Search results for: Numerical
2333 Some Results on the Generalized Higher Rank Numerical Ranges
Authors: Mohsen Zahraei
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In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for Є > 0, the notion of Birkhoff-James approximate orthogonality sets for Є−higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural generalization of the standard higher rank numerical ranges.Keywords: Rank−k numerical range, isometry, numerical range, rectangular matrix polynomials.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 15722332 A First Course in Numerical Methods with “Mathematica“
Authors: Andrei A. Kolyshkin
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In the present paper some recommendations for the use of software package “Mathematica" in a basic numerical analysis course are presented. The methods which are covered in the course include solution of systems of linear equations, nonlinear equations and systems of nonlinear equations, numerical integration, interpolation and solution of ordinary differential equations. A set of individual assignments developed for the course covering all the topics is discussed in detail.Keywords: Numerical methods, "Mathematica", e-learning.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 36532331 Performance Comparison and Analysis of Different Schemes and Limiters
Authors: Wang Wen-long, Li Hua, Pan Sha
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Eight difference schemes and five limiters are applied to numerical computation of Riemann problem. The resolution of discontinuities of each scheme produced is compared. Numerical dissipation and its estimation are discussed. The result shows that the numerical dissipation of each scheme is vital to improve scheme-s accuracy and stability. MUSCL methodology is an effective approach to increase computational efficiency and resolution. Limiter should be selected appropriately by balancing compressive and diffusive performance.
Keywords: Scheme; Limiter, Numerical simulation, Riemannproblem.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 24722330 Stabilization of the Bernoulli-Euler Plate Equation: Numerical Analysis
Authors: Carla E. O. de Moraes, Gladson O. Antunes, Mauro A. Rincon
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The aim of this paper is to study the internal stabilization of the Bernoulli-Euler equation numerically. For this, we consider a square plate subjected to a feedback/damping force distributed only in a subdomain. An algorithm for obtaining an approximate solution to this problem was proposed and implemented. The numerical method used was the Finite Difference Method. Numerical simulations were performed and showed the behavior of the solution, confirming the theoretical results that have already been proved in the literature. In addition, we studied the validation of the numerical scheme proposed, followed by an analysis of the numerical error; and we conducted a study on the decay of the energy associated.
Keywords: Bernoulli-Euler Plate Equation, Numerical Simulations, Stability, Energy Decay, Finite Difference Method.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 20282329 Analytical and Numerical Approaches in Coagulation of Particles
Authors: Bilal Barakeh
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In this paper we discuss the effect of unbounded particle interaction operator on particle growth and we study how this can address the choice of appropriate time steps of the numerical simulation. We provide also rigorous mathematical proofs showing that large particles become dominating with increasing time while small particles contribute negligibly. Second, we discuss the efficiency of the algorithm by performing numerical simulations tests and by comparing the simulated solutions with some known analytic solutions to the Smoluchowski equation.
Keywords: Stochastic processes, coagulation of particles, numerical scheme.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 14892328 Numerical Methods versus Bjerksund and Stensland Approximations for American Options Pricing
Authors: Marasovic Branka, Aljinovic Zdravka, Poklepovic Tea
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Numerical methods like binomial and trinomial trees and finite difference methods can be used to price a wide range of options contracts for which there are no known analytical solutions. American options are the most famous of that kind of options. Besides numerical methods, American options can be valued with the approximation formulas, like Bjerksund-Stensland formulas from 1993 and 2002. When the value of American option is approximated by Bjerksund-Stensland formulas, the computer time spent to carry out that calculation is very short. The computer time spent using numerical methods can vary from less than one second to several minutes or even hours. However to be able to conduct a comparative analysis of numerical methods and Bjerksund-Stensland formulas, we will limit computer calculation time of numerical method to less than one second. Therefore, we ask the question: Which method will be most accurate at nearly the same computer calculation time?
Keywords: Bjerksund and Stensland approximations, Computational analysis, Finance, Options pricing, Numerical methods.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 60532327 Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations
Authors: Fuziyah Ishak, Siti Norazura Ahmad
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Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.
