High-order Polynomial Research Articles

Quasi-static modal analysis for reduced order modeling of geometrically nonlinear structures

When modeling dynamic structures, the quasi-static modal analysis (QSMA) approach seeks to very accurately resolve the force-displacement behavior of each individual mode but neglects dynamic modal coupling, while many other reduced order modeling (ROM) methods take an opposite view, capturing static and dynamic modal coupling but using polynomials that limit the fidelity with which the force-displacement behavior of each individual mode can be captured. This work contrasts these two approaches both theoretically and by applying them to geometrically nonlinear structures of varying complexity. The comparison reveals a potential deficiency in the QSMA approach in which the secant approximation that was typically used to estimate the effective natural frequency from the load-displacement curves is found to be inaccurate for some of the strongly nonlinear structures considered in this work. Conversely, the examples demonstrate that the low-order polynomial typically employed by ROM methods such as implicit condensation and expansion (ICE) is often insufficient to describe the force-displacement behavior with adequate fidelity when a single mode is used. These examples prompt the creation of a new method that is a hybrid between QSMA and ICE, which fits a high order polynomial to the modal response curve of a single mode. Several case studies show that the resulting single-mode ICE ROM, here dubbed a SICE-ROM, can capture the resonant behavior in the vicinity of the mode very accurately, but to do so the force-displacement relationship must be captured with extremely high accuracy. These comparisons highlight the importance of static coupling in these structures, suggesting that it is often far more important than dynamic coupling even in cases where the latter had been previously thought to be very important. The geometrically nonlinear structures studied in this work include FE models of a flat beam and an exhaust cover plate and a highly nonlinear curved beam where the Riks method is used to obtain the force-displacement curves through the snap-through regime.

Efficient projection kernels for discontinuous Galerkin simulations of disperse multiphase flows on arbitrary curved elements

In this work, we develop projection kernels for Euler-Lagrange point-particle simulations of disperse multiphase flows on arbitrary curved elements. These kernels are employed in a high-order discontinuous Galerkin framework for projecting the action of the particles to the Eulerian mesh. Instead of commonly used isotropic kernels, such as a Gaussian-type kernel, we construct an anisotropic polynomial-based smoothing function that preserves the compactness of the discontinuous Galerkin method on high-aspect-ratio elements and maintains an acceptable computational cost. At the same time, it mitigates inaccuracies associated with larger particles and numerical instabilities arising from the Dirac delta function and low-order kernels. Specifically, the geometric mapping of a physical element to the reference element is exploited to construct a kernel that is elliptical in 2D and ellipsoidal in 3D. We also develop a strategy to conserve interphase transfer near boundaries, particularly on curved elements. This strategy employs a high-order polynomial approximation to appropriately rescale the interphase source terms in an efficient manner. The compatibility of the proposed methodology with different types of meshes is investigated. We then apply it to a number of particle-laden flow configurations, including supersonic dusty flow over a flat plate, moving shocks interacting with clouds of particles, and hypersonic dusty flows over blunt bodies.

Abstract P191: A Large Cohort Study Predicting the Temporal Course and Rate of Improvement in Stroke Patients Admitted to Inpatient Rehabilitation Facilities

Introduction: We sought to predict the course and rate of functional improvement and length of stay (LOS) in patients with ischemic and hemorrhagic strokes admitted to inpatient rehabilitation facilities(IRF). Aim: To study the course and associating factors affecting functional outcomes among a large cohort of stroke patients admitted to IRFs. Methods: The cohort consists of stroke patients admitted to 5 IRFs in Houston, between 4/17-8/19. Higher order polynomial (quartic, cubic, quadratic) regressions were fitted to predict the temporal relationship between FIM score improvement and LOS, and based on goodness of fit statistics, cubic polynomial fit was selected. Effects of interactions were tested and later dropped from the final model because of non-statistical significance. Models were adjusted for age, gender, stroke type (hemorrhagic vs ischemic), and stroke severity based on NIHSS. Results: The demographics of patients are presented in table 1. Among 679 patients, the univariate analyses reflected that age (F=24.2, p <0.001), admission NIHSS score (F=67, p<0.001), stroke type (F=25.2, p <0.001), and admission FIM (F=283, P<0.001) were significant factors predicting IRF LOS. History of previous stroke, diabetes, hypertension, and hyperlipidemia did not have any significant effects on LOS. In multiple regression model, age at onset (β=-0.18, p<0.001), hemorrhagic vs ischemic stroke (β=3.02, p <0.01) were significant predictors of total FIM change score when adjusting for stroke severity and gender. Total FIM change score was positively correlated with LOS at IRF with a steep improvement in the first 15 days of IRF stay which plateaued afterwards in patients with 1-30 days of IRF stay (Fig. 2). Conclusion: Patient’s age, ischemic vs hemorrhagic stroke were the most significant predictors when deriving the relationship between total FIM score and LOS while adjusting for stroke severity and patient’s gender, which plateaued after 2 weeks of IRF stay.

