Analytical Formulas Research Articles

Optimization on frequency constraints with FFT using automatic differentiation on hybrid ODE applications

PurposeThis study aims to optimize electrical systems represented by ordinary differential equations and events, using their frequency spectrum is an important purpose for designers, especially to calculate harmonics.Design/methodology/approachThis paper presents a methodology to achieve this, by using a gradient-based optimization algorithm. The paper proposes to use a time simulation of the electrical system, and then to compute its frequency spectrum in the optimization loop.FindingsThe paper shows how to proceed efficiently to compute the frequency spectrum of an electrical system to include it in an optimization loop. Derivatives of the frequency spectrum such as the optimization inputs can also be calculated. This is possible even if the sized system behavior cannot be defined a priori, e.g. when there are static converters or electrical devices with natural switching.Originality/valueUsing an efficient sequential quadratic programming optimizer, automatic differentiation is used to compute the model gradients. Frequency spectrum derivatives with respect to the optimization inputs are calculated by an analytical formula. The methodology uses a “white-box” approach so that automatic differentiation and the differential equations simulator can be used, unlike most state-of-the-art simulators.

Numerical Calculation and Analytic Formulas of Yarkovsky Effect Parameter of Circumsolar Asteroids

Numerical Calculation and Analytic Formulas of Yarkovsky Effect Parameter of Circumsolar Asteroids

Features of determining the estimated length of positive arches made of laminated wood

The issue of loss of stability of arched structures is very important in engineering practice and design. This issue is especially key from the point of view of reliability and safety and economic losses, since structures of this type are usually used in buildings with a consequence class of at least CC2. [1].Laminated timber arches are long-span com-pression-bent elements in which the loss of stability plays a major role in determining the cross-sectional dimensions. In contrast to beams, an arch has a much larger ratio between the length of the element and its cross section, which in turn increases the risk of loss of stability. The loss of stability of an arch and its cor-rectly determined design length are directly related. The design length is defined as the value at which an arch can lose its stability under the influence of external loads. The main aspects of this relationship are: the value of the design length of the arch; mechanisms of loss of stabil-ity; dependence on the material and shape; ac-curacy of engineering calculations.It should be noted that recommendations for determining the design length of an arch are contained exclusively in the methodological lit-erature and outdated regulatory documents, such as [7]. In the latest current regulatory doc-ument [8], this issue is resolved by checking the strength and stability of the element, without limiting the flexibility. Since the flexibility of the element depends on its design length, the is-sue of determining this parameter remains open and relies directly on the design engineer, or rather, on his experience and competence. This issue is very relevant, given that a minor error in determining the design length significantly affects the results of testing compressed-bent elements.As a result of the work carried out, we analyzed the determination of the design lengths of hollow arches with different shapes and types of loading using two calculation methods: analytical formulas and the finite element method using the LIRA-SAPR-2019 software. Based on the calculations, recommendations for determining the design lengths of arches are given.

Mixed noise and posterior estimation with conditional deepGEM

We develop an algorithm for jointly estimating the posterior and the noise parameters in Bayesian inverse problems, which is motivated by indirect measurements and applications from nanometrology with a mixed noise model. We propose to solve the problem by an expectation maximization (EM) algorithm. Based on the current noise parameters, we learn in the E-step a conditional normalizing flow that approximates the posterior. In the M-step, we propose to find the noise parameter updates again by an EM algorithm, which has analytical formulas. We compare the training of the conditional normalizing flow with the forward and reverse Kullback–Leibler divergence, and show that our model is able to incorporate information from many measurements, unlike previous approaches.