Keywords: Accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 16382326 3-D Numerical Model for Wave-Induced Seabed Response around an Offshore Pipeline
Authors: Zuodong Liang, Dong-Sheng Jeng
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Seabed instability around an offshore pipeline is one of key factors that need to be considered in the design of offshore infrastructures. Unlike previous investigations, a three-dimensional numerical model for the wave-induced soil response around an offshore pipeline is proposed in this paper. The numerical model was first validated with 2-D experimental data available in the literature. Then, a parametric study will be carried out to examine the effects of wave, seabed characteristics and confirmation of pipeline. Numerical examples demonstrate significant influence of wave obliquity on the wave-induced pore pressures and the resultant seabed liquefaction around the pipeline, which cannot be observed in 2-D numerical simulation.Keywords: Pore pressure, 3D wave model, seabed liquefaction, pipeline.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 10232325 Numerical Inverse Laplace Transform Using Chebyshev Polynomial
Authors: Vinod Mishra, Dimple Rani
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In this paper, numerical approximate Laplace transform inversion algorithm based on Chebyshev polynomial of second kind is developed using odd cosine series. The technique has been tested for three different functions to work efficiently. The illustrations show that the new developed numerical inverse Laplace transform is very much close to the classical analytic inverse Laplace transform.
Keywords: Chebyshev polynomial, Numerical inverse Laplace transform, Odd cosine series.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 13912324 Numerical Simulation of Thermoreversible Polymer Gel Filtration
Authors: Said F. Urmancheev, Victor N. Kireev, Svetlana F. Khizbullina
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This paper presents results of numerical simulation of filtration of abnormal thermoviscous fluid on an example of thermo reversible polymer gel.
Keywords: Abnormal thermoviscous fluid, filtration, numerical simulation.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 14462323 Fin Spacing Effect of the Tube Fin Heat Exchanger at the Floor Heating Convector
Authors: F. Lemfeld, K. Frana
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This article deals with numerical simulation of the floor heating convector in 3D. Numerical simulation is focused on cooling mode of the floor heating convector. Geometrical model represents section of the heat exchanger – two fins with the gap between, pipes are not involved. Two types of fin are examined – sinusoidal and angular shape with different fin spacing. Results of fin spacing in case of constant Reynolds number are presented. For the numerical simulation was used commercial software Ansys Fluent.Keywords: fin spacing, cooling output, floor heating convector, numerical simulation.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 19502322 Design Methodology through Risk Assessment of Massive Water Retaining Structures
Authors: A. Rouili
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In the present paper the results of a numerical study are presented, numerical models were developed to simulate the behaviour of vertical massive dikes. The proposed models were developed according to the geometry, boundary conditions, loading conditions and initial conditions of a physical model taken as reference. The results obtained were compared to the experimental data. As far as the overall behaviour, the displacements and the failure mechanisms of the dikes is concerned, the numerical results were in good agreement with the experimental results, which clearly indicates a good quality of numerical modelling. The validated numerical models were used in a parametric study were the displacements and failure mechanisms were fully investigated. Out of the results obtained, some conclusions and recommendations related to the design of massive dikes are proposed.
Keywords: Water conservation, dikes, risk assessment and numerical modelling.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 15392321 Spline Basis Neural Network Algorithm for Numerical Integration
Authors: Lina Yan, Jingjing Di, Ke Wang
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A new basis function neural network algorithm is proposed for numerical integration. The main idea is to construct neural network model based on spline basis functions, which is used to approximate the integrand by training neural network weights. The convergence theorem of the neural network algorithm, the theorem for numerical integration and one corollary are presented and proved. The numerical examples, compared with other methods, show that the algorithm is effective and has the characteristics such as high precision and the integrand not required known. Thus, the algorithm presented in this paper can be widely applied in many engineering fields.
