Search results for: Lorenz equation.

Authors: V. Ghadamyari, F. Samadi, F. Kowsary

Abstract:

An inverse geometry problem is solved to predict an unknown irregular boundary profile. The aim is to minimize the objective function, which is the difference between real and computed temperatures, using three different versions of Conjugate Gradient Method. The gradient of the objective function, considered necessary in this method, obtained as a result of solving the adjoint equation. The abilities of three versions of Conjugate Gradient Method in predicting the boundary profile are compared using a numerical algorithm based on the method. The predicted shapes show that due to its convergence rate and accuracy of predicted values, the Powell-Beale version of the method is more effective than the Fletcher-Reeves and Polak –Ribiere versions.

Keywords: Boundary elements, Conjugate Gradient Method, Inverse Geometry Problem, Sensitivity equation.

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Authors: M. Akbari, S. Sadodin

Abstract:

In this paper, the melting of a semi-infinite body as a result of a moving laser beam has been studied. Because the Fourier heat transfer equation at short times and large dimensions does not have sufficient accuracy; a non-Fourier form of heat transfer equation has been used. Due to the fact that the beam is moving in x direction, the temperature distribution and the melting pool shape are not asymmetric. As a result, the problem is a transient threedimensional problem. Therefore, thermophysical properties such as heat conductivity coefficient, density and heat capacity are functions of temperature and material states. The enthalpy technique, used for the solution of phase change problems, has been used in an explicit finite volume form for the hyperbolic heat transfer equation. This technique has been used to calculate the transient temperature distribution in the semi-infinite body and the growth rate of the melt pool. In order to validate the numerical results, comparisons were made with experimental data. Finally, the results of this paper were compared with similar problem that has used the Fourier theory. The comparison shows the influence of infinite speed of heat propagation in Fourier theory on the temperature distribution and the melt pool size.

Keywords: Non-Fourier, Enthalpy technique, Melt pool, Radiational boundary condition

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Authors: Yiqin Lin, Liang Bao, Yimin Wei

Abstract:

In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XBT +CDT = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projection subspaces is outlined. We show that the approximate solution is an exact solution of a perturbed Sylvester matrix equation. Moreover, exact expression for the norm of residual is derived and results on finite termination and convergence are presented. Some numerical examples are presented to illustrate the effectiveness of the proposed method.

Keywords: Arnoldi process, Krylov subspace, Iterative method, Sylvester equation, Dissipative matrix.

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Authors: Arti Vaish, Harish Parthasarathy

Abstract:

The scalar wave equation for a potential in a curved space time, i.e., the Laplace-Beltrami equation has been studied in this work. An action principle is used to derive a finite element algorithm for determining the modes of propagation inside a waveguide of arbitrary shape. Generalizing this idea, the Maxwell theory in a curved space time determines a set of linear partial differential equations for the four electromagnetic potentials given by the metric of space-time. Similar to the Einstein-s formulation of the field equations of gravitation, these equations are also derived from an action principle. In this paper, the expressions for the action functional of the electromagnetic field have been derived in the presence of gravitational field.

Keywords: General theory of relativity, electromagnetism, metric tensor, Maxwells equations, test functions, finite element method.

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Authors: Djemai Bara, Mohamed Faouzi Mahboub, Djamila Bennaceur-Doumaz

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In this work, the role of the preformed plasma created on the front face of a target, irradiated by a high intensity short pulse laser, in the framework of ion acceleration process, modeled by Target Normal Sheath Acceleration (TNSA) mechanism, is studied. This plasma is composed of cold ions governed by fluid equations and non-thermal & trapped with densities represented by a "Cairns-Gurevich" equation. The self-similar solution of the equations shows that electronic trapping and the presence of non-thermal electrons in the pre-plasma are both responsible in ion acceleration as long as the proportion of energetic electrons is not too high. In the case where the majority of electrons are energetic, the electrons are accelerated directly by the ponderomotive force of the laser without the intermediate of an accelerating plasma wave.

Keywords: Cairns-Gurevich Equation, ion acceleration, plasma expansion, pre-plasma.

