Search results for: boundary value analysis

Authors: Sen Yung Lee, Chih Cheng Huang, Te Wen Tu

Abstract:

A solution methodology without using integral transformation is proposed to develop analytical solutions for transient heat conduction in nonuniform hollow cylinders with time-dependent boundary condition at the outer surface. It is shown that if the thermal conductivity and the specific heat of the medium are in arbitrary polynomial function forms, the closed solutions of the system can be developed. The influence of physical properties on the temperature distribution of the system is studied. A numerical example is given to illustrate the efficiency and the accuracy of the solution methodology.

Keywords: Analytical solution, nonuniform hollow cylinder, time-dependent boundary condition, transient heat conduction.

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Authors: M. Foroutan, H. Dalayeli, M. Sadeghian

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In this paper processes including large deformations of a rubber with hyperelastic material behavior are simulated by the RKPM method. Due to the loss of kronecker delta properties in the mesh less shape functions, the imposition of essential boundary conditions consumes significant CPU time in mesh free computations. In this work transformation method is used for imposition of essential boundary conditions. A RKPM material shape function is used in this analysis. The support of the material shape functions covers the same set of particles during material deformation and hence the transformation matrix is formed only once at the initial stages. A computer program in MATLAB is developed for simulations.

Keywords: RKPM, large deformations, transformation, essentialboundary conditions.

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Authors: Chinwendu. B. Eleje, Udechukwu P. Egbuhuzor

Abstract:

One of the difficulties encountered in solving nonlinear Boundary Value Problems (BVP) by many researchers is finding approximated solutions with minimum deviations from the exact solutions without so much rigor and complications. In this paper, we propose an approach to solve a two point BVP which involves a combination of Taylor series expansion method and Newton Raphson method. Furthermore, the fourth and sixth order approximated solutions are obtained and we compare their relative error and rate of convergence to the exact solution. Finally, some numerical simulations are presented to show the behavior of the solution and its derivatives.

Keywords: Newton Raphson method, non-linear boundary value problem, Taylor series approximation, Michaelis-Menten equation.

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Authors: Dimitrios Zarpalas, Anastasios Zafeiropoulos, Petros Daras, Nicos Maglaveras

Abstract:

Segmentation techniques based on Active Contour Models have been strongly benefited from the use of prior information during their evolution. Shape prior information is captured from a training set and is introduced in the optimization procedure to restrict the evolution into allowable shapes. In this way, the evolution converges onto regions even with weak boundaries. Although significant effort has been devoted on different ways of capturing and analyzing prior information, very little thought has been devoted on the way of combining image information with prior information. This paper focuses on a more natural way of incorporating the prior information in the level set framework. For proof of concept the method is applied on hippocampus segmentation in T1-MR images. Hippocampus segmentation is a very challenging task, due to the multivariate surrounding region and the missing boundary with the neighboring amygdala, whose intensities are identical. The proposed method, mimics the human segmentation way and thus shows enhancements in the segmentation accuracy.

Keywords: Medical imaging & processing, Brain MRI segmentation, hippocampus segmentation, hippocampus-amygdala missingboundary, weak boundary segmentation, region based segmentation, prior information, local weighting scheme in level sets, spatialdistribution of labels, gradient distribution on boundary.

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Authors: G. Revathi, P. Saikrishnan

Abstract:

An analysis is performed to study the influence of nonuniform double slot suction on a steady laminar boundary layer flow over a rotating sphere when fluid properties such as viscosity and Prandtl number are inverse linear functions of temperature. Nonsimilar solutions have been obtained from the starting point of the streamwise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the streamwise co-ordinate, at the edges of the slot and at the point of separation have been overcome by applying an implicit finite difference scheme in combination with the quasi-linearization technique and an appropriate selection of the finer step sizes along the stream-wise direction. The present investigation shows that the point of ordinary separation can be delayed by nonuniform double slot suction if the mass transfer rate is increased and also if the slots are positioned further downstream. In addition, the investigation reveals that double slot suction is found to be more effective compared to a single slot suction in delaying ordinary separation. As rotation parameter increase the point of separation moves upstream direction.

Keywords: Boundary layer, suction, mass transfer, rotating sphere.

