High-order Polynomial Research Articles

Loads and hull-pressure measurements on a generic submarine in different stages of model build-up

This is an experimental study of loads and surface pressure on a generic BB2 submarine in different stages of model build-up. The build-up is from a bare hull to a fully-appended hull consisting of a casing, 4 aft control surfaces in an X-form configuration, a vertical sail fin and 2 horizontal sailplanes. The test is carried out in the Defence Science and Technology Group low-speed wind tunnel on a 2 m long hull at a freestream velocity of 60 m/s, where the hull-length Reynolds number is 8 million. The load trends are established for incidence angles up to 12∘, where the measured forces are up to 89 N for pure pitch and 163 N for pure yaw. The drag force is up to 14 N. Overall, the load trends are well approximated by high-order polynomials. From constructed pressure-gradient maps of the hull, the effect of model build-up can be observed in the changing pressure footprint and interpreted by using flow topology.

Characterization of diffractive bifocal intraocular lenses

Multifocal intraocular lenses incorporate a variety of design considerations, including dimensioning of the base monofocal shape and the diffraction grating. While studying three different lens models, we present a practical approach for mathematical modelling and evaluation of these geometries. Contrary to typical lens measurement methods, non-contact measurements were performed on the Alcon SN6AD1, HumanOptics MS 612 DAY and the AMO ZMA00 lenses using a confocal microscope. Subsequent data processing includes centering, tilting correction, filtering and an algorithmic decomposition into a conic and polynomial part and the diffraction grating. Lastly, evaluation of fitting parameters and grating shape is done to allow for inferences about further optical properties. Results and analysis show the confocal microscope to be a suitable imaging method for lens measurements. The processing of this data enables the reconstruction of the annular diffraction grating over the complete lens diameter. Apodization, near addition and diffraction efficiency characteristics are found utilizing the grating shape. Additionally, near-optical axis curvature, asphericity and higher order polynomials are identified qualitatively from the reconstruction of the monofocal base form. Derived properties also include the lens optical base and addition power. By making use of the surface geometries, as well as the lens’ material and thickness, a full lens model can be created for further studies. In summary, our analytical approach enables the insight to various intraocular lens design decisions. Furthermore, this procedure is suitable for lens model creation for research and simulation.

Experimental characterization of piezoelectric and elastic operational field-dependent non-linearities of multilayer benders for resonant driving of bone-conduction hearing aids via a dual-actuators side-by-side device

A soft piezoceramic multilayer (ML) bending thin and thick dual-actuators side-by-side device, for use in bone-conduction hearing aids (BCHA), was characterized experimentally. For this purpose, first, quasi-static (0.1 Hz square wave) transverse displacements (deflections) of each ML bender were measured using a laser vibrometer (LV) under unimorph and bimorph drivings at various non-amplified peak (p) voltages (1Vp–13Vp). The resulting peak displacement–voltage curves were found clearly nonlinear, indicating a pronounced piezoelectric (PE) field-dependent nonlinearity (FDNL). Then, experimental electric impedance analyses were conducted for each ML bender, using an impedance analyzer, under 1Vp unimorph driving for the whole audio frequency range (20 Hz–20 kHz). The identified three modes’ superposed impedance magnitudes and phases showed their increasing deviation to the frequency scale left with increasing the modes’ order, indicating a possible softening elastic FDNL, as confirmed later by harmonic frequency response LV measurements at 1Vp. Finally, the harmonic frequency responses, for the above audio frequency range, were measured for each ML bender using the LV under unimorph and bimorph drivings at various non-amplified voltages (1Vp–10Vp). Here, beside the PE FDNL, the frequency response functions showed clearly a pronounced softening elastic FDNL. Consequently, with increasing input voltages, the identified two modes’ displacement amplitudes were nonlinearly increasing, while resonant frequencies and corresponding quality factors were nonlinearly decreasing. The three parameters-voltage curves were found to fit well (R2 ≈ 1) with high-order polynomials. These experimental results contradict the widely assumed displacement–potential linearity of middle ear or BCHA implantable PE vibrators of open literature available types (with or without shim bimorphs, unimorphs or diaphragms, and stacks) under similar targeted driving voltage and frequency ranges. The ML benders type quasi-static, frequency response and electric impedance characterizations, as well as the dual actuators side-by-side device, are proposed here for the first time for hearing aid applications.

Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus

<abstract><p>The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.</p></abstract>

ДЕКОМПОЗИЦІЯ МОДЕЛІ НЕСТАЦІОНАРНОГО ВИПАДКОВОГО ПРОЦЕСУ

In the work, the peculiarity of the task of constructing the decomposition of a non-stationary model of a random process is studied using the examples of the analysis of two types of real data. The peculiarities of the decomposition of the model of a non-stationary random process are considered. It consists in the separate definition of an arbitrary trend, a seasonal component and a stationary component. This allows solving, for example, forecasting tasks. This task means several steps ahead in predicting the values of a random process. Some problems that arise when using the autoregression-integrated moving average (ARIMA) model for the decomposition of the model of a random non-stationary process are shown. The results of research on real data are given. They demonstrate the process of decomposition of a model of a non-stationary random process. The analyzed classes of random non-stationary processes with a trend allow us to solve two main problems with acceptable accuracy. The first task involves the decomposition of a model of a random non-stationary process. It includes the calculation of an arbitrary trend, a quasi-periodic seasonal component and a stationary component of the process. The second task involves the actual forecasting of a non-stationary time series using the ARIMA model. In the ARIMA model, it is assumed that the trends are deterministic and are linear, quadratic or higher order polynomials. The seasonal component is periodic and has equal counts in each period, which can be removed through the subtraction period. If the seasonal component does not satisfies these requirements, then when it is removed by difference operators, problems arise and the ARIMA model may lose accuracy.

Seeking Velocity-Free Consensus for Multi-Agent Systems With Nonuniform Communication and Measurement Delays

This paper addresses the consensus control problem of nonuniform delayed multi-agent systems without velocity measurements, and further obtains an explicit expression for the delay margin condition (critical time delay). Firstly, the passivity-based control technique is introduced to design the nominal consensus controller, which can enhance the robustness of the system by injecting additional relative damping. Then, the system states and the virtual state of the damping term are incrementally generalized into a third-order equation. Based on the frequency domain method, this equation was transformed into a high-order polynomial. Thus, the consensus conditions of the delayed multi-agent system are obtained according to the monotonicity of the polynomial. In detail, the conditions are presented in the form of an upper bound on the time delays and inequality sets of controller parameters. Finally, the effectiveness of the proposed controller and its corresponding consensus conditions are verified by designing the numerical simulation experiments.

A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

<p style='text-indent:20px;'>Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.</p>

Research on Dynamic Path Exploration and Planning Method of Quadrotor Unmanned Aerial Vehicle in Complex Environment

As a research hotspot in the field of mobile robots in recent years, UAV(Unmanned Aerial Vehicle) has been improving its autonomous performance. Compared with traditional UAV, UAV has many advantages, such as vertical lifting, strong maneuverability and convenient operation. However, the dynamic path planning of quadrotor UAV, as a typical autonomous ability of mobile robot, has always been the research focus in the field of UAV. In this regard, this paper will try to study the dynamic path exploration planning method of quadrotor UAV in complex environment. In the complex environment model, the comparative simulation from the aspects of calculation time, calculation complexity and success rate shows that the selection of high-order polynomials can ensure high-order continuity, but it will also increase the calculation time and complexity. Compared with CF algorithm, the calculation time in this paper is 57% lower on average and the success rate is 17.6% higher on average, which shows that this algorithm is more suitable for online flight path planning of quadrotor. Comparatively speaking, the success rate of the other two methods decreases with the increase of environmental complexity, and the overall calculation time is longer.