Проблемы динамического моделирования ввода в действие жилых домов в постсоветских странах

The article is devoted to the description of procedures of economic and mathematical modeling of trends in the field of housing construction taking into account the peculiarities of various countries of the post-Soviet space. The results of analysis of well-known scientific publications on forecasting the dynamics of housing market indicators are presented. It has been shown that most domestic and foreign scientists as the most effective methods of modeling these indicators consider methods of analyzing time trends, in which polynomials of high (in some cases up to the fourth degree) order are used to approximate the available retrospective data. Other common approaches to solving this problem are the use of short-term forecasting based on moving average algorithms, as well as the use of the SARIMA model, which takes into account the trend and seasonal wave. The article shows that these methods do not fully take into account the profound changes in the construction complexes of the post-Soviet states caused by the significant structural transformation of their socio-economic systems. The authors proposed to use econometric models based on regressions with dummy variables to model the main indicators of housing construction, taking into account the complex structure of the external and internal environment of national construction complexes. It has been shown that in a significant number of practical situations, a fairly simple but effective way to take into account the components of the time series of the indicators under consideration in one complex model is to use the model of "change in growth (fall)" when choosing the time of the beginning (end) of a crisis situation as a characteristic point. The results of modeling the main indicators of housing construction for various countries of the post-Soviet space showed that the proposed model when constructing the medium-term forecast allows taking into account the situation component of the analyzed time series.

Nonlinear weighting process in ghost-cell immersed boundary methods for compressible flow

Computational challenges arise for the immersed boundary method (IBM) when dealing with compressible flow, where discontinuous and smoothly varying flow regions appear near the immersed boundary. The conventional ghost-cell IBM provides inaccurate results for smoothly varying regions when a low-order interpolation is used, or it suffers from a numerical instability for the discontinuous flow when a high-order interpolation is used. In this study, a new ghost-cell approach, nonlinear-weighted IBM (NWIBM), is developed to address the abovementioned issues. Inspired by a variety of weighted nonoscillatory interpolation methods, this work combines the high- and low-order polynomials during ghost-cell value estimation to enforce proper boundary conditions in the immersed boundary. A multidimensional smoothness indicator is designed to evaluate flow discontinuities. The nonlinear weighting obtained from the smoothness indicator makes the high-order polynomial dominant and the low-order polynomial negligible in the smoothly varying region, and vice versa in the discontinuous region. The enhanced performance and applicability of the proposed method were validated through various numerical tests in compressible flow. It was demonstrated that the NWIBM provides more stable and more accurate numerical solutions compared with conventional ghost-cell approaches.

An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method. Error estimates are provided, and the numerical results in function interpolation, integration and some benchmark problems in uncertainty quantification are used to compare different collocation schemes.

The structure of algebraic solitons and compactons in the generalized Korteweg–de Vries equation

We analyze the main properties of soliton solutions to the generalized KdV equation ut+[F(u)]x+uxxx=0, where the leading term F(u)∼quα, α>0, q∈R. The far field of such solitons may have three options. For q>0 and α>1 the analysis re-confirmed that all traveling solitons have “light” exponentially decaying tails and propagate to the right. If q<0 and α<1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated Fu and α>1 (e.g., the Gardner equation) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of Fu is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if Fu represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.

