Numerical Coefficients Research Articles

On the Theoretical Prediction of Microalgae Growth for Parallel Flow

The established microalgae growth models are semi-empirical or considerable fitting coefficients exist currently. Therefore, the ability of the model prediction is reduced by the numerous fitting coefficients. Furthermore, the predicted results of the established models are dependent on the size of the photobioreactor (PBR), light intensity, flow and concentration field. The growth mechanism of microalgae has not clearly understood in PBR cultivation. It is difficult to predict the microalgae growth by theoretical methods, owing to the aforementioned factors. We developed an exploratory bridging microalgae growth model to predict the microalgae growth rate in PBRs by using the nondimensional method which is effectively in fluid dynamics and heat transfer. The analytical solution of the growth rate was obtained for the parallel flow. The nondimensional growth rate expressed as function of Reynolds number and Schmidt number, which can be used for arbitrary parallel flow due to the solution was expressed as nondimensional quantities. The theoretically predicted growth rate is compared with the experimentally measured microalgae growth rate on the order of magnitude. The nondimensional method successfully applied to the microalgae growth problem for the first time. The general nondimensional solution can unify the numerous experimental data for different laboratory conditions, and give a direction for the disorder of the microalgae growth problem. The nondimensional solution may be useful to explain the growth mechanism of microalgae and design large-scale PBRs for microalgae biofuel production. The significance of the work is to give a theoretical foundation and methodology of biological theory of microalgae growth.

ОЦІНЮВАННЯ ВИМІРЮВАЛЬНИХ СИГНАЛІВ НА ОСНОВІ ТРИКУТНОЇ ЧАС-ІМПУЛЬСНОЇ МОДУЛЯЦІЇ ДЛЯ КОНТРОЛЮ ТЕХНІЧНОГО СТАНУ ЗАСОБІВ ЗВ’ЯЗКУ ТА ПЕРЕДАЧІ ДАНИХ

A method is proposed for triangular modulation of data on the temporal parameters of the carrier, followed by narrow-band filtering of measuring signals for monitoring the technical condition of communication and transmission facilities, which, compared with the binary modulation method, provides a decrease in the harmonic coefficient of the envelope at approximately the same, fairly small error in setting the amplitude modulation coefficient of the amplitude-modulated signal. The analysis of the obtained relations for the amplitude modulation coefficient of measuring signals with a triangular modulation law showed the following. Unipolar and bipolar carrier pulse sequences lead to the same values of the amplitude modulation coefficient of the generated amplitude-modulated signal, but with a bipolar carrier pulse sequence, the amplitude of the output signal doubles with the same amplitude of carrier pulses (without pulses). With phase modulation, the value of the amplitude modulation coefficient is much less than with width modulation. At the same time, two-way width modulation provides twice the value of the amplitude modulation coefficient than the modulation of the front or cutoff of the carrier pulses. This statement, however, is valid only for small values of the modulation coefficients proportional to the modulation coefficients of the temporal parameters of the carrier pulses with the corresponding numerical coefficients depending on the type of modulation. If the amplitude modulation coefficients are not small, then they are determined by more complex dependencies, but also more accurate, allowing to reduce the methodological error. In this case, for the required values of the amplitude modulation coefficient, the time modulation parameters can be calculated in advance or using the built-in microprocessor and stored in the memory of the measuring device (calibrator).

INTERPOLATION OF TABULAR FUNCTIONS FROM ONE INDEPENDENT VARIABLE USING THE TAYLOR POLYNOMIAL

A method of local interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree in arbitrarily located interpolation nodes has been developed. This makes it possible to calculate intermediate values of tabular functions between interpolation nodes. The conducted analysis of the latest research and publications in the field of interpolation of tabular functions showed that the main part of the research is a strict theory of interpolation, i.e. clarification of its fundamental mathematical provisions. Some features of the interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree are considered, namely: the solution algorithm and mathematical formulation of the interpolation problem are given; its formalized notation is given, as well as the matrix notation of interpolation procedures for certain values of the argument. A scalar algorithm for solving the problem of interpolation of tabular functions from one independent variable using the Taylor polynomial of the 2nd, 3rd, and 4th degrees has been developed. The simplicity and clarity of this algorithm is one of its advantages, but the algorithm is inconvenient for software implementation. The mathematical formulation of the problem of interpolation of tabular functions in terms of matrix algebra is given. The interpolation task is reduced to performing the following actions: based on the values of nodal points known from the table, it is necessary to calculate the Taylor matrix of the nth degree; based on the function values specified in the table a column vector of interpolation nodes should be formed; solve a linear system of algebraic equations, the root of which is the numerical coefficients of the Taylor polynomial of the nth degree. A method of calculating the coefficients of the interpolant, given by the Taylor polynomial of the nth degree for one independent variable has been developed. The essence of the method reduces to the product of the matrix, inverse of the Taylor matrix, which is determined by the nodal points of the tabular function, by a column vector containing the values of the interpolation nodes. Specific examples demonstrate the peculiarities of calculating the interpolant coefficients of the 2nd, 3rd and 4th degrees for one independent variable, and for each of them the interpolated value of the function at a given point is calculated. Calculations were performed in the Excel environment, which by analogy can be successfully implemented in any other computing environment.