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Authors: Tan K. B., Norhashidah Hj. M. Ali

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In this article, we aim to discuss the formulation of two explicit group iterative finite difference methods for time-dependent two dimensional Burger-s problem on a variable mesh. For the non-linear problems, the discretization leads to a non-linear system whose Jacobian is a tridiagonal matrix. We discuss the Newton-s explicit group iterative methods for a general Burger-s equation. The proposed explicit group methods are derived from the standard point and rotated point Crank-Nicolson finite difference schemes. Their computational complexity analysis is discussed. Numerical results are given to justify the feasibility of these two proposed iterative methods.

Keywords: Standard point Crank-Nicolson (CN), Rotated point Crank-Nicolson (RCN), Explicit Group (EG), Explicit Decoupled Group (EDG).

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Authors: Zhenying Hong, Guangwei Yuan, Xuedong Fu, Shulin Yang

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In this paper, the typical exponential method, diamond difference and modified time discrete scheme is researched for self adaptive time step. The second-order time evolution scheme is applied to time-dependent spherical neutron transport equation by discrete ordinates method. The numerical results show that second-order time evolution scheme associated exponential method has some good properties. The time differential curve about neutron current is more smooth than that of exponential method and diamond difference and modified time discrete scheme.

Keywords: Exponential method, diamond difference, modified time discrete scheme, second-order time evolution scheme.

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Authors: Pranav Ravikumar, Arunabha Bera, Vijay K. Mehra, Anand Kumar

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This paper solves the Non Linear Schrodinger Equation using the Split Step Fourier method for modeling an optical fiber. The model generates a complex wave of optical pulses and using the results obtained two graphs namely Loss versus Wavelength and Dispersion versus Wavelength are generated. Taking Chromatic Dispersion and Polarization Mode Dispersion losses into account, the graphs generated are compared with the graphs formulated by JDS Uniphase Corporation which uses standard values of dispersion for optical fibers. The graphs generated when compared with the JDS Uniphase Corporation plots were found to be more or less similar thus verifying that the model proposed is right. MATLAB software was used for doing the modeling.

Keywords: Modulation, Non Linear Schrodinger Equation, Optical fiber, Split Step Fourier Method.

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Authors: Punit Kumar, Niraj Kumar

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The influence of transverse surface roughness on EHL characteristics has been investigated numerically using an extensive set of full EHL line contact simulations for shear-thinning lubricants under pure sliding condition. The shear-thinning behavior of lubricant is modeled using Carreau viscosity equation along with Doolittle-Tait equation for lubricant compressibility. The surface roughness is assumed to be sinusoidal and it is present on the stationary surface. It is found that surface roughness causes sharp pressure peaks along with reduction in central and minimum film thickness. With increasing amplitude of surface roughness, the minimum film thickness decreases much more rapidly as compared to the central film thickness.

Keywords: EHL, Carreau, Shear-thinning, Surface Roughness, Amplitude, Wavelength.

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Authors: Chien-Hua Lee, Cheng-Yi Chen

Abstract:

The stability test problem for homogeneous large-scale perturbed bilinear time-delay systems subjected to constrained inputs is considered in this paper. Both nonlinear uncertainties and interval systems are discussed. By utilizing the Lyapunove equation approach associated with linear algebraic techniques, several delay-independent criteria are presented to guarantee the robust stability of the overall systems. The main feature of the presented results is that although the Lyapunov stability theorem is used, they do not involve any Lyapunov equation which may be unsolvable. Furthermore, it is seen the proposed schemes can be applied to solve the stability analysis problem of large-scale time-delay systems.

Keywords: homogeneous bilinear system, constrained input, time-delay, uncertainty, transient response, decay rate.

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Authors: Hassan Manouzi, Taous-Meriem Laleg-Kirati

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In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Itô chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm alternative. To reduce the computational complexity of this system, we shall use a decomposition method using the Wiener-Itô chaos expansion. Once the approximation of the solution is performed using the finite element method for example, the statistics of the numerical solution can be easily evaluated.

Keywords: Eigenvalue problem, Wick product, SPDEs, finite element, Wiener-Itô chaos expansion.