Unsteady solute dispersion of electro-osmotic flow of micropolar fluid in a rectangular microchannel

This study scrutinizes the two-dimensional concentration distribution for a solute cloud containing a micropolar fluid in a rectangular microchannel under the influence of an applied electric field. The concentration distribution is obtained up to second order approximation using Mei's homogenization method. Analytical formulas are derived for dispersion coefficient, mean and two-dimensional concentration distributions. This study also includes the analytical expressions for electric potential, velocity, and microrotation profiles. This study discusses the impact of coupling number, couple stress parameter, electric double layer thickness, and Péclet number on solute concentration distribution. The results of fluid velocity and dispersion coefficient are validated with available works in the literature. The non-Newtonian parameter and electric double layer thickness are shown to have a significant impact on dispersion. Our study reveals that concentration distribution rises but spreading of solute reduces when the coupling number increases. This is also true when the Debye length decreases. It is also obtained that the solute spreads more in the Newtonian fluid case compared to the micropolar fluid case. Finally, coupling number and electric double layer thickness show a symmetric pattern to the indicator function for the transverse concentration variation rate. The findings of this work have broad implications in deoxyribonucleic acid analysis, chemical mixing, and separation.

Coherence of the frequency-difference autoproduct deduced from high-frequency acoustic fields scattered from a rough sea surfacea).

The prevalence of random scattering from a rough ocean surface increases with increasing χ=kh cos θ, where k is the acoustic wavenumber, h is the root-mean-square surface height, and θ is the incidence angle. Generally, when χ≫1, coherence between incident and surface-scattered fields is lost. However, such coherence may be recovered when χ≫1 by considering the frequency-difference autoproduct of the surface-scattered field, a quadratic product of complex fields at nearby frequencies. Herein, the autoproduct's coherent reflection coefficient for χ> 20 is determined from surface-scattered sound fields obtained from 50 independent realizations of the rough ocean surface measured in pelagic waters off the coast of California in January 1992. The recordings were made with a source at a depth of 147 m that broadcasted 30 and 40 kHz signals to a single receiver 576 m away at depth of 66 m. An analytic formula for the coherent reflection coefficient of the frequency-difference autoproduct, based on the Kirchhoff approximation and a Gaussian surface autocorrelation function, compares favorably with measurements. Improved agreement with the single-receiver measurements is possible via a minor adjustment to the surface autocorrelation length. The adjustment identified here matches that determined previously from horizontal spatial coherence estimates utilizing the experiment's eight-element receiving array.

Inverse Kinematics Analysis of a 2-DOF Stabilized Platform using Simulink and Simscape

Abstract This paper presents a parallel manipulator with two universal-prismatic-universal (UPU)-type active limbs and one revolute-spherical (RS)-type passive limb which is used as a stabilizing platform based on the shipboard helicopter’s landing process and its kinematics are studied systematically. First, a 3D model of this stabilized platform is constructed, and its degrees of freedom are analyzed. Second, analytic formulas for solving the inverse kinematics and calculating the required lengths of each limb are derived and modeled using Simulink®. Third, a Simscape™ model is constructed for the proposed design, and finally, the comparison of the results between the two models is done.

Analytical perturbations of relativistic images in Kerr space-time

Light rays passing very close to black holes may wind several times before escaping. For any given electromagnetic source around the black hole, a distant observer would thus observe two infinite sequences of images on either side of the black hole. These images are generated by light rays performing an increasing numbers of loops. The strong deflection limit provides a simple analytic formalism to describe such higher order images for spherically symmetric metrics, while for axially symmetric black holes one typically resorts to numerical approaches. Here we present the leading order perturbation to higher order images when the black hole spin is turned on. We show that the images slide around the black hole shadow as an effect of space-time dragging. We derive analytical formulae for their shifts and the perturbation of their time delays. We also discuss how such simple analytical formulae for images by Kerr black holes can be of great help in many applications.

An analytical method to estimate the accuracy representation index of circular and elliptical disc models for rectangular fractures with application to seepage analysis

Constructing a suitable discrete fracture network (DFN) to represent rock fractures proves vital for tackling engineering projects involving rock masses. Among the commonly used models for constructing DFN, the circular disc model (CDM) and elliptical disc model (EDM) stand out. The selection between these models has been facilitated by the introduction of the Accuracy Representation Index (ARI), which quantitatively assesses their suitability. The existing methods for calculating ARI based on Monte Carlo simulations are time-consuming. In order to quickly estimate the ARI for CDM and EDM concerning any length-width ratio (kr) of rectangular fracture, the paper derives analytical formulas based on the geometric relationships between circles and ellipses with rectangles. These formulas show that: (a) for the CDM, ARI is only related to kr, and its value decreases as kr increases; (b) for the EDM, ARI is dependent on the relationship between kr and the long-short axis length ratio (ke) of the ellipse, and it has been theoretically proven that ARI reaches its maximum value of 0.91 when kr is equal to ke. Moreover, we apply ARI to the study of rock mass seepage characteristics, and the results show that ARI could reflect the error rate of permeability of the CDM and EDM for rock masses with rectangular fractures.