A Comparative Study for Compressible Turbulence Models

The incompressible models for the pressure-strain correlation are unable to correctly predict the turbulence flows evolving with significant compressibility. Huang and Fu use a damping function of the turbulent Mach number to modify two numerical coefficients of the incompressible model for the pressure strain developed by Launder, Reece and Rodi. This model predicts the spreading rate and the shear stress behavior in compressible turbulent mixing well. However, the model does not show the well-known compressibility effects on the compressible homogenous shear flow. In the present work, the model of Huang-Fu is revised, all resulting model coefficients become dependent on the turbulent Mach number, the gradient Mach number and the convective Mach number. The proposed model is tested in different compressible turbulent homogeneous shear flow and mixing layers cases. In general, the predicted results from the proposed model are in an acceptable agreement with DNS and experiment data.

Model Order Diminution of Discrete Interval Systems Using Kharitonov Polynomials

In this research proposal, diminution of higher order (HO) discrete interval system (DIS) is accomplished by utilizing Kharitonov polynomials. The DIS is firstly, transformed into continuous interval system (CIS). The Markov-parameters (MPs) and time-moments (TMs) are exploited for determination of approximated models. The ascertainment of model order diminution (MOD) of DISs is done by Routh-Padé approximation. The Routh table is utilized to obtain the denominator of approximated model. The unknown numerator coefficients of desired approximated model are determined by matching MPs and TMs of DISs and desired model. This whole procedure of MOD is elucidated with the help of one test illustration in which third order system is reduced to first order model as well as second order model. To prove applicability of the proposed method, impulse, step and Bode responses are plotted for both system and model. For relative comparison, time-domain specifications of proposed model are tabulated for both upper and lower limits. Further, performance indices are specified for dissimilarities between responses of system and model. The obtained results depict the effectiveness and efficacy for the proposed method.

Functionalized Data Operator Model for System Analysis and Forecasting

A deconvolution data processing is developed for obtaining a Functionalized Data Operator (FDO) model that is trained to approximate past and present, input-output data relations. The FDO model is designed to predict future output features for deviated input vectors from any expected, feared of conceivable, future input for optimum control, forecast, or early-warning hazard evaluation. The linearized FDO provides fast analytical, input-output solution in matrix equation form. If the FDO is invertible, the necessary input for a desired output may be explicitly evaluated. A numerical example is presented for FDO model identification and hazard evaluation for methane inflow into the working face in an underground mine: First, a Physics-Based Operator (PBO) model to match monitored data. Second, FDO models are identified for matching the observed, short-term variations with time in the measured data of methane inflow, varying model parameters and simplifications following the parsimony concept of Occam’s Razor. The numerical coefficients of the PBO and FDO models are found to differ by two to three orders of magnitude for methane release as a function of short-time barometric pressure variations. As being data-driven, the significantly different results from an FDO versus PBO model is either an indication of methane release processes poorly understood and modeled in PBO, missing some physics for the pressure spikes; or of problems in the monitored data fluctuations, erroneously sampled with time; or of false correlation. Either way, the FDO model is originated from the functionalized form of the monitored data, and its result is considered experimentally significant within the specified RMS error of model matching.