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Authors: Stephen Kirkup

Abstract:

This paper discusses the implementation of the boundary element method (BEM) on an Excel spreadsheet and how it can be used in teaching vector calculus and simulation. There are two separate spreadheets, within which Laplace equation is solved by the BEM in two dimensions (LIBEM2) and axisymmetric three dimensions (LBEMA). The main algorithms are implemented in the associated programming language within Excel, Visual Basic for Applications (VBA). The BEM only requires a boundary mesh and hence it is a relatively accessible method. The BEM in the open spreadsheet environment is demonstrated as being useful as an aid to teaching and learning. The application of the BEM implemented on a spreadsheet for educational purposes in introductory vector calculus and simulation is explored. The development of assignment work is discussed, and sample results from student work are given. The spreadsheets were found to be useful tools in developing the students’ understanding of vector calculus and in simulating heat conduction.

Keywords: Boundary element method, laplace equation, vector calculus, simulation, education.

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Authors: Karandeep Singh, Baibing Li

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Traffic density, an indicator of traffic conditions, is one of the most critical characteristics to Intelligent Transport Systems (ITS). This paper investigates recursive traffic density estimation using the information provided from inductive loop detectors. On the basis of the phenomenological relationship between speed and density, the existing studies incorporate a state space model and update the density estimate using vehicular speed observations via the extended Kalman filter, where an approximation is made because of the linearization of the nonlinear observation equation. In practice, this may lead to substantial estimation errors. This paper incorporates a suitable transformation to deal with the nonlinear observation equation so that the approximation is avoided when using Kalman filter to estimate the traffic density. A numerical study is conducted. It is shown that the developed method outperforms the existing methods for traffic density estimation.

Keywords: Density estimation, Kalman filter, speed-densityrelationship, Traffic surveillance.

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Authors: Amir Hossein Daei Sorkhabi, Farid Vakili Tahami

Abstract:

In this paper, creep constitutive equations of base (Parent) and weld materials of the weldment for cold-drawn 304L stainless steel have been obtained experimentally. For this purpose, test samples have been generated from cold drawn bars and weld material according to the ASTM standard. The creep behavior and properties have been examined for these materials by conducting uniaxial creep tests. Constant temperatures and constant load uni-axial creep tests have been carried out at two high temperatures, 680 and 720 oC, subjected to constant loads, which produce initial stresses ranging from 240 to 360 MPa. The experimental data have been used to obtain the creep constitutive parameters using numerical optimization techniques.

Keywords: Creep, Constitutive equation, Cold-drawn 304L stainless steel, Weld, Base material

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Authors: Zebiri Chemseddine, Benabdelaziz Fatiha

Abstract:

The effect of a chiral bianisotropic substrate on the complex resonant frequency of a rectangular microstrip resonator has been studied on the basis of the integral equation formulation. The analysis is based on numerical resolution of the integral equation using Galerkin procedure for moment method in the spectral domain. This work aim first to study the effect of the chirality of a bianisotopic substrate upon the resonant frequency and the half power bandwidth, second the effect of a magnetic anisotropy via an asymptotic approach for very weak substrate upon the resonant frequency and the half power bandwidth has been investigated. The obtained results are compared with previously published work [11-9], they were in good agreement.

Keywords: Microstrip antenna, bianisotropic media, resonant frequency, moment method.

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Authors: Gopu Sreenivasulu, Bimlesh Kumar, Achanta Ramakrishna Rao

Abstract:

To define or predict incipient motion in an alluvial channel, most of the investigators use a standard or modified form of Shields- diagram. Shields- diagram does give a process to determine the incipient motion parameters but an iterative one. To design properly (without iteration), one should have another equation for resistance. Absence of a universal resistance equation also magnifies the difficulties in defining the model. Neural network technique, which is particularly useful in modeling a complex processes, is presented as a tool complimentary to modeling incipient motion. Present work develops a neural network model employing the RBF network to predict the average velocity u and water depth y based on the experimental data on incipient condition. Based on the model, design curves have been presented for the field application.

Keywords: Incipient motion, Prediction error, Radial-Basisfunction, Sediment transport, Shields' diagram.

