Simple Analytical Formula Research Articles

Simple analytic expressions of Fresnel diffraction patterns at a straight strip and slit for Gaussian wave illumination

Diffraction of light on an opaque thin strip and a slit is considered, providing simple analytic formulas describing Fresnel diffraction patterns for Gaussian-beam illumination. The presented equations provide a simple way to carry out diffraction simulations. Evolution of the intensity distribution of a scattered wave through a rectilinear strip from the near to far field is also shown.

Secular spin-axis dynamics of exoplanets

Context. Seasonal variations and climate stability of a planet are very sensitive to the planet obliquity and its evolution. This is of particular interest for the emergence and sustainability of land-based life, but orbital and rotational parameters of exoplanets are still poorly constrained. Numerical explorations usually realised in this situation are therefore in heavy contrast with the uncertain nature of the available data. Aims. We aim to provide an analytical formulation of the long-term spin-axis dynamics of exoplanets, linking it directly to physical and dynamical parameters, but still giving precise quantitative results if the parameters are well known. Together with bounds for the poorly constrained parameters of exoplanets, this analysis is designed to enable a quick and straightforward exploration of the spin-axis dynamics. Methods. The long-term orbital solution is decomposed into quasi-periodic series and the spin-axis Hamiltonian is expanded in powers of eccentricity and inclination. Chaotic zones are measured by the resonance overlap criterion. Bounds for the poorly known parameters of exoplanets are obtained from physical grounds (rotational breakup) and dynamical considerations (equipartition of the angular momentum deficit). Results. This method gives accurate results when the orbital evolution is well known. The detailed structure of the chaotic zones for the solar system planets can be retrieved from simple analytical formulas. For less-constrained planetary systems, the maximal extent of the chaotic regions can be computed, requiring only the mass, the semi-major axis, and the eccentricity of the planets present in the system. Additionally, some estimated bounds of the precession constant allow to classify which observed exoplanets are necessarily out of major spin-orbit secular resonances (unless the precession rate is affected by the presence of massive satellites).

Theoretical, numerical, and experimental investigation of smoke dynamics in high-rise buildings

Smoke kills more people than the associated fire and thus predicting smoke spreading inside high-rise buildings is of paramount importance to structural and safety engineers. Here, the velocity, temperature, and concentration fields in large-scale turbulent smoke plumes were predicted using classical self-similar turbulent plume theory, which assumes a point fire source under open-air conditions. Turbulent fires of various heat release rates in a confined space were also simulated numerically using Fire Dynamics Simulator (FDS), which was verified against experimental data before being used to validate the analytical plume jet results. The agreement between analytical, numerical, and experimental results was good. This demonstrates for the first time that for realistic, wide shafts, analytical results from self-similar theory of free turbulent plumes are as accurate as the numerical simulations and appropriately describe experimental data. This allows engineers to avoid lengthy, cumbersome numerical simulations to estimate the consequences of smoke spreading in high-rise buildings using simple analytical formulae. In addition, parametric studies were conducted using plume theory for building heights up to 500 m and heat release rates up to 500 MW. Smoke velocity, temperature, and concentration fields described smoke evolution at different heights.

An experimental, numerical and analytical study on deformation mechanisms of web girders in a collision or grounding incident

ABSTRACTThis paper presents an experimental, numerical, and analytical study on the crushing of web girders, to provide insights into the behaviour of a ship’s structure in a collision or grounding incident. Web girders, which are plated structures subjected to in-plane loads in an accident, develop a variety of deformation mechanics to absorb impact energy. The web stiffeners on them provide additional resistance. However, studies devoted to the contributions of longitudinal and transverse stiffeners to the load-carrying capacity of web girders are scarce. It remains a technical challenge to quantify the crushing strength of stiffened web girders, particularly the extent by which longitudinal and transverse web stiffeners can increase the load-carrying capacity of web girders in a collision or grounding incident. To further research on the behaviour of web girders in a collision or grounding incident, a new series of laboratory experiments was conducted, where four scaled double hull specimens were crushed in a quasi-static manner. The specimens included one unstiffened web girder, one longitudinally stiffened web girder, and two transversely stiffened web girders. Experiments of this nature provide a good basis for comparing different designs of web girders, and they have been seldom reported in publicly available literature. The aim of our work is to provide experimental data suitable for a comparative study. To supplement the laboratory experiments, nonlinear finite element method (FEM) analyses were also conducted to provide additional information that was not possible to obtain solely from experimental measurements. Certain existing analytical models were reviewed, and their predictions were compared with the experimental results. While all these simplified analytical formulae provide reasonable estimates, their predictions deviate more from the experimental results when the web girders are stiffened. Finally, the typically applied approach of equivalent plate thickness is not optimal, particularly when the web stiffeners are in line with the striking indenter.