3D full-time anisotropic TEM modelling using a mixed BDF2/SAI method

Abstract Transient electromagnetic (TEM) data are affected by resistivity anisotropy, which should be considered in 3D modelling. The influence of anisotropy on full-time response is the main focus of this research. For spatial discretisation of an anisotropic model, the mimetic finite volume approach was applied. The accuracy of the shift-and-invert (SAI) Krylov subspace approach and the two-step backward differentiation formula (BDF2) for modelling 3D full-time electromagnetic data has been demonstrated. However, both algorithms require time-consuming calculations. The SAI technique requires a number of projection subspace constructions, whereas the BDF2 algorithm necessitates numerous coefficient matrix decompositions. We proposed a novel mixed BDF2/SAI algorithm in this paper, which combines the advantages of the two algorithms. The on-time response is computed using BDF2, while the off-time response is computed using the SAI-Krylov subspace method. The forward results of a 1D model with a half-sine waveform demonstrated that the new algorithm is accurate and faster than both the BDF2 algorithm and the SAI algorithm. During the full-time period, the forward results of a 3D anisotropic model with half-sine waveform show that abnormal responses can be induced. It was shown that the relative abnormal of ${{{\bf b}}_{\boldsymbol{z}}}$ is higher during the on-time period, while the relative abnormal of $\partial {{{\bf b}}_{\boldsymbol{z}}}/\partial t$ is higher during the off-time period. Furthermore, the change in relative anomaly is more obvious as the anisotropic block rotates around the x-axis. And the larger the rotation angle, the larger the relative anomaly.

New Relation between the Coefficients of a Rational Function and the intersection points with Oblique Asymptotes in a Vector Equations

It’s clear that Vieta’s formula relates the coefficients of polynomial to the sum and product of their roots. In this paper for the first time, we introduce a vector equations relate the coefficients of numerator and denominator of rational functions to the sum and product of intersection points with oblique asymptotes. Furthermore, previously we learned how to find Oblique asymptote of rational functions by Long division, but in this paper we introduce new easier method for finding oblique asymptotes of rational functions with the aid of determinant of a matrix.

A resolution criterion based on characteristic time-scales for MHD simulations of molecular clouds

ABSTRACT We investigate the effect of numerical magnetic diffusion in magnetohydrodynamic (MHD) simulations of magnetically supported molecular clouds. To this end, we have performed numerical studies on adaptive mesh isothermal simulations of marginally subcritical molecular clouds. We find that simulations with low and intermediate resolutions collapse, contrary to what is theoretically expected. However, the simulation with the highest numerical resolution oscillates around an equilibrium state without collapsing. In order to quantify the numerical diffusion of the magnetic field, we ran a second suit of current-sheet simulations in which the numerical magnetic diffusion coefficient can be directly measured and computed the corresponding diffusion times at various numerical resolutions. On this basis, we propose a criterion for the resolution of magnetic fields in MHD simulations based on requiring that the diffusion time to be larger than the characteristic time-scale of the physical process responsible for the dynamic evolution of the structure.

Extracting bigravity from string theory

The origin of the graviton from string theory is well understood: it corresponds to a massless state in closed string spectra, whose low-energy effective action, as extracted from string scattering amplitudes, is that of Einstein-Hilbert. In this work, we explore the possibility of such a string-theoretic emergence of ghost-free bimetric theory, a recently proposed theory that involves two dynamical metrics, that around particular backgrounds propagates the graviton and a massive spin-2 field, which has been argued to be a viable dark matter candidate. By choosing to identify the latter with a massive spin-2 state of open string spectra, we compute tree-level three-point string scattering amplitudes that describe interactions of the massive spin-2 with itself and with the graviton. With the mass of the external legs depending on the string scale, we discover that extracting the corresponding low-energy effective actions in four spacetime dimensions is a subtle but consistent process and proceed to appropriately compare them with bimetric theory. Our findings consist in establishing that string and bimetric theory provide to lowest order the same set of two-derivative terms describing the interactions of the massive spin-2 with itself and with the graviton, albeit up to numerical coefficient discrepancies, a fact that we analyze and interpret. We conclude with a mention of future investigations.

Design of Controller for Large-Scale Uncertain Systems Adopting Affine Arithmetic

In this article presents a new controller design technique for large scale uncertain systems. The new controller is designed through a reduced order system from certain high order system. In the projected reduction method, the numerator coefficients are obtained with γ-table while the denominator polynomial is achieved with δ - table. An optimised reduced order model is derived with minimum value of ISE. In this proposed method gives better stability of the reduced order model, if consider the original stable high order system. A PID controller is designed for the high order original systems through its proposed reducer order model. Consider several numerical examples are available in the literatureto illustrate the usefulness of the proposed method andit gives sufficientoutcome.

Applying the scalability apparatus to estimate the thermal efficiency of a single finned tube

The paper explores the possibility of scaling the integral parameters of the cooling system operation using the working elements with the length from 0.01 to 0.5 m. Numerical solution of the conjugate problem of external aeromechanics, internal hydrodynamics and heat exchange is carried out. Parametric studies of the cooling process and aerodynamic resistance of finned tubular elements of various lengths are performed. As a result of generalization and unification of the computational experimental data, a numerical coefficient has been obtained to calculate the necessary integral characteristics for a cooling element of the assigned length.

