Rational Polynomial Model Research Articles

A reduced-order improved rational polynomial method for viscoelastically damped structures considering ambient temperature effect

This study focuses on the development of a framework for the seismic analysis of viscoelastically damped structures to investigate the ambient temperature effect on seismic responses. The improved rational polynomial model is used to describe the temperature dependence of the complex stiffness of viscoelastic (VE) dampers. The free‐interface component mode synthesis method is employed to improve the computing efficiency. Numerical simulations of a 20-storey benchmark building with added VE dampers under historical earthquakes are presented to illustrate the ambient temperature effect. The results show that the proposed framework is feasible in the seismic analysis of viscoelastically damped structures considering ambient temperature effect. The floor displacement and interstory drift reductions decrease with the increase of the ambient temperature, while fluctuations of the floor acceleration and base shear reductions with the ambient temperature are complicated due to coupled influences of spectrum characteristics of the earthquake, dynamic characteristics of the structure, and temperature dependence of VE dampers. The VE damper with a strong temperature dependence may not be effective in reducing acceleration responses and base shear of structures under specific earthquakes at high and low ambient temperatures.

Effect of frequency dependence on the seismic performance of linear viscoelastic base-isolated structures

The purpose is to address two issues arising from the frequency dependence of viscoelastic (VE) isolation bearings implemented in base-isolated structures (BIS): (i) how the frequency dependence affects the structural response; and (ii) which method is appropriate for the VE BIS. Effects of the frequency dependence on the absolute acceleration transmissibility are analyzed based on a two-degree-of-freedom system. Three generic algorithms: (i) constant-parameter approximation (CPA), (ii) mode superposition iteration, and (iii) modified rational polynomial approximate (RPA) are used to calculate the seismic responses. Effects of the frequency dependence on the seismic responses of the VE BIS subjected to the historic earthquakes are numerically analyzed through a five-story frame structure. The results show that the frequency dependence of the stiffness can broaden the absolute acceleration transmissibility curve of the VE BIS. The seismic responses in general are underestimated when the frequency dependence is not well considered, and the response errors are influenced by the frequency dependence, excitation characteristics and structural dynamic property. The RPA method takes into account the frequency dependence of VE isolation bearings completely through the rational polynomial model. The CPA and mode superposition iteration methods can only consider frequency dependence effect partially, and the mode superposition iteration method performs better.

Robust Fast Electromagnetic Optimization of SIW Filters Using Model-Based Deviation Estimation and Jacobian Matrix Update

In this paper, a simple fast electromagnetic (EM) optimization method for substrate integrated waveguide (SIW) filters is presented. The feature parameter deviations are estimated by direct response approximation based on the feature parameter model with few times quasi-Newton method. The parameter extraction process is more concise than the parameter identification techniques based on the rational polynomial model and can be used for lossless networks as well as lossy networks. The dimension correction is based on the Jacobian matrix which relates the physical dimensions and the feature parameters. An analytical general mapping model for SIW filters is extracted and applied to generate and update the Jacobian matrix for the first time. The extracted model can be directly used for SIW filter design without repeated extraction. This helps to minimize the number of EM simulations and avoid extra work for model extraction. Using general mapping model can overcome the robustness issue introduced by Broyden update methods. The number of EM simulations required to meet the specifications can be significantly reduced. The effectivity and robustness of the proposed method is validated by two design examples: a 10-order SIW Chebyshev filter and a 5-order SIW cross-coupled filter. The proposed method exhibits low complexity, good stability, good convergence speed and the ability to avoid being trapped in local minimum.

Digital Estimation and Compensation of Analog Errors in Frequency-Interleaved ADCs

A novel digital compensation scheme for measuring, estimating and correcting linear weakly time-varying analog errors in frequency-interleaved analog-to-digital converters (FI-ADCs) is presented. This method features three important improvements over existing approaches: First, the Wigner–Ville distribution (WVD) is used to better estimate the nonstationary analog channel frequency response (ACFR) spectrum. Secondly, the estimated ACFR spectrum is approximated with a rational polynomial model using the [Formula: see text]-norm metric. The corresponding [Formula: see text]-norm minimization problem is solved using a primal-relaxed dual global optimization (PRD-GOP) method. Thirdly, the digital compensation circuitry is designed utilizing a preconditioned biconjugate gradient stabilized (BICGSTAB) algorithm that yields a computationally efficient solution. Numerical experimentations have been conducted and the outcomes validate the feasibility and superior performance of this proposed method.

