Full Polynomial Research Articles

A Method to Identify the Nonlinear Stiffness Characteristics of an Elastic Continuum Mechanism

The humanoid robot David is equipped with a novel robotic neck based on an elastic continuum mechanism (ECM). To realize a model-based motion control, the six dimensional stiffness characteristics needs to be known. This letter presents an approach to experimentally identify the stiffness characteristic using a robot manipulator to deflect the ECM and measure the Cartesian wrenches and Cartesian poses with external sensors. A three-step process is proposed to establish Cartesian wrench and pose pairs experimentally. The process consists of a simulation step, to select a good model, a second step that extracts effective poses from workspace which are sampled experimentally and the third step, the pose sampling procedure in which the robot drives the ECM to these effective poses. A full cubic polynomial regression model is adopted based on simulation data to fit the stiffness characteristics. To extract the poses to be sampled in the experiments, two different approaches are evaluated and compared to ensure a well-posed identification. The identification process on the hardware is performed by using Cartesian impedance and inverse kinematics control in combination to comply with the physical constraints imposed by the ECM.

Discovery of statistical equivalence classes using computer algebra

Discrete statistical models supported on labeled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation which is a notion formalized in this paper. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.

An efficient multifidelity [formula omitted]-minimization method for sparse polynomial chaos

The Polynomial Chaos Expansion (PCE) methodology is widely used for uncertainty quantification of stochastic problems. The computational cost of PCE increases exponentially with the number of input uncertain variables (known as curse of dimensionality). Therefore, use of PCE for uncertainty quantification of industrial applications with large number of uncertain variables is challenging. In this paper, a novel methodology is presented for efficient uncertainty quantification of stochastic problems with large number of input random variables. The proposed method is based on PCE with combination of ℓ1-minimization and multifidelity methods. The developed method employs the ℓ1-minimization method to recover important coefficients of PCE using low-fidelity computations. The low-fidelity evaluations should be accurate enough to capture the physical trends well. After that the multifidelity PCE method is utilized to correct a subset of recovered coefficients using high-fidelity computations. A threshold parameter is defined in order to select the subset of recovered coefficients to be corrected. Two challenging analytical and CFD test cases namely, the Ackley function and the transonic RAE2822 airfoil with combined operational and geometrical uncertainties are considered to examine the performance of the methodology. It is shown that the proposed method can reproduce accurate results with much lower computational cost than the classical full Polynomial Chaos (PC), and ℓ1-minimization methods. It is observed that for the considered examples, the present method can achieve comparable accuracy with respect to the full PC and the ℓ1-minimization methods with significantly lower number of samples.

Kohnert tableaux and a lifting of quasi-Schur functions

We introduce the quasi-key basis of the polynomial ring which contains the quasi-Schur polynomials of Haglund, Luoto, Mason and van Willigenburg. We prove that stable limits of quasi-key polynomials are quasi-Schur functions, thus lifting the quasi-Schur basis of quasisymmetric polynomials to the full polynomial ring. The new tool we introduce for this purpose is the combinatorial model of Kohnert tableaux. We use this model to prove that key polynomials expand positively in quasi-key polynomials which in turn expand positively in fundamental slide polynomials introduced earlier by the authors. We give simple combinatorial formulas for these expansions in terms of Kohnert tableaux, lifting the parallel expansions of a Schur function into quasi-Schur functions into fundamental quasisymmetric functions.

Optimization investigation on configuration parameters of serrated fin in plate-fin heat exchanger based on fluid structure interaction analysis

The comprehensive performance of serrated fin in plate-fin heat exchangers (PFHEs) is numerically studied based on fluid structure interaction (FSI) analysis in this paper. Based on Full 2nd-Order Polynomial response surface (RS) and analysis of variance (ANOVA), the effects and second order interaction effects of the fin height, fin space, fin thickness and fin interrupted length on heat transfer, flow resistance and stress are quantitatively and thoroughly assessed. The results show that the heat transfer is the most sensitive to fin interrupted length, and the flow resistance and the maximum stress are the most sensitive to fin thickness. The interaction effect of the fin space and fin interrupted length is the strongest when the j factor is set as objective function, while the interaction effect of the fin space and fin thickness is the strongest when the f factor or maximum stress is set as objective function. Based on Full 2nd-Order Polynomial RS, Multi-Objective Genetic Algorithm (MOGA) is applied to optimize the fin structure comprehensively, with enhancing heat transfer, decreasing pressure drop and stress set as objectives. To demonstrate the effectiveness of optimized structures, a comprasion between the original design and optimized structures is performed. The results show that the JF factor of optimized structures 2 and 3 increases by 23.0% and 19.7% respectively, and the maximum stress decreases by 5.8% and 15.2% respectively. The MOGA offers theorectical guidance for optimization design of PFHE.