Harmful algal blooms (red tide): a review of causes, impacts and approaches to monitoring and prediction

Red tide, an impermanent natural phenomenon including harmful algal blooms, causes changing the color of the sea generally to red or almost brown, and has a serious impact on environment along the coast and aquatic ecosystem. Due to recent extensive steady harmful algal blooms events that cause adverse impacts on human healthsome, aquaculture and tourism industry, and the entire economy of the coastal region, the need of society for realizing these phenomena is much greater than the past. In the recent decades, consideration of algal blooms and determination of bloom-former species and fundamental researches about dynamics of blooms are increased worldwide. Development in technology has increased our abilities in monitoring oceans and has provided new opportunities for blooms identification as well as defining the biological, physical and chemical parameters that lead to algae beginning, expansion and disappearance. In spite of these rapidly developing observational capabilities, harmful algal blooms’ proceedings will carry on to be undersampled for the foreseeable future, due to their spacious spaced and temporal coverage. Therefore, reliance on models to help interpreting observations is necessary. To watch red tides, there are different methods: field observation and using sampling data, satellite-based studies, laboratory studies, modeling. (This item includes complex numerical models, conceptual models, simple analytic formula, semi-empirical models and aggregated box models or zero-dimensional models.) This paper proposes different observation and prediction methods ever used worldwide.

Calibrating the Young’s modulus of soft materials with surface tilt angle measured by atomic force microscopy

We investigated the apparent Young’s modulus, Em, of soft materials with the surface tilt angle, as measured by colloidal probe atomic force microscopy (AFM). The AFM measurements of soft polymer hydrogels and natural unfertilized eggs showed a clear universal behavior of Em as a function of the tilt angle, θ, of the sample surface at the local contact area. We found that the observed θ dependence of Em was well fitted with a simple modified Hertz contact model, in which the pressure distribution in the contact area follows the conventional Hertz contact model, the vertical component of the net force balances the loading force, and the remaining lateral component of the net force is approximately ignored. We demonstrated how a simple analytical formula derived from the modified Hertz contact model can calibrate the Em values for single cells in a spherical embryo and for single isolated cells and a confluent cell monolayer adhered on flat substrates.

Diffusion-influenced reactions on non-spherical partially absorbing axisymmetric surfaces.

The calculation of the diffusion-controlled reaction rate for partially absorbing, non-spherical boundaries presents a formidable problem of broad relevance. In this paper we take the reference case of a spherical boundary and work out a perturbative approach to get a simple analytical formula for the first-order correction to the diffusive flux onto a non-spherical partially absorbing surface of revolution. To assess the range of validity of this formula, we derive exact and approximate expressions for the reaction rate in the case of partially absorbing prolate and oblate spheroids. We also present numerical solutions by a finite-element method that extend the validity analysis beyond spheroidal shapes. Our perturbative solution provides a handy way to quantify the effect of non-sphericity on the rate of capture in the general case of partial surface reactivity.

Action of Shocks Generated in Solid Targets by Dense Plasma Focus Devices and at Pulsed Laser Irradiation

Shock waves actions upon materials perspective for using in future thermonuclear fusion reactors are investigated experimentally and by means of numerical modelling. The shocks are generated by powerful streams of plasma and fast ions in Dense Plasma Focus devices as well as by irradiation with a laser operating in a Q-spoiled mode. Power flux densities of these streams on the targets’ surfaces are in the range 1014 – 1016 W/m2. They are used for tests of the above materials, and their influences are compared for a number of substances. It is shown that in the above-mentioned identical conditions Plasma Foci generate shock waves with amplitudes of approximately two times higher than that for the laser case. Fronts of the shocks are formed here faster than at the laser irradiation. A simple analytical formula for calculations of the amplitudes of shock waves in radiation material science experiments provided with Dense Plasma Focus devices are advanced.