A practical method for determining the rheology of drilling fluid

A new mathematical model for analyzing the rheology of a drilling fluid was developed based on a nonlinear function fit of measured data of shear stress and shear rate of the drilling mud used during the drilling process of petroleum and geothermal wells. This mathematical model was obtained from a rational polynomial model proposed for determining of the viscosity behavior. We proposed an analytical approximation to allow a simple fit that approximately describes the rheology of a drilling mud obtaining squared correlation coefficients greater than 0.9995 for the fitted series. The logarithmic function and rational functions obtained describe the shear stress and viscosity behavior, respectively. The fitting process was automated using a simple Wolfram language script. The result is a simple, accurate and practical method to describe the rheology of a drilling mud. This model was compared against most used models, applied to a set of 87 experimental data series, had a good agreement in 81% of the analyzed series and being the best choice in 48.3% of all cases. The method has the advantages of using the Wolfram script with common measurements made during a drilling process and can be applied in either the petroleum or geothermal industries to analyze the drilling mud rheology.

Geopositioning precision analysis of multiple image triangulation using LROC NAC lunar images

With continuing data acquisition from lunar orbiters, many areas of the lunar surface have been observed multiple times under different viewing conditions. This raises an issue regarding how to select the best stereo sets and achieve the highest 3D geopositioning precision in such areas. This paper presents a comprehensive analysis of the geopositioning precision of multiple image triangulation using Lunar Reconnaissance Orbiter Camera (LROC) Narrow Angle Camera (NAC) images. Seven and nine LROC NAC images are used, respectively, for the Apollo-11 and Chang’e-3 landing sites. The photogrammetric methods are developed based on a rigorous camera sensor model and rational polynomial model. Experiments with different combinations of dual images are performed for comparisons at both sites. The results demonstrate that the geopositioning precision, especially the height precision, is improved as the convergence angle increases from near 0°–50°. More importantly, we find that as the convergence angle of a stereo pair increases, the image matching precision decreases approximately linearly, which makes the geopositioning less precise. We also find that the shadow-tip distance and the aspect ratio have roughly linear effect on degrading matching quality. So these two factors also need to be considered when selecting image pairs for stereo mapping. The geopositioning precision is mainly controlled by the convergence angle when it is less than about 10° while the image matching error plays a more critical role when the convergence angle is greater than 10°. Experiments with multiple images indicate that utilizing more images produces higher precision than almost all dual-image models; meanwhile, using fewer images can produce better precision than using all available images together. A progressive selection method is proposed to find the best image combination for maximum precision and effectiveness.

Improved method for estimating static formation temperatures in geothermal and petroleum wells

This paper describes a mathematical method for estimating static formation temperatures using a function based on extrapolation of the measured data in the thermal recovery process of geothermal and petroleum wells. This method uses the least squares fit of a rational polynomial model to describe the transient temperature data and also provides an analytical expression of thermal recovery processes. This expression gives the asymptotic value for final temperature. We then apply this method for estimating static formation temperatures of a geothermal and petroleum drilled well.In this work we have developed the equations of least squares fit of rational polynomial function and used the equation to estimate the uncertainties of its coefficients. This fitting procedure allows us to obtain a mathematical model that represents the thermal recovery process and to extrapolate the value of final temperature. This is a practical method to estimate static formation temperature, having the advantages that the thermal properties of the drilling mud and its formation are not required; it can also be used in the geothermal and petroleum industries.