Bernstein–Walsh Theory Associated to Convex Bodies and Applications to Multivariate Approximation Theory

We prove a version of the Bernstein–Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full polynomial space associated to a convex body P. As a consequence, we validate and clarify some observations of Trefethen in multivariate approximation theory.

Probabilistic assessment of reinforced soil wall performance using response surface method

The paper demonstrates the use of the response surface method (RSM) to carry out probabilistic assessment of selected performance features of geosynthetic-reinforced segmental retaining walls under operational conditions. The method substantially reduces the number of Monte Carlo simulations required to carry out probabilistic analysis of numerical models with a large number of problem parameters. A numerical model is verified using performance data from three physical full-scale laboratory walls, and is then used to generate a large synthetic database of numerical results for maximum wall facing deformation, maximum reinforcement connection strains and maximum reinforcement strains in the backfill. Closed-form solutions (performance functions) are formulated using a full quadratic polynomial, which is a common feature of the RSM. The coefficient values of the closed-form solutions are found using a least squares method to give good agreement between the RSM equations and numerical results. The RSM equations for each performance function are used to carry out Monte Carlo simulations and the results presented as cumulative distribution functions. The resulting cumulative distribution functions can be used for quantitative reliability-based assessment of wall facing deformations and reinforcement strains under operational conditions for walls matching the physical test walls used in the initial numerical model verification.

Effects of operational and geometrical uncertainties on heat transfer and pressure drop of ribbed passages

In this paper uncertainty quantification of turbulent flow and heat transfer through ribbed channels of gas turbine blades is presented and discussed. The uncertainty quantification method is first validated using the Ishigami function and then applied to turbulent flow and heat transfer through a ribbed pipe. Combination of four operational and geometrical uncertain variables (Re, Pr, P/h and D/h) are considered. Results for local Nusselt number, average Nusselt number and non-dimensional pressure drop using a polynomial chaos expansion of order 3 are presented and discussed. Uncertainty quantification results for the local Nusselt number showed that the regions located on top and immediately after the rib are more sensitive to the uncertain input variables. To tackle the curse of dimensionality, the sparse polynomial chaos expansion is further employed in this study. Results for quantities of interest showed that the sparse polynomial chaos method reduces the computational cost by at least 50% in comparison to the full polynomial chaos expansion method. Furthermore, results show that the sparse PCE works better for the non-dimensional pressure drop than the average Nusselt number. Results of sensitivity analysis for non-dimensional pressure-drop showed that, in agreement with the empirical correlations, this quantity is independent of Reynolds number. Also, results of sensitivity analysis for local Nusselt number showed that generally the major variability of Nusselt number is due to the blockage ratio.

Precision of full polynomial response surface designs on models with missing coefficients

The precision of using full polynomial response surface designs on models with missing coefficients (reduced models) is studied using efficiency measures. The loss in D- and G-efficiency of constructed first-order exact designs is minimized for the model with missing interaction coefficient. However, higher losses in D- and G-efficiency are recorded when constructed second-order exact designs are used on the model with missing interaction coefficient with few exceptions showing preferences for using the designs on the reduced model. Lower condition numbers are observed for the designs under the first-order reduced models thus indicating that the N-point exact designs are closer to being orthogonal for the reduced model than for the full model. Perfect orthoganality is achieved at design sizes 4 and 8. In fact, N-point exact designs of multiples of N=4 show perfect orthoganality when defined either for the full or reduced first-order models. In comparison to a design with perfect orthoganality, the second-order designs are far from being orthogonal.

Polynomial Harmonic Decompositions

Abstract For real polynomials in two indeterminates a classical polynomial harmonic decomposition (cf. (1) below) is extended from square-norm divisors to conic ones. The main result is then applied to obtain a full polynomial harmonic decomposition, and to solve a Dirichlet problem with polynomial boundary data.

Efficient high degree polynomial root finding using GPU

Polynomials are mathematical algebraic structures that play a great role in science and engineering. Finding the roots of high degree polynomials is computationally demanding. In this paper, we present the results of a parallel implementation of the Ehrlich–Aberth algorithm for the root finding problem for high degree polynomials on GPUs using CUDA and on multi-core processors using OpenMP. The main result we achieved is to solve high degree polynomials (up to 1,000,000) efficiently. We also compare the Ehrlich–Aberth method and the Durand–Kerner one on both full and sparse polynomials. Accordingly, our second result is that the first method is much faster and more efficient. Last, but not least, an original proof of the convergence of the asynchronous implementation for the EA method is produced.

Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams

We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize the quasi-shuffle product and shuffle product, respectively. We use this to give a Littlewood–Richardson rule for expanding a product of Schubert polynomials into fundamental slide polynomials and to give formulas for products of Stanley symmetric functions in terms of Schubert structure constants.

Experimental Optimization of High-Strength Self-Compacting Concrete Based on D-Optimal Design

AbstractBecause of the high variability in the material content of high-strength self-compacting concrete (HSSCC), a huge number of trial sets (experiments) are often essential to find the optimum mixture proportions needed to achieve desired HSSCC properties. In the current investigation, an effort was made to considerably reduce the number of such experiments and to study the influence of different mixture parameters on the properties of HSSCC using a response surface method coupled with D-optimal design of experiment (DOE). In this study, water-binder ratio, cement content, fine-aggregate percentage, fly ash content and superplasticizer content were chosen as input variables, with compressive strength, passing ability, segregation, and manufacturing cost as responses. For each response, a full quadratic model and second-order polynomial equation was developed using central composite design (CCD) and D-optimal design. Analysis of variance (ANOVA) was carried out to check adequacy. Responses were analyze...

Supercritical fluid extraction of omega-3 from Dracocephalum kotschyi seed oil: Process optimization and oil properties

This paper describes the extraction of linolenic acid (ω-3) from Dracocephalum kotschyi seed using supercritical fluid extraction (SFE) for the first time. Response surface methodology (RSM) based on central composite design (CCD) was used to determine the effects of pressure (100–220bar), temperature (35–55°C), particle size (0.15–1.35mm) and dynamic time (10–130min) on the yield of linolenic acid (ω-3) in supercritical carbon dioxide (SC-CO2). The RSM analysis showed that the experimental data was adequately fitted to a full quadratic polynomial model. The results were evaluated using the different statistical analyses such as determination coefficient (R2=0.9795), adjusted determination coefficient (R2adj.=0.9703), predicted determination coefficient (R2pred.=0.9466) and absolute average relative deviation (AARD=2.81) to investigate statistical significance of the developed polynomial equation. For all experiments, fatty acid composition and also properties of oil such as total phenol, antioxidant activity, thermo-gravimetric (TG) analysis as well as oil stability were determined. Furthermore, qualitative analysis of organic compounds was carried out by Fourier transform infrared (FTIR) spectroscopy. According to the obtained results, high quality oil extracted via SC-CO2 technique may find potential applications in medical, pharmaceutical and food industries.

Surrogate modeling of a 3D scramjet-powered hypersonic vehicle based on screening method IFFD

The emphasis of this paper lies in the development of an efficient approach to reproduce the behaviors of the scramjet-powered hypersonic system with high fidelity. The modeling of the dual-mode scramjet powered hypersonic vehicle dynamics with shock interaction, Ram-to-Scram transition, and finite rate chemistry reaction is firstly introduced. The structure of surrogate model is identified with the implement of iterative fractional factorial design (IFFD). In order to declare the reliability of the surrogate models, ν-gap metric is applied to distinguish the difference among these surrogate models in terms of closed-loop performance. The results show that the influence of Mach number on the aerodynamics should not be overlooked, and the effect of propulsion system to the aerodynamic pitch moment is dramatic. Further, the partial Kriging model appears to have the closest plants throughout the flight envelope compared with the full Kriging model and polynomials model. Nevertheless, considering the briefness of analytical expression, the polynomials model may be an alternative approach for design-oriented modeling.

Study on the mapping of dark matter clustering from real space to redshift space

The mapping of dark matter clustering from real space to redshift space introduces the anisotropic property to the measured density power spectrum in redshift space, known as the redshift space distortion effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to indefinite cross correlations between the density and velocity fields, and the Finger-of-God effect due to the randomness of the peculiar velocity field. Whilst the full higher order polynomials remain unknown, the other systematics can be controlled consistently within the same order truncation in the expansion of the mapping formula, as shown in this paper. The systematic due to the unknown non-linear density and velocity fields is removed by separately measuring all terms in the expansion directly using simulations. The uncertainty caused by the velocity randomness is controlled by splitting the FoG term into two pieces, 1) the ``one-point" FoG term being independent of the separation vector between two different points, and 2) the ``correlated" FoG term appearing as an indefinite polynomials which is expanded in the same order as all other perturbative polynomials. Using 100 realizations of simulations, we find that the Gaussian FoG function with only one scale-independent free parameter works quite well, and that our new mapping formulation accurately reproduces the observed 2-dimensional density power spectrum in redshift space at the smallest scales by far, up to k∼ 0.2 Mpc−1, considering the resolution of future experiments.

