Search results for: Non-linear Riks Analysis

Authors: A.H. Akhaveissy

Abstract:

Nonlinear finite element method and Serendipity eight nodes element are used for determining of ground surface settlement due to tunneling. Linear element with elastic behavior is used for modeling of lining. Modified Generalized plasticity model with nonassociated flow rule is applied for analysis of a tunnel in Sao Paulo – Brazil. The tunnel had analyzed by Lades- model with 16 parameters. In this work modified Generalized Plasticity is used with 10 parameters, also Mohr-Coulomb model is used to analysis the tunnel. The results show good agreement with observed results of field data by modified Generalized Plasticity model than other models. The obtained result by Mohr-Coulomb model shows less settlement than other model due to excavation.

Keywords: Non-associated flow rule, Generalized plasticity, tunnel excavation, Excavation method.

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Authors: C. V. Pietreanu, S. E. Zaharia, C. Dinu

Abstract:

Hazard assessment and risks quantification are key components for estimating the impact of existing regulations. But since regulatory compliance cannot cover all risks in aviation, the authors point out that by studying causal factors and eliminating uncertainty, an accurate analysis can be outlined. The research debuts by making delimitations on notions, as confusion on the terms over time has reflected in less rigorous analysis. Throughout this paper, it will be emphasized the fact that the variation in human performance and organizational factors represent the biggest threat from an operational perspective. Therefore, advanced risk assessment methods analyzed by the authors aim to understand vulnerabilities of the system given by a nonlinear behavior. Ultimately, the mathematical modeling of existing hazards and risks by eliminating uncertainty implies establishing an optimal solution (i.e. risk minimization).

Keywords: Control, human factor, optimization, risk management, uncertainty.

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Authors: Mohammed Farag, Mina Attari, S. Andrew Gadsden, Saeid R. Habibi

Abstract:

Battery state of charge (SOC) estimation is an important parameter as it measures the total amount of electrical energy stored at a current time. The SOC percentage acts as a fuel gauge if it is compared with a conventional vehicle. Estimating the SOC is, therefore, essential for monitoring the amount of useful life remaining in the battery system. This paper looks at the implementation of three nonlinear estimation strategies for Li-Ion battery SOC estimation. One of the most common behavioral battery models is the one state hysteresis (OSH) model. The extended Kalman filter (EKF), the smooth variable structure filter (SVSF), and the time-varying smoothing boundary layer SVSF are applied on this model, and the results are compared.

Keywords: State of charge estimation, battery modeling, one-state hysteresis, filtering and estimation.

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Authors: Seong-Ho Song, Ki-Seob Kim, Seop-Hyeong Park, Seon-Woo Lee

Abstract:

In this paper, the data correction algorithm is suggested when the environmental air temperature varies. To correct the infrared data in this paper, the initial temperature or the initial infrared image data is used so that a target source system may not be necessary. The temperature data obtained from infrared detector show nonlinear property depending on the surface temperature. In order to handle this nonlinear property, Taylor series approach is adopted. It is shown that the proposed algorithm can reduce the influence of environmental temperature on the components in the board. The main advantage of this algorithm is to use only the initial temperature of the components on the board rather than using other reference device such as black body sources in order to get reference temperatures.

Keywords: Infrared camera, Temperature Data compensation, Environmental Ambient Temperature, Electric Component

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Authors: Keigo Matsuura, Isao Tomita

Abstract:

We have studied a method to widen the spectrum of optical pulses that pass through an InGaAsP waveguide for application to broadband optical communication. In particular, we have investigated the competitive effect between spectral broadening arising from nonlinear refraction (optical Kerr effect) and shrinking due to two photon absorption in the InGaAsP waveguide with χ(3) nonlinearity. The shrunk spectrum recovers broadening by the enhancement effect of the nonlinear refractive index near the bandgap of InGaAsP with a bandgap wavelength of 1490 nm. The broadened spectral width at around 1525 nm (196.7 THz) becomes 10.7 times wider than that at around 1560 nm (192.3 THz) without the enhancement effect, where amplified optical pulses with a pulse width of ∼ 2 ps and a peak power of 10 W propagate through a 1-cm-long InGaAsP waveguide with a cross-section of 4 (μm)2.