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Authors: M. A. Talha, M. Osman Gani, M. Ferdows

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This paper focuses on the study of two dimensional magnetohydrodynamic (MHD) steady incompressible viscous Williamson nanofluid with exponential internal heat generation containing gyrotactic microorganism over a stretching sheet. The governing equations and auxiliary conditions are reduced to a set of non-linear coupled differential equations with the appropriate boundary conditions using similarity transformation. The transformed equations are solved numerically through spectral relaxation method. The influences of various parameters such as Williamson parameter γ, power constant λ, Prandtl number P_r, magnetic field parameter M, Peclet number P_e, Lewis number Le, Bioconvection Lewis number Lb, Brownian motion parameter Nb, thermophoresis parameter Nt, and bioconvection constant σ are studied to obtain the momentum, heat, mass and microorganism distributions. Moment, heat, mass and gyrotactic microorganism profiles are explored through graphs and tables. We computed the heat transfer rate, mass flux rate and the density number of the motile microorganism near the surface. Our numerical results are in better agreement in comparison with existing calculations. The Residual error of our obtained solutions is determined in order to see the convergence rate against iteration. Faster convergence is achieved when internal heat generation is absent. The effect of magnetic parameter M decreases the momentum boundary layer thickness but increases the thermal boundary layer thickness. It is apparent that bioconvection Lewis number and bioconvection parameter has a pronounced effect on microorganism boundary. Increasing brownian motion parameter and Lewis number decreases the thermal boundary layer. Furthermore, magnetic field parameter and thermophoresis parameter has an induced effect on concentration profiles.

Keywords: Convection flow, internal heat generation, similarity, spectral method, numerical analysis, Williamson nanofluid.

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Authors: O. Miraliyari

Abstract:

This article attempts to analyze functionally graded beam thermal buckling along with piezoelectric layers applying based on the third order shearing deformation theory considering various boundary conditions. The beam properties are assumed to vary continuously from the lower surface to the upper surface of the beam. The equilibrium equations are derived using the total potential energy equations, Euler equations, piezoelectric material constitutive equations and third order shear deformation theory assumptions. In order to fulfill such an aim, at first functionally graded beam with piezoelectric layers applying the third order shearing deformation theory along with clamped -clamped boundary conditions are thoroughly analyzed, and then following making sure of the correctness of all the equations, the very same beam is analyzed with piezoelectric layers through simply-simply and simply-clamped boundary conditions. In this article buckling critical temperature for functionally graded beam is derived in two different ways, without piezoelectric layer and with piezoelectric layer and the results are compared together. Finally, all the conclusions obtained will be compared and contrasted with the same samples in the same and distinguished conditions through tables and charts. It would be noteworthy that in this article, the software MAPLE has been applied in order to do the numeral calculations.

Keywords: Thermal buckling, functionally graded beam, piezoelectric layer, various boundary conditions.

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Authors: M.R.Isvandzibaei, P.J.Awasare

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In this paper a study on the vibration of thin cylindrical shells with ring supports and made of functionally graded materials (FGMs) composed of stainless steel and nickel is presented. Material properties vary along the thickness direction of the shell according to volume fraction power law. The cylindrical shells have ring supports which are arbitrarily placed along the shell and impose zero lateral deflections. The study is carried out based on third order shear deformation shell theory (T.S.D.T). The analysis is carried out using Hamilton-s principle. The governing equations of motion of FGM cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.

Keywords: Vibration, FGM, Cylindrical shell, Hamilton'sprinciple, Ring support.

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Authors: Murtaza Adil, Sen Yang, Zhou Chao, Song Xiaoping

Abstract:

The morphotropic phase boundary (MPB) recently has attracted constant interest in ferromagnetic systems for obtaining enhanced large magnetoelastic response. In the present study, structural and magnetoelastic properties of MPB involved ferromagnetic Tb_1-xGd_xFe₂ (0≤x≤1) system has been investigated. The change of easy magnetic direction from <111> to <100> with increasing x up MPB composition of x=0.9 is detected by step-scanned [440] synchrotron X-ray diffraction reflections. The Gd substitution for Tb changes the composition for the anisotropy compensation near MPB composition of x=0.9, which was confirmed by the analysis of detailed scanned XRD, magnetization curves and the calculation of the first anisotropy constant K₁. The spin configuration diagram accompanied with different crystal structures for Tb_1-xGd_xFe₂ was designed. The calculated first anisotropy constant K₁ shows a minimum value at MPB composition of x=0.9. In addition, the large ratio between magnetostriction, and the absolute values of the first anisotropy constant │λ_S∕K₁│ appears at MPB composition, which makes it a potential material for magnetostrictive application. Based on experimental results, a theoretically approach was also proposed to signify that the facilitated magnetization rotation and enhanced magnetoelastic effect near MPB composition are a consequence of the anisotropic flattening of free energy of ferromagnetic crystal. Our work specifies the universal existence of MPB in ferromagnetic materials which is important for substantial improvement of magnetic and magnetostrictive properties and may provide a new route to develop advanced functional materials.

Keywords: Free energy, lattice distortion, magnetic anisotropy, magnetostriction, morphotropic phase boundary.

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Authors: Indu Saini, Phool Singh, Vikas Malik

Abstract:

A novel method based on Genetic Algorithm to solve the boundary value problems (BVPs) of the Falkner–Skan equation over a semi-infinite interval has been presented. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method based on Genetic Algorithm to transform the BVP into initial value problems (IVPs). Genetic Algorithm is used to calculate shooting angle. The initial value problems arisen during shooting are computed by Runge-Kutta Fehlberg method. The numerical solutions obtained by the present method are in agreement with those obtained by previous authors.

Keywords: Boundary Layer Flow, Falkner–Skan equation, Genetic Algorithm, Shooting method.

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Authors: M.R.Isvandzibaei, E.Bidokh, M.R.Alinaghizadeh, A.Nasirian, A.Moarrefzadeh

Abstract:

In this paper a study on the vibration of thin cylindrical shells with ring supports and made of functionally graded materials (FGMs) composed of stainless steel and nickel is presented. Material properties vary along the thickness direction of the shell according to volume fraction power law. The cylindrical shells have ring supports which are arbitrarily placed along the shell and impose zero lateral deflections. The study is carried out based on third order shear deformation shell theory (T.S.D.T). The analysis is carried out using Hamilton-s principle. The governing equations of motion of FGM cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.

Keywords: Vibration, FGM, Cylindrical shell, Hamilton'sprinciple, Ring support.

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Authors: Roslinda Nazar, Amin Noor, Khamisah Jafar, Ioan Pop

Abstract:

In this paper, the steady laminar three-dimensional boundary layer flow and heat transfer of a copper (Cu)-water nanofluid in the vicinity of a permeable shrinking flat surface in an otherwise quiescent fluid is studied. The nanofluid mathematical model in which the effect of the nanoparticle volume fraction is taken into account is considered. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation which is then solved numerically using the function bvp4c from Matlab. Dual solutions (upper and lower branch solutions) are found for the similarity boundary layer equations for a certain range of the suction parameter. A stability analysis has been performed to show which branch solutions are stable and physically realizable. The numerical results for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles are obtained, presented and discussed in detail for a range of various governing parameters.

Keywords: Heat Transfer, Nanofluid, Shrinking Surface, Stability Analysis, Three-Dimensional Flow.

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Authors: A. F. Khadrawi

Abstract:

The transient hydrodynamics and thermal behaviors of fluid flow in open-ended vertical parallel-plate porous microchannel are investigated semi-analytically under the effect of the hyperbolic heat conduction model. The model that combines both the continuum approach and the possibility of slip at the boundary is adopted in the study. The Effects of Knudsen number , Darcy number , and thermal relaxation time  on the microchannel hydrodynamics and thermal behaviors are investigated using the hyperbolic heat conduction models. It is found that as  increases the slip in the hydrodynamic and thermal boundary condition increases. This slip in the hydrodynamic boundary condition increases as  increases. Also, the slip in the thermal boundary condition increases as  decreases especially the early stage of time.

Keywords: free convection, hyperbolic heat conduction, macroscopic heat conduction models in microchannel, porous media, vertical microchannel, microchannel thermal, hydrodynamic behavior.