A study on the geometric effects of a concrete filling body in remaining roadways with fully mechanised caving

Abstract To address the problem of the concrete filling body (CFB) force failing to reach test strength in remaining roadways, the weakening effects due to aspect ratio and dimensional parameters on the actual CFB strength were investigated in this study. The geometric effects of CFB (including hoop and size effects) as well as the geometric effect coefficient determination method were analysed. Through laboratory tests and PFC numerical simulations, the hoop and size effect coefficients of the CFB in the Gaohe Coal Mine were studied. Furthermore, the calculation equations of actual strength and bearing capacity of the CFB were derived. Regarding the filling body failure and coal deformation in the remaining roadway located at the No. W1319 working face, the actual bearing capacity of CFB and surrounding rock stability during secondary exploitation were theoretically studied. The investigation suggests the adoption of a grouting reinforcement scheme for surrounding rock. The field applications performed have demonstrated that the deformation control effect in the remaining-roadway surrounding rock was effectively improved during second mining and the filling body beside the roadway suffered no additional damage. Studying the geometric effect of CFB can provide some theoretical guidance and industrial significance to accurately identify the filling body strength and reduce the failure risk of surrounding rock in remaining roadways.

Numerical Study on the Hydrodynamic Characteristics of a Double-Row Floating Breakwater Composed of a Pontoon and an Airbag

By adding a cylindrical airbag on the leeward side of a cuboid pontoon, a new-type double-row floating breakwater is designed to improve the wave attenuation performance, and its hydrodynamic characteristics are studied through numerical simulations. First, based on the smoothed particle hydrodynamics (SPH) method, a numerical model used to simulate the interaction between waves and moored floating bodies is built. The fluid motion is governed by the Navier–Stokes equations. The motion of the floating body is computed according to Newton’s second law. The modified dynamic boundary condition is employed to treat the solid boundary. The lumped-mass method is adopted to implement the mooring system. Then, two physical model experiments on waves interaction with cuboid and dual cylindrical floating pontoons are reproduced. By comparing the experimental and numerical wave transmission coefficients, wave reflection coefficients, response amplitude operators and mooring force, the reliability of the numerical model is validated. Finally, the validated numerical model is applied to study the influence of separation distance and wave parameters on the hydrodynamic characteristics of the double-row floating breakwater. The results indicate that the optimal separation distance between pontoon and airbag is 0.75 times the wavelength. At such separation distance and within the concerned 1–4 m wave heights and 4–7 s wave periods, the pontoon-airbag system presents better wave attenuation performance than a single pontoon. This improvement weakens as wave height increases while it strengthens as the wave period increases. In addition, the double-row floating breakwater is more effective in a high-wave regime than in a low-wave regime. In the case of short waves, attention should be paid to the stability and mooring reliability of the seaward pontoon, while in the case of long waves, care needs to be taken of the leeward airbag.

Anomalous couplings in Higgs-boson pair production at approximate NNLO QCD

We combine NLO predictions with full top-quark mass dependence with approximate NNLO predictions for Higgs-boson pair production in gluon fusion, including the possibility to vary coupling parameters within a non-linear Effective Field Theory framework containing five anomalous couplings for this process. We study the impact of the anomalous couplings on various observables, and present Higgs-pair invariant-mass distributions at seven benchmark points characterising different mhh shape types. We also provide numerical coefficients for the approximate NNLO cross section as a function of the anomalous couplings at sqrt{s} = 14 TeV.

АНАЛИЗ УСТОЙЧИВОСТИ ЖЕСТКИХ СИСТЕМ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

A method for analyzing stability in the sense of Lyapunov for systems of ordinary differentialequations is proposed. The method is based on stability criteria in the form of necessary and sufficientconditions obtained on the basis of vector-matrix transformations of difference numericalintegration schemes. The varieties of criteria in multiplicative, additive and matrix form are presented.The design of the criteria implies the possibility of their programmatic realization. To increasethe reliability of the stability analysis, the approximations of the solution included in theconstruction of the criteria are based on piecewise interpolation approximation by Lagrange polynomialsconverted to a form with numerical coefficients. A programming and numerical experimentis carried out to analyze the stability of the Belousov-Jabotinsky periodic reaction model,which belongs to the class of rigid systems, under given initial conditions. The analysis is carriedout on the basis of the presented criteria and the results of the program clearly determine the natureof the stability in real time. Based on the results of the experiment, it can be argued that replacingthe difference approximations of the solution with piecewise interpolation approximationsincreases the reliability of the stability analysis, reduces the study time, and makes it possible todetermine the asymptotic properties of the solution. In general, the proposed approach is an alternativeto the methods of the qualitative theory of differential equations and makes it possible toreliably determine the stability of rigid systems of ordinary differential equations in real time.