Develop a Model to Study the Energy Distribution of Cascades of Atomic Collisions

In this paper, based on the solution of the Boltzmann kinetic equation, we determine the energy distribution function describing the steady-state deceleration of the cascade of moving atoms taking into account their multiplication at the power interaction potential U∼1/rn.A new approach to the solution of the kinetic equation based on the extended concept of primary knocked-on atoms (PKA) is used for its calculation. One of the advantages of using the power interaction potential is that in this case it is possible to obtain simple analytical formulas for the distribution function of the cascade of slowing-down atoms taking into account their multiplication and demonstrate the simplicity and convenience of the proposed new approach to the solution of the kinetic equation. On the other hand, based on the obtained results it is possible to estimate the accuracy of various approximate solutions. It is shown that this approach will be applicable to other interatomic interaction potentials, if the average PKA energy loss in individual collisions decreases monotonically with decreasing energy, and the relative PKA energy loss in individual collisions will be small.

Высоковольтные диффузионные диоды с резким восстановлением. I. Численное моделирование

The operation of diffusion step recovery diodes (dSRD) as current interrupters in high-power nanosecond pulse generators was studied in detail by the methods of numerical computer simulation for the first time. One of the necessary conditions that minimize the loss in dSRD was specified. Dependences of pre-pulse voltage, front duration, amplitude and duration of pulse formed on the active load, and the energy of switching losses in the dSRD on the device area and the break current density are obtained. It is shown that simulation results can be described by simple analytical formulas obtained in the second part of the work, with an accuracy of 10-20% if the pulse amplitude does not exceed the avalanche breakdown voltage of the dSRD. The generalized figure of merit which can be used for optimization of dSRD parameters, a mode of its operation and comparison of efficiency of current interrupters of various types was offered.

Design of Artificial-Material-Based Antennas Using Inverse Scattering Techniques

A new approach to the design of graded artificial materials (GAMs)-based devices is proposed by exploiting the inverse scattering framework as a synthesis tool. The introduced general methodology can be applied to arbitrary far-field specifications, thus allowing the design of noncanonical devices. In particular, two different strategies are developed for GAMs based on either graded refractive index (GAM $_{R}$ ) or graded filling factor (GAM $_{F}$ ). In both strategies, the inverse scattering problem is solved by a proper reformulation of the contrast source inversion method, wherein a proper rescaling of the amplitude of the primary sources is also used. In particular, in the first strategy, the GAM $_{F}$ is obtained by exploiting homogenization theories. In the second strategy, the GAM $_{R}$ is synthesized by exploiting a suitable representation basis for the unknown contrast function, and then simple analytical formulas are used to determine the corresponding GAM $_{F}$ . The proposed approach is assessed through the synthesis of an antenna generating a $\Sigma /\Delta $ reconfigurable pattern.

Simple guidelines to predict self-phase modulation patterns

We present a simple approach to predict the main features of optical spectra affected by self-phase modulation (SPM), which is based on regarding the spectrum modification as an interference effect. A two-wave interference model is found sufficient to describe the SPM-broadened spectra of initially transform-limited or up-chirped pulses, whereas a third wave should be included in the model for initially down-chirped pulses. Simple analytical formulae are derived, which accurately predict the positions of the outermost peaks of the spectra.

