Correction Polynomial Research Articles

Optimised correction polynomial functions for the Flux Reconstruction method in time-harmonic electromagnetism

The Flux Reconstruction (FR) method is classically used in the Computational Fluid Dynamics field. However, its use for the simulation of electromagnetic wave propagation is not as developed yet. Following on from the development of a priori error estimates for the 1D wave equations, we introduce optimisation problems to allow an adaptation of the FR correction polynomial functions to the discretisation parameters. Showing notable accuracy gains in 1D, especially in the preasymptotic regime, we generalise this procedure to the 3D Maxwell’s equations, leading to similar interesting possibilities to reduce the computational cost for a given accuracy.

A novel motion-reconstruction method for inertial sensors with constraints

Motion reconstruction for rigid bodies and rigid-body frames using data from inertial measurement units (IMUs) is a challenging task. Position and orientation determination by means of IMUs is erroneous, as deterministic and stochastic errors accumulate over time. The former of which errors can be minimized by standard calibration approaches, however, sensor calibration with respect to a common reference coordinate system to correct misalignment, has not been fully addressed yet. The latter stochastic errors are mostly reduced using sensor fusion. In this paper, we present a novel motion-reconstruction method utilizing optimization to correct measured IMU data by means of correction polynomials to minimize the deviation of motion constraints. In addition, we perform gyrometer and accelerometer calibration with an industrial manipulator to address deterministic IMU errors, especially misalignment. To evaluate the performance of the novel methods, two types of experiments, one at constant orientation and another with simultaneous translation and rotation, were conducted utilizing the manipulator. The experiments were repeated for five individual IMUs successively. Application of the calibration and optimization methods yielded an average decrease of 95% in the maximum position error compared to the results of common motion reconstruction. Moreover, the average position error over the measurement duration decreased by nearly 90%. The proposed method is applicable to velocity, position, and orientation constraints for every experiment that starts and ends at standstill.

General Modulation Transfer Function Fitting Equation for Human Eye with Contact Lens

In this research, a general equation was derived to represent the modulation transfer function (MTF) of the human eye with the presence of the contact lenses (CL) and during the vision with monochrome or multi-wavelength (polychromatic) light. This equation was used to fit the MTF's curves for the eye with pupil diameters (EPD) (2.5, 4, 6, and 8 mm). The fitting equation accuracy was revealed by using the standard deviation (STD). The results indicated that the image MTF is sensitive to (EPD). Moreover, eye vision degrades as (EPD) increases due to the increased spherical and chromatic aberrations. The new equation consists of two exponential terms in addition to a correction polynomial function (H) to fit the EPD effect. The results showed that H function with power two is sufficient to give a good accuracy when the EPD is less than 4 mm, while H with power five is necessary when the EPD is larger or equal to 5 mm. The equation gives a high accuracy also in the case of vision with white light for the eye with a contact lens.

Adaptive multi-fidelity sparse polynomial chaos-Kriging metamodeling for global approximation of aerodynamic data

The multi-fidelity metamodeling method can dramatically improve the efficiency of metamodeling for computationally expensive engineering problems when multiple levels of fidelity data are available. In this paper, an efficient and novel adaptive multi-fidelity sparse polynomial chaos-Kriging (AMF-PCK) metamodeling method is proposed for accurate global approximation. This approach, by first using low-fidelity computations, builds the PCK model as a model trend for the high-fidelity function and captures the relative importance of the significant sparse polynomial bases selected by least angle regression (LAR). Then, by using high-fidelity model evaluations, the developed method utilizes the trend information to adaptively refine a scaling PCK model using an adaptive correction polynomial expansion-Gaussian process modeling. Here, the most relevant sparse polynomial basis set and the optimal correction expansion are adaptively identified and constructed based on a devised nested leave-one-out cross-validation-based LAR procedure. As a result, the optimal AMF-PCK metamodel is adaptively established, which combines advantages of high flexibility and strong nonlinear modeling ability. Moreover, an adaptive sequential sampling approach is specially developed to further improve the multi-fidelity metamodeling efficiency. The developed method is evaluated by several benchmark functions and two practically challenging transonic aerodynamic modeling applications. A comprehensive comparison with the popular hierarchical Kriging, universal Kriging, and LAR-PCK approaches demonstrates that the proposed method is the most efficient and provides the best global approximation accuracy, with particular superiority for quantities of interest in the multimodal and highly nonlinear landscape. This novel method is very promising for efficient uncertainty analysis and surrogate-based optimization of expensive engineering problems.