A New GNSS TEC Neural Network Prediction Algorithm With the Data Fusion of Physical Observation

This work models and predicts anomalous jumps in GNSS total electron content (TEC) caused by geomagnetic disturbance or solar radiation by introducing F10.7 and regional Dst indices, which are used to characterize solar radiation and geomagnetic disturbance status, respectively. We then introduced long-distance constraint stations to restrain ionospheric variations during high solar activity, which reduced the influence of unmodelled factors on prediction accuracy. In our experimental work, Global Ionosphere Map (GIM) TEC data from 11 stations located at different latitudes were used for comparison during maximum (2014) and minimum (2018) of solar cycle 24. Sample data for >27 days were used for learning, and then ionospheric changes for the whole year were predicted using the sliding window method. The results showed that, the standard deviation for the global ionospheric prediction products could be guaranteed to be within 2 TECU. The proximity station constraint method proposed in this paper and the Dst high-order polynomial correction method during geomagnetic disturbance control the prediction error within -4.4 TECU, while the change of adjacent days is -51.4 TECU in 2014. The new algorithm reduced the TEC jump effect in predictions, and the prediction of ionospheric products provided by this algorithm can meet the basic requirements of ionospheric delay correction in high-precision positioning.

Increasing the flexibility of mixed models by using fractional polynomials

The class of regression models incorporating Fractional Polynomials (FPs), proposed by Royston and colleagues in the 1990’s, has been extensively studied and shown to be fruitful in the presence of nonlinearity between the response variable and continuous covariates. FP functions provide an alternative to higher-order polynomials and splines for dealing with lack-of-fit. Mixed models may also benefit from this class of curves in the presence of non-linearity. The inclusion of FP functions into the structure oflinear mixed models has been previously explored, though for simple layouts, e.g. a single covariate in the random intercept model. This paper proposes a general strategy for model-building and variable selection that takes advantage of the FPs within the framework of linear mixed models. Application of the methodto three data sets from the literature, known for violating the linearity assumption, illustrates that it is possible to solve the problem of lack-of-fit by using fewer terms in the model than the usual approach of fitting higher-order polynomials.

Mandatory lane-changing decision making and planning in emergency situations

Abstract By considering mandatory lane-changing as a collision avoidance measure, this paper presented the corresponding lane-change decision making and trajectory planning algorithm under an emergency scenario. Different from the traditional algorithm in which lane-change decision making and trajectory planning are separated, they are here coupled in a proposed algorithm and the related parameters are dynamically adjusted in the whole process. In addition to lane-change collision avoidance feasibility analysis, lane-change time instance and duration time are obtained by solving the constrained convex quadratic optimization programme. By taking lane-change time instance and duration time as inputs, the algorithm then proceeded to propose a kinematic model-based high-order polynomial lane change trajectory. By giving the simulation result compassion with the related algorithm, it is proved that the proposed algorithm has a good robustness and high efficiency.

Symmetries and Dynamics of Generalized Biquaternionic Julia Sets Defined by Various Polynomials

Higher-dimensional hypercomplex fractal sets are getting more and more attention because of the discovery of more and more interesting properties and visual aesthetics. In this study, the attention was focused on generalized biquaternionic Julia sets and a generalization of classical Julia sets, defined by power and monic higher-order polynomials. Despite complex and quaternionic Julia sets, their biquaternionic analogues are still not well investigated. The performed morphological analysis of 3D projections of these sets allowed for definition of symmetries, limit shapes, and similarities with other fractal sets of this class. Visual observations were confirmed by stability analysis for initial cycles, which confirm similarities with the complex, bicomplex, and quaternionic Julia sets, as well as manifested differences between the considered formulations of representing polynomials.