Accurate semiempirical analytical formulas for spontaneous polarization by crystallographic parameters of SrTiO3-BaTiO3 system by ab initio calculations

Spontaneous polarizations (PS’s) of BaTiO3 and SrTiO3 under various conditions are calculated ab initio using different exchange-correlation functionals. The extensive theoretical sets of PS vs. ion positions are found to lie on a single curve, despite the chemical differences and the wide variations of PS and lattice parameters. This uncovers accurate simple analytical formulas of PS of SrTiO3-BaTiO3 system expressed by ion positions; a single formula predicts both macroscopic and atomic-scale PS of SrTiO3, BaTiO3 and SrTiO3-BaTiO alloys. The accuracy of the formula is demonstrated by the application to experiments, BaTiO3-SrTiO3 (-CaTiO3) alloys, Sr4Ti4O12 with PS // a-axis, a parallel domain, and a headon domain. In addition, the present results verify empirically that oxygen displacement is the primary identifier and the origin of PS of SrTiO3 and BaTiO3 and indicate that BaTiO3 and SrTiO3 may transforms into new state by an extremely large strain, e.g., −3%. Furthermore, the earlier prediction of headon domain without aid of defects was confirmed. The present procedures for finding formulas can be applied to other materials.

Grid of upper atmosphere models for 1–40 M⊕ planets: application to CoRoT-7 b and HD 219134 b,c

There is growing observational and theoretical evidence suggesting that atmospheric escape is a key driver of planetary evolution. Commonly, planetary evolution models employ simple analytic formulae (e.g. energy limited escape) that are often inaccurate, and more detailed physical models of atmospheric loss usually only give snapshots of an atmosphere’s structure and are difficult to use for evolutionary studies. To overcome this problem, we have upgraded and employed an existing upper atmosphere hydrodynamic code to produce a large grid of about 7000 models covering planets with masses 1–39 M⊕ with hydrogen-dominated atmospheres and orbiting late-type stars. The modelled planets have equilibrium temperatures ranging between 300 and 2000 K. For each considered stellar mass, we account for three different values of the high-energy stellar flux (i.e. low, moderate, and high activity). For each computed model, we derived the atmospheric temperature, number density, bulk velocity, X-ray and EUV (XUV) volume heating rates, and abundance of the considered species as a function of distance from the planetary centre. From these quantities, we estimate the positions of the maximum dissociation and ionisation, the mass-loss rate, and the effective radius of the XUV absorption. We show that our results are in good agreement with previously published studies employing similar codes. We further present an interpolation routine capable to extract the modelling output parameters for any planet lying within the grid boundaries. We used the grid to identify the connection between the system parameters and the resulting atmospheric properties. We finally applied the grid and the interpolation routine to estimate atmospheric evolutionary tracks for the close-in, high-density planets CoRoT-7 b and HD 219134 b,c. Assuming that the planets ever accreted primary, hydrogen-dominated atmospheres, we find that the three planets must have lost them within a few Myr.

Distribution and magnitude of stress due to lateral variation of gravitational potential energy between Indian lowland and Tibetan plateau

Magnitudes of differential stress in the lithosphere, especially in the crust, are still disputed. Earthquake-based stress drop estimates indicate median values ca. 180 MPa, corresponding to a friction angle of ca. 10° to maintain the topographic relief between lowland and plateau for >10 Ma. The relative contribution of crustal strength to total lithospheric strength varies considerably laterally. In the region between lowland and plateau and inside the plateau the depth-integrated crustal strength is approximately equal to the depth-integrated strength of the mantle lithosphere. Simple analytical formulae predicting the lateral variation of depth-integrated stresses agree with numerically calculated stress fields, which show both the accuracy of the numerical results and the applicability of simple, rheology-independent, analytical predictions to highly variable, rheology-dependent stress fields. Our results indicate that (1) crustal strength can be locally equal to mantle lithosphere strength and that (2) crustal stresses must be at least one order of magnitude larger than median stress drops in order to support the plateau relief over a duration of ca. 10 Ma.