Sparse polynomial interpolation based on diversification

We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari (1988) for interpolating polynomials over fields with characteristic zero, we develop a new Monte Carlo algorithm over finite fields by doing additional probes. To interpolate a polynomial $$f \in {\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$$ with a partial degree bound D and a term bound T, our new algorithm costs $${O^ \sim }(nT{\log ^2}q + nT\sqrt D \log q)$$ bit operations and uses 2(n + 1)T probes to the black box. If q ⩾ O(nT2D), it has constant success rate to return the correct polynomial. Compared with the previous algorithms over general finite field, our algorithm has better complexity in the parameters n, T and D, and is the first one to achieve the complexity of fractional power about D, while keeping linear in n and T. A key technique is a randomization which makes all the coefficients of the unknown polynomial distinguishable, producing a diverse polynomial. This approach, called diversification, was proposed by Giesbrecht and Roche (2011). Our algorithm interpolates each variable independently by using O(T) probes, and then uses the diversification to correlate terms in different images. At last, we get the exponents by solving the discrete logarithms and obtain the coefficients by solving a linear system. We have implemented our algorithm in Maple. Experimental results show that our algorithm can be applied to the sparse polynomials with large degree within a few minutes. We also analyze the success rate of the algorithm.

Prediction of wine sensory properties using mid-infrared spectra of Cabernet Sauvignon and Chardonnay grape berries and wines

The study determined optimal parameters to four preprocessing techniques for mid-infrared (MIR) spectra of wines and grape berry homogenates and tested MIR’s ability to model sensory properties of research Cabernet Sauvignon and Chardonnay wines. Savitsky-Golay (SG) derivative, smoothing points, and polynomial order, and extended multiplicative signal correction (EMSC) polynomial were investigated as preprocessing techniques at 2, 2, 5, and 3 levels, respectively, all in combination. Preprocessed data were analysed with partial least squares regression (PLS) to model the wine sensory data and the regression coefficients of PLS calibration models (R2) were further analysed with multivariate analysis of variance (MANOVA). SG transformations were significant factors from the MANOVA that influenced R2, while EMSC did not. Overall, PLSR models that predicted wine sensory characteristics gave a poor to moderate R2. Consistently predicting wine sensory attributes within a variety and across vintages is challenging, regardless of using grape or wine spectra as predictors.

A polynomial-division based correction model for camera calibration: a large comparative study

These days, algorithms which enable correction of an image aberration such us barrel or pillow distortion can be found in a large number of scientific papers. Some sorts of algorithms have universal character, others can be used only with a certain type of optical systems. This paper compares several types of methods which were designed to remove an image distortion. The methods which are based on so called Polynomial Division Model (PDM) are only considered. On the grounds of extensive experimental results, a method which uses the polynomial division model was proposed. The method brought significantly better results in the correction process. It does not matter if the images are loaded with a pillow or a barrel distortion. The image correction calculation procedure is from a general point of view, very time demanding. The calculation with use of the PDM is fast, and does not need a large number of members of the correction polynomial. This rule is also valid for an image which is loaded with large distortion e.g., for Fish-Eye object-lenses - a newly proposed PDM preserves all good features of previous algorithms which are based on the PDM and extends them. All the results which are recorded in the experimental section show that the PDM is superior and more effective in comparison to other tested methods.

Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods

Characteristic boundary conditions for the Navier-Stokes equations (NSCBC) are implemented for the first time with discontinuous spectral methods, namely the spectral difference and flux reconstruction. The implementation makes use of the resolution by these methods of the strong form of the Navier-Stokes equations by applying these conditions through a flux balance regularization which takes the form of a generalized element-compact correction polynomial. It is shown to be at least as effective as similar implementations in finite volume solvers, and sustains arbitrarily-high orders of accuracy on hexahedral-based unstructured meshes. Further, Navier-Stokes time-domain impedance boundary conditions are derived and implemented as a NSCBC sub-class. They account for the diffusive process at the wall and are shown to properly resolve the broadband response of porous materials under normal and grazing flow conditions. The ability of these NSCBC in preventing the appearance of spurious reflections at the boundaries is demonstrated through a varied series of bench-marking simulations. They effectively shield the inner computational domain from any far-field unphysical contamination. Overall, this work enables the use of strong discontinuous spectral methods to study unsteady problems on complex geometries.