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Authors: R. Faiez, M. Mashhoudi, F. Najafi

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Implicit in most large-scale numerical analyses of the crystal growth from the melt is the assumption that the shape and position of the phase boundary are determined by the transport phenomena coupled strongly to the melt hydrodynamics. In the present numerical study, the interface shape-effect on the convective interactions in a Czochralski oxide melt is described. It was demonstrated that thermocapillary flow affects inversely the phase boundaries of distinct shapes. The inhomogenity of heat flux and the location of the stagnation point at the crystallization front were investigated. The forced convection effect on the point displacement at the boundary found to be much stronger for the flat plate interface compared to the cone-shaped one with and without the Marangoni flow.

Keywords: Computer simulation, fluid flow, interface shape, thermocapillary effect.

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Authors: Gabriela R. Fernandes, Danilo H. Konda, Luiz C. F. Sanches

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In this work, the plate bending formulation of the boundary element method - BEM, based on the Reissner?s hypothesis, is extended to the analysis of plates reinforced by beams taking into account the membrane effects. The formulation is derived by assuming a zoned body where each sub-region defines a beam or a slab and all of them are represented by a chosen reference surface. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composed structure as a single body. In order to reduce the number of degrees of freedom, the problem values defined on the interfaces are written in terms of their values on the beam axis. Initially are derived separated equations for the bending and stretching problems, but in the final system of equations the two problems are coupled and can not be treated separately. Finally are presented some numerical examples whose analytical results are known to show the accuracy of the proposed model.

Keywords: Boundary elements, Building floor structures, Platebending.

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Authors: Hailong Zhu, Zhaoxiang Li

Abstract:

In this paper we investigate numerically positive solutions of the equation -Δu = λuq+up with Dirichlet boundary condition in a boundary domain ╬® for λ > 0 and 0 < q < 1 < p < 2*, we will compute and visualize the range of λ, this problem achieves a numerical solution.

Keywords: positive solutions, concave-convex, sub-super solution method, pseudo arclength method.

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Authors: Xin Luo, Jin Huang, Pan Cheng

Abstract:

By means of Sidi-Israeli’s quadrature rules, mechanical quadrature methods (MQMs) for solving the first kind boundary integral equations (BIEs) of steady state Stokes problem are presented. The convergence of numerical solutions by MQMs is proved based on Anselone’s collective compact and asymptotical compact theory, and the asymptotic expansions with the odd powers of the errors are provided, which implies that the accuracy of the approximations by MQMs possesses high accuracy order O (h3). Finally, the numerical examples show the efficiency of our methods.

Keywords: Stokes problem, boundary integral equation, mechanical quadrature methods, asymptotic expansions.

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Authors: B. I. Yun

Abstract:

A dual-reciprocity boundary element method is presented for the numerical solution of a class of axisymmetric elastodynamic problems. The domain integrals that arise in the integrodifferential formulation are converted to line integrals by using the dual-reciprocity method together suitably constructed interpolating functions. The second order time derivatives of the displacement in the governing partial differential equations are suppressed by using Laplace transformation. In the Laplace transform domain, the problem under consideration is eventually reduced to solving a system of linear algebraic equations. Once the linear algebraic equations are solved, the displacement and stress fields in the physical domain can be recovered by using a numerical technique for inverting Laplace transforms.

Keywords: Axisymmetric elasticity, boundary element method, dual-reciprocity method, Laplace transform.

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Authors: Feng Ge, Tiecheng Liu, Song Wang, Joachim Stahl

Abstract:

This paper presents a novel template-based method to detect objects of interest from real images by shape matching. To locate a target object that has a similar shape to a given template boundary, the proposed method integrates three components: contour grouping, partial shape matching, and boundary verification. In the first component, low-level image features, including edges and corners, are grouped into a set of perceptually salient closed contours using an extended ratio-contour algorithm. In the second component, we develop a partial shape matching algorithm to identify the fractions of detected contours that partly match given template boundaries. Specifically, we represent template boundaries and detected contours using landmarks, and apply a greedy algorithm to search the matched landmark subsequences. For each matched fraction between a template and a detected contour, we estimate an affine transform that transforms the whole template into a hypothetic boundary. In the third component, we provide an efficient algorithm based on oriented edge lists to determine the target boundary from the hypothetic boundaries by checking each of them against image edges. We evaluate the proposed method on recognizing and localizing 12 template leaves in a data set of real images with clutter back-grounds, illumination variations, occlusions, and image noises. The experiments demonstrate the high performance of our proposed method1.