Thermomechanical buckling and wrinkling characteristics of softcore sandwich panels with CNT reinforced composite face sheets

This article explores the local and global stability of CNT reinforced composite (CNTRC) sandwich shell panels consist of softcore material. The shell panel's face sheets are reinforced with functionally graded nanotube particles. The effective properties of CNTRC face sheets are determined considering the extended rule of mixture technique, and a multi-layer approach is used to examine the local and global stability phenomena. The material properties of the face sheets are assumed to be temperature-dependent. The formulation account for the core compressibility effect by considering higher-order polynomial terms for transverse core displacement. The governing equations for each layer of the sandwich panel are derived by minimizing the total potential energy of the system. The Galerkin's technique is adopted to obtain a set of non-linear algebraic equations from the derived partial differential equations. Global buckling and wrinkling loads of sandwich shell panels under thermomechanical loadings are obtained by solving standard eigenvalue problem. The nonlinear responses for sandwich shell panels subjected to mechanical and thermal stresses are traced through the arc-length approach and iterative approach, respectively. The obtained responses elucidate the influence of CNT gradation, volume fractions and core thickness on the local and global stability behaviour of CNTRC sandwich shell panels.

Noise-Robust ISAR Translational Motion Compensation via HLPT-GSCFT

Translational motion compensation is a prerequisite of inverse synthetic aperture radar (ISAR) imaging. Translational motion compensation for datasets with low signal-to-noise ratio (SNR) is important but challenging. In this work, we proposed a noise-robust translational motion compensation method based on high-order local polynomial transform–generalized scaled Fourier transform (HLPT-GSCFT). We first model the translational motion as a fourth-order polynomial according to order-of-magnitude analysis, and then design HLPT-GSCFT for translation parameter estimation and parametric translational motion compensation. Specifically, HLPT is designed to estimate the acceleration and third-order acceleration of the translational motion and GSCFT is introduced to estimate the second-order acceleration. Both HLPT and GSCFT have a strong ability for cross-term suppression. In addition, we use a minimum weighted entropy algorithm to estimate the velocity of the translational motion, which can improve the noise robustness of the parameter estimation. Experimental results based on a measured dataset prove that the proposed method is effective and noise-robust.

A fully-implicit hybridized discontinuous Galerkin spectral element method for two phase flow in petroleum reservoirs

A hybridized discontinuous Galerkin method with high order spectral element discretization is presented for modeling multiphase immiscible 2D fluid flow in petroleum reservoir simulations. The time dependent nonlinear governing equations for reservoir simulation are presented, and a fully coupled implicit time stepping procedure is implemented using a high order W-Rosenbrock algorithm. For incompressible fluids considered in the present study, the governing PDE's include an elliptic PDE for the pressure field and a parabolic convective diffusion equation for the saturation (water volume fraction) in a two phase oil-water system. The PDE systems are coupled through the highly nonlinear relative permeability and capillary pressure functions relating the fluid velocities of individual species to the individual species pressures and saturations. With HDG/spectral element (HDGSEM) spatial discretization of the coupled governing equations, the system constitutes an index 1 differential algebraic equation. It is shown that the W-Rosenbrock algorithm in combination with the HDGSEM method yields an efficient and effective solution procedure which demonstrates robust, high order temporal accuracy.Numerous physical tests cases are presented demonstrating that the versatility of this approach is sufficient to address the challenges in modeling heterogeneous permeability fields and geologic features of arbitrary shape with discontinuous permeability functions. It is also capable of resolving the sharp time dependent displacement fronts in oil-water systems encountered in secondary oil recovery. It is demonstrated that the high order polynomial basis functions (up to 12th order) used in the HDGSEM algorithm have the ability to resolve complex geometry and achieve accurate results utilizing a simple Cartesian mesh for non-conforming geological features. Eliminating the need to re-grid and conform the mesh to the boundaries of different geological features greatly simplifies the workflow for petroleum engineers.

A high order positivity-preserving polynomial projection remapping method

In this paper, we develop a high-order positivity-preserving polynomial projection remapping method based on the L2 projection for the discontinuous Galerkin (DG) scheme. Combined with the Lagrangian type DG scheme and the rezoning strategies, we present an indirect arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method. By clipping precisely the intersections between the old distorted mesh and the new rezoned mesh, our remapping method is high-order accurate and has no limitation for the mesh movements, so it is suitable for the large deformable problems. A positivity-preserving limiter is also added for the physical variables in computational fluid dynamics without losing the original high-order accuracy and conservation. A multi-resolution weighted essentially non-oscillatory (WENO) limiter is adopted to overcome numerical oscillations and it can keep the original high-order accuracy in the smooth region. This WENO limiter combines several different degrees of polynomials which are the local L2 projections of the original polynomial with nonlinear weights calculated by their smoothness, therefore, it is highly parallel efficient. The properties of positivity-preserving, non-oscillation and high-order accuracy of the remapping method will be shown by a variety of numerical experiments on one, two and three dimensional unstructured meshes. The performance of the ALE-DG scheme with rezoning and remapping is also tested for the Euler system in one and two dimensions.