Fast maximum-likelihood identification of modal parameters with uncertainty intervals: A modal model-based formulation

A modal estimation method is presented, which estimates the parameters of the modal model directly instead of identifying a rational fraction polynomial model. The method also gives the uncertainty on the estimated parameters (i.e. poles, mode shapes, participation factors, lower and upper residual terms) without using linearization formulas which are needed when identifying a rational fraction polynomial model first. The proposed estimator belongs to the class of maximum likelihood estimators (MLE). The key challenges behind introducing this approach are to keep the benefits of the well-known poly-reference Least-squares Complex Frequency-domain (pLSCF) estimator—commercially known as PolyMAX—while giving other additional features like improved estimates in cases of high noise level and weakly exited modes together with a proper handling of the uncertainty on the measured data. The proposed method can be considered as an add-on for the pLSCF method since it starts from the initially estimated modal model by the pLSCF method. Our approach has been optimized to reduce the computation time and the memory requirements. The algorithm is evaluated and compared with two published algorithms by means of Monte-Carlo simulation as well as experimental measurements.

E. coli Transport through Surface-Connected Biopores Identified from Smoke Injection Tests

Macropores are the primary mechanism by which fecal bacteria from surface-applied manure can be transported into subsurface drains or shallow groundwater bypassing the soil matrix. Limited research has been performed investigating fecal bacteria transport through specific macropores identified in the field. The objective of this research was to better understand how fecal bacteria, using Escherichia coli (E. coli) as an indicator organism, are transported through naturally occurring macropores and potential interactions between the macropore and soil matrix domains in the field under controlled experimental conditions. Direct injection/infiltration tests were conducted in two naturally occurring, surface-connected macropores (biopores) that penetrated to the subsurface drain depth, as identified by smoke tests. Data included total drain flow rate (baseflow rate and biopore flow rate), biopore inflow rate, and Rhodamine WT and E. coli concentrations in the drains. Analysis techniques included determining increases in subsurface drain flow rates due to the infiltration tests and percentage of the injected concentration reaching the subsurface drains after dilution with the drain baseflow. In the absence of data for mechanistic models, empirically based rational polynomial models were compared to the more commonly utilized lognormal distribution for modeling the load rate breakthrough curves. Load estimates were derived from integrated forms of these empirical functions, and percent reductions were calculated for Rhodamine WT and E. coli. Peak total drain flow rates increased nearly two-fold due to direct injection into the biopores. Less than 25% of the initial concentrations injected into the biopores reached the drain after dilution with the baseflow in the drain. Lognormal distributions best fit the Rhodamine WT load rate breakthrough curves (R2 = 0.99 for both biopores) and the E. coli load rate data for one of the biopores (R2 = 0.98). A rational fractional polynomial model that tailed off more slowly best fit the E. coli load rate data for the other biopore (R2 = 0.98). Approximately one log reduction was estimated for E. coli loads due to interaction with the soil profile as water flowed through the tortuous path of the biopores; in other words, the soil surrounding the biopore filtered approximately 90% of the E. coli load that entered the biopore compared to approximately 75% for Rhodamine WT. Considering that applied animal manure can contain millions of bacteria per mL, high concentrations and loads are still possible in the subsurface drain flow if macropores are present.

Technical Note: Rational Polynomial Functions for Modeling E. coli and Bromide Breakthrough

Fecal bacteria peak concentrations and breakthrough times as affected by preferential flow to subsurface (tile) drainage systems following irrigation or rainfall are important when assessing the risk of contamination. Process-based, convective-dispersive modeling of microbial transport through preferential flow has been conducted. Likewise, regression modeling has been used to study solute transport (e.g., nitrate) under agricultural systems and can have advantages over process-based modeling, such as fewer or easier to determine parameters and easier determination of confidence intervals. However, empirical models (e.g., regression) have only rarely been used to investigate microbial transport. In addition, the selection of time response curves to empirically model simple, right skewed, single breakthrough events from field or laboratory data is generally an arbitrary choice and often considers only conventional distribution-shaped response curves, such as lognormal distributions. In this study, we evaluate four rational polynomial functions for modeling bromide and E. coli data from a single breakthrough event from a tile-drained field near Nashua, Iowa. Bromide and liquid swine manure were applied to the plot immediately prior to 42 mm of overhead sprinkler irrigation. E. coli and bromide concentrations were determined in subsurface drainage water samples collected for the next 24 h. Nonlinear iteratively re-weighted least squares regression procedures were used to model the breakthrough data. The maximum event value, time of occurrence, and event total were estimated from the parameters for each model. Selection of the best model was based on multiple performance criteria. A simple rational polynomial with a linear factor in the numerator and quadratic form in the denominator was the overall best choice for E. coli (R² = 0.92). A related fractional order form also known as the Gunary model was the best choice for bromide (R² = 0.93). In comparison, the more commonly assumed lognormal distribution described only 78% of the variation in E. coli and 68% of the variation in bromide, with a weighted mean square error 3.0 to 4.6 times larger than each selected rational polynomial model. In this experiment, the chosen models clearly tracked E. coli and bromide distribution better than the lognormal model.