Smooth minimum time trajectory planning with minimal feed fluctuation

Smooth and effective feedrate profile is very important for high-speed and high-precision machining. In this paper, a two-stage feedrate scheduling scheme (TFSS) is proposed to efficiently generate the smooth feedrate profile. The minimum time trajectory planning (MTTP) is firstly employed to generate the initial feedrate profile considering constraints of both the kinematics and the chord error. To improve the computational efficiency, the MTTP is transformed into a problem of linear programming by applying direct transcription method. Then, all the sharp corners along the initial feedrate profile are identified, especially those that are interacted by each other. The jerk-limited method of eliminating interaction (JMEI) at sharp corners is applied to smooth the final feedrate profile. Finally, the improved feed correction polynomials are applied to reduce the fluctuations owing to non-arc-length parameterization. To illustrate the validity and rationality of the proposed scheme, two curves were employed to verify the proposed method. The simulation and experimental results demonstrate the efficiency of the proposed scheme for the nonuniform rational basis spline (NURBS) tool paths.

Threshold stability curves for a nonlinear rotor-bearing system

Design parameters of fluid film bearings affect the stability of rotor-bearing systems, which can be characterized using so called threshold speed and a threshold curve (depending on a shaft speed and a chosen design parameter). The paper brings a detailed study of the shape of the threshold curve for a rigid rotor supported on a plain bearing. A novel comparison of stability threshold curves obtained employing various approaches to the hydrodynamic force modelling for all relevant bearing parameters is presented. Hydrodynamics forces are solved using four approaches: the infinitely short (IS) approximation, the infinitely short approximation extended by multiplicative correction polynomials for finite-length journal bearings (IScor), the finite differences method and the finite elements method. The influence of run-up time and acceleration is also investigated. All obtained threshold curves are compared to the analytical expressions for the threshold curves calculated for the IS approximation.

Dynamic coefficients and stability analysis of finite-length journal bearings considering approximate analytical solutions of the Reynolds equation

The paper presents linearized coefficients for finite-length journal bearings based on the HD forces with multiplicative correction polynomials applied to the infinitely short and the infinitely long solutions under the assumption of π-film boundary condition. Based on the linear theory, stability thresholds and their dependence on the length-to-diameter ratio are investigated. Moreover, using the numerical continuation method the change of the generalized Hopf bifurcation which separates regions with subcritical and supercritical Hopf is documented. All results are compared with those of the solution obtained using the finite differences method.

Influence of Scheimpflug condition on measurements of a scanning laser line sensor for 3D imaging

Satisfying the Scheimpflug condition in a rotational scanning 3D triangulation sensor system can be challenging. By violating this condition an reproducible measurement error is generated, which is caused by the out of focus projected laser point on the detector. In this work a ray-tracing simulation of the scanning system is performed, in order to determine potential aberrations. With the reference measurement of a flat surface the error due to violation of the Scheimpflug condition is obtained. The error is systematic and reproducible and can be compensated by a correction polynomial, with coefficients determined by using the least squares method. Experimental results reveal, that the measurement error can be significantly reduced by the correction polynomial.

Effect of various analytical descriptions of hydrodynamic forces on dynamics of turbochargers supported by floating ring bearings

Abstract The scope of this paper is to provide a general model of a turbocharger using approximate analytical solutions of the hydrodynamic forces in oil films improved by correction polynomials. This approach brings two main advantages: the considered analytical solutions hold for wide range of bearing's length-to-diameter ratios and it is not restricted to infinitely short or infinitely long journal bearing approximations whose assumptions are often not satisfied. The usage of the closed-form approximate analytical solution of the Reynolds equation is more time efficient compared to the usage of common numerical methods and enables to effectively solve complex problems like parametric studies. The influence of various models of hydrodynamic forces on the turbochargers behaviour is studied on two different model systems.

Multi-fidelity non-intrusive polynomial chaos based on regression

In this paper we present a multi-fidelity (MF) extension of non-intrusive polynomial chaos based on regression (point collocation) for uncertainty quantification purposes. The proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients. Due to its usage of regression to estimate the coefficients, the present method offers high flexibility in the sampling and generation of the polynomial basis. The method takes advantage of a nested sampling plan to create the samples for the low-fidelity (LF) and correction expansions where the high-fidelity (HF) samples are a subset of the LF ones. To build the polynomial basis, a total order or hyperbolic truncation strategy is used with a highly flexible combination of the LF and correction polynomial expansions. The method is demonstrated on some artificial test problems and aerodynamic problems of the Euler flow around an airfoil and common three-dimensional research models. In order to derive the strategies for successful MF approximation, the effect of the correlation and the errors between the LF and HF functions is also studied. The results show that high correlation and moderately low errors are important to improve the MF approximation’s accuracy. On a common research model problem, the MF approach with partially-converged simulations as the LF samples can successfully reduce the computational cost to about 40% for similar accuracy compared to an approach using a single HF expansion.