Phase-unwrapping method based on local polynomial models and a maximum a posteriori model correction.

Recently, a theory on local polynomial approximations for phase-unwrapping algorithms, considering a state space analysis, has been proposed in Appl. Opt.56, 29 (2017)APOPAI0003-693510.1364/AO.56.000029. Although this work is a suitable methodology to deal with relatively low signal to noise ratios observed in the wrapped phase, the methodology has been developed only for local-polynomial phase models of order 1. The resultant proposal is an interesting Kalman filter approach for estimating the coefficient or state vectors of these local plane models. Thus, motivated by this approach and simple Bayesian theory, and considering our previous research on local polynomial models up to the third order [Appl. Opt.58, 436 (2019)APOPAI0003-693510.1364/AO.58.000436], we propose an equivalent methodology based on a simple maximum a posteriori estimation, but considering a different state space: difference vectors of coefficients for the current high-order polynomial models. Specific estimations of the covariance matrices for difference vectors, as well as noise covariance matrices involved with the correct estimation of coefficient vectors, are proposed and reconstructions with synthetic and real data are provided.

Two-Variable Type 2 Poly-Fubini Polynomials

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

Stability analysis of delayed neural networks based on a relaxed delay-product-type Lyapunov functional

This paper deals with the stability analysis of delayed neural networks. A necessary and sufficient condition for positivity or negativity of the high-order polynomial over a finite interval is derived. An appropriate Lyapunov–Krasovskii functional (LKF), including a relax delay-product-type Lyapunov functional, is constructed. The necessary and sufficient condition of the polynomial inequality and a relaxed delay-product-type Lyapunov functional are employed to derive less conservative stability criteria. Finally, three commonly used numerical examples are presented to demonstrate the effectiveness and less conservativeness of the proposed method.

Discrete Phase Shifters-Based Hybrid Precoding for Full-Duplex mmWave Relaying Systems

This paper considers the hybrid precoding design with quantized phase shifters (PSs) for full-duplex millimeter-wave relaying systems, in which the relay node is equipped with PSs of different resolution levels. The joint optimization of relay hybrid precoding and discrete PSs is a high-dimensional non-convex problem with both higher-order polynomial and modulus constraints. To solve this challenging problem, the optimization function under the imperfect channel state information (CSI) is reformulated into a low-complexity norm-minimization form equation. Then, a relaxed algorithm based on the alternating direction method of multipliers and quantization steps is proposed to obtain the coupling approximate solutions for the above optimized equation. And the optimality of analytical solutions is guaranteed based on the imposed termination criterion and necessary conditions. Furthermore, given the requirement of arbitrary quantization, the trade-off between spectral efficiency (SE) and energy efficiency (EE) is investigated under the imperfect CSI. Numerical results verify that the hybrid precoding with low-resolution PSs can achieve excellent SE close to the full-resolution one but with a considerably lighter burden of energy consumption. Moreover, the transmission power plays a positive role in compensating quantization loss, but the excessive value will bring the reduction of EE.

High-precision shape approximation low-thrust trajectory optimization method satisfying bi-objective index

The shape approximation method has been proven to be rapid and practicable in resolving low-thrust trajectory; however, it still faces the challenges of large deviation from the optimal solution and inability to satisfy the specific flight time and fuel mass constraints. In this paper, a modified shape approximation low-thrust model is presented, and a novel constrained optimization algorithm is developed to solve this problem. The proposed method aims at settling the bi-objective optimization orbit involving the twin objectives of minimum flight time and low fuel consumption and enhancing the accuracy of optimized orbit. In particular, a transformed high-order polynomial model based on finite Fourier series is proposed, which can be characterized as a multi-constraint optimization problem. Then, a novel optimization algorithm is specifically developed to optimize the large-scale multi-constraint dynamical equations of shape trajectory. The key performance indicators of the index include minimum flight time, low fuel consumption and bi-objective optimization of the two. Simulation results prove that this approach possesses both the high precision achievable by numerical methods and low computational complexity offered by shape approximation techniques. Besides, the Pareto front of the fuel-time bi-objective optimization orbit is firstly introduced to analyze an intact optimal solution set. Furthermore, we have demonstrated that our proposed approach is appropriate to generate the preliminary orbit for pseudo-spectral method.