Accurate phase expansion on reference planes in grating projection profilometry

A novel phase expansion technology based on a binary quadratic rational polynomial model is proposed. Firstly, the unwrapped phase distribution on reference planes parallel with a zero reference plane in a different grating project profilometry measurement system is analyzed and synthesized. Then a binary quadratic rational polynomial model is proposed to express the uneven phase distribution on every reference plane. The least-squares iterative estimation method based on a first-order Taylor series expansion is employed to obtain the polynomial coefficients. Finally, accurate phase expansion of reference planes is implemented according to the rational polynomial model. The results of the experiments proved that the method proposed in this paper can recover the phase distribution on the full field from parts reference planes, and the measurement errors such as the geometrical parameter error, system structure error, real reference plane profile error and random error can be decreased.

Generalized Hydraulic Calculation Method Using Rational Polynomial Model

A generalized hydraulic calculation technique is presented that requires, as input, only the raw rheological data for the fluid and the values of the relevant hydraulic parameters. The method utilizes a powerful new rheological model called the Rational Polynomial model, which is capable of accurately representing the rheogram of virtually any time-independent fluid. Predictions of the new method are compared with published experimental data gathered from large-scale flow loops. The tested cases include laminar and turbulent flow for a variety of drilling fluids in both pipes and concentric annuli.

Rheological evaluation of non-Newtonian Mexican nixtamalised maize and dry processed masa flours

An experimental programme for evaluating the rheological properties of thirteen maize flour samples collected from several Mexican commercial producers was conducted. Measurements of shear stress and shear rate were performed under calibrated conditions using a rotational Haake viscometer. A comprehensive analysis of the rheometric data obtained during the experimental runs was carried out for studying the actual non-Newtonian behaviour of the maize flour pastes. Non-Newtonian rheological models based on the classical Power Law, the Robertson–Stiff and the Rational Polynomial were used for fitting the rheometric data over a wide range of shear rates (24–3200 s −1). The best description of the non-Newtonian behaviour in most of the maize flour pastes was obtained using the Rational Polynomial Model. Details of the experimental methodology and the results obtained in the rheological evaluation are outlined.

Computer modelling and estimation of recruitment patterns of non-branching coral colonies at three sites in the Wakatobi Marine Park, S.E. Sulawesi, Indonesia; implications for coral reef conservation

