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The purpose of this paper is to evaluate the performance of a numerical method for the solution to shallow-water equations on a staggered grid, in simulations for shear instabilities at two convective Froude numbers.

The simulations start from a small perturbation to a base flow with a hyperbolic-tangent velocity profile. The subsequent development of the shear instabilities is studied from the simulations using a number of flux-limiting schemes, including the second-order MINMOD, the third-order ULTRA-QUICK and the fifth-order WENO schemes for the spatial interpolation of the nonlinear fluxes. The fourth-order Runge-Kutta method advances the simulation in time.

The simulations determine two parameters: the fractional growth rate of the linear instabilities; and the vorticity thickness of the first nonlinear peak. Grid refinement using 32, 64, 128, 256 and 512 nodes over one wave length determines the exact values by extrapolation and the computational error for the parameters. It also determines the overall order of convergence for each of the flux-limiting schemes used in the numerical simulations.

The four-digit accuracy of the numerical simulations presented in this paper are comparable to analytical solutions. The development of this reliable numerical simulation method has paved the way for further study of the instabilities in shear flows that radiate waves.

The paper presents a mathematical model for the hysteresis phenomenon in a multi-winding single-phase core type transformer. The set of loop differential equations was developed for Kth winding transformer model where the flux linkages of each winding includes a flux common Φ to all windings as function of magneto motive force Θ of all windings. The purpose of this paper is to first determine a hysteresis nonlinearity involved in Φ(Θ) function using modified Preisach theory and second to develop new analytical formula of Preisach distribution function (PDF).

It is assumed in this paper that flux linkage characteristics Ψ(i) of each winding have nonlinear component due to the magnetization characteristic of the steel core and sum of linear components due to the self and mutual leakage fluxes. This nonlinear component of Ψ(i) characteristic can be expressed as a flux common Φ to all windings vs ampere-turns Θ of all windings. The nonlinear flux linkage characteristics Ψ(i) of the tested transformer are calculated from the set of measured terminal voltages and terminal currents. To simulate magnetic behavior of the iron core the feedback scalar Preisach model of hysteresis is proposed which gives more accurate predictions than classical model. For this hysteresis model the PDF and feedback function are needed. The intend of this paper is to find these function as an analytical formulas which are convenient for numerical simulations. For identification of the PDF and feedback function parameters of the considered iron core of tested transformer the Levenberg-Marquardt optimization algorithm was used.

The flux common to all windings is calculated by integrating the induced voltages of the appropriate windings. In this paper the PDF is proposed as a functional series including two dimensional Gauss expressions. In order to proper approximation of hysteresis nonlinearity of the tested iron core the first three terms of functional series of the PDF have been used. In the optimization algorithm only initial and descending limiting hysteresis curves Φ(Θ) were utilized. The feedback function for proposed hysteresis model is assumed as third-order polynomial. The hysteresis model has been successfully validated by comparing the calculated and measured results of Φ(Θ) hysteresis curves. This hysteresis model can be used in transient and steady state simulations of tested transformer taking into account the hysteresis phenomenon. The developed hysteresis model can be also used for analysis of the influence of remnant flux on the operation of tested transformer especially in transient states.

In this paper the feedback Preisach hysteresis model is involved in the flux common to all windings vs ampere-turns of all windings. The new PDF is proposed as functional series including two dimensional Gauss expressions. For tested transformer the three first terms of this functional series may be used for proper approximation of hysteresis nonlinearities.

To obtain error estimates for 3D consistent boundary‐flux approximations.

Isoparametric approach is used for constructing finite‐element approximations.

This research study presents a convergence analysis of 3D boundary‐flux approximations. Error estimates are proved for the approximate solutions of the problem under consideration.

General results for a consistent boundary‐flux problem are obtained for all 3D domains with Lipschitz‐continuous boundary. This investigation will be continued studying combined effect of curved boundaries and isoparametric numerical integration. An optimal refined strategy with respect to algorithmic aspects for solving 3D boundary‐flux problem also will be considered.

The obtained results enable engineers to calculate the flux across the curved boundaries using finite element method (FEM).

The paper presents an isoparametric finite‐element method for a 3D consistent boundary‐flux problem in domains with complex geometry. The work is addressed to the possible‐related fields of interest of postgraduate students and specialists in fluid mechanics and numerical analysis.

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

To propose and adaptive nonlinear controller for adjustable speed sensorless induction motor drive, using a novel adaptive rotor flux observer. The adaptive flux observer scheme in this paper provides the simultaneous estimation of the rotor speed, rotor resistance and stator resistance.

The IM rotor speed and rotor flux controllers are designed based on combination of input‐output feedback linearizing, linear optimal feedback control and sliding‐mode (SM) control methods. In addition a novel adaptive rotor flux observer is designed based on Lyapunov theory. The proposed control method is tested by simulation and experimental results.

The composite rotor speed and rotor flux observer in combination with adaptive rotor flux scheme guarantees a perfect speed, torque and flux tracking control for the IM sensorless drive.

The proposed control method has a drawback in the IM low speed operating region. Additional research may be able to solve this problem as well as should analyze the sensitivity of the IM drive system performance with respect to variation of the system controller and adaptive flux observer gains. In addition, this research should also analyze the influence of sampling rate, truncation errors, measurement noise, simplifying model assumption and magnetic saturation.

The proposed control method can be used for adaptive and robust control of the IM drive where an optimal efficiency is desired subject to the variable load torque demand.

Based on Lyapunov theory, a novel adaptive rotor flux observer is introduced in which the rotor speed, rotor resistance and stator resistance are treated as the unknown constant parameters.

This paper aims to provide a symplectic conservation numerical analysis method for the study of nonlinear LC circuit.