Sorption of DDT from synthetic aqueous solution by eucalyptus barkusing response surface methodology

Low-cost biosorbents are considered as a promising alternative for conventional adsorbents to remove organic pollutant from water and waste water. This study examines the efficiency of Eucalyptus bark as adsorbent for the adsorption of 1,1,1-trichloro-2,2-bis(p-chlorophenyl)ethane (DDT). Four parameter pH, contact time, initial concentration and dosing are optimized by response surface methodology. A second-order full polynomial approximation model was applied to evaluate the results of the optimization studies. To achieve this, 30 experimental runs were developed according to central composite design and analyzed with the aid of a software package. Result shows 93.38% of DDT adsorption at optimum conditions. The quadratic model was found to be maximum adjusted R20.7285453, and predicted R20.589665values excluding cubic model.

A novel Rys quadrature algorithm for use in the calculation of electron repulsion integrals

The half-range Rys polynomials, Rn(x), are orthonormal with respect to weight function w(x)=e-cx2 on the interval x∈[0,1] and defined with the set of coefficients, αn and βn, in the three term recurrence relation for the polynomials. Full range Rys polynomials, Jn(x), are orthonormal with respect to w(x)=e-cx2 on the interval x∈[-1,1]. They are defined with the set of β̂n recurrence coefficients as α̂n=0. The Gauss-Rys quadrature defined with the Rys polynomials are used to evaluate electron repulsion integrals in quantum chemistry computer codes. The present paper proposes a new algorithm for the efficient computation of the Rys quadrature weights and points versus the parameter c in the weight function. The method is based on the full range Rys polynomials and a novel method for the calculation of the positive quadrature points and related weights.

DEFINITION OF CALCULATION DISTURBANCE FOR AUTOMATIC CONTROL SYSTEMS PRESSURE PUMPING STATIONS TRUNK OIL PIPELINES

The automatic control system (ACS) pressure pumping station must ensure testing of the estimated disturbance associated with the shutdown of the pump at the next pump station (PS), or start one pump at the previous PS [1,2]. Estimated disturbance occurs as the result of several processes: - the occurrence of the initial perturbation, education waves flow and pressure at the station, where there was a start-up or shutdown of the pump; - the spread of this disturbance through the pipeline section between the point of origin of the perturbation and the station where the ACS pressure and attenuation as we move, reducing the amplitude and rise time. The wave, which reached ACS stantsiis pressure is called the incoming wave; - the interaction of the incident wave with a pumping station. The regulatory system should have such a speed that it creates a pressure wave had the opposite sign of the slope of the front and not less than the rise time estimated disturbance. In this paper, the expression for determining the initial disturbance that occurs when the pump is turned off at an intermediate transfer station without the tank farm. Pressure pump performance approximated full second-order polynomial (all the coefficients are different from zero). To calculate the attenuation slope coefficient of wave front defined linearization of unsteady flow of fluid through the hydraulic gradient value of the derivative of consumption. To determine the minimum length of the decay is taken to be 50 km. The interaction of the incoming wave is formed with a pumping station to the station reflected pressure wave and flow, moving along the conduit in a direction opposite to the direction of motion of the oncoming wave. The relationship of the pressure waves and flow rate determined by the formula «Zhukovsky». The equation for determining the slope of the front of the reflected wave at a certain steepness of the wave front the pressure considering the nonlinearity of the discharge characteristics of the station. Calculations to determine the calculated perturbations for pipeline DN 500 - 1200 mm.

Testing Fit Patterns with Polynomial Regression Models

Fit hypotheses, also labeled ‘congruence’, ‘discrepancy’, or ‘congruity’ hypotheses, contain the notion that an outcome is optimal when two predictor variables match well, while incongruent/discrepant combinations of the predictors lead to a suboptimal outcome. Previous statistical frameworks for analyzing fit hypotheses emphasized the necessity of commensurable scales, which means that both predictors must be measured on the same content dimension and on the same numerical scale. In some research areas, however, it is impossible to achieve scale equivalence, because the predictors have to be measured with different methods, such as explicit attitudes (e.g., questionnaires) and implicit attitudes (e.g., reaction time task). In this paper, I differentiate numerical congruence from fit patterns, a concept that does not depend on the notion of commensurability, and hence can be applied to fit hypotheses with incommensurable scales. Polynomial regression can be used to test for the presence of a fit pattern in empirical data. I propose several new regression models for testing fit patterns which are statistically simpler and conceptually more meaningful than a full polynomial model. An R package is introduced which provides user-friendly functions for the computation, visualization, and model comparison of several fit patterns. An empirical example on implicit/explicit motive fit demonstrates the usage of the new methods.