Keywords: Object detection, shape matching, contour grouping.

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Authors: A. Gherbi, L. Dahmani, A. Boudjemia

Abstract:

This paper presents the Finite Element Method (FEM) for analyzing the failure pattern of rectangular slab with various edge conditions. Non-Linear static analysis is carried out using ANSYS 15 Software. Using SOLID65 solid elements, the compressive crushing of concrete is facilitated using plasticity algorithm, while the concrete cracking in tension zone is accommodated by the nonlinear material model. Smeared reinforcement is used and introduced as a percentage of steel embedded in concrete slab. The behavior of the analyzed concrete slab has been observed in terms of the crack pattern and displacement for various loading and boundary conditions. The finite element results are also compared with the experimental data. One of the other objectives of the present study is to show how similar the crack path found by ANSYS program to those observed for the yield line analysis. The smeared reinforcement method is found to be more practical especially for the layered elements like concrete slabs. The value of this method is that it does not require explicit modeling of the rebar, and thus a much coarser mesh can be defined.

Keywords: ANSYS, cracking pattern, displacements, RC Slab, smeared reinforcement.

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Authors: Ho Phu TRAN, Frédéric PLOURDE

Abstract:

The ongoing effort to develop an in-house compressible solver with multi-disciplinary physics is presented in this paper. Basic compressible solver combined with IBM technique provides us an effective numerical tool able to tackle the physics phenomena and especially physic phenomena involved in Solid Rocket Motors (SRMs). Main principles are introduced step by step describing its implementation. This paper sheds light on the whole potentiality of our proposed numerical model and we strongly believe a way to introduce multi-physics mechanisms strongly coupled is opened to ablation in nozzle, fluid/structure interaction and burning propellant surface with time.

Keywords: Compressible Flow, Immersed Boundary Method, Multi-disciplinary physics, Solid Rocket Motors.

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Authors: Xin Luo, Jin Huang, Chuan-Long Wang

Abstract:

The mechanical quadrature methods for solving the boundary integral equations of the anisotropic Darcy-s equations with Dirichlet conditions in smooth domains are presented. By applying the collectively compact theory, we prove the convergence and stability of approximate solutions. The asymptotic expansions for the error show that the methods converge with the order O (h3), where h is the mesh size. Based on these analysis, extrapolation methods can be introduced to achieve a higher convergence rate O (h5). An a posterior asymptotic error representation is derived in order to construct self-adaptive algorithms. Finally, the numerical experiments show the efficiency of our methods.

Keywords: Darcy's equation, anisotropic, mechanical quadrature methods, extrapolation methods, a posteriori error estimate.

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Authors: Mostafa Ghayour, Mohammad Sadegh Golabi

Abstract:

In the recent years, functionally gradient materials (FGMs) have gained considerable attention in the high temperature environment applications. In this paper, free vibration of thin functionally graded cylindrical shell with hole composed of stainless steel and zirconia is studied. The mechanical properties vary smoothly and continuously from one surface to the other according to a volume fraction power-law distribution. The Influence of shell geometrical parameters, variations of volume fractions and boundary conditions on natural frequency is considered. The equations of motion are based on strains-displacement relations from Love-s shell theory and Rayleigh method. The results have been obtained for natural frequencies of cylindrical shell with holes for different shape, number and location in this paper.

Keywords: Functionally gradient material, Vibration, various boundary conditions, cylindrical shells.

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Authors: Huei Chu Weng, Chien-Hung Liu

Abstract:

This paper presents a study on the effect of second-order slip and jump on forced convection through a long isothermally heated or cooled planar microchannel. The fully developed solutions of thermal flow fields are analytically obtained on the basis of the second-order Maxwell-Burnett slip and Smoluchowski jump boundary conditions. Results reveal that the second-order term in the Karniadakis slip boundary condition is found to contribute a negative velocity slip and then to lead to a higher pressure drop as well as a higher fluid temperature for the heated-wall case or to a lower fluid temperature for the cooled-wall case. These findings are contrary to predictions made by the Deissler model. In addition, the role of second-order slip becomes more significant when the Knudsen number increases.

Keywords: Microfluidics, forced convection, gas rarefaction, second-order boundary conditions.