A correction method for power skiving of cylindrical gears lead modification

The latest research clearly demonstrates the excellent capability of the gear skiving technology. To improve the twisted flanks of cylindrical gears when performing a lead modification in skiving process, a correction method is proposed. First, a kinematic model of power skiving and correction tooth surface are established based on the engagement principle of crossed helical gear. The phenomenon of flank twist in the skiving process is investigated and the counter lines of modified error on both flanks are obtained. Then, a correction method for cylindrical gears lead modification is proposed to correct the twisted tooth flanks by optimizing the cutter profile. In this method, the cutter profile is formulated as B-spline curves using a curve fitting technique, and a sensitivity matrix and correction polynomial coefficients are derived. Thus, the skiving tooth flank can be approximated to the desired tooth flank by adjusting the coefficients of the polynomials. Finally, a numerical example shows that the optimization method of cutter profile can effectively reduce the modified errors as well as improve the modification twist in gear skiving process.

A Simplified Formulation of the Flux Reconstruction Method

The flux reconstruction (FR) methodology has proved to be an attractive approach to obtaining high-order solutions to hyperbolic partial differential equations. However, the utilization of somewhat arbitrarily defined correction polynomials in the application of these schemes, while adding some flexibility, detracts from their ease of implementation and computational efficiency. This paper describes a simplified formuation of the flux reconstruction method that replaces the application of correction polynomials with a single Lagrange interpolation operation. A proof of the algebraic equivalence of this scheme to the FR formulation of the nodal discontinuous Galerkin (DG) method provided that the interior solution points are placed at the zeros of a corresponding Legendre polynomial is presented. Next, a proof of linear stability for this formulation is given. Subsequently, von Neumann analysis is carried out on the new formulation to identify a range of linearly stable schemes achieved by variations of the interior solution point locations. This analysis leads to the discovery of linearly stable schemes with greater formal order of accuracy than the DG method.

Efficient Fitting of the Feed Correction Polynomial for Real-Time Spline Interpolation

Efficient Fitting of the Feed Correction Polynomial for Real-Time Spline Interpolation

Camera Calibration Using Direct Mapping and Adaptive Metaheuristics

Every optical system is influenced with a set of aberrations. Some of them are corrected by the manufacturer of the optical system, other aberrations must be corrected additionally e.g. using methods of digital image correction. The presented paper describes a method which enables correction of a barrel or pillow distortion of an image of small range using the direct-mapping technique. The proposed method uses an algorithm of adaptive metaheuristics marked as jDE and a polynomial representation of the image distortion. The advantage of connecting direct mapping technique and adaptive metaheuristics is primarily high speed of finding basic coefficients of the correction polynomial; in our case it is polynomial of degree up to 7 and with a selectable combination of the used polynomial degrees. The disadvantage of the direct-mapping method is the necessity of reconstruction of the missing pixels in the final image, which is here solved very effectively by using optimizer jDE again. Using the proposed method it is possible to obtain high quality image free of barrel or pillow distortion. If the focus length of the used optical system is constant, values of the correction coefficients and positions of individual recovered pixels stay invariable as well

Specification Search for Identifying the Correct Mean Trajectory in Polynomial Latent Growth Models

This study investigated the optimal strategy for model specification search under the latent growth modeling (LGM) framework, specifically on searching for the correct polynomial mean or average growth model when there is no a priori hypothesized model in the absence of theory. In this simulation study, the effectiveness of different starting models on the search of the true mean growth model was investigated in terms of the mean and within-subject variance-covariance (V-C) structure model. The results showed that specifying the most complex (i.e., unstructured) within-subject V-C structure with the use of LRT, ΔAIC, and ΔBIC achieved the highest recovery rate (>85%) of the true mean trajectory. Implications of the findings and limitations are discussed.

Digital calibration of inter-stage nonlinear errors in pipelined SAR ADCs

Two digital calibration techniques to linearize the residue amplifier in pipelined SAR ADCs are presented. The proposed techniques utilize a single, one-bit pseudorandom noise (PN) to simultaneously identify all coefficients of a correction polynomial. Behavioral simulation results demonstrate the effectiveness of the two proposed techniques, in which the SNDR and SFDR of a 12-bit pipelined SAR ADC are improved from 54 and 69 dB to 72 and 100 dB, respectively. Some circuit design details are included for the PN injection circuit as well as the digital calibration logic.