The flux linkage control type nonlinear inductance model is adopted, and the LC circuit can be converted into the Hamiltonian system by introducing the electric charge as the state variable of the flux linkage. The nonlinear Hamiltonian matrix equation can be solved by perturbation method, which can be written as the sum of linear and nonlinear terms. Firstly, the linear part can be solved exactly. On this basis, the nonlinear part is analyzed by the canonical transformation. Then, the coefficient matrix of the obtained equation is still a Hamiltonian matrix, so symplectic conservation is achieved.

Numerical results reveal that the method proposed has strong stability, high precision and efficiency, and it has great advantages in long-term simulations.

This method provides a novel and effective way in studying the nonlinear LC circuit.

The purpose of this paper is to create a reliable nonlinear magnetic equivalent circuit (NMEC) of the hybrid magnetic bearing (HMB). Commonly used magnetic equivalent circuits of HMB omit a saturation effect of the magnetic material as well as the leakage and fringing flux. It results in imprecise modelling of the magnetic field distribution. On the other hand, only 3D finite element analysis (FEA) can be used to precisely simulate the magnetic field in this type of the magnetic bearing. The proposed NMEC incorporates the saturation effect of the magnetic material, as well as the leakage and fringing flux.

The magnetic equivalent circuit of presented HMB is proposed to obtain a reliable model that ensures short calculation time. Developed NMEC incorporates the phenomena as the saturation effect, as well as the leakage and fringing flux. The reluctance of the air gap that includes the fringing flux was calculated using 3D FEA. Kirchhoffs’ laws were used to create a set of nonlinear equations that were iteratively solved by Broyden’s method.

Incorporating into NMEC of the HMB a saturation effect of the magnetic material, as well as the leakage and fringing flux, resulted in the accurate model that was in good agreement with 3 D finite element model and the real object. The developed NMEC offers the calculation time in the range of miliseconds, therefore can be successfully used in the engineering design instead of the FEM.

Presented NMEC can be considered as a fundamental model that can be successfully used for accurate and fast simulation of the HMB. Proposed NMEC includes considerable factors that decide about the model accuracy such as the saturation effect of the ferromagnetic material and the leakage and fringing flux. The developed NMEC can be used in the optimization procedures and for simulations of dynamic responses.

The purpose of this study is to develop a new method of lines for one-dimensional (1D) advection-reaction-diffusion (ADR) equations that is conservative and provides piecewise analytical solutions in space, compare it with other finite-difference discretizations and assess the effects of advection and reaction on both 1D and two-dimensional (2D) problems.

A conservative method of lines based on the piecewise analytical integration of the two-point boundary value problems that result from the local solution of the advection-diffusion operator subject to the continuity of the dependent variables and their fluxes at the control volume boundaries is presented. The method results in nonlinear first-order, ordinary differential equations in time for the nodal values of the dependent variables at three adjacent grid points and triangular mass and source matrices, reduces to the well-known exponentially fitted techniques for constant coefficients and equally spaced grids and provides continuous solutions in space.

The conservative method of lines presented here results in three-point finite difference equations for the nodal values, implicitly treats the advection and diffusion terms and is unconditionally stable if the reaction terms are implicitly treated. The method is shown to be more accurate than other three-point, exponentially fitted methods for nonlinear problems with interior and/or boundary layers and/or source/reaction terms. The effects of linear advection in 1D reacting flow problems indicates that the wave front steepens as it approaches the downstream boundary, whereas its back corresponds to a translation of the initial conditions; for nonlinear advection, the wave front exhibits steepening but the wave back shows a linear dependence on space. For a system of two nonlinearly coupled, 2D ADR equations, it is shown that a counter-clockwise rotating vortical field stretches the spiral whose tip drifts about the center of the domain, whereas a clock-wise rotating one compresses the wave and thickens its arms.

A new, conservative method of lines that implicitly treats the advection and diffusion terms and provides piecewise-exponential solutions in space is presented and applied to some 1D and 2D advection reactions.

This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository.

The nonlinear two-point flux approximation is used to discretize the diffusion flux and has a fixed stencil. The cell-vertex unknowns are applied to define the auxiliary unknowns and can be interpolated by the cell-centered unknowns. The approximation of convection flux is based on the second-order upwind method with a slope limiter.

Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

A new positivity-preserving nonlinear finite volume scheme is proposed to simulate the far-field model used in the geological radioactive waste repository. Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

The majority of three‐phase dynamic transformer models used in commercially available electric power system transient simulation programs offer only saturated three‐phase transformer models built from three single‐phase transformer models. This paper sets out to deal with the modelling and transient analysis of a saturated three‐limb core‐type transformer.

Three iron core models I‐III are given by the current‐dependent characteristics of flux linkages. In the first model, these characteristics are given by a set of piecewise linear functions, which include saturation. In the second model, the piecewise linear functions are replaced by the measured nonlinear characteristic. The more complex third model is given by a set of measured flux linkage characteristics.

The behaviour of transformers used in electric power applications depends considerably on the properties of magnetically nonlinear iron core. The best agreement between the calculated and measured results is obtained by use of the most complex iron core model III, which takes into account magnetic cross‐couplings between different limbs, caused by saturation.

Measurement of the current‐dependent flux linkage characteristics of the 0.4 kV, 3.5 kVA laboratory transformer requires corresponding excitation of windings by three independent linear amplifiers. Current‐dependent flux linkage characteristics of the larger power transformer can be determined either by similar measurement with linear amplifiers of an appropriate power or by extracting them from the calculated magnetic field, which is done by the finite element method.

A three‐phase dynamic transformer model with the obtained iron core model III is suitable for the numerical analysis of nonsymmetric transient states in power systems.

This paper presents a three‐phase dynamic transformer model with an original iron core model III, which accounts for magnetic cross‐couplings between different limbs, caused by saturation.