Fifth-order Polynomial Research Articles

RBF meshless method for large deflection of thin plates with immovable edges

An efficient meshless formulation is presented for large deflection of thin plates with immovable edges. In this method, a fifth-order polynomial radial basis function (RBF) is used to approximate the solution variables. The governing equations are formulated in terms of the three displacement components u, v and w. The solution is obtained by satisfying three coupled partial differential equations and their boundary conditions inside the domain and over the boundary of the plate, respectively. The collocation procedure produces a system of coupled non-linear algebraic equations, which are solved using an incremental-iterative procedure. The numerical efficiency of the proposed method is illustrated through numerical examples.

A high-order finite volume remapping scheme for nonuniform grids: The piecewise quartic method (PQM)

A hierarchy of one-dimensional high-order remapping schemes is presented and their performance with respect to accuracy and convergence rate investigated. The schemes are also compared based on remapping experiments in closed domains. The piecewise quartic method (PQM) is presented, based on fifth-order accurate piecewise polynomials, and is motivated by the need to significantly improve hybrid coordinate systems of ocean climate models, which require the remapping to be conservative, monotonic and highly accurate. A limiter for this scheme is fully described that never decreases the polynomial degree, except at the location of extrema. We assess the use of high-order explicit and implicit (i.e., compact) estimates for the edge values and slopes needed to build the piecewise polynomials in both piecewise parabolic method (PPM) and PQM. It is shown that all limited PQM schemes perform significantly better than limited PPM schemes and that PQM schemes are much more cost-effective.

A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes

This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes. In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials. The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered. The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme. Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems.

Numerical simulation of wave propagation in 2-D fractured media: scattering attenuation at different stages of the growth of a fracture population

SUMMARY In this paper, we systematically examine the multiple scattering process of seismic waves at consecutive stages of the evolution of 2-D fracture population. Synthetic seismograms are computed using the pseudo-spectral method for elastic wave propagation, where spatial derivations are computed using fast Fourier transforms and time derivatives are computed using second-order finite differences. The grid sizes are 2560 × 2560 with 1 m interval and a Ricker wavelet with a peak frequency of 30 Hz is used (or equivalently a wavelength of 10 m for the P-wave velocity of 3000 m s ‐1 used in our modelling). Fracture patterns are generated using a 2-D cellular automaton model of rupture with healing to account for clustering and anisotropy in the fracture growth process. The cellular automation model takes into account the discontinuous and segmented nature of a fracture population, and reproduces in the statistical sense the intermediate stages of fracture growths. To estimate the frequency-dependence of scattering attenuation (quantified by the inverse quality factor Q −1 ) at different stages of the fracture evolution, we use the spectral ratio method. Variations of Q −1 with frequency are then fitted to a polynomial of order up to 8 for each state of the fracture evolution as we do not want to make an assumption about how Q −1 should depend on frequency or scales. This allows us to determine the nature of the frequency-dependence of scattering attenuation as a function of fracture evolution. Our results confirm, as expected, the dependence of scattering attenuation on frequency, and the fifth-order polynomial seems to fit the measured attenuation from synthetic seismograms better. In addition, the inverse quality factor Q −1 is shown to be linearly dependent on fracture density, reaching a maximum when fracture density is the highest. In summary, our numerical results confirm that scattering attenuation has a complex dependence on frequency, and measurements of attenuations may be potentially used to characterize spatial distributions of fracture networks in particular, the scale distributions.

Numerical Evaluation of Contemporary Low-Speed Wind Tunnel Contraction Designs

A potential flow and viscous flow solver have been coupled to produce a robust computational tool useful for the design of low-speed wind tunnel contractions. After validation against published numerical and experimental wind tunnel data, the method is used to evaluate recently proposed contraction shapes from the literature. The results show that, on balance, a fifth-order polynomial provides a good design solution. Newly proposed shapes will either improve available flow area at the expense of contraction outlet flow uniformity or vice versa.

Analysis and calculation of current-flow amplitude spectrum of a microwave mixer diode at poly-harmonic excitation

A new approximating function in the form of a fifth-order polynomial is obtained for the current-voltage characteristic of a mixer diode with Schottky barrier and with compensation of potential barrier of p-n-junction. Mathematical expressions are set up and relative levels calculated for the amplitude spectrum of harmonic and combinative components of the current in a mixer diode excited by four independent signals with different frequencies. The conditions for minimum of the combinative components are formulated for a wide dynamic range of input signals.