A numerical scheme for advection dominated problems based on a Lagrange interpolation

Applying conventional numerical techniques for solving advection dominated problems, especially in the presence of sharp gradients, produces inaccurate results. This paper presents a robust numerical scheme for approximating the advection equation. The proposed scheme is based on the method of characteristics and Lagrange interpolation. The accuracy and validity of the developed model have been verified through several test cases. The comparison between numerical outcomes and the exact solutions revealed the great capability of the proposed scheme to simulate the sharp gradient of the initial distribution and the rotating velocity field. Besides, it indicated deploying higher-order polynomials can lead to obtaining an accurate solution for the advection dominated problem. The stability analysis of different Lagrange interpolation polynomials has been carried out by comparing both amplification factor and relative phase error of the polynomials under different Courant numbers. The results can serve as a prominent indication for choosing a perfect algebraic polynomial interpolation to solve advection dominated problems.

A non-linear cubic spline layerwise time domain spectral finite element for the analysis of impacts on sandwich structures

The present work addresses the non-linear impact dynamic response of sandwich composite plates with Hertzian contact interaction between the impactor and the plate. Initially, a well-established Cubic Spline Layerwise Theory is employed for the precise description of complex through the thickness kinematics of sandwich composite plates. Furthermore, the expressions of Green Lagrange non-linear strain terms, along with the Deshpande-Fleck flow rule for polymeric crushable foams are presented for non-linear explicit impact dynamics. The material and geometric non-linearities as well as the layerwise mechanics, are integrated into a Time Domain Spectral Element which possesses the attributes of high-order Lagrangian polynomial shape functions and node collocation with the integration points due to the Gauss-Legendre-Lobbato integration scheme. The aforementioned numerical package provides a fast and accurate non-linear explicit integration dynamic module for the prediction of the impact response of sandwich plates, demonstrating the universality and validity of using Hertzian contact in sandwich and laminated composite structures. The proposed novel non-linear numerical model is correlated with experimental results and also with a high-fidelity three-dimensional solid finite element model in terms of accuracy and computationally efficiency.

New algorithm to calculate the cutter location points (CL) optimizing the chordal tolerance for 2 ½ axis milling of free form surfaces

Even though the high order polynomial can be used to describe a free form tool path for CNC milling, traditional linear interpolation (G01) is the most used today. As investigated in a previous work, different CAM systems calculate different points (CC) to mill the same surface under the same tolerance and cutting conditions. The current work presents a new approach to generate NC codes optimizing the use of the chordal tolerance for 2½ milling. The experimental results shown that the NC codes generated by the proposed method reduced in about 60% the milling time by reducing the CC points, without losing the accuracy.

Judder Suppression of Friction Transmission Device with &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;∞&lt;/sub&gt; Control

Judder is well known as a vibration that occurs in a dry clutch. In order to maintain the ride quality and quietness of dry clutches, research on judder analysis and mechanical design to suppress judder is actively pursued. This paper describes the design and implementation of robust control techniques as an approach to suppressing judder during gear shifting in traction-controlled transmissions. This paper is concerned with transmissions that exhibit strongly nonlinear frictional characteristics that can only be approximated by higher order polynomials. Specifically, the coefficient of friction varies significantly nonlinearly with the rotational speed during gear shifting. In this paper, the controller is designed by expressing this nonlinear property as the uncertainty of the frequency. Simulation and bench test results are presented to validate the validity of such an approach. The results show that judder-free shifting is possible with the proposed approach. Furthermore, the requirements of three controllers are presented by analyzing the robust controllers.