Analysis of the effects of inclination angle, nanoparticle volume fraction and its size on forced convection from an inclined elliptic cylinder in aqueous nanofluids

In this work, a detailed numerical investigation has been carried out to study the forced convection heat transfer from an inclined elliptic cylinder of a fixed aspect ratio of 0.5 immersed in a streaming water-based Al2O3 nanofluid using the two-phase Buongiorno’s model. In particular, this study presents extensive numerical results on how the cylinder inclination angle, nanoparticle volume fraction and its size are going to influence the local flow, temperature and nanoparticle concentration fields in the vicinity of the cylinder surface as well as the gross engineering parameters like the drag coefficient and Nusselt number over the following ranges of conditions as: Reynolds number, $$0.01 \le Re \le 40$$ ; cylinder inclination angle, $$0 \le \lambda \le 90$$ ; nanoparticle volume fraction, $$0 \le \phi \le 0.06$$ and two nanoparticle sizes (dnp), namely 30 nm and 60 nm. It has been found that both the Nusselt number and drag force increase with $$\phi$$ , but decrease with dnp under otherwise identical conditions. On the other hand, at a particular value of Re, $$\phi$$ and dnp, the value of the average Nusselt number increases with the increasing values of $$\lambda$$ , whereas the value of the drag coefficient decreases. Finally, from an application standpoint, a simple analytical formula for the average Nusselt number is provided as a function of Re, $$\phi$$ and $$\lambda$$ which will facilitate the interpolation of the present results for the intermediate values of these governing parameters.

Analytical formulas for gear body-induced tooth deflections of spur gears considering structure coupling effect

As a result of gear body flexibility, fillet foundation stiffness is defined as the meshing force needed for unit gear body-induced tooth deflections. However, it has long been overestimated in double-tooth engagement region in TAM (traditional analytical model), in which the structure coupling effect is ignored. Based on the elastic circular ring theory of Muskhelishvili, simplified analytical formulas are derived to conveniently calculate the gear body-induced tooth deflections considering structure coupling effect, where cubic and parabolic stress distributions along the junction arc are adopted. To prove the effectiveness of the proposed formulas, the gear body-induced tooth deflections obtained by the proposed formulas are compared with those by FEM (finite element method). The results show that proposed formulas can offer a relatively accurate estimation for gear body-induced tooth deflections.

Pulse-shape effects in fifth-order multidimensional optical spectroscopy

We study how the effects of finite pulse durations modify the fifth-order two-quantum two-dimensional (2Q-2D) and three-dimensional (3D) optical spectra. Instead of directly calculating the time domain polarization, the equivalent approach in frequency domain significantly simplifies the calculations and delivers exact expressions for the 2Q-2D and 3D spectra. Our results minimize the number of numerical integrations needed. We also derive simple analytical formulas for practically important situations when the laser pulses are short on the timescale of nuclear motion as well as short (or long) on the timescale of electronic dephasing. These formulas evaluate the measured spectra by multiplying the spectral response function by a factor depending on the spectral pulse envelopes.

Asymptotic variability of (multilevel) multirater kappa coefficients.

Agreement studies are of paramount importance in various scientific domains. When several observers classify objects on categorical scales, agreement can be quantified through multirater kappa coefficients. In most statistical packages, the standard error of these coefficients is only available under the null hypothesis that the coefficient is equal to zero, preventing the construction of confidence intervals in the general case. The aim of this paper is triple. First, simple analytic formulae for the standard error of multirater kappa coefficients will be given in the general case. Second, these formulae will be extended to the case of multilevel data structures. The formulae are based on simple matrix algebra and are implemented in the R package “multiagree”. Third, guidelines on the choice between the different mulitrater kappa coefficients will be provided.

Method for Radiation Center Position Measurements of a Combined Antenna in the Pulsed Mode

This paper proposes a method for measuring the position of radiation centers for an ultra-wideband combined antenna (CA) in the pulsed mode. The radiation centers for E- and H-planes of the antenna are determined. Antenna rotation in these planes, with respect to an arbitrary axis, results in the time delays (advances) of the recorded receiving pulses for various angles. The purpose of the measurements is to find the rotation axes providing minimum time delays of the radiated pulse registration at the receiving antenna for the largest possible range of observation angles. These rotation axes pass through local areas (radiation centers) in the antenna volume. Simple analytic formulas refining the position of the radiation center from an arbitrary measurement are proposed. In the simulations, a code based on the FDTD method was used. The CA has at least two radiation centers. Radiation center of the H-plane is located at 1/3–1/2 of the antenna length counting from its rear wall, slightly below the antenna’s half height. Radiation center of the E-plane is located near the aperture plane of the antenna, also slightly below the antenna’s half height.