Keywords: InGaAsP Waveguide, χ(3) Nonlinearity, Spectral Broadening.

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Authors: Jose D. Herrera, Mario A. Rios

Abstract:

This paper deals with the coordinated tuning of the Power System Stabilizer (PSS) controller and Power Oscillation Damping (POD) Controller of Flexible AC Transmission System (FACTS) in a multi-machine power systems. The coordinated tuning is based on the critical eigenvalues of the power system and a model reduction technique where the Hankel Singular Value method is applied. Through the linearized system model and the parameter-constrained nonlinear optimization algorithm, it can compute the parameters of both controllers. Moreover, the parameters are optimized simultaneously obtaining the gains of both controllers. Then, the nonlinear simulation to observe the time response of the controller is performed.

Keywords: Balanced realization, controllability Grammian, electromechanical oscillations, FACTS, Hankel singular values, observability Grammian, POD, PSS.

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Authors: Ramandeep Behl, S. S. Motsa

Abstract:

The main focus of this manuscript is to provide a highly efficient two-point sixth-order family of super-Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. Each member of the proposed family requires two evaluations of the given function and two evaluations of the first-order derivative per iteration. By using Mathematica-9 with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm t he t heoretical d evelopment. From their basins of attraction, it has been observed that the proposed methods have better stability and robustness as compared to the other sixth-order methods available in the literature.

Keywords: Basins of attraction, nonlinear equations, simple roots, Super-Halley.

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Authors: Reza Abazari, Rasool Abazari

Abstract:

In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.

Keywords: Coupled Korteweg-de Vries(KdV) equation, Coupled Burgers equation, Coupled Schrödinger equation, differential transformation method.

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Authors: L. Bikulciene, E. Venskaityte, G. Jarusevicius

Abstract:

Nonstationary and nonlinear signals generated by living complex systems defy traditional mechanistic approaches, which are based on homeostasis. Previous our studies have shown that the evaluation of the interactions of physiological signals by using special analysis methods is suitable for observation of physiological processes. It is demonstrated the possibility of using deep physiological model, based on the interpretation of the changes of the human body’s functional states combined with an application of the analytical method based on matrix theory for the physiological signals analysis, which was applied on high risk cardiac patients. It is shown that evaluation of cardiac signals interactions show peculiar for each individual functional changes at the onset of hemodynamic restoration procedure. Therefore, we suggest that the alterations of functional state of the body, after patients overcome surgery can be complemented by the data received from the suggested approach of the evaluation of functional variables’ interactions.

Keywords: Cardiac diseases, Complex systems theory, ECG analysis, matrix analysis.

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Authors: Magdi M. Nabi, Ding-Li Yu

Abstract:

Control of a semi-batch polymerization reactor using an adaptive radial basis function (RBF) neural network method is investigated in this paper. A neural network inverse model is used to estimate the valve position of the reactor; this method can identify the controlled system with the RBF neural network identifier. The weights of the adaptive PID controller are timely adjusted based on the identification of the plant and self-learning capability of RBFNN. A PID controller is used in the feedback control to regulate the actual temperature by compensating the neural network inverse model output. Simulation results show that the proposed control has strong adaptability, robustness and satisfactory control performance and the nonlinear system is achieved.

Keywords: Chylla-Haase polymerization reactor, RBF neural networks, feed-forward and feedback control.

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Authors: Farhad Namdari, Reza Sedaghati

Abstract:

Economic dispatch (ED) has been considered to be one of the key functions in electric power system operation which can help to build up effective generating management plans. The practical ED problem has non-smooth cost function with nonlinear constraints which make it difficult to be effectively solved. This paper presents a novel heuristic and efficient optimization approach based on the new Bat algorithm (BA) to solve the practical non-smooth economic dispatch problem. The proposed algorithm easily takes care of different constraints. In addition, two newly introduced modifications method is developed to improve the variety of the bat population when increasing the convergence speed simultaneously. The simulation results obtained by the proposed algorithms are compared with the results obtained using other recently develop methods available in the literature.

Keywords: Non-smooth, economic dispatch, bat-inspired, nonlinear practical constraints, modified bat algorithm.

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Authors: S. Farzi

Abstract:

Recently, a lot of attention has been devoted to advanced techniques of system modeling. PNN(polynomial neural network) is a GMDH-type algorithm (Group Method of Data Handling) which is one of the useful method for modeling nonlinear systems but PNN performance depends strongly on the number of input variables and the order of polynomial which are determined by trial and error. In this paper, we introduce GPNN (genetic polynomial neural network) to improve the performance of PNN. GPNN determines the number of input variables and the order of all neurons with GA (genetic algorithm). We use GA to search between all possible values for the number of input variables and the order of polynomial. GPNN performance is obtained by two nonlinear systems. the quadratic equation and the time series Dow Jones stock index are two case studies for obtaining the GPNN performance.

Keywords: GMDH, GPNN, GA, PNN.

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Authors: Viet-Hung Truong, Seung-Eock Kim

Abstract:

This paper presents a robust software package for practical advanced analysis of space steel framed structures. The pre- and post-processors of the presented software package are coded in the C++ programming language while the solver is written by using the FORTRAN programming language. A user-friendly graphical interface of the presented software is developed to facilitate the modeling process and result interpretation of the problem. The solver employs the stability functions for capturing the second-order effects to minimize modeling and computational time. Both the plastic-hinge and fiber-hinge beam-column elements are available in the presented software. The generalized displacement control method is adopted to solve the nonlinear equilibrium equations.

Keywords: Advanced analysis, beam-column, fiber-hinge, plastic hinge, steel frame.

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Authors: A. Thabet, M. Boutayeb, M. N. Abdelkrim

Abstract:

This paper investigates a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation (DAE) models using the extended Kalman filter. The method involves the use of a transformation from a DAE to ordinary differential equation (ODE). A relevant dynamic power system model using decoupled techniques will be proposed. The estimation technique consists of a state estimator based on the EKF technique as well as the local stability analysis. High performances are illustrated through a simulation study applied on IEEE 13 buses test system.

Keywords: Power system, Dynamic decoupled model, Extended Kalman Filter, Convergence analysis, Time computing.

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Authors: H. Bensouilah, H. Boucherit, M. Lahmar

Abstract:

A theoretical investigation on the effects of both steady-state and dynamic deformations of the foils on the dynamic performance characteristics of a self-acting air foil journal bearing operating under small harmonic vibrations is proposed. To take into account the dynamic deformations of foils, the perturbation method is used for determining the gas-film stiffness and damping coefficients for given values of excitation frequency, compressibility number, and compliance factor of the bump foil. The nonlinear stationary Reynolds’ equation is solved by means of the Galerkins’ finite element formulation while the finite differences method are used to solve the first order complex dynamic equations resulting from the perturbation of the nonlinear transient compressible Reynolds’ equation. The stiffness of a bump is uniformly distributed throughout the bearing surface (generation I bearing). It was found that the dynamic properties of the compliant finite length journal bearing are significantly affected by the compliance of foils especially whenthe dynamic deformation of foils is considered in addition to the static one by applying the principle of superposition.

Keywords: Elasto-aerodynamic lubrication, Air foil bearing, Steady-state deformation, Dynamic deformation, Stiffness and damping coefficients, Perturbation method, Fluid-structure interaction, Galerk infinite element method, Finite difference method.

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Authors: Al-Kashif M., Abdel-Mooty M., Fahmy E., Abou Zeid M., Haroun M.

Abstract:

Sandwich panels are widely used in the construction industry for their ease of assembly, light weight and efficient thermal performance. They are composed of two RC thin outer layers separated by an insulating inner layer. In this research the inner insulating layer is made of lightweight Autoclaved Aerated Concrete (AAC) blocks which has good thermal insulation properties and yet possess reasonable mechanical strength. The shear strength of the AAC infill is relied upon to replace the traditionally used insulating foam and to provide the shear capacity of the panel. A comprehensive experimental program was conducted on full scale sandwich panels subjected to bending. In this paper, detailed numerical modeling of the tested sandwich panels is reported. Nonlinear 3-D finite element modeling of the composite action of the sandwich panel is developed using ANSYS. Solid elements with different crashing and cracking capabilities and different constitutive laws were selected for the concrete and the AAC. Contact interface elements are used in this research to adequately model the shear transfer at the interface between the different layers. The numerical results showed good correlation with the experimental ones indicating the adequacy of the model in estimating the loading capacity of panels.

Keywords: Autoclaved Aerated Concrete, Concrete Sandwich Panels, Finite Element Modeling.

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Authors: U. Mutman, S. B. Coskun

Abstract:

In this study Homotopy Perturbation Method (HPM) is employed to investigate free vibration of an Euler beam with variable stiffness resting on an elastic foundation. HPM is an easy-to-use and very efficient technique for the solution of linear or nonlinear problems. HPM produces analytical approximate expression which is continuous in the solution domain. This work shows that HPM is a promising method for free vibration analysis of nonuniform Euler beams on elastic foundation. Several case problems have been solved by using the technique and solutions have been compared with those available in the literature.

Keywords: Homotopy Perturbation Method, Elastic Foundation, Vibration, Beam

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Authors: Karandeep Singh, Baibing Li

Abstract:

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: H. Ghanbarpour Asl, S. H. Pourtakdoust

Abstract:

Extended Kalman Filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, not only it has difficulties arising from linearization but also many times it becomes numerically unstable because of computer round off errors that occur in the process of its implementation. To overcome linearization limitations, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. Kalman filter that uses UT for calculation of the first two statistical moments is called Unscented Kalman Filter (UKF). Square-root form of UKF (SRUKF) developed by Rudolph van der Merwe and Eric Wan to achieve numerical stability and guarantee positive semi-definiteness of the Kalman filter covariances. This paper develops another implementation of SR-UKF for sequential update measurement equation, and also derives a new UD covariance factorization filter for the implementation of UKF. This filter is equivalent to UKF but is computationally more efficient.

Keywords: Unscented Kalman filter, Square-root unscentedKalman filter, UD covariance factorization, Target tracking.

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Authors: Jeong-Hwan Choi, Hyo-Sang Park, Tae-Hyung Lee

Abstract:

The transmission tower is one of the crucial lifeline structures in a modern society, and it needs to be protected against extreme loading conditions. However, the transmission tower is a very complex structure and, therefore, it is very difficult to simulate the actual damage and the collapse behavior of the tower structure. In this study, the actual collapse behavior of the transmission tower due to lateral loading conditions such as wind load is evaluated through the computational simulation. For that, a progressive collapse procedure is applied to the simulation. In this procedure, after running the simulation, if a member of the tower structure fails, the failed member is removed and the simulation run again. The 154kV transmission tower is selected for this study. The simulation is performed by nonlinear static analysis procedure, namely pushover analysis, using OpenSEES, an earthquake simulation platform. Three-dimensional finite element models of those towers are developed.

Keywords: Transmission tower, OpenSEES, pushover analysis, progressive collapse.

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Authors: M. Palizvan, M. T. Abadi, M. H. Sadr

Abstract:

Due to variations in damage mechanisms in the microscale, the behavior of fiber-reinforced composites is nonlinear and difficult to model. To make use of computational advantages, homogenization method is applied to the micro-scale model in order to minimize the cost at the expense of detail of local microscale phenomena. In this paper, the effective stiffness is calculated using the homogenization of nonlinear behavior of a composite representative volume element (RVE) containing fiber-matrix debonding. The damage modes for the RVE are considered by using cohesive elements and contacts for the cohesive behavior of the interface between fiber and matrix. To predict more realistic responses of composite materials, different random distributions of fibers are proposed besides square and hexagonal arrays. It was shown that in some cases, there is quite different damage behavior in different fiber distributions. A comprehensive comparison has been made between different graphs.

Keywords: Homogenization, cohesive zone model, fiber-matrix debonding, RVE.

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Authors: Gilberto Gonzalez-A, Noe Barrera-G

Abstract:

An analysis of a synchronous generator in a bond graph approach is proposed. This bond graph allows to determine the simplified models of the system by using singular perturbations. Firstly, the nonlinear bond graph of the generator is linearized. Then, the slow and fast state equations by applying singular perturbations are obtained. Also, a bond graph to get the quasi-steady state of the slow dynamic is proposed. In order to verify the effectiveness of the singularly perturbed models, simulation results of the complete system and reduced models are shown.

Keywords: Bond graph modelling, synchronous generator, singular perturbations

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Authors: Han-Soo Kim, Jae-Gyun Ahn, Hyo-Seung Ahn

Abstract:

Though nonlinear dynamic analysis using a specialized hydro-code such as AUTODYN is accurate and useful tool for progressive collapse assessment of a multi-story building subjected to blast load, it takes too much time to be applied to a practical simulation of progressive collapse of a tall building. In this paper, blast analysis of a RC frame structure using a simplified model with Reinforcement Contact technique provided in Ansys Workbench was introduced and investigated on its accuracy. Even though the simplified model has a fraction of elements of the detailed model, the simplified model with this modeling technique shows similar structural behavior under the blast load to the detailed model. The proposed modeling method can be effectively applied to blast loading progressive collapse analysis of a RC frame structure.

Keywords: Autodyn, Blast Load, Progressive Collapse, Reinforcement Contact.

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Authors: Atul Krishna Banik

Abstract:

Nonlinear response behaviour of a cracked RC beam under harmonic excitation is analysed to investigate various instability phenomena like, bifurcation, jump phenomena etc. The nonlinearity of the system arises due to opening and closing of the cracks in the RC beam and is modelled as a cubic polynomial. In order to trace different branches at the bifurcation point on the response curve (amplitude versus frequency of excitation plot), an arc length continuation technique along with the incremental harmonic balance (IHBC) method is employed. The stability of the solution is investigated by the Floquet theory using Hsu-s scheme. The periodic solutions obtained by the IHBC method are compared with these obtained by the numerical integration of the equation of motion. Characteristics of solutions fold bifurcation, jump phenomena and from stable to unstable zones are identified.

Keywords: Incremental harmonic balance, arc-length continuation, bifurcation, jump phenomena.

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Authors: P. Ramana , K. Alice Mary, M. Surya Kalavathi, M. Phani Kumar

Abstract:

This paper discusses two observers, which are used for the estimation of parameters of PMSM. Former one, reduced order observer, which is used to estimate the inaccessible parameters of PMSM. Later one, full order observer, which is used to estimate all the parameters of PMSM even though some of the parameters are directly available for measurement, so as to meet with the insensitivity to the parameter variation. However, the state space model contains some nonlinear terms i.e. the product of different state variables. The asymptotic state observer, which approximately reconstructs the state vector for linear systems without uncertainties, was presented by Luenberger. In this work, a modified form of such an observer is used by including a non-linear term involving the speed. So, both the observers are designed in the framework of nonlinear control; their stability and rate of convergence is discussed.

Keywords: Permanent magnet synchronous motor, Mathematicalmodelling, Rotor reference frame, parameter estimation, Luenbergerobserver, reduced order observer, full order observer

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Authors: Mohamed A. Shanan, Ashraf H. El-Zanaty, Kamal G. Metwally

Abstract:

This paper presents the effect of concrete compressive strength and rectangularity ratio on strength and ductility of normal and high strength reinforced concrete columns confined with transverse steel under axial compressive loading. Nineteen normal strength concrete rectangular columns with different variables tested in this research were used to study the effect of concrete compressive strength and rectangularity ratio on strength and ductility of columns. The paper also presents a nonlinear finite element analysis for these specimens and another twenty high strength concrete square columns tested by other researchers using ANSYS 15 finite element software. The results indicate that the axial force – axial strain relationship obtained from the analytical model using ANSYS are in good agreement with the experimental data. The comparison shows that the ANSYS is capable of modeling and predicting the actual nonlinear behavior of confined normal and high-strength concrete columns under concentric loading. The maximum applied load and the maximum strain have also been confirmed to be satisfactory. Depending on this agreement between the experimental and analytical results, a parametric numerical study was conducted by ANSYS 15 to clarify and evaluate the effect of each variable on strength and ductility of the columns.

Keywords: ANSYS, concrete compressive strength effect, ductility, rectangularity ratio, strength.

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Authors: Madhu Aneja, Sapna Sharma

Abstract:

The water-based bioconvection of a nanofluid containing motile gyrotactic micro-organisms over nonlinear inclined stretching sheet has been investigated. The governing nonlinear boundary layer equations of the model are reduced to a system of ordinary differential equations via Oberbeck-Boussinesq approximation and similarity transformations. Further, the modified set of equations with associated boundary conditions are solved using Finite Element Method. The impact of various pertinent parameters on the velocity, temperature, nanoparticles concentration, density of motile micro-organisms profiles are obtained and analyzed in details. The results show that with the increase in angle of inclination δ, velocity decreases while temperature, nanoparticles concentration, a density of motile micro-organisms increases. Additionally, the skin friction coefficient, Nusselt number, Sherwood number, density number are computed for various thermophysical parameters. It is noticed that increasing Brownian motion and thermophoresis parameter leads to an increase in temperature of fluid which results in a reduction in Nusselt number. On the contrary, Sherwood number rises with an increase in Brownian motion and thermophoresis parameter. The findings have been validated by comparing the results of special cases with existing studies.

Keywords: Bioconvection, inclined stretching sheet, Gyrotactic micro-organisms, Brownian motion, thermophoresis, finite element method.

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Authors: I. A. Farhat

Abstract:

The The dynamic economic dispatch (DED) problem is one of the complex constrained optimization problems that have nonlinear, con-convex and non-smooth objective functions. The purpose of the DED is to determine the optimal economic operation of the committed units while meeting the load demand. Associated to this constrained problem there exist highly nonlinear and non-convex practical constraints to be satisfied. Therefore, classical and derivative-based methods are likely not to converge to an optimal or near optimal solution to such a dynamic and large-scale problem. In this paper, an Artificial Immune System technique (AIS) is implemented and applied to solve the DED problem considering the transmission power losses and the valve-point effects in addition to the other operational constraints. To demonstrate the effectiveness of the proposed technique, two case studies are considered. The results obtained using the AIS are compared to those obtained by other methods reported in the literature and found better.

Keywords: Artificial Immune System (AIS), Dynamic Economic Dispatch (DED).

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Authors: Liang Chen

Abstract:

A family of improved secant-like method is proposed in this paper. Further, the analysis of the convergence shows that this method has super-linear convergence. Efficiency are demonstrated by numerical experiments when the choice of α is correct.

Keywords: Nonlinear equations, Secant method, Convergence order, Secant-like method.

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Authors: I. A. Farhat

Abstract:

The dynamic economic dispatch (DED) problem is one of the complex constrained optimization problems that have nonlinear, con-convex and non-smooth objective functions. The purpose of the DED is to determine the optimal economic operation of the committed units while meeting the load demand.  Associated to this constrained problem there exist highly nonlinear and non-convex practical constraints to be satisfied. Therefore, classical and derivative-based methods are likely not to converge to an optimal or near optimal solution to such a dynamic and large-scale problem. In this paper, an Artificial Immune System technique (AIS) is implemented and applied to solve the DED problem considering the transmission power losses and the valve-point effects in addition to the other operational constraints. To demonstrate the effectiveness of the proposed technique, two case studies are considered. The results obtained using the AIS are compared to those obtained by other methods reported in the literature and found better.

Keywords: Artificial Immune System (AIS), Dynamic Economic Dispatch (DED).

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