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Authors: Dara Singh, Keikhosrow Firouzbakhsh, Mohammad Taghi Ahmadian

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In this paper, conventional laser Keratoplasty surgeries in the human eye are studied. For this purpose, a validated 3D finite volume model of the human eye is introduced. In this model the fluid flow has also been considered. The discretized domain of the human eye incorporates a bio-heat transfer equation coupled with a Boussinesq equation. Both continuous and pulsed lasers have been modeled and the results are compared. Moreover, two different conventional surgical positions that are upright and recumbent are compared for these laser therapies. The simulation results show that in these conventional surgeries, the temperature rises above the critical values at the laser insertion areas. However, due to the short duration and the localized nature, the potential damages are restricted to very small regions and can be ignored. The conclusion is that the present day lasers are acceptably safe to the human eye.

Keywords: Eye, heat-transfer, keratoplasty laser, surgery.

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Authors: Hailong Zhu, Zhaoxiang Li, Kejun Zhuang

Abstract:

By means of Contractor Iteration Method, we solve and visualize the Lane-Emden(-Fowler) equation Δu + up = 0, in Ω, u = 0, on ∂Ω. It is shown that the present method converges quadratically as Newton’s method and the computation of Contractor Iteration Method is cheaper than the Newton’s method.

Keywords: Positive solutions, newton's method, contractor iteration method, Eigenpairs.

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Authors: Rana Khalid Naeem, Waseem Ahmed Khan, Muhammad Akhtar, Asif Mansoor

Abstract:

The Navier Stokes Equations (NSE) for an incompressible fluid of variable viscosity in the presence of an unknown external force in Von-Mises system x,\ are transformed, and some new exact solutions for a class of flows characterized by equation y f x a\b for an arbitrary state equation are determined, where f x is a function, \ the stream function, a z 0 and b are the arbitrary constants. In three, out of four cases, the function f x is arbitrary, and the solutions are the solutions of the flow equations for all the flows characterized by the equationy f x a\b. Streamline patterns for some forms of f x in unbounded and bounded regions are given.

Keywords: Bounded and unbounded region, Exact solution, Navier Stokes equations, Streamline pattern, Variable viscosity, Von- Mises system

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Authors: Arunroong Wongkungwan

Abstract:

This study aimed to investigate the influence of selected antecedents, which were tourists’ satisfaction towards attractions in Bangkok, perceived value of the attractions, feelings of engagement with the attractions, acquaintance with the attractions, push factors, pull factors and motivation to seek novelty, on foreign tourist’s loyalty towards tourist attractions in Bangkok. By using multi stage sampling technique, 400 international tourists were sampled. After that, Semi Structural Equation Model was utilized in the analysis stage by LISREL. The Semi Structural Equation Model of the selected antecedents of tourist’s loyalty attractions had a correlation with the empirical data through the following statistical descriptions: Chi- square = 3.43, df = 4, P- value = 0.48893; RMSEA = 0.000; CFI = 1.00; CN = 1539.75; RMR = 0.0022; GFI = 1.00 and AGFI = 0.98. The findings indicated that all antecedents were able together to predict the loyalty of the foreign tourists who visited Bangkok at 73 percent.

Keywords: Antecedents, Loyalty, Foreign Tourists, Tourist Attractions, Bangkok.

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Authors: Xie Kefeng, Zhang He

Abstract:

For the stability and control demand of offshore small floating platform, a 2-HUS/U parallel mechanism was presented as offshore platform. Inverse kinematics was obtained by institutional constraint equation, and the dynamic model of offshore 2-HUS/U parallel platform was derived based on rigid body’s Lagrangian method. The equivalent moment of inertia, damping and driving force/torque variation of offshore 2-HUS/U parallel platform were analyzed. A numerical example shows that, for parallel platform of given motion, system’s equivalent inertia changes 1.25 times maximally. During the movement of platform, they change dramatically with the system configuration and have coupling characteristics. The maximum equivalent drive torque is 800 N. At the same time, the curve of platform’s driving force/torque is smooth and has good sine features. The control system needs to be adjusted according to kinetic equation during stability and control and it provides a basis for the optimization of control system.

Keywords: 2-HUS/U platform, Dynamics, Lagrange, Parallel platform.