Does Education Really Cause Domestic Violence? Replication and Reappraisal of "for Better or for Worse? Education and the Prevalence of Domestic Violence in Turkey"

Using the 2008 Turkish National Survey of Domestic Violence against Women, Erten and Keskin (2018, henceforth EK), published in AEJ–Applied Economics, find that women's education increases the psychological violence and financial control behavior that they face from their partners. The authors also claim that the incidence of financial control behavior rises because women become more likely to be employed—supporting the instrumental violence hypothesis. They present this evidence only for women who live in what they call areas during childhood. EK's findings are an artifact of the way the authors create two key variables: the variable that classifies women into rural vs. urban childhood location and the variable measuring financial control behavior. EK misclassify the variable on childhood rural status. We find that once this variable is defined properly, the evidence for all their findings vanishes. EK make use of two of the three variables related to financial control behavior in the dataset. We show that using all three variables—or any other combination of two of the three variables—generates no evidence of a policy effect on financial control behavior. Even after ignoring these problems, the evidence EK provide in their paper is highly specification sensitive and the standard checks of the continuity assumption in RDD fail for their key outcomes. Moreover, the results obtained from the analysis of urban areas—not provided in their paper—are inconsistent with the instrumental violence hypothesis. In addition, EK claim—using RDD graphs with high-order polynomials but no estimation results—that the policy has no effect on men's schooling, contrary to the findings of the previous literature. However, we show a clear and substantial policy effect on men's schooling, resulting in the failure of their exclusion restriction assumption.

Rate-Distortion Optimized Encoding for Deep Image Compression

Deep-learned variational auto-encoders (VAE) have shown remarkable capabilities for lossy image compression. These neural networks typically employ non-linear convolutional layers for finding a compressible representation of the input image. Advanced techniques such as vector quantization, context-adaptive arithmetic coding and variable-rate compression have been implemented in these auto-encoders. Notably, these networks rely on an end-to-end approach, which fundamentally differs from hybrid, block-based video coding systems. Therefore, signal-dependent encoder optimizations have not been thoroughly investigated for VAEs yet. However, rate-distortion optimized encoding heavily determines the compression performance of state-of-the-art video codecs. Designing such optimizations for non-linear, multi-layered networks requires to understand the relationship between the quantization, the bit allocation of the features and the distortion. Therefore, this paper examines the rate-distortion performance of a variable-rate VAE. In particular, one demonstrates that the trained encoder network typically finds features with a near-optimal bit allocation across the channels. Furthermore, one approximates the relationship between distortion and quantization by a higher-order polynomial, whose coefficients can be robustly estimated. Based on these considerations, the authors investigate an encoding algorithm for the Lagrange optimization, which significantly improves the coding efficiency.

Homoclinic Breathing Wave Solutions and High-Order Rogue Wave Solutions of (3+1)-Dimensional Variable Coefficient Kudryashov-Sinelshchikov Equations

Based on the Hirota bilinear method, the homoclinic breathing wave solutions to the (3+1)-dimensional variable coefficient Kudryashov-Sinelshchikov (K-S) equations were obtained by means of the extended homoclinic breathing test method. Homoclinic breathing waves with different structures were given through selection of appropriate values for the parameters of the solution, and the rogue wave solutions to the equation were derived under the limit of the periodicity of the homoclinic breathing wave solutions. Finally, a special high-order polynomial was constructed as a test function to obtain the 1st-order and the 2nd-order rogue wave solutions.

Bucketed PCA Neural Networks with Neurons Mirroring Signals

The bucketed PCA neural network (PCA-NN) with transforms is developed here in an effort to benchmark deep neural networks (DNN's), for problems on supervised classification. Most classical PCA models apply PCA to the entire training data set to establish a reductive representation and then employ non-network tools such as high-order polynomial classifiers. In contrast, the bucketed PCA-NN applies PCA to individual buckets which are constructed in two consecutive phases, as well as retains a genuine architecture of a neural network. This facilitates a fair apple-to-apple comparison to DNN's, esp. to reveal that a major chunk of accuracy achieved by many impressive DNN's could possibly be explained by the bucketed PCA-NN (e.g., 96% out of 98% for the MNIST data set as an example). Compared with most DNN's, the three building blocks of the bucketed PCA-NN are easier to comprehend conceptually - PCA, transforms, and bucketing for error correction. Furthermore, unlike the somewhat quasi-random neurons ubiquitously observed in DNN's, the PCA neurons resemble or mirror the input signals and are more straightforward to decipher as a result.