We have studied growth and estimated recruitment of massive coral colonies at three sites, Kaledupa, Hoga and Sampela, separated by about 1.5 km in the Wakatobi Marine National Park, S.E. Sulawesi, Indonesia. There was significantly higher species richness ( P<0.05), coral cover ( P<0.05) and rugosity ( P<0.01) at Kaledupa than at Sampela. A model for coral reef growth has been developed based on a rational polynomial function: dx/ dt= a 1W+a 2W 2+a 3W 3+…a nW n 1+b 1W+b 2W 2+b 3W 3+…b mW m where d x/d t is an index of coral growth with time; W is the variable (for example, coral weight, coral length or coral area), up to the power of n in the numerator and m in the denominator; a 1…… a n and b 1… b m are constants. The values for n and m represent the degree of the polynomial, and can relate to the morphology of the coral. The model was used to simulate typical coral growth curves, and tested using published data obtained by weighing coral colonies underwater in reefs on the south-west coast of Curaçao [‘Neth. J. Sea Res. 10 (1976) 285’]. The model proved an accurate fit to the data, and parameters were obtained for a number of coral species. Surface area data was obtained on over 1200 massive corals at three different sites in the Wakatobi Marine National Park, S.E. Sulawesi, Indonesia. The year of an individual's recruitment was calculated from knowledge of the growth rate modified by application of the rational polynomial model. The estimated pattern of recruitment was variable, with little numbers of massive corals settling and growing before 1950 at the heavily used site, Sampela, relative to the reef site with little or no human use, Kaledupa, and the intermediate site, Hoga. There was a significantly greater sedimentation rate at Sampela than at either Kaledupa ( P<0.0001) or Hoga ( P<0.0005). The relative mean abundance of fish families present at the reef crests at the three sites, determined using digital video photography, did not correlate with sedimentation rates, underwater visibility or lack of large non-branching coral colonies. Radial growth rates of three genera of non-branching corals were significantly lower at Sampela than at Kaledupa or at Hoga, and there was a high correlation ( r=0.89) between radial growth rates and underwater visibility. Porites spp. was the most abundant coral over all the sites and at all depths followed by Favites ( P<0.04) and Favia spp. ( P<0.03). Colony ages of Porites corals were significantly lower at the 5 m reef flat on the Sampela reef than at the same depth on both other reefs ( P<0.005). At Sampela, only 2.8% of corals on the 5 m reef crest are of a size to have survived from before 1950. The Scleractinian coral community of Sampela is severely impacted by depositing sediments which can lead to the suffocation of corals, whilst also decreasing light penetration resulting in decreased growth and calcification rates. The net loss of material from Sampela, if not checked, could result in the loss of this protective barrier which would be to the detriment of the sublittoral sand flats and hence the Sampela village.

CORRELATION OF RHEOMETRIC DATA AND HYDRAULIC CALCULATIONS USING RATIONAL POLYNOMIALS

A rational polynomial function, i.e., the ratio of two polynomials, is suggested as a general correlating equation for rheometric data. Using data from the literature, it is shown that the rational-polynomial model is capable of accurately representing the rheogram of virtually any time-independent fluid. The utility of the rational-polynomial model in hydraulic calculations is demonstrated using data for a variety of drilling fluids. It is shown that the pressure drop in circular pipes is more accurately predicted using the rational-polynomial model than with the standard fluid models used in the petroleum industry.

Orthonormal lattice filter—a multistage multichannel estimation technique

The orthonormal lattice filter algorithm is a unified recursive technique to estimate the least‐squares coefficients of a broad class of linear models. It consists of a sequence of two recursions. First, the Gram‐Schmidt orthonormalization procedure estimates a set of inner products from multiple signal vectors, for example, multichannel sampled observations. A lattice type network structure, suitable for adaptive applications, implements the Gram‐Schmidt procedure. Second, a recursion introduced in this paper transforms the inner products into model coefficients. In contrast with conventional algorithms, this algorithm does not require the covariance matrix to design the filters but directly filters the observations. We consider a number of special cases of the multichannel model, for example the rational polynomial model, the autoregressive model, and the autoregressive‐moving average model with known and unknown inputs. In addition, we introduce the autoregressive‐orthonormal model for the unknown input case. We also transform the coefficients of the autoregressive‐orthonormal model to estimate the poles and zeros of a signal transfer function and thereby obtain the amplitude and phase spectra of the signal. For each model, we appropriately simplify the general orthonormal lattice filter algorithm and also the filter network structure. It turns out that for the autoregressive model the simplified algorithm is the Itakura‐Saito algorithm which is a direct version of the Levinson recursion and the simplified structure is a lattice type ladder. With the example of a noisy truncated signal, we compare the actual and estimated values of (a) the coefficients of a known‐input autoregressive‐moving average model, and (b) the autoregressive‐orthonormal power density spectrum of the signal.