Keywords: Numerical integration, Spline basis function, Neural network algorithm
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 29182320 On a New Inverse Polynomial Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations
Authors: R. B. Ogunrinde
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This paper presents the development, analysis and implementation of an inverse polynomial numerical method which is well suitable for solving initial value problems in first order ordinary differential equations with applications to sample problems. We also present some basic concepts and fundamental theories which are vital to the analysis of the scheme. We analyzed the consistency, convergence, and stability properties of the scheme. Numerical experiments were carried out and the results compared with the theoretical or exact solution and the algorithm was later coded using MATLAB programming language.Keywords: Differential equations, Numerical, Initial value problem, Polynomials.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 17582319 An Eulerian Numerical Method and its Application to Explosion Problems
Authors: Li Hao, Yan Zhang, Jingan Cui
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The Eulerian numerical method is proposed to analyze the explosion in tunnel. Based on this method, an original software M-MMIC2D is developed by Cµ program language. With this software, the explosion problem in the tunnel with three expansion-chambers is numerically simulated, and the results are found to be in full agreement with the observed experimental data.Keywords: Eulerian method, numerical simulation, shock wave, tunnel
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 14442318 Role of Association Rule Mining in Numerical Data Analysis
Authors: Sudhir Jagtap, Kodge B. G., Shinde G. N., Devshette P. M
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Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. The numerical data analysis became key process in research and development of all the fields [6]. In this paper we have made an attempt to analyze the specified numerical patterns with reference to the association rule mining techniques with minimum confidence and minimum support mining criteria. The extracted rules and analyzed results are graphically demonstrated. Association rules are a simple but very useful form of data mining that describe the probabilistic co-occurrence of certain events within a database [7]. They were originally designed to analyze market-basket data, in which the likelihood of items being purchased together within the same transactions are analyzed.Keywords: Numerical data analysis, Data Mining, Association Rule Mining
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 28522317 Numerical Simulation of High Pressure Hydrogen Emerges to Air
Authors: Mohamed H. Elhsnawi, Mesbah M. Salem, Saleh B. Mohamed
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Numerical simulation performed to investigate the behavior of the high pressure hydrogen jetting of air. High pressure hydrogen (30–40 MPa) was injected to air at atmospheric pressure through 2mm orifice. Numerical simulations were performed with Kiva3V code with 2D axisymmetric geometry. Numerical simulations showed that auto ignition of high pressure hydrogen to air are possible due to molecular diffusion. Auto ignition was predicted at hydrogen-air contact surface due to mass and energy exchange between high temperature hydrogen and air heated by shock wave.
Keywords: Spontaneous Ignition, Diffusion Ignition, Hydrogen ignition, Hydrogen Jet.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 18932316 The Impact of Modeling Method of Moisture Emission from the Swimming Pool on the Accuracy of Numerical Calculations of Air Parameters in Ventilated Natatorium
Authors: Piotr Ciuman, Barbara Lipska
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The aim of presented research was to improve numerical predictions of air parameters distribution in the actual natatorium by the selection of calculation formula of mass flux of moisture emitted from the pool. Selected correlation should ensure the best compliance of numerical results with the measurements' results of these parameters in the facility. The numerical model of the natatorium was developed, for which boundary conditions were prepared on the basis of measurements' results carried out in the actual facility. Numerical calculations were carried out with the use of ANSYS CFX software, with six formulas being implemented, which in various ways made the moisture emission dependent on water surface temperature and air parameters in the natatorium. The results of calculations with the use of these formulas were compared for air parameters' distributions: Specific humidity, velocity and temperature in the facility. For the selection of the best formula, numerical results of these parameters in occupied zone were validated by comparison with the measurements' results carried out at selected points of this zone.
Keywords: Experimental validation, indoor swimming pool, moisture emission, natatorium, numerical calculations, CFD, thermal and humidity conditions, ventilation.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 14892315 The Ratios between the Spectral Norm, the Numerical Radius and the Spectral Radius
Authors: Kui Du
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Recently, Uhlig [Numer. Algorithms, 52(3):335-353, 2009] proposed open questions about the ratios between the spectral norm, the numerical radius and the spectral radius of a square matrix. In this note, we provide some observations to answer these questions.