Bootstrapping cascaded random matrix models: Correlations in permutations of matrix products.

Random matrix theory is a useful tool in the study of the physics of multiple scattering systems, often striking a balance between computation speed and physical rigour. Propagation of waves through thick disordered media, as arises, for example, in optical scattering or electron transport, typically necessitates cascading of multiple random matrices drawn from an underlying ensemble for thin media, greatly increasing the computational burden. Here we propose a dual pool based bootstrapping approach to speed up statistical studies of scattering in thick random media. We examine how potential matrix reuse in a pool based approach can impact statistical estimates of population averages. Specifically, we discuss how both bias and additional variance in the sample mean estimator are introduced through bootstrapping. In the diffusive scattering regime, the extra estimator variance is shown to originate from samples in which cascaded transfer matrices are permuted matrix products. Through analysis of the combinatorics and cycle structure of permutations we quantify the resulting correlations. Proofs of several analytic formulas enumerating the frequency with which correlations of different strengths occur are derived. Extension to the ballistic regime is briefly considered.

On the sound field of an oscillating elliptical disk in free space with an open and closed back.

An oscillating elliptical disk in free space has a uniform surface velocity distribution and is therefore a rigid sound radiator. The radiated sound also represents that scattered from a stationary disk in the presence of a normal plane wave. Using the Fourier (Hankel) Green's function in cylindrical coordinates, together with the monopole Kirchhoff-Helmholtz boundary integral, rigorous analytical formulas are derived for calculating the surface pressure distribution, radiation impedance, on-axis pressure, and, directivity pattern of an elliptical open-back disk in free space, and these are plotted over a range of normalized frequencies. The directivity is compared with that of a rigid circular disk in free space, and asymptotic low-frequency approximations are derived for the radiation impedance and on-axis pressure. Using the Gutin concept, the radiated sound is combined with that from a rigid elliptical disk in an infinite baffle to obtain the radiation impedance, on-axis pressure, and, directivity pattern of a closed-back elliptical disk in free space. The latter has a uniform velocity distribution on the front surface and zero velocity over the entire back surface.

Modeling the longitudinal wave in a nanorod based on a novel theory of elastic waves with surface effects

This paper investigates the impact of surface effects on the propagation behavior of longitudinal waves in a nanorod. A theoretical model has been established on the basis of a newly proposed theory of elastic waves with surface effects. The surface effects comprise two components: the effect of surface energy and the effect of surface inertia. An analytical formula for the longitudinal wave velocity of a nanorod has been derived. Two inherent lengths at nanoscale have been deduced to characterize these two types of surface effects. The results indicate that the longitudinal wave in a nanorod is still nondispersive. However, an attractive phenomenon uncovered is that when the size of a rod reduces to the inherent lengths at nanoscale, the longitudinal wave velocity becomes size-dependent due to the effects of surface energy and surface inertia. The former increases the longitudinal wave velocity, whereas the latter decreases it. This can be understood as the former equivalently increasing the stiffness of the nanorod, whereas the latter enhancing its effective density. On the other hand, when the rod is at the macroscale, the longitudinal wave velocity degenerates to the classical velocity for a macroscopic rod without any surface effects. The current findings not only enhance our understanding of the size-dependent wave velocity of longitudinal waves in nanorods but also facilitate precisely designing the elastic wave nanodevices.

