Search results for: point estimate method

Authors: Akash Rathee, Harish Parthasarathy

Abstract:

In this paper, a frequency-variation based method has been proposed for transistor parameter estimation in a commonemitter transistor amplifier circuit. We design an algorithm to estimate the transistor parameters, based on noisy measurements of the output voltage when the input voltage is a sine wave of variable frequency and constant amplitude. The common emitter amplifier circuit has been modelled using the transistor Ebers-Moll equations and the perturbation technique has been used for separating the linear and nonlinear parts of the Ebers-Moll equations. This model of the amplifier has been used to determine the amplitude of the output sinusoid as a function of the frequency and the parameter vector. Then, applying the proposed method to the frequency components, the transistor parameters have been estimated. As compared to the conventional time-domain least squares method, the proposed method requires much less data storage and it results in more accurate parameter estimation, as it exploits the information in the time and frequency domain, simultaneously. The proposed method can be utilized for parameter estimation of an analog device in its operating range of frequencies, as it uses data collected from different frequencies output signals for parameter estimation.

Keywords: Perturbation Technique, Parameter estimation, frequency-variation based method.

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Authors: Y. T. Tsai, Jin H. Huang

Abstract:

The sound pressure level (SPL) of the moving-coil loudspeaker (MCL) is often simulated and analyzed using the lumped parameter model. However, the SPL of a MCL cannot be simulated precisely in the high frequency region, because the value of cone effective area is changed due to the geometry variation in different mode shapes, it is also related to affect the acoustic radiation mass and resistance. Herein, the paper presents the inverse method which has a high ability to measure the value of cone effective area in various frequency points, also can estimate the MCL electroacoustic parameters simultaneously. The proposed inverse method comprises the direct problem, adjoint problem, and sensitivity problem in collaboration with nonlinear conjugate gradient method. Estimated values from the inverse method are validated experimentally which compared with the measured SPL curve result. Results presented in this paper not only improve the accuracy of lumped parameter model but also provide the valuable information on loudspeaker cone design.

Keywords: Inverse problem, cone effective area, loudspeaker, nonlinear conjugate gradient method.

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Authors: S. Golmohammadi, M. Noorian Bidgoli

Abstract:

Because the status and mechanical parameters of discontinuities in the rock mass are included in the calculations, various methods of rock engineering classification are often used as a starting point for the design of different types of structures. The Q-system is one of the most frequently used methods for stability analysis and determination of support systems of underground structures in rock, including tunnel. In this method, six main parameters of the rock mass, namely, the Rock Quality Designation (RQD), joint set number (Jn), joint roughness number (Jr), joint alteration number (Ja), joint water parameter (Jw) and Stress Reduction Factor (SRF) are required. In this regard, in order to achieve a reasonable and optimal design, identifying the effective parameters for the stability of the mentioned structures is one of the most important goals and the most necessary actions in rock engineering. Therefore, it is necessary to study the relationships between the parameters of a system and how they interact with each other and, ultimately, the whole system. In this research, it has been attempted to determine the most effective parameters (key parameters) from the six parameters of rock mass in the Q-system using the Rock Engineering System (RES) method to improve the relationships between the parameters in the calculation of the Q value. The RES system is, in fact, a method by which one can determine the degree of cause and effect of a system's parameters by making an interaction matrix. In this research, the geomechanical data collected from the water conveyor tunnel of Azad Dam were used to make the interaction matrix of the Q-system. For this purpose, instead of using the conventional methods that are always accompanied by defects such as uncertainty, the Q-system interaction matrix is coded using a technique that is actually a statistical analysis of the data and determining the correlation coefficient between them. So, the effect of each parameter on the system is evaluated with greater certainty. The results of this study show that the formed interaction matrix provides a reasonable estimate of the effective parameters in the Q-system. Among the six parameters of the Q-system, the SRF and Jr parameters have the maximum and minimum impact on the system, respectively, and also the RQD and Jw parameters have the maximum and minimum impact on the system, respectively. Therefore, by developing this method, we can obtain a more accurate relation to the rock mass classification by weighting the required parameters in the Q-system.

Keywords: Q-system, Rock Engineering System, statistical analysis, rock mass, tunnel.

