear fibers

Abstract The free vibration analysis of symmetrically laminated composite circular plates with curvilinear fibers is performed using the first-order shear deformation theory along with a curved hierarchical square finite element. The blending function method is used to describe accurately the geometry of the circular plate. The hierarchical shape functions are constructed from Legendre orthogonal polynomials. The element stiffness and mass matrices are integrated numerically by means of the Gauss-Legendre quadrature. The equations of motion are derived using Lagrange’s method. Results for the fundamental frequency are obtained for clamped and soft simply supported laminated composite circular plates with E-glass, graphite, and boron curvilinear fibers in epoxy matrices. The element is validated by means of the convergence test and comparison with published data for isotropic and laminated composite circular plates with rectilinear fibers. Contour plots of frequency as a function of fiber orientation angles for laminated composite circular plates with curvilinear fibers are presented. The fiber material and boundary conditions are shown to influence the distribution of frequency throughout the design space. Frequency curves as a function of fiber orientation angles for the first five modes of laminated composite circular plates with curvilinear fibers are also presented. They reveal that none of the first five modes of clamped and soft simply supported laminates is affected by crossing but modes 3 and 4 of clamped graphite/epoxy and boron/epoxy laminates are affected by veering.