An improved lumped model for freezing of a freely suspended supercooled water droplet in air stream

This work deals with the mathematical modeling of the transient freezing process of a supercooled water droplet in a cold air stream. The aim is to develop a simple yet accurate lumped-differential model for the energy balance for a freely suspended water droplet undergoing solidification, that allows for cost effective computations of the temperatures and freezing front evolution along the whole process. The complete freezing process was described by four distinct stages, namely, supercooling, recalescence, solidification, and cooling. At each stage, the Coupled Integral Equations Approach (CIEA) is employed, which reduces the partial differential equation for the temperature distribution within the spherical droplet into coupled ordinary differential equations for dimensionless boundary temperatures and the moving interface position. The resulting lumped-differential model is expected to offer improved accuracy with respect to the classical lumped system analysis, since boundary conditions are accounted for in the averaging process through Hermite approximations for integrals. The results of the CIEA were verified using a recently advanced accurate hybrid numerical-analytical solution through the Generalized Integral Transform Technique (GITT), for the full partial differential formulation, and comparisons with numerical and experimental results from the literature. After verification and validation of the proposed model, a parametric analysis is implemented, for different conditions of airflow velocity and droplet radius, which lead to variations in the Biot numbers that allow to inspect for their influence on the accuracy of the improved lumped-differential formulation.

Solution of structural mechanic’s problems by machine learning

This article proposes analysis procedure of structural mechanic’s problem as integral formulation. The analysis procedure was proposed as stressed-based analysis procedure as before plying the procedure, it is required to define stress distribution within the structural body by proper modelling and structural idealization assumptions. The methodology can suitable be applied for finding the solution of engineering applications with required accuracy. The methodology exploits the unfolded part of the structural analysis problems which were not so easy to solve such as geometric and material nonlinearity together with simple integration technique [11]. It has already unfolded the misery of physically exploiting plastic behaviour structures before the start of fracture of elastic materials [13]. The formulation is integral formulation rather than differential formulation in which whole stress –strain behaviour is utilised in the analysis procedure by using neural network as regression tool. In this article, one dimensional problem of uniaxial bar and plane strain axis symmetric problem of cylinder subjected to internal pressure is solved. The results are compared with the classical differential formulation or linear theory.

Strayfield calculation for micromagnetic simulations using true periodic boundary conditions

We present methods for calculating the strayfield in finite element and finite difference micromagnetic simulations using true periodic boundary conditions. In contrast to pseudo periodic boundary conditions, which are widely used in micromagnetic codes, the presented methods eliminate the shape anisotropy originating from the outer boundary. This is a crucial feature when studying the influence of the microstructure on the performance of composite materials, which is demonstrated by hysteresis calculations of soft magnetic structures that are operated in a closed magnetic loop configuration. The applied differential formulation is perfectly suited for the application of true periodic boundary conditions. The finite difference equations can be solved by a highly efficient Fast Fourier Transform method.

Hybrid integral transform analysis of supercooled droplets solidification

The freezing phenomena in supercooled liquid droplets are important for many engineering applications. For instance, a theoretical model of this phenomenon can offer insights for tailoring surface coatings and for achieving icephobicity to reduce ice adhesion and accretion. In this work, a mathematical model and hybrid numerical–analytical solutions are developed for the freezing of a supercooled droplet immersed in a cold air stream, subjected to the three main transport phenomena at the interface between the droplet and the surroundings: convective heat transfer, convective mass transfer and thermal radiation. Error-controlled hybrid solutions are obtained through the extension of the generalized integral transform technique to the transient partial differential formulation of this moving boundary heat transfer problem. The nonlinear boundary condition for the interface temperature is directly accounted for by the choice of a nonlinear eigenfunction expansion base. Also, the nonlinear equation of motion for the freezing front is solved together with the ordinary differential system for the integral transformed temperatures. After comparisons of the solution with previously reported numerical and experimental results, the influence of the related physical parameters on the droplet temperatures and freezing time is critically analysed.

OSCILLATIONS DESCRIBED BY ANALOGUES OF VAN DER POL AND RAYLEIGH EQUATIONS

The paper deals with the modes of steady-state quasilinear self-oscillations described by the analogues of Van der Pol and Rayleigh differential equations. The differential formulation of the energy balance method is applied for studying the motion. The conditions for the equations to describe quasilinear self-oscillations with the amplitude independent of the initial conditions are derived in the form of inequalities. The formulae for computing this amplitude using the table of gamma functions are proposed. The steady-state mode of the self-oscillations is proved to be stable in contrast to the static equilibrium which appears unstable. The inequalities are also obtained which guarantee the equations of the type considered to describe the damped free oscillations about the zero equilibrium or the oscillations which build up resulting in the loss of system stability. These forms of motion depend on the initial conditions. For small initial deflections, which are less than the threshold value, the oscillations decay whereas for large once they build up. The dynamical system, which is stable in small, is unstable in large. The impact of the constant component of the resistance force on the oscillatory process is also studied. It is shown to cause the shift of the position about which the steady-state self-oscillations occur but not to influence their amplitude or frequency, which is the result of the linear elasticity of the system. The special cases are separated, when the computational formulae proposed become the results previously known. The analytical studies are followed by numerical solution of the respective Cauchy problem. By comparing the results obtained by the two methods we substantiate the adequacy of the computational formulae obtained.