Studying the breakdown electric field in uniform and non-uniform air gaps

High voltage is essential in power grids, but it inevitably leads to high electrical stress and the associated risk of electrical discharges. Due to the complexity of the phenomena involved in electrical discharges, there are no analytical formulas for predicting the electric field strength at which they initiate, so experimental data and numerical methods are required for this purpose. According to many sources, electrical discharges can occur in air at normal pressure and temperature when the electric field strength is approximately 3 kV mm−1 or greater. This paper analyzes and discusses this threshold in detail by examining relevant electrode geometries used in high voltage applications from experimental data found in the scientific literature and using 2D finite element analysis simulations. Uniform, quasi-uniform, and non-uniform field gaps are analyzed to help students draw conclusions and gain insight into the nature of gas breakdown and the applicability of the 3 kV mm−1 threshold. The approach proposed in this paper is well suited for a practical session or group project for undergraduate or even graduate courses. Despite the important effects and design implications of electrical discharges on high voltage equipment, apparatus and systems, this topic is often not covered in sufficient detail in regular courses.

High harmonic generation from an atom in a squeezed-vacuum environment

We investigate high harmonic generation (HHG) of an atom in the presence of squeezed vacuum of a single harmonic mode. Based on a fully quantum time-dependent Schrödinger equation, we derive an analytical formula for the harmonic amplitude. Our simulations of the HHG spectrum with this formula show that the harmonic amplitude of the corresponding squeezed mode can undergo significant changes with different parameters of the squeezed vacuum. Using the time-frequency analysis method, the physics underlying the effects of the vacuum quantum fluctuation (VQF) on the harmonic generation is revealed, which is found to be consistent with the explanation of Fermi's golden rule. Our work establishes a profound connection between harmonic generation and VQF, and may provide an unconventional approach to manipulating harmonic emission. Published by the American Physical Society 2024

Mass Spectrum of Noncharmed and Charmed Meson States in Extended Linear-Sigma Model

The mass spectrum of different meson particles is generated using an effective Lagrangian of the extended linear-sigma model (eLSM) for scalar and pseudoscalar meson fields and quark flavors, up, down, strange, and charm. Analytical formulas for the masses of scalar, pseudoscalar, vector, and axialvector meson states are derived assuming global chiral symmetry. The various eLSM parameters are analytically deduced and numerically computed. This enables accurate estimations of the masses of sixteen noncharmed and thirteen charmed meson states at vanishing temperature. The comparison of these results to a recent compilation of the particle data group (PDG) allows us to draw the conclusion that the masses of sixteen noncharmed and thirteen charmed meson states calculated in the eLSM are in good agreement with the PDG. This shows that the eLSM, with its configurations and parameters, is an effective theoretical framework for determining the mass spectra of various noncharmed and charmed meson states, particularly at vanishing temperature.

Assessment of flight range and transmission of video information when monitoring emergency situations from an unmanned aircraft in difficult meteorological conditions

The purpose of research. Modern unmanned aerial vehicles of various types provide the ability to perform various tasks of operational information collection, monitoring the state of the environment, technological facilities and territories, updating data on these facilities, as well as for reconnaissance and monitoring their condition. The purpose of the study is to establish the maximum flight range and distance necessary for the organization of a communication channel through which the video stream and flight control commands using the KAM-16 Turbo ¾ method will be transmitted between the ground control complex and the unmanned aerial vehicle in a difficult meteorological situation.Methods are based on the basics of radio electronics, diagnostics and forecasting of the technical condition of aircraft. The methods of multicriteria analysis, parametric and structural synthesis were used. The principles of transmitting video images from drones used to monitor emergency situations were studied. A critical assessment of the maximum flight range of an unmanned aerial vehicle in a difficult meteorological situation was carried out.Results. Analytical formulas for determining the energy potential of a data transmission channel in line of sight are presented. Graphs of the dependence of the energy reserve in the data transmission channel between the unmanned aerial vehicle and the ground control system are presented, which make it possible to determine the maximum distance for transmitting video images in Full HD quality in the 2.4 GHz band. Calculations were carried out on the basis of complex analytical expressions and graphical dependences of the flight range of a micro UAV on the speed of tailwind and headwind were constructed. The maximum headwind speed is calculated, which does not allow the use of micro UAVs in blizzard conditions.Conclusion. A promising area of study of the use of UAV in emergency monitoring are micro-class copters capable of detecting victims by transmitting video information from a thermal imager during search and rescue operations in difficult weather conditions, such as snowstorms.

Efficient parameter estimation for ODE models of cellular processes using semi-quantitative data.

Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. The parameters of these models are commonly estimated from experimental data. Yet, experimental data generated from different techniques do not provide direct information about the state of the system but a nonlinear (monotonic) transformation of it. For such semi-quantitative data, when this transformation is unknown, it is not apparent how the model simulations and the experimental data can be compared. We propose a versatile spline-based approach for the integration of a broad spectrum of semi-quantitative data into parameter estimation. We derive analytical formulas for the gradients of the hierarchical objective function and show that this substantially increases the estimation efficiency. Subsequently, we demonstrate that the method allows for the reliable discovery of unknown measurement transformations. Furthermore, we show that this approach can significantly improve the parameter inference based on semi-quantitative data in comparison to available methods. Modelers can easily apply our method by using our implementation in the open-source Python Parameter EStimation TOolbox (pyPESTO) available at https://github.com/ICB-DCM/pyPESTO.

An Improved Near-State Pulse-Width Modulation with Low Switching Loss for a Permanent Magnet Synchronous Machine Drive System

Common-mode voltage (CMV) leads to the shaft voltage and shaft current by coupling the capacitor network in the permanent magnet synchronous machine (PMSM), which affects the reliability of the whole motor drive system. Based on the low-CMV modulation strategy for the PMSM drive system, this paper proposed an improved near-state pulse-width modulation (NSPWM) on switching loss. First, the generation mechanism for the switching signals of NSPWM was analyzed, and it was observed that there exists one phase of switches in an inactive state for every sector. Then, to reduce the switching loss of the NSPWM, this paper proposed an improved NSPWM modulation strategy based on power factor angle to adjust switching action, which ensures the switching tubes that have the biggest conduction current have no switching action. In addition, the switching loss analytic formula of the NSPWM was derived to prove the correctness of the proposed method for optimizing switching loss. Finally, the proposed modulation strategy was carried out in the simulation and experimental platform. Under the premise of good steady and dynamic performance, the results show that the proposed modulation strategy has less switching loss.

Impact of solar cell materials on the energy performance of SPVT air collector

The impact of roof-integrated semitransparent photovoltaic thermal (SPVT) air collectors connected in series for cold climates has been assessed using a new model established in this research. Numerical calculations have been performed using analytical formulae derived for the temperatures of the room air, roof, and solar cells. These findings have led to evaluations of the thermal, electrical, and overall exergy of various solar cell materials (Single-crystalline, Multi-crystalline silicon, Nanocrystalline -Silicon, Amorphous silicon, Cadmium Telluride and copper indium gallium selenide). According to the findings, amorphous silicon has the maximum peak solar cell electrical efficiency of 10.8 % and single-crystalline has the minimum peak efficiency of 7.0 % for the proposed system. Further, performance analysis has also been done based on different packing factor and mass flow rates.

Learning a stable approximation of an existing but unknown inverse mapping: application to the half-time circular Radon transform

Supervised deep learning-based methods have inspired a new wave of image reconstruction methods that implicitly learn effective regularization strategies from a set of training data. While they hold potential for improving image quality, they have also raised concerns regarding their robustness. Instabilities can manifest when learned methods are applied to find approximate solutions to ill-posed image reconstruction problems for which a unique and stable inverse mapping does not exist, which is a typical use case. In this study, we investigate the performance of supervised deep learning-based image reconstruction in an alternate use case in which a stable inverse mapping is known to exist but is not yet analytically available in closed form. For such problems, a deep learning-based method can learn a stable approximation of the unknown inverse mapping that generalizes well to data that differ significantly from the training set. The learned approximation of the inverse mapping eliminates the need to employ an implicit (optimization-based) reconstruction method and can potentially yield insights into the unknown analytic inverse formula. The specific problem addressed is image reconstruction from a particular case of radially truncated circular Radon transform (CRT) data, referred to as ‘half-time’ measurement data. For the half-time image reconstruction problem, we develop and investigate a learned filtered backprojection method that employs a convolutional neural network to approximate the unknown filtering operation. We demonstrate that this method behaves stably and readily generalizes to data that differ significantly from training data. The developed method may find application to wave-based imaging modalities that include photoacoustic computed tomography.