Numerical Computing and Graphics for the Power Method Transformation UsingMathematica

This paper provides the requisite information and description of software that perform numerical computations and graphics for the power method polynomial transformation. The software developed is written in the Mathematica 5.2 package PowerMethod.m and is associated with fifth-order polynomials that are used for simulating univariate and multivariate non-normal distributions. The package is flexible enough to allow a user the choice to model theoretical pdfs, empirical data, or a user’s own selected distribution(s). The primary functions perform the following (a) compute standardized cumulants and polynomial coecients, (b) ensure that polynomial transformations yield valid pdfs, and (c) graph power method pdfs and cdfs. Other functions compute cumulative probabilities, modes, trimmed means, intermediate correlations, or perform the graphics associated with fitting power method pdfs to either empirical or theoretical distributions. Numerical examples and Monte Carlo results are provided to demonstrate and validate the use of the software package. The notebook Demo.nb is also provided as a guide for user of the power method.

On the gap in the spectra of surface-layer atmospheric turbulence

A large set of tower data was used to identify the gap that separates small-scale turbulence and mesoscale structures in the cospectra of surface fluxes. The cospectra were obtained using a multi-resolution decomposition algorithm. The gap time scale τg was found by fitting a fifth-order polynomial to the cospectra and identifying special points occurring after the peak at small scales. In unstable conditions (day) τg was found to fall as the mean wind speed increased, while no such dependence was observed in stable conditions (night). The gap scale was found to change very weakly with stability both in moderately stable and moderately unstable conditions, with a sharp drop from about 1100 to 250 s occurring in near-neutral conditions. The vertical fluxes computed at different averaging intervals were found to correlate exceptionally well with each other, the scatter being somewhat larger during the night. Although considerable discrepancy may occur for individual records, when averaged over 10 months, the difference in the flux estimated at 7 to 109 min intervals never exceeded 4%, which is comparable or less than the instrumental error.

Study on the High-Sealing of Roots Rotor With Variable Trochoid Ratio

Vacuum pump plays an important role in the vacuum system. In this paper, a new method is proposed to design the Roots rotor that was the trochoidal curve with variable trochoid ratio. The fifth-order polynomial and cubic spline functions are used to generate smoothly trochoid ratio functions and thus a regular addendum curve for Roots rotor will be generated. Based on the addendum curve, the tooth profile for the complete Roots rotor is derived by the gearing theory. In order to improve the sealing between the rotor tip and the chamber, a design procedure is also developed for roots rotor with high sealing and high area efficiency. In addition, the undercutting equation is used to study the constraints of design parameters and the center of curvature is used to check the occurrence of the carryover on the tooth profile. Finally, some design examples are presented to show the feasibility of the proposed method.

Study on high volumetric efficiency of the Roots rotor profile with variable trochoid ratio

Vacuum pumps play an important role in vacuum systems. The method proposed in this article, which uses the extended cycloid curve with a variable trochoid ratio, can be utilized to improve pump performance. The third-order and the fifth-order polynomials are used, and their boundary conditions are designed to generate a smooth trochoid ratio function. Hence, a regular addendum curve for the Roots rotor is generated. The tooth profile for the complete Roots rotor is derived from the addendum curve. Undercutting and carryover analyses are used to verify the feasibility of the designed method. Finally, a design flow on how to achieve high volumetric efficiency for a vacuum system is presented.

Genetic parameters for various random regression models to describe total sperm cells per ejaculate over the reproductive lifetime of boars

The objective of this study was to model the variances and covariances of total sperm cells per ejaculate (TSC) over the reproductive lifetime of AI boars. Data from boars (n = 834) selected for AI were provided by Smithfield Premium Genetics. The total numbers of records and animals were 19,629 and 1,736, respectively. Parameters were estimated for TSC by age of boar classification with a random regression model using the Simplex method and DxMRR procedures. The model included breed, collector, and year-season as fixed effects. Random effects were additive genetic, permanent environmental effect of boar, and residual. Observations were removed when the number of data at a given age of boar classification was < 10 records. Preliminary evaluations showed the best fit with fifth-order polynomials, indicating that the best model would have fifth-order fixed regression and fifth-order random regressions for animal and permanent environmental effects. Random regression models were fitted to evaluate all combinations of first- through seventh-order polynomial covariance functions. Goodness of fit for the models was tested using Akaike's Information Criterion and the Schwarz Criterion. The maximum log likelihood value was observed for sixth-, fifth-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. However, the best fit as determined by Akaike's Information Criterion and the Schwarz Criterion was by fitting sixth-, fourth-, and seventh-order polynomials; and fourth-, second-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. Heritability estimates for TSC ranged from 0.27 to 0.48 across age of boar classifications. In addition, heritability for TSC tended to increase with age of boar classification.