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Authors: Saeed Sarabadan, Mehran Nikarya, Kouroah Parand

Abstract:

The magnetohydrodynamic (MHD) Falkner-Skan equations appear in study of laminar boundary layers flow over a wedge in presence of a transverse magnetic field. The partial differential equations of boundary layer problems in presence of a transverse magnetic field are reduced to MHD Falkner-Skan equation by similarity solution methods. This is a nonlinear ordinary differential equation. In this paper, we solve this equation via spectral collocation method based on Bessel functions of the first kind. In this approach, we reduce the solution of the nonlinear MHD Falkner-Skan equation to a solution of a nonlinear algebraic equations system. Then, the resulting system is solved by Newton method. We discuss obtained solution by studying the behavior of boundary layer flow in terms of skin friction, velocity, various amounts of magnetic field and angle of wedge. Finally, the results are compared with other methods mentioned in literature. We can conclude that the presented method has better accuracy than others.

Keywords: MHD Falkner-Skan, nonlinear ODE, spectral collocation method, Bessel functions, skin friction, velocity.

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Authors: Francois T. Pronk, Steven J. Hulshoff

Abstract:

A study of various turbulent inflow generation methods was performed to compare their relative effectiveness for LES computations of turbulent boundary layers. This study confirmed the quality of the turbulent information produced by the family of recycling and rescaling methods which take information from within the computational domain. Furthermore, more general inflow methods also proved applicable to such simulations, with a precursor-like inflow and a random inflow augmented with forcing planes showing promising results.

Keywords: Boundary layer, Flat plate, Inflow modeling, LES

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Authors: M.R.Alinaghizadehand, M.R.Isvandzibaei

Abstract:

Study of the vibration cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is important. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of boundary conditions on the natural frequencies of the FG cylindrical shell. The study is carried out using third order shear deformation shell theory. The governing equations of motion of FG cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of constituent volume fractions and the effects of clampedclamped boundary conditions.

Keywords: Vibration, FGM, Cylindrical shell, Hamilton's principle.

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Authors: Chenguang Guo, Lei Chen, Yanlong Zhang

Abstract:

This paper describes a novel optimized JTAG interface circuit between a JTAG controller and target IC. Being able to access JTAG using only one or two pins, this circuit does not change the original boundary scanning test frequency of target IC. Compared with the traditional JTAG interface which based on IEEE std. 1149.1, this reduced pin technology is more applicability in pin limited devices, and it is easier to control the scale of target IC for the designer.

Keywords: Boundary scan, JTAG interface, Test frequency, Reduced pin

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Authors: E. Popov, N. Yorino, Y. Zoka, Y. Sasaki, H. Sugihara

Abstract:

This paper discusses the performance of critical trajectory method (CTrj) for power system transient stability analysis under various loading settings and heavy fault condition. The method obtains Controlling Unstable Equilibrium Point (CUEP) which is essential for estimation of power system stability margins. The CUEP is computed by applying the CTrjto the boundary controlling unstable equilibrium point (BCU) method. The Proposed method computes a trajectory on the stability boundary that starts from the exit point and reaches CUEP under certain assumptions. The robustness and effectiveness of the method are demonstrated via six power system models and five loading conditions. As benchmark is used conventional simulation method whereas the performance is compared with and BCU Shadowing method.

Keywords: Power system, Transient stability, Critical trajectory method, Energy function method.

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Authors: Rabah Haoui

Abstract:

The aim of this work is to analyze a viscous flow in the axisymmetric nozzle taken into account the mesh size both in the free stream and into the boundary layer. The resolution of the Navier- Stokes equations is realized by using the finite volume method to determine the supersonic flow parameters at the exit of convergingdiverging nozzle. The numerical technique uses the Flux Vector Splitting method of Van Leer. Here, adequate time stepping parameter, along with CFL coefficient and mesh size level is selected to ensure numerical convergence. The effect of the boundary layer thickness is significant at the exit of the nozzle. The best solution is obtained with using a very fine grid, especially near the wall, where we have a strong variation of velocity, temperature and shear stress. This study enabled us to confirm that the determination of boundary layer thickness can be obtained only if the size of the mesh is lower than a certain value limits given by our calculations.

Keywords: Supersonic flow, viscous flow, finite volume, nozzle

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