Search results for: System of Fredholm and Volterra integral equations.

Authors: Seul-Kee Kim, Chi-Seung Lee, Myung-Hyun Kim, Jae-Myung Lee

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In this study, the hydrogen transport phenomenon was numerically evaluated by using hydrogen-enhanced localized plasticity (HELP) mechanisms. Two dominant governing equations, namely, the hydrogen transport model and the elasto-plastic model, were introduced. In addition, the implicitly formulated equations of the governing equations were implemented into ABAQUS UMAT user-defined subroutines. The simulation results were compared to published results to validate the proposed method.

Keywords: Hydrogen-enhanced localized plasticity (HELP), Hydrogen embrittlement, Hydrogen transport analysis, ABAQUS UMAT, Finite element method (FEM).

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Authors: Kalyan Chatterjee

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The optimal design of PI controller for Automatic Generation Control in two area is presented in this paper. The concept of Dual mode control is applied in the PI controller, such that the proportional mode is made active when the rate of change of the error is sufficiently larger than a specified limit otherwise switched to the integral mode. A digital simulation is used in conjunction with the Hooke-Jeeve’s optimization technique to determine the optimum parameters (individual gain of proportional and integral controller) of the PI controller. Integrated Square of the Error (ISE), Integrated Time multiplied by Absolute Error(ITAE) , and Integrated Absolute Error(IAE) performance indices are considered to measure the appropriateness of the designed controller.  The proposed controller are tested for a two area single nonreheat thermal system considering the practical aspect of the problem such as Deadband and Generation Rate Constraint(GRC). Simulation results show that  dual mode with optimized values of the gains improved the control performance than the commonly used Variable Structure .

Keywords: Load Frequency Control, Area Control Error(ACE), Dual Mode PI Controller, Performance Index

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Authors: Markus G. Ortner, Christian Magele, Klaus Krischan

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Knowledge about the magnetic quantities in a magnetic circuit is always of great interest. On the one hand, this information is needed for the simulation of a transformer. On the other hand, parameter studies are more reliable, if the magnetic quantities are derived from a well established model. One possibility to model the 3-phase transformer is by using a magnetic equivalent circuit (MEC). Though this is a well known system, it is often not an easy task to set up such a model for a large number of lumped elements which additionally includes the nonlinear characteristic of the magnetic material. Here we show the setup of a solver for a MEC and the results of the calculation in comparison to measurements taken. The equations of the MEC are based on a rearranged system of the nodal analysis. Thus it is possible to achieve a minimum number of equations, and a clear and simple structure. Hence, it is uncomplicated in its handling and it supports the iteration process. Additional helpful tasks are implemented within the solver to enhance the performance. The electric circuit is described by an electric equivalent circuit (EEC). Our results for the 3-phase transformer demonstrate the computational efficiency of the solver, and show the benefit of the application of a MEC.

Keywords: Inrush current, magnetic equivalent circuit, nonlinear behavior, transformer.

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Authors: Consalva J. Msigwa, Beda J. Kundy, Bakari M. M. Mwinyiwiwa

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In this paper the reference current for Voltage Source Converter (VSC) of the Shunt Active Power Filter (SAPF) is generated using Synchronous Reference Frame method, incorporating the PI controller with anti-windup scheme. The proposed method improves the harmonic filtering by compensating the winding up phenomenon caused by the integral term of the PI controller. Using Reference Frame Transformation, the current is transformed from om a - b - c stationery frame to rotating 0 - d - q frame. Using the PI controller, the current in the 0 - d - q frame is controlled to get the desired reference signal. A controller with integral action combined with an actuator that becomes saturated can give some undesirable effects. If the control error is so large that the integrator saturates the actuator, the feedback path becomes ineffective because the actuator will remain saturated even if the process output changes. The integrator being an unstable system may then integrate to a very large value, the phenomenon known as integrator windup. Implementing the integrator anti-windup circuit turns off the integrator action when the actuator saturates, hence improving the performance of the SAPF and dynamically compensating harmonics in the power network. In this paper the system performance is examined with Shunt Active Power Filter simulation model.

Keywords: Phase Locked Loop (PLL), Voltage SourceConverter (VSC), Shunt Active Power Filter (SAPF), PI, Pulse WidthModulation (PWM).

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Authors: Udomsak Thongkrajay, Padej Pao-La-Or, Thanatchai Kulworawanichpong

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Installation of power compensation equipment in some cases places additional buses into the system. Therefore, a total number of power flow equations and voltage unknowns increase due to additional locations of installed devices. In this circumstance, power flow calculation is more complicated. It may result in a computational convergence problem. This paper presents a power flow calculation by using Newton