Keywords: Spectral norm, Numerical radius, Spectral radius, Ratios
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 18162314 Numerical Investigation of Wave Interaction with Double Vertical Slotted Walls
Authors: H. Ahmed, A. Schlenkhoff
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Recently, permeable breakwaters have been suggested to overcome the disadvantages of fully protection breakwaters. These protection structures have minor impacts on the coastal environment and neighboring beaches where they provide a more economical protection from waves and currents. For regular waves, a numerical model is used (FLOW-3D, VOF) to investigate the hydraulic performance of a permeable breakwater. The model of permeable breakwater consists of a pair of identical vertical slotted walls with an impermeable upper and lower part, where the draft is a decimal multiple of the total depth. The middle part is permeable with a porosity of 50%. The second barrier is located at distant of 0.5 and 1.5 of the water depth from the first one. The numerical model is validated by comparisons with previous laboratory data and semi-analytical results of the same model. A good agreement between the numerical results and both laboratory data and semi-analytical results has been shown and the results indicate the applicability of the numerical model to reproduce most of the important features of the interaction. Through the numerical investigation, the friction factor of the model is carefully discussed.
Keywords: Coastal structures, permeable breakwater, slotted wall, numerical model, energy dissipation coefficient.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 22382313 A Study on the Heading of Spur Gears: Numerical Analysis and Experiments
Authors: M.Zadshakouyan, E.Abdi Sobbouhi, H.Jafarzadeh
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In this study, the precision heading process of spur gears has been investigated by means of numerical analysis. The effect of some parameters such as teeth number and module on the forming force and material flow were presented. The simulation works were performed rigid-plastic finite element method using DEFORM 3D software. In order to validate the estimated numerical results, they were compared with those obtained experimentally during heading of spur gear using lead as a model material. Results showed that the optimum number of gear teeth is between 10 to 20, that is because of being the specific pressure in its minimum value.Keywords: Heading, spur gear, numerical analysis, experiments.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 19452312 Solution of First kind Fredholm Integral Equation by Sinc Function
Authors: Khosrow Maleknejad, Reza Mollapourasl, Parvin Torabi, Mahdiyeh Alizadeh,
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Sinc-collocation scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this paper, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.Keywords: Integral equation, Fredholm type, Collocation method, Sinc approximation.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 27422311 Sinc-Galerkin Method for the Solution of Problems in Calculus of Variations
Authors: M. Zarebnia, N. Aliniya
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In this paper, a numerical solution based on sinc functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.Keywords: Calculus of variation; Sinc functions; Galerkin; Numerical method
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 19482310 Numerical Simulation of Punching Shear of Flat Plates with Low Reinforcement
Authors: Fatema-Tuz-Zahura, Raquib Ahsan
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Punching shear failure is usually the governing failure mode of flat plate structures. Punching failure is brittle in nature which induces more vulnerability to this type of structure. In the present study, a 3D finite element model of a flat plate with low reinforcement ratio and without any transverse reinforcement has been developed. Punching shear stress and the deflection data were obtained on the surface of the flat plate as well as through the thickness of the model from numerical simulations. The obtained data were compared with the experimental results. Variation of punching stress with respect to deflection as obtained from numerical results is found to be in good agreement with the experimental results; the range of variation of punching stress is within 5%. The numerical simulation shows an early and gradual onset of nonlinearity, whereas the same is late and abrupt as observed in the experimental results. The range of variation of punching stress for different slab thicknesses between experimental and numerical results is less than 15%. The developed numerical model is useful to complement available punching test series performed in the past. The results obtained from the numerical model will be helpful for designing retrofitting schemes of flat plates.Keywords: Flat plate, finite element model, punching shear, reinforcement ratio.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 14182309 Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations
Authors: M. Zarebnia, M. Hoshyar, M. Sedaghati
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In this paper, a numerical solution based on nonpolynomial cubic spline functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.Keywords: Calculus of variation; Non-polynomial spline functions; Numerical method
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 19702308 Groundwater Seepage Estimation into Amirkabir Tunnel Using Analytical Methods and DEM and SGR Method
Authors: Hadi Farhadian, Homayoon Katibeh
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In this paper, groundwater seepage into Amirkabir tunnel has been estimated using analytical and numerical methods for 14 different sections of the tunnel. Site Groundwater Rating (SGR) method also has been performed for qualitative and quantitative classification of the tunnel sections. The obtained results of above mentioned methods were compared together. The study shows reasonable accordance with results of the all methods unless for two sections of tunnel. In these two sections there are some significant discrepancies between numerical and analytical results mainly originated from model geometry and high overburden. SGR and the analytical and numerical calculations, confirm high concentration of seepage inflow in fault zones. Maximum seepage flow into tunnel has been estimated 0.425 lit/sec/m using analytical method and 0.628 lit/sec/m using numerical method occured in crashed zone. Based on SGR method, six sections of 14 sections in Amirkabir tunnel axis are found to be in "No Risk" class that is supported by the analytical and numerical seepage value of less than 0.04 lit/sec/m.