A Novel Computational Method to Identify/Analyze Hysteresis Loops of Hard Magnetic Materials

In this study, a novel computational method capable of reproducing hysteresis loops of hard magnetic materials is proposed. It is conceptually based on the classical Preisach model but has a completely different approach in the modeling of the hysteron effect. Indeed, the change in magnetization caused by a single hysteron is compared here to the change in velocity of two disk-shaped solids elastically colliding with each other rather than the effect of ideal geometrical entities giving rise to so-called Barkhausen jumps. This allowed us to obtain a simple differential formulation for the global magnetization equation with a significant improvement in terms of computational performance. A sensitivity analysis on the parameters of the proposed method has indeed shown the capability to model a large class of hysteresis loops. Moreover, the proposed method permits modeling of the temperature effect on magnetization of neodymium magnets, which is a key point for the design of electrical machines. Therefore, application of the proposed method to the hysteresis loop of a real NdFeB magnet has been proven to be very accurate and efficient for a large temperature range.

Reduction of the Diffusion Equations of Ion Channel Clusters with Virtually No Increase in the Error Bound

A stochastic differential formulation for the collective dynamics of ion channel clusters in excitable membranes is developed from the so-called “reduced strong diffusion formulation”. In this error bound optimizing reduced formulation, the potassium channel states [Formula: see text] and [Formula: see text], and, the sodium channel states [Formula: see text] and [Formula: see text] are the retained states; consequently, the formulation accommodates only four channel variables and five white noises. The accuracy of the formulation is tested over the standard deviations and autocorrelation times of the channel density fluctuations. The findings are seen to be virtually identical to the corresponding results from the exact microscopic Markov simulations. The formulation arises as the most accurate model with that structural simplicity, thus making it an important model for both analytic analyses and numerical simulations in the study of finite-sized membranes.

Finite Element Study for Magnetohydrodynamic (MHD) Tangent Hyperbolic Nanofluid Flow over a Faster/Slower Stretching Wedge with Activation Energy

The below work comprises the unsteady flow and enhanced thermal transportation for Carreau nanofluids across a stretching wedge. In addition, heat source, magnetic field, thermal radiation, activation energy, and convective boundary conditions are considered. Suitable similarity functions use to transmuted partial differential formulation into the ordinary differential form, which is solved numerically by the finite element method and coded in Matlab script. Parametric computations are made for faster stretch and slowly stretch to the surface of the wedge. The progressing value of parameter A (unsteadiness), material law index ϵ, and wedge angle reduce the flow velocity. The temperature in the boundary layer region rises directly with exceeding values of thermophoresis parameter Nt, Hartman number, Brownian motion parameter Nb, ϵ, Biot number Bi and radiation parameter Rd. The volume fraction of nanoparticles rises with activation energy parameter EE, but it receded against chemical reaction parameter Ω, and Lewis number Le. The reliability and validity of the current numerical solution are ascertained by establishing convergence criteria and agreement with existing specific solutions.

Multiscale Modeling of Composite Materials with DECM Approach: Shape Effect of Inclusions

This paper addresses the study of the stress field in composites continua with the multiscale approach of the DECM (Discrete Element modeling with the Cell Method). The analysis focuses on composites consisting of a matrix with inclusions of various shapes, to investigate whether and how the shape of the inclusions changes the stress field. The purpose is to provide a numerical explanation for some of the main failure mechanisms of concrete, which is precisely a composite consisting of a cement-based matrix and aggregates of various shapes. Actually, while extensive experimental campaigns detailed the shape effect of concrete aggregates in the past, so far it has not been possible to model the stress field within the inclusions and on the interfaces accurately. The reason for this lies in the limits of the differential formulation, which is the basis of the most commonly used numerical methods. The Cell Method (CM), on the contrary, is an algebraic method that provides descriptions up to the micro-scale, independently of the presence of rheological discontinuities or concentrated sources. This makes the CM useful for describing the shape effect of the inclusions, on the micro-scale. When used together with a multiscale approach, it also models the macro-scale behavior of periodic composite continua, without losing accuracy on the micro-scale. The DECM uses discrete elements precisely to provide the CM with a multiscale approach.

MESOSCALE METEOROLOGICAL MODEL TSUNM3 FOR THE STUDY AND FORECAST OF METEOROLOGICAL PARAMETERS OF THE ATMOSPHERIC SURFACE LAYER OVER A MAJOR POPULATION CENTER

The paper describes the mathematical formulation and numerical method of the TSUNM3 high-resolution mesoscale meteorological model being developed at Tomsk State University. The model is nonhydrostatic and includes three-dimensional nonstationary equations of hydrothermodynamics of the atmospheric boundary layer with parameterization of turbulence, moisture microphysics, long-wave and short-wave (solar) radiation, and advective and latent heat flows in the atmosphere and at the boundary of its interaction with the underlying surface. The numerical algorithm is constructed using structured grids with uniform spacing in horizontal directions and condensing to the Earth surface in the vertical direction. When approximating the differential formulation of the problem, the finite volume method with the second order approximation in the spatial variables is used. Explicit-implicit approximations in time (Adams–Bashforth and Crank–Nicolson) are used to achieve second-order accuracy in time. The paper presents results of numerical forecasting of the main meteorological parameters of the atmosphere (temperature, humidity, wind speed and direction) and precipitation in different seasons in the Siberian region. The models were tested with the help of observations obtained using the Volna-4M sodar, MTR-5 temperature profile meter, and Meteo-2 ultrasonic weather stations of the Atmosfera Collective Use Center. The improved TSUNM3 model is shown to adequately reflect the precipitation time and intensity. However, in some cases, the times of its beginning and end do not always coincide, the difference can reach several hours. The precipitation phase state is reflected reliably. Over 70% of precipitation cases are confirmed by numerical calculations. The model satisfactorily predicts temperature and humidity characteristics. The quality of the precipitation forecast model is comparable to the modern mesoscale models, such as the Weather Research and Forecasting (WRF) model.