On algorithms testing positivity of real symmetric polynomials

We show that positivity (≥0) on mathbb{R}_{+}^{n} and on mathbb{R}^{n} of real symmetric polynomials of degree at most p in nge 2 variables is solvable by algorithms running in polynomial time in the number n of variables. For real symmetric quartics, we find discriminants which lead to the efficient algorithms QE4+ and QE4 running in O(n) time. We describe the Maple implementation of both algorithms, which are then used not only for testing concrete inequalities (with given numerical coefficients and number of variables), but also for proving symbolic inequalities.

Non-linear dynamic simulations of two-body wave energy converters via identification of viscous drag coefficients of different shapes of the submerged body based on numerical wave tank CFD simulation

Two-body point absorbers are useful for wave energy conversion in high energetic and high wave period sea locations. Analysis and simulation show that the two-body point absorbers have high efficiencies which makes them a potential candidate to promote wave energy harvesting technology towards commercialization and implementation. The submerged body of the two-body point absorbers has a large effect on their performance, hence the question arises: which geometric shape of the submerged body will lead to the best performance of the two-body wave energy harvesters? This paper presents a comprehensive study that analyses the different shapes of the submerged bodies based on the numerical wave tank simulated viscous drag coefficient and hydrodynamic property parameters of the oscillating device. A set of transient and three-dimensional computational fluid dynamic simulations of a numerical wave tank were carried out in Ansys Fluent to determine the viscous drag coefficients of the different submerged body shapes. And then the boundary element hydrodynamic simulation and linear dynamic frequency-domain modeling have been applied to identify the optimized PTO damping. Finally, non-linear time-domain modeling has been applied with the identified viscous damping drag coefficient and PTO damping coefficients to predict and compare the power output performance of different large-scale devices with different shapes of submerged bodies. It is found that each submerged body shape is suitable for a different range of wave frequencies, as one must establish a balance between the hydrodynamic properties and the viscous drag of the submerged body based on the sea location. It is also found that the typical viscous drag coefficient values used in the past studies underestimate the drag coefficient, and the linearized model widely adopted in frequency domain models is non-accurate for the devices with large viscous drag forces.

Wave-optics simulation of dynamic speckle: II. In an image plane.

This two-part paper demonstrates the use of wave-optics simulations to model the effects of dynamic speckle. In Part II, we formulate closed-form expressions for the analytical irradiance correlation coefficient, specifically in the image plane of an optical system. These expressions are for square, circular, and Gaussian limiting apertures and four different modes of extended-object motion, including in-plane and out-of-plane translation and rotation. Using a phase-screen approach, we then simulate the equivalent scattering from an optically rough extended object, where we assume that the surface heights are uniformly distributed and delta correlated from grid point to grid point. For comparison to the analytical irradiance correlation coefficient, we also calculate the numerical irradiance correlation coefficient from the dynamic speckle after propagation from the simulated object plane to the simulated image plane. Overall, the analytical and numerical results definitely demonstrate that, relative to theory, the dynamic speckle in the simulated image plane is properly correlated from one frame to the next. Such validated wave-optics simulations provide the framework needed to model more sophisticated setups and obtain accurate results for system-level studies.

Wave-optics simulation of dynamic speckle: I. In a pupil plane.

This two-part paper demonstrates the use of wave-optics simulations to model the effects of dynamic speckle. In Part I, we formulate closed-form expressions for the analytical irradiance correlation coefficient, specifically in the pupil plane of an optical system. These expressions are for square, circular, and Gaussian scattering spots and four different modes of extended-object motion, including in-plane and out-of-plane translation and rotation. Using a phase-screen approach, we then simulate the equivalent scattering from an optically rough extended object, where we assume that the surface heights are uniformly distributed and delta correlated from grid point to grid point. For comparison to the analytical irradiance correlation coefficient, we also calculate the numerical irradiance correlation coefficient from the dynamic speckle after propagation from the simulated object plane to the simulated pupil plane. Overall, the analytical and numerical results definitely demonstrate that, relative to theory, the dynamic speckle in the simulated pupil plane is properly correlated from one frame to the next. Such validated wave-optics simulations provide the framework needed to model more sophisticated setups and obtain accurate results for system-level studies.