Keywords: Water Seepage, Amirkabir Tunnel, Analytical Method, DEM, SGR.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 39462307 A Numerical Investigation on the Dynamic Stall of a Wind Turbine Section Using Different Turbulent Models
Authors: S. A. Ahmadi, S. Sharif, R. Jamshidi
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In this article, the flow behavior around a NACA 0012 airfoil which is oscillating with different Reynolds numbers and in various amplitudes has been investigated numerically. Numerical simulations have been performed with ANSYS software. First, the 2- D geometry has been studied in different Reynolds numbers and angles of attack with various numerical methods in its static condition. This analysis was to choose the best turbulent model and comparing the grids to have the optimum one for dynamic simulations. Because the analysis was to study the blades of wind turbines, the Reynolds numbers were not arbitrary. They were in the range of 9.71e5 to 22.65e5. The angle of attack was in the range of -41.81° to 41.81°. By choosing the forward wind speed as the independent parameter, the others like Reynolds and the amplitude of the oscillation would be known automatically. The results show that the SST turbulent model is the best choice that leads the least numerical error with respect the experimental ones. Also, a dynamic stall phenomenon is more probable at lower wind speeds in which the lift force is less.
Keywords: Dynamic stall, Numerical simulation, Wind turbine, Turbulent Model
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 19962306 Numerical Approximation to the Performance of CUSUM Charts for EMA (1) Process
Authors: K. Petcharat, Y. Areepong, S. Sukparungsri, G. Mititelu
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These paper, we approximate the average run length (ARL) for CUSUM chart when observation are an exponential first order moving average sequence (EMA1). We used Gauss-Legendre numerical scheme for integral equations (IE) method for approximate ARL0 and ARL1, where ARL in control and out of control, respectively. We compared the results from IE method and exact solution such that the two methods perform good agreement.Keywords: Cumulative Sum Chart, Moving Average Observation, Average Run Length, Numerical Approximations.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 21522305 Dissipation of Higher Mode using Numerical Integration Algorithm in Dynamic Analysis
Authors: Jin Sup Kim, Woo Young Jung, Minho Kwon
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In general dynamic analyses, lower mode response is of interest, however the higher modes of spatially discretized equations generally do not represent the real behavior and not affects to global response much. Some implicit algorithms, therefore, are introduced to filter out the high-frequency modes using intended numerical error. The objective of this study is to introduce the P-method and PC α-method to compare that with dissipation method and Newmark method through the stability analysis and numerical example. PC α-method gives more accuracy than other methods because it based on the α-method inherits the superior properties of the implicit α-method. In finite element analysis, the PC α-method is more useful than other methods because it is the explicit scheme and it achieves the second order accuracy and numerical damping simultaneously.Keywords: Dynamic, α-Method, P-Method, PC α-Method, Newmark method.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 30652304 On Constructing a Cubically Convergent Numerical Method for Multiple Roots
Authors: Young Hee Geum
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We propose the numerical method defined by
xn+1 = xn − λ[f(xn − μh(xn))/]f'(xn) , n ∈ N,
and determine the control parameter λ and μ to converge cubically. In addition, we derive the asymptotic error constant. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-precision computability.
Keywords: Asymptotic error constant, iterative method , multiple root, root-finding.
Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1495