Evolutionary Learning Acquisition of Optimal Joint Angle Trajectories of Flexible Robot Arm

This paper proposes the evolutionary learning acquisition method of the optimal joint angle trajectories of a flexible robot arm using the genetic algorithm is proposed, and the effects of the optimal joint angle trajectories obtained by the present evolutionary learning acquisition method on the residual vibration reduction are ascertained numerically and experimentally. In the construction of the evolutionary learning acquisition algorithm of the optimal joint angle trajectories, the joint angular velocity curves are depicted with fifth-order polynomials, and, by considering the boundary and constraint conditions, they are expressed by four parameters. Then, the residual vibrations of the flexible robot arm are expressed as a function of the chromosome consisting of four parameters, namely, four genes, and a fitness function of the genetic algorithm for the residual vibration reduction is defined. Furthermore, the numerical calculations have been carried out, and it is confirmed that the residual vibrations almost disappear. Moreover, the experimental results are demonstrated, and the usefulness of the present evolutionary learning acquisition method of the optimal joint angle trajectories of the flexible robot arm using the genetic algorithm is ascertained experimentally.

Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil

A single fluid model of sheet/cloud cavitation is developed and applied to a NACA0015 hydrofoil. First, a cavity formation model is set up, based on a three-dimensional (3D) non-cavitation model of Navier–Stokes equations with a large eddy simulation (LES) scheme for weakly compressible flows. A fifth-order polynomial curve is adopted to describe the relationship between density coefficient ratio and pressure coefficient when cavitation occurs. The Navier–Stokes equations including cavitation bubble clusters are solved using the finite-volume approach with time-marching scheme, and MacCormack’s explicit-corrector scheme is adopted. Simulations are carried out in a 3D field acting on a hydrofoil NACA0015 at angles of attack 4°, 8° and 20°, with cavitation numbers σ = 1.0, 1.5 and 2.0, Re = 10 6 , and a 360 × 63 × 29 meshing system. We study time-dependent sheet/cloud cavitation structures, caused by the interaction of viscous objects, such as vortices, and cavitation bubbles. At small angles of attack (4°), the sheet cavity is relatively stable just by oscillating in size at the accumulation stage; at 8° it has a tendency to break away from the upper foil section, with the cloud cavitation structure becoming apparent; at 20°, the flow separates fully from the leading edge of the hydrofoil, and the vortex cavitation occurs. Comparisons with other studies, carried out mainly in the context of flow patterns on which prior experiments and simulations were done, demonstrate the power of our model. Overall, it can snapshot the collapse of cloud cavitation, and allow a study of flow patterns and their instabilities, such as “crescent-shaped regions.”

Improving Godunov-type reconstructions for simulation of vortex-dominated flows

A systematic Fourier accuracy analysis is performed to examine the numerical diffusion inherent in a Godunov-type reconstruction, including both the reconstruction of the solution within each cell and the computation of the derivative terms of the reconstruction. It is found that compared with the more popular fifth-order polynomial fit of the interface values, a piecewise quadratic reconstruction of the solution with more accurate slope and curvature, especially those computed by compact difference schemes, is much less dissipative. Therefore, further given in the paper is a general framework to make a piecewise quadratic reconstruction free of numerical oscillations around the shocks. The improved accuracy and robustness of the resulting Godunov-type schemes for simulation of vortex-dominated flows are demonstrated with the numerical results of several carefully selected cases, including vortex convection and shock–vortex interaction.

Detection of Different Shapes of Lactation Curve for Milk Yield in Dairy Cattle by Empirical Mathematical Models

The study of relationships between mathematical properties of functions used to model lactation curves is usually limited to the evaluation of the goodness of fit. Problems related to the existence of different lactation curve shapes are usually neglected or solved drastically by considering shapes markedly different from the standard as biologically atypical. A deeper investigation could yield useful indications for developing technical tools aimed at modifying the lactation curve in a desirable fashion. Relationships between mathematical properties and lactation curve shapes were analyzed by fitting several common functions (Wood incomplete gamma, Wilmink's exponential, Ali and Schaeffer's polynomial regression, and fifth-order Legendre polynomials) to 229,518 test-day records belonging to 27,837 lactations of Italian Simmental cows. Among the best fits (adjusted r2 higher than 0.75), the 3-parameter models (Wood and Wilmink) were able to detect 2 main groups of curve shape: standard and atypical. Five-parameter models (Ali and Schaeffer function and the Legendre polynomials) were able to recognize a larger number of curve shapes. The higher flexibility of 5-parameter models was accompanied by increased sensitivity to local random variation as evidenced by the bias in estimated test-day yields at the beginning and end of lactation (border effect). Meaning of parameters, range of their values and of their (co) variances are clearly different among groups of curves. Our results suggest that analysis based on comparisons between parameter values and (co)variances should be done carefully. Comparisons among parameter values and (co)variances could yield more robust, reliable, and easy to interpret results if performed within groups based on curve shape.