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Authors: M. Breška, I. Peruš, V. Stankovski

Abstract:

The number of Ground Motion Prediction Equations (GMPEs) used for predicting peak ground acceleration (PGA) and the number of earthquake recordings that have been used for fitting these equations has increased in the past decades. The current PF-L database contains 3550 recordings. Since the GMPEs frequently model the peak ground acceleration the goal of the present study was to refit a selection of 44 of the existing equation models for PGA in light of the latest data. The algorithm Levenberg-Marquardt was used for fitting the coefficients of the equations and the results are evaluated both quantitatively by presenting the root mean squared error (RMSE) and qualitatively by drawing graphs of the five best fitted equations. The RMSE was found to be as low as 0.08 for the best equation models. The newly estimated coefficients vary from the values published in the original works.

Keywords: Ground Motion Prediction Equations, Levenberg-Marquardt algorithm, refitting PF-L database.

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Authors: Ömer Aktaş, Olga A. Suvorova, Oleg Tretyakov

Abstract:

The boundary value problem on non-canonical and arbitrary shaped contour is solved with a numerically effective method called Analytical Regularization Method (ARM) to calculate propagation parameters. As a result of regularization, the equation of first kind is reduced to the infinite system of the linear algebraic equations of the second kind in the space of L₂. This equation can be solved numerically for desired accuracy by using truncation method. The parameters as cut-off wavenumber and cut-off frequency are used in waveguide evolutionary equations of electromagnetic theory in time-domain to illustrate the real-valued TM fields with lossy and lossless media.

Keywords: Arbitrary cross section waveguide, analytical regularization method, evolutionary equations of electromagnetic theory of time-domain, TM field.

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Authors: K. Yathish, K. G. Binu, B. S. Shenoy, D. S. Rao, R. Pai

Abstract:

Load carrying capacity of an oil lubricated two-axial groove journal bearing is simulated by taking into account the viscosity variations in lubricant due to the addition of TiO2 nanoparticles as lubricant additive. Shear viscosities of TiO2 nanoparticle dispersions in oil are measured for various nanoparticle additive concentrations. The viscosity model derived from the experimental viscosities is employed in a modified Reynolds equation to obtain the pressure profiles and load carrying capacity of two-axial groove journal bearing. Results reveal an increase in load carrying capacity of bearings operating on nanoparticle dispersions as compared to plain oil.

Keywords: Journal bearing, TiO2 nanoparticles, viscosity model, Reynolds equation, load carrying capacity.

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Authors: A. Nicknam, N. Eftekhari, A. Mazarei, M. Ganjvar

Abstract:

This article presents an alternative collapse capacity intensity measure in the three elements form which is influenced by the spectral ordinates at periods longer than that of the first mode period at near and far source sites. A parameter, denoted by β, is defined by which the spectral ordinate effects, up to the effective period (2T1), on the intensity measure are taken into account. The methodology permits to meet the hazard-levelled target extreme event in the probabilistic and deterministic forms. A MATLAB code is developed involving OpenSees to calculate the collapse capacities of the 8 archetype RC structures having 2 to 20 stories for regression process. The incremental dynamic analysis (IDA) method is used to calculate the structure’s collapse values accounting for the element stiffness and strength deterioration. The general near field set presented by FEMA is used in a series of performing nonlinear analyses. 8 linear relationships are developed for the 8structutres leading to the correlation coefficient up to 0.93. A collapse capacity near field prediction equation is developed taking into account the results of regression processes obtained from the 8 structures. The proposed prediction equation is validated against a set of actual near field records leading to a good agreement. Implementation of the proposed equation to the four archetype RC structures demonstrated different collapse capacities at near field site compared to those of FEMA. The reasons of differences are believed to be due to accounting for the spectral shape effects.

Keywords: Collapse capacity, fragility analysis, spectral shape effects, IDA method.

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Authors: Masamitsu Chikaraishi, Mutsuto Kawahara

Abstract:

This paper focuses on a technique for identifying the geological boundary of the ground strata in front of a tunnel excavation site using the first order adjoint method based on the optimal control theory. The geological boundary is defined as the boundary which is different layers of elastic modulus. At tunnel excavations, it is important to presume the ground situation ahead of the cutting face beforehand. Excavating into weak strata or fault fracture zones may cause extension of the construction work and human suffering. A theory for determining the geological boundary of the ground in a numerical manner is investigated, employing excavating blasts and its vibration waves as the observation references. According to the optimal control theory, the performance function described by the square sum of the residuals between computed and observed velocities is minimized. The boundary layer is determined by minimizing the performance function. The elastic analysis governed by the Navier equation is carried out, assuming the ground as an elastic body with linear viscous damping. To identify the boundary, the gradient of the performance function with respect to the geological boundary can be calculated using the adjoint equation. The weighed gradient method is effectively applied to the minimization algorithm. To solve the governing and adjoint equations, the Galerkin finite element method and the average acceleration method are employed for the spatial and temporal discretizations, respectively. Based on the method presented in this paper, the different boundary of three strata can be identified. For the numerical studies, the Suemune tunnel excavation site is employed. At first, the blasting force is identified in order to perform the accuracy improvement of analysis. We identify the geological boundary after the estimation of blasting force. With this identification procedure, the numerical analysis results which almost correspond with the observation data were provided.

Keywords: Parameter identification, finite element method, average acceleration method, first order adjoint equation method, weighted gradient method, geological boundary, navier equation, optimal control theory.

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Authors: Kazuo Komatsu, Hitoshi Takata

Abstract:

The objective of this study is to propose an observer design for nonlinear systems by using an augmented linear system derived by application of a formal linearization method. A given nonlinear differential equation is linearized by the formal linearization method which is based on Taylor expansion considering up to the higher order terms, and a measurement equation is transformed into an augmented linear one. To this augmented dimensional linear system, a linear estimation theory is applied and a nonlinear observer is derived. As an application of this method, an estimation problem of transient state of electric power systems is studied, and its numerical experiments indicate that this observer design shows remarkable performances for nonlinear systems.

Keywords: nonlinear system, augmented linear system, nonlinear observer, formal linearization, electric power system.

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Authors: Azlina Abu Bakar, Abdul Fatah Wan Sidek

Abstract:

Well-being has been given special emphasis in quality of life. It involves living a meaningful, life satisfaction, stability and happiness in life. Well-being also concerns the satisfaction of physical, psychological, social needs and demands of an individual. The purpose of this study was to validate three-factor measurement model of well-being using structural equation modeling (SEM). The conceptions of well-being measured such dimensions as physical, psychological and social well-being. This study was done based on a total sample of 650 adolescents from east-coast of peninsular Malaysia. The Well-Being Scales which was adapted from [1] was used in this study. The items were hypothesized a priori to have nonzero loadings on all dimensions in the model. The findings of the SEM demonstrated that it is a good fitting model which the proposed model fits the driving theory; (x2df = 1.268; GFI = .994; CFI = .998; TLI= .996; p = .255; RMSEA = .021). Composite reliability (CR) was .93 and average variance extracted (AVE) was 58%. The model in this study fits with the sample of data and well-being is important to bring sustainable development to the mainstream.

Keywords: Adolescence, Structural Equation Modeling, Sustainable Development, Well-Being.

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Authors: Marian Gaiceanu, Emil Rosu

Abstract:

In this paper the optimal control strategy for Permanent Magnet Synchronous Motor (PMSM) based drive system is presented. The designed full optimal control is available for speed operating range up to base speed. The optimal voltage space-vector assures input energy reduction and stator loss minimization, maintaining the output energy in the same limits with the conventional PMSM electrical drive. The optimal control with three components is based on the energetically criteria and it is applicable in numerical version, being a nonrecursive solution. The simulation results confirm the increased efficiency of the optimal PMSM drive. The properties of the optimal voltage space vector are shown.

Keywords: Matlab/Simulink, optimal control, permanent magnet synchronous motor, Riccati equation, space vector PWM

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Authors: B. K. Gill, Himani Sharma, V. K. Rattan

Abstract:

Experimental data of refractive index, excess molar volume and viscosity of binary mixture of morpholine with cumene over the whole composition range at 298.15 K, 303.15 K, 308.15 K and normal atmospheric pressure have been measured. The experimental data were used to compute the density, deviation in molar refraction, deviation in viscosity and excess Gibbs free energy of activation as a function of composition. The experimental viscosity data have been correlated with empirical equations like Grunberg- Nissan, Herric correlation and three body McAllister’s equation. The excess thermodynamic properties were fitted to Redlich-Kister polynomial equation. The variation of these properties with composition and temperature of the binary mixtures are discussed in terms of intermolecular interactions.

Keywords: Cumene, excess Gibbs free energy, excess molar volume, morpholine.

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