Power Optimization Distribution Method for Fuel Cell System Cluster Comprehensively Considering System Economy

In high-power applications, the use of fuel cell system cluster (FCSC) is increasingly valued. However, most of the existing strategies emphasis in optimizing a certain control objective (e.g., efficiency or consumption), but do not fully consider the impact of other factors such as stack degradation. To address this research gap, this article proposes a power allocation method related to the system economy, which considers multiple factors that affect the operating cost of FCSC. The efficiency and hydrogen consumption are analyzed, and two high-order polynomials are used to represent these two curves. Besides, considering that the fuel cell (FC) output characteristics are variable, an online identification algorithm is adopted to update the system parameters in real-time. Moreover, in the formulated cost function, stack degradation is also considered. What is more, to ensure that the FCs can operate in the high efficiency range as much as possible, we also add constraints to limit the FCs output power. The effectiveness of the proposed method has been verified on a hardware-in-the-loop simulation platform. Compared with the equal distribution strategy, the proposed method can effectively ameliorate the economy of the system. Furthermore, the proposed method has been experimentally validated and its real-time operation ability has been highlighted on an actual FCSC.

Root Locus of Some High-order Polynomials

For a third-order complex polynomial with its roots bounded by a circle, Marden’s theorem provides a useful way to evaluate the locus of its critical points. We construct a fourth-order complex polynomial by restricting its roots to ellipse and hyperbola. The locus of its critical points can be expressed by a formula. In addition, if the roots of complex polynomials are not restricted, their critical points will not occupy some regions.

NLFEM WITH HIGHER ORDER INTERPOLATION FUNCTION FOR EFFICIENT ANALYSIS OF IRREGULAR DOMAIN

This study proposes a new approach to developing a more efficient numerical technique by coupling the non-uniform rational B-spline (NURBS) with the higher-order polynomial basis functions under the framework of the Finite Element Method (FEM). In this technique, denoted as NURBS-Lagrange FEM (NLFEM), the NURBS basis functions are employed to represent the geometry of the problem domain, while the Lagrange interpolation functions are employed for the higher-order polynomial functions to interpolate the field variables. The NURBS is a mathematical model which provides a numerically stable algorithm to exactly represent all conic sections, and the Lagrange interpolation function allows for higher-order basis functions resulting in a faster convergence rate of analysis. By taking advantage of both models, the objective of this study is to propose a new approach, i.e., NLFEM, which can improve the accuracy of the analysis of the irregular domain with more efficient consumption of computer resources. A steady heat transfer formulation for a curved boundary problem is presented to demonstrate the validity and accuracy of the developed technique. The performance is verified against converged solutions obtained using higher-order FEM (FEM/Q9) and NURBS-Augmented FEM (NAFEM). The presented result shows that the NLFEM provides a favorable comparison against other methods. The converged solution is achieved 20% faster than the FEM/Q9 and 80 % faster than the NAFEM. This highlights the potential of the NLFEM as a new approach in numerical techniques for solving problems with irregular boundaries.

High-order Polynomial Comparison and Masking Lattice-based Encryption

The main protection against side-channel attacks consists in computing every function with multiple shares via the masking countermeasure. For IND-CCA secure lattice-based encryption schemes, the masking of the decryption algorithm requires the high-order computation of a polynomial comparison. In this paper, we describe and evaluate a number of different techniques for such high-order comparison, always with a security proof in the ISW probing model. As an application, we describe the full high-order masking of the NIST standard Kyber, with a concrete implementation on ARM Cortex M architecture, and a t-test evaluation.