Evidence for Domain-specific Recognition of SK and Kv Channels by MTX and HsTx1 Scorpion Toxins

Maurotoxin (MTX) and HsTx1 are two scorpion toxins belonging to the alpha-KTx6 structural family. These 34-residue toxins, cross-linked by four disulfide bridges, share 59% sequence identity and fold along the classical alpha/beta scaffold. Despite these structural similarities, they fully differ in their pharmacological profiles. MTX is highly active on small (SK) and intermediate (IK) conductance Ca(2+)-activated (K(+)) channels and on voltage-gated Kv1.2 channel, whereas HsTx1 potently blocks voltage-gated Kv1.1 and Kv1.3 channels only. Here, we designed and chemically produced MTX-HsTx1, a chimera of both toxins that contains the N-terminal helical region of MTX (sequence 1-16) and the C-terminal beta-sheet region of HsTx1 (sequence 17-34). The three-dimensional structure of the peptide in solution was solved by (1)H NMR. MTX-HsTx1 displays the activity of MTX on SK channel, whereas it exhibits the pharmacological profile of HsTx1 on Kv1.1, Kv1.2, Kv1.3, and IK channels. These data demonstrate that the helical region of MTX exerts a key role in SK channel recognition, whereas the beta-sheet region of HsTx1 is crucial for activity on all other channel types tested.

Performance Analysis of Multicarrier Systems in the Presence of Smooth Nonlinearity

Most of the performance studies devoted to the analysis of multicarrier systems in the presence of nonlinear distortion assume that the multicarrier signal can be modeled as a random Gaussian process regardless of modulation format, guard interval duration, and number of subcarriers. On the contrary, we present an analysis based on the discrete model of the multicarrier signal. It is shown that the discrete model provides more accurate signal-to-distortion ratio and out-of-band spectrum prediction if the number of subcarriers is small. Only the effect of third-order nonlinearity is studied; however results can also be extended to the fifth-order polynomial model. Simulation results are provided which confirm analytical derivations.

Cosmographic Evaluation of the Deceleration Parameter Using Type Ia Supernova Data

The apparent magnitude-redshift data of Type Ia supernovae (SNe Ia) call for modifications in the standard model energy densities. Under the circumstance that this modification cannot be limited to the addition of a mere cosmological constant, a serious situation has emerged in cosmology in which the energy densities in the universe have become largely speculative. In this situation, an equation of state of the form p = wρ itself is not well motivated. In this paper, we argue that the reasonable remaining option is to make a model-independent analysis of SNe data without reference to the energy densities. In this basically kinematic approach, we limit ourselves to the observationally justifiable assumptions of homogeneity and isotropy, i.e., to the assumption that the universe has a Robertson-Walker metric. This cosmographic approach is historically the original one in cosmology. We perform the analysis by expanding the scale factor into a fifth-order polynomial, an assumption that can be further generalized to any order. The present expansion rates h, q0, r0, etc., are evaluated by computing the marginal likelihoods for these parameters. These values are relevant since any cosmological solution would ultimately need to explain them.

Using a subsurface drip irrigation system to measure crop water use

Procedures are presented for determining crop water use and crop coefficients for a row crop, using a neutron scattering probe with an efficient subsurface drip irrigation system. One procedure is called the slope-projection method, and the other is called a covariance procedure. Field tests were conducted with full-season, narrow-row cotton (Gossypium hirsutum L.) on a well-drained, sandy soil in a semiarid environment over a 5-year period. The goal was to improve automated irrigation scheduling, by relating evapotranspiration (ET) to growing degree days (GDD). The result, using a Penman–Monteith reference ET, was an average midseason crop coefficient of 1.11, with a standard error of 0.056. With class A pan evaporation as the reference ET, the average midseason crop coefficient was 0.877, with a standard error of 0.029. A fifth-order polynomial for the pan-based crop coefficient as a function of GDD was programmed into a controller and used successfully to irrigate a field automatically for one season.

A Method for Simulating Correlated Non-Normal Systems of Linear Statistical Equations

A procedure is derived for simulating correlated non-normal systems of linear statistical equations. The method is based on fifth-order polynomial transformations to generate multivariate non-normal distributions. The procedure allows for the simultaneous control of the correlated non-normal (a) stochastic disturbance distributions, (b) independent variables, and (c) dependent and independent variables for each equation throughout a system. A numerical example is provided to demonstrate the procedure. The results of a Monte Carlo simulation are provided to confirm that the proposed method generates the specified standardized cumulants and correlations.