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Authors: Yuxin Chen, Qingyun Di, C. Dinis da Gama

Abstract:

In projects like waterpower, transportation and mining, etc., proving up the rock-mass structure and hidden tectonic to estimate the geological body-s activity is very important. Integrating the seismic results, drilling and trenching data, CSAMT method was carried out at a planning dame site in southwest China to evaluate the stability of a deformation. 2D and imitated 3D inversion resistivity results of CSAMT method were analyzed. The results indicated that CSAMT was an effective method for defining an outline of deformation body to several hundred meters deep; the Lung Pan Deformation was stable in natural conditions; but uncertain after the future reservoir was impounded. This research presents a good case study of the fine surveying and research on complex geological structure and hidden tectonic in engineering project.

Keywords: CSAMT Surveying, Deformation Stability.

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Authors: A. Abbasi Moshaii, F. Najafi

Abstract:

In this paper a new model for center of motion creating is proposed. This new method uses cables. So, it is very useful in robots because it is light and has easy assembling process. In the robots which need to be in touch with some things this method is so useful. It will be described in the following. The accuracy of the idea is proved by two experiments. This system could be used in the robots which need a fixed point in the contact with some things and make a circular motion.

Keywords: Center of Motion, Robotic cables, permanent touching.

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Authors: A. Seidl, Th. Schmidt

Abstract:

Meshless Finite Element Methods, namely element-free Galerkin and point-interpolation method were implemented and tested concerning their applicability to typical engineering problems like electrical fields and structural mechanics. A class-structure was developed which allows a consistent implementation of these methods together with classical FEM in a common framework. Strengths and weaknesses of the methods under investigation are discussed. As a result of this work joint usage of meshless methods together with classical Finite Elements are recommended.

Keywords: Finite Elements, meshless, element-free Galerkin, point-interpolation.

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Authors: Dimitrios J. Dimitriou

Abstract:

It is widely recognised that the assets portfolio development is helping to enhance economic growth, productivity and competitiveness. While numerous studies and reports certify the positive effect of investments in large infrastructure investments on the local economy, still, the methodology to estimate the contribution in economic development is a challenging issue for researchers and economists. The key question is how to estimate those economic impacts in each economic system. This paper provides a compact and applicable methodological framework providing quantitative results in terms of the overall jobs and income generated into the project life cycle. According to a deterministic mathematical approach, the key variables and the modelling framework are presented. The numerical case study highlights key results for a new motorway project in Greece, which is experienced economic stress for many years, providing the opportunity for comparisons with similar cases.

Keywords: Quantitative modelling, economic impact; large transport infrastructure; economic assessment.

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Authors: Orose Rugchati, Kanita Thanacharoenchanaphas

Abstract:

This research studied the simulation of increased ambient ozone to estimate nutrient content and genetic changes in two Thai soybean cultivars (Chiang Mai 60 and Srisumrong 1). Ozone stress conditions affected proteins and lipids. It was found that proteins decreased, but lipids increased. Srisumrong 1 cultivars were more sensitive to ozone stress than Chiang Mai 60 cultivars. The effect of ozone stress conditions on plant phenotype and genotype was analyzed using the AFLP technique for the 2 Thai soybean cultivars (Chiang Mai 60 and Srisumrong 1).

Keywords: simulation, ambient ozone estimate, nutrient content, genetic changes , Thai soybean

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Authors: Shunsuke Miyoshi, Yasuyuki Nogami, Takuya Kusaka, Nariyoshi Yamai

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Elliptic curve discrete logarithm problem(ECDLP) is one of problems on which the security of pairing-based cryptography is based. This paper considers Pollard’s rho method to evaluate the security of ECDLP on Barreto-Naehrig(BN) curve that is an efficient pairing-friendly curve. Some techniques are proposed to make the rho method efficient. Especially, the group structure on BN curve, distinguished point method, and Montgomery trick are well-known techniques. This paper applies these techniques and shows its optimization. According to the experimental results for which a large-scale parallel system with MySQL is applied, 94-bit ECDLP was solved about 28 hours by parallelizing 71 computers.

Keywords: Pollard’s rho method, BN curve, Montgomery multiplication.

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Authors: Alireza Gholami, Amir H. D. Markazi

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In this paper, an observer-based direct adaptive fuzzy sliding mode (OAFSM) algorithm is proposed. In the proposed algorithm, the zero-input dynamics of the plant could be unknown. The input connection matrix is used to combine the sliding surfaces of individual subsystems, and an adaptive fuzzy algorithm is used to estimate an equivalent sliding mode control input directly. The fuzzy membership functions, which were determined by time consuming try and error processes in previous works, are adjusted by adaptive algorithms. The other advantage of the proposed controller is that the input gain matrix is not limited to be diagonal, i.e. the plant could be over/under actuated provided that controllability and observability are preserved. An observer is constructed to directly estimate the state tracking error, and the nonlinear part of the observer is constructed by an adaptive fuzzy algorithm. The main advantage of the proposed observer is that, the measured outputs is not limited to the first entry of a canonical-form state vector. The closed-loop stability of the proposed method is proved using a Lyapunov-based approach. The proposed method is applied numerically on a multi-link robot manipulator, which verifies the performance of the closed-loop control. Moreover, the performance of the proposed algorithm is compared with some conventional control algorithms.

Keywords: Adaptive algorithm, fuzzy systems, membership functions, observer.

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Authors: Vandna Jowaheer, Naushad A. Mamode Khan

Abstract:

Com Poisson distribution is capable of modeling the count responses irrespective of their mean variance relation and the parameters of this distribution when fitted to a simple cross sectional data can be efficiently estimated using maximum likelihood (ML) method. In the regression setup, however, ML estimation of the parameters of the Com Poisson based generalized linear model is computationally intensive. In this paper, we propose to use quasilikelihood (QL) approach to estimate the effect of the covariates on the Com Poisson counts and investigate the performance of this method with respect to the ML method. QL estimates are consistent and almost as efficient as ML estimates. The simulation studies show that the efficiency loss in the estimation of all the parameters using QL approach as compared to ML approach is quite negligible, whereas QL approach is lesser involving than ML approach.

Keywords: Com Poisson, Cross-sectional, Maximum Likelihood, Quasi likelihood

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Authors: S. Rao Rayapudi

Abstract:

Economic Load Dispatch (ELD) is a method of determining the most efficient, low-cost and reliable operation of a power system by dispatching available electricity generation resources to supply load on the system. The primary objective of economic dispatch is to minimize total cost of generation while honoring operational constraints of available generation resources. In this paper an intelligent water drop (IWD) algorithm has been proposed to solve ELD problem with an objective of minimizing the total cost of generation. Intelligent water drop algorithm is a swarm-based natureinspired optimization algorithm, which has been inspired from natural rivers. A natural river often finds good paths among lots of possible paths in its ways from source to destination and finally find almost optimal path to their destination. These ideas are embedded into the proposed algorithm for solving economic load dispatch problem. The main advantage of the proposed technique is easy is implement and capable of finding feasible near global optimal solution with less computational effort. In order to illustrate the effectiveness of the proposed method, it has been tested on 6-unit and 20-unit test systems with incremental fuel cost functions taking into account the valve point-point loading effects. Numerical results shows that the proposed method has good convergence property and better in quality of solution than other algorithms reported in recent literature.

Keywords: Economic load dispatch, Transmission loss, Optimization, Valve point loading, Intelligent Water Drop Algorithm.

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Authors: Ewa Dostatni, Jacek Diakun, Damian Grajewski, Radosław Wichniarek, Anna Karwasz

Abstract:

In the paper the method of product analysis from recycling point of view has been described. The analysis bases on set of measures that assess a product from the point of view of final stages of its lifecycle. It was assumed that such analysis will be performed at the design phase – in order to conduct such analysis the computer system that aids the designer during the design process has been developed. The structure of the computer tool, based on agent technology, and example results has been also included in the paper.

Keywords: Agent technology, PLM, design for environment, ecodesign.

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Authors: S. A. Nta

Abstract:

The study assessed the levels of some heavy metals in the contaminated soil from a point source using pollution indices to measure the extent of pollution. The soil used was sandy-loam in texture. The contaminant used was landfill leachate, introduced as a point source through an entry point positioned at the center of top layer of the soil tank. Samples were collected after 50 days and analyzed for heavy metal (Zn, Ni, Cu and Cd) using standard methods. The mean concentration of Ni ranged from 5.55-2.65 mg/kg, Zn 3.67-0.85 mg/kg, Cu 1.60-0.93 mg/kg and Cd 1.60-0.15 mg/kg. The richness of metals was in decreasing order: Ni > Zn > Cu > Cd. The metals concentration was found to be maximum at 0.25 m radial distance from the point of leachate application. The geo-accumulation index (Igeo) studied revealed that all the metals recovered at 0.25 and 0.50 m radial distance and at 0.15, 0.30, 0.45 and 0.60 m depth from the point of application of leachate fall under unpolluted to moderately polluted range. Ecological risk assessment showed high ecological risk index with values higher than RI > 300. The RI shows that the ecological risk in this study was mostly contributed by Cd ranging from 9-96.

Keywords: Ecological risk, assessment, heavy metals, test soils, landfill leachate.

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Authors: D. Shobha Rani, M. Muralidhar

Abstract:

Solar photovoltaic (SPV) power systems can be employed as electrical power sources to meet the daily residential energy needs of rural areas that have no access to grid systems. In view of this, a standalone SPV powered air cooling system is proposed in this paper, which constitutes a dc-dc boost converter, two voltage source inverters (VSI) connected to two brushless dc (BLDC) motors which are coupled to a centrifugal water pump and a fan blower. A simple and efficient Maximum Power Point Tracking (MPPT) technique based on Silver Mean Method (SMM) is utilized in this paper. The air cooling system is developed and simulated using the MATLAB / Simulink environment considering the dynamic and steady state variation in the solar irradiance.

Keywords: Boost converter, solar photovoltaic array, voltage source inverter, brushless DC motor, solar irradiance, Maximum Power Point Tracking, Silver Mean Method.

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Authors: Ahcene Habbi, Yassine Boudouaoui

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This paper deals with the problem of automatic rule generation for fuzzy systems design. The proposed approach is based on hybrid artificial bee colony (ABC) optimization and weighted least squares (LS) method and aims to find the structure and parameters of fuzzy systems simultaneously. More precisely, two ABC based fuzzy modeling strategies are presented and compared. The first strategy uses global optimization to learn fuzzy models, the second one hybridizes ABC and weighted least squares estimate method. The performances of the proposed ABC and ABC-LS fuzzy modeling strategies are evaluated on complex modeling problems and compared to other advanced modeling methods.

Keywords: Automatic design, learning, fuzzy rules, hybrid, swarm optimization.

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Authors: Davod Khojasteh Salkuyeh

Abstract:

An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.

Keywords: Ordinary differential equations, optimal stepsize, error, stochastic arithmetic, CESTAC, CADNA.

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Authors: M. Altekin, R. F. Yükseler

Abstract:

Geometrically nonlinear axisymmetric bending of a shallow spherical shell with a point support at the apex under linearly varying axisymmetric load was investigated numerically. The edge of the shell was assumed to be simply supported or clamped. The solution was obtained by the finite difference and the Newton-Raphson methods. The thickness of the shell was considered to be uniform and the material was assumed to be homogeneous and isotropic. Sensitivity analysis was made for two geometrical parameters. The accuracy of the algorithm was checked by comparing the deflection with the solution of point supported circular plates and good agreement was obtained.

Keywords: Bending, nonlinear, plate, point support, shell.

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Authors: Ahmad Attiq Al-Ogaibi, Ahmad Sharieh, Moh’d Belal Al-Zoubi, R. Bremananth

Abstract:

In pattern clustering, nearest neighborhood point computation is a challenging issue for many applications in the area of research such as Remote Sensing, Computer Vision, Pattern Recognition and Statistical Imaging. Nearest neighborhood computation is an essential computation for providing sufficient classification among the volume of pixels (voxels) in order to localize the active-region-of-interests (AROI). Furthermore, it is needed to compute spatial metric relationships of diverse area of imaging based on the applications of pattern recognition. In this paper, we propose a new methodology for finding the nearest neighbor point, depending on making a virtually grid of a hexagon cells, then locate every point beneath them. An algorithm is suggested for minimizing the computation and increasing the turnaround time of the process. The nearest neighbor query points Φ are fetched by seeking fashion of hexagon holistic. Seeking will be repeated until an AROI Φ is to be expected. If any point Υ is located then searching starts in the nearest hexagons in a circular way. The First hexagon is considered be level 0 (L0) and the surrounded hexagons is level 1 (L1). If Υ is located in L1, then search starts in the next level (L2) to ensure that Υ is the nearest neighbor for Φ. Based on the result and experimental results, we found that the proposed method has an advantage over the traditional methods in terms of minimizing the time complexity required for searching the neighbors, in turn, efficiency of classification will be improved sufficiently.

Keywords: Hexagon cells, k-nearest neighbors, Nearest Neighbor, Pattern recognition, Query pattern, Virtually grid

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Authors: Min Sun, Jing Liu

Abstract:

In this article, a new inexact alternating direction method(ADM) is proposed for solving a class of variational inequality problems. At each iteration, the new method firstly solves the resulting subproblems of ADM approximately to generate an temporal point ˜xk, and then the multiplier yk is updated to get the new iterate yk+1. In order to get xk+1, we adopt a new descent direction which is simple compared with the existing prediction-correction type ADMs. For the inexact ADM, the resulting proximal subproblem has closedform solution when the proximal parameter and inexact term are chosen appropriately. We show the efficiency of the inexact ADM numerically by some preliminary numerical experiments.

Keywords: variational inequality problems, alternating direction method, global convergence

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Authors: Aki Happonen, Adrian Burian, Erwin Hemming

Abstract:

Fixed-point simulation results are used for the performance measure of inverting matrices by Cholesky decomposition. The fixed-point Cholesky decomposition algorithm is implemented using a fixed-point reconfigurable processing element. The reconfigurable processing element provides all mathematical operations required by Cholesky decomposition. The fixed-point word length analysis is based on simulations using different condition numbers and different matrix sizes. Simulation results show that 16 bits word length gives sufficient performance for small matrices with low condition number. Larger matrices and higher condition numbers require more dynamic range for a fixedpoint implementation.

Keywords: Cholesky Decomposition, Fixed-point, Matrix inversion, Reconfigurable processing.

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Authors: Khaled M. EL-Naggar

Abstract:

This paper presents a fast and efficient on-line technique for estimating impedance of unbalanced loads in power systems. The proposed technique is an application of a discrete timedynamic filter based on stochastic estimation theory which is suitable for estimating parameters in noisy environment. The algorithm uses sets of digital samples of the distorted voltage and current waveforms of the non-linear load to estimate the harmonic contents of these two signal. The non-linear load impedance is then calculated from these contents. The method is tested using practical data. Results are reported and compared with those obtained using the conventional least error squares technique. In addition to the very accurate results obtained, the method can detect and reject bad measurements. This can be considered as a very important advantage over the conventional static estimation methods such as the least error square method.

Keywords: Estimation, identification, Harmonics, Dynamic Filter.

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Authors: S. Khamsawang, S. Jiriwibhakorn

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This paper proposes an application of the differential evolution (DE) algorithm for solving the economic dispatch problem (ED). Furthermore, the regenerating population procedure added to the conventional DE in order to improve escaping the local minimum solution. To test performance of DE algorithm, three thermal generating units with valve-point loading effects is used for testing. Moreover, investigating the DE parameters is presented. The simulation results show that the DE algorithm, which had been adjusted parameters, is better convergent time than other optimization methods.

Keywords: Differential evolution, Economic dispatch problem, Valve-point loading effect, Optimization method.

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Authors: S. K. Pillai, M. K. Jeyakumar

Abstract:

Estimation of model parameters is necessary to predict the behavior of a system. Model parameters are estimated using optimization criteria. Most algorithms use historical data to estimate model parameters. The known target values (actual) and the output produced by the model are compared. The differences between the two form the basis to estimate the parameters. In order to compare different models developed using the same data different criteria are used. The data obtained for short scale projects are used here. We consider software effort estimation problem using radial basis function network. The accuracy comparison is made using various existing criteria for one and two predictors. Then, we propose a new criterion based on linear least squares for evaluation and compared the results of one and two predictors. We have considered another data set and evaluated prediction accuracy using the new criterion. The new criterion is easy to comprehend compared to single statistic. Although software effort estimation is considered, this method is applicable for any modeling and prediction.

Keywords: Software effort estimation, accuracy, Radial Basis Function, linear least squares.

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Authors: E. E. El-Kholy, Ahamed Kalas, Mahmoud Fauzy, M. El-Shahat Dessouki, Abdou. M. El-Refay, Mohammed El-Zefery

Abstract:

Direct Torque Control (DTC) is an AC drive control method especially designed to provide fast and robust responses. In this paper a progressive algorithm for direct torque control of threephase induction drive system supplied by photovoltaic arrays using voltage source inverter to control motor torque and flux with maximum power point tracking at different level of insolation is presented. Experimental results of the new DTC method obtained by an experimental rapid prototype system for drives are presented. Simulation and experimental results confirm that the proposed system gives quick, robust torque and speed responses at constant switching frequencies.

Keywords: Photovoltaic (PV) array, direct torque control (DTC), constant switching frequency, induction motor, maximum power point tracking (MPPT).

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Authors: Jiaji Zhou, Le Liang, Long Chen

Abstract:

Geographic Profiling has successfully assisted investigations for serial crimes. Considering the multi-cluster feature of serial criminal spots, we propose a Multi-point Centrography model as a natural extension of Single-point Centrography for geographic profiling. K-means clustering is first performed on the data samples and then Single-point Centrography is adopted to derive a probability distribution on each cluster. Finally, a weighted combinations of each distribution is formed to make next-crime spot prediction. Experimental study on real cases demonstrates the effectiveness of our proposed model.

Keywords: Geographic profiling, Centrography model, K-means algorithm

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Authors: Sam Teek Ling, Norhashidah Hj. Mohd. Ali

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We investigate the formulation and implementation of new explicit group iterative methods in solving the two-dimensional Poisson equation with Dirichlet boundary conditions. The methods are derived from a fourth order compact nine point finite difference discretization. The methods are compared with the existing second order standard five point formula to show the dramatic improvement in computed accuracy. Numerical experiments are presented to illustrate the effectiveness of the proposed methods.

Keywords: Explicit group iterative method, finite difference, fourth order compact, Poisson equation.

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Authors: F. Maamar, M. Sartor

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The present study focuses on methods allowing a convenient and quick calculation of the SIFs in order to predict the static adhesive strength of bonded joints. A new SIF calculation method is proposed, based on the stresses obtained from a FE model at a reference point located in the adhesive layer at equal distance of the free-edge and of the two interfaces. It is shown that, even limiting ourselves to the two main modes, i.e. the opening and the shearing modes, and using the values of the stresses resulting from a low detailed FE model, an efficient calculation of the peeling stress at adhesive-substrate corners can be obtained by this way. The proposed method is interesting in that it can be the basis of a prediction tool that will allow the designer to quickly evaluate the SIFs characterizing a particular application without developing a detailed analysis.

Keywords: Adhesive layer, bounded joints, free-edge corner, stress intensity factor.

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Authors: Wiesław Szulczewski, Wojciech Jakubowski, Andrzej Żyromski, Małgorzata Biniak-Pieróg

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The aim of the paper was to elaborate a novel calculator BasWilCalc, that allows to estimate the actual amount of biomass on the basket willow plantations. The proposed method is based on the results of field experiment conducted during years  2011-2013 on basket willow plantation in the south-western part of Poland. As input data the results of destructive measurements of the diameter, length and weight of willow stems and non-destructive biometric measurements of diameter in the middle of stems and their length during the growing season performed at weekly intervals were used. Performed analysis enabled to develop the algorithm which, due to the fact that energy plantations are of known and constant planting structure, allows to estimate the actual amount of willow basket biomass on the plantation with a given probability and accuracy specified by the model, based on the number of stems measured and the age of the plantation.

Keywords: Basket willow (Salix viminalis) biomass, biometric measurements, yield, biomass calculator.

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Authors: Vai Kuong Sin, Chon Kit Chio

Abstract:

This paper investigates the nature of the development of two-dimensional laminar flow of an incompressible fluid at the reversed stagnation-point. ". In this study, we revisit the problem of reversed stagnation-point flow over a flat plate. Proudman and Johnson (1962) first studied the flow and obtained an asymptotic solution by neglecting the viscous terms. This is no true in neglecting the viscous terms within the total flow field. In particular it is pointed out that for a plate impulsively accelerated from rest to a constant velocity V0 that a similarity solution to the self-similar ODE is obtained which is noteworthy completely analytical.

Keywords: reversed stagnation-point flow, similarity solutions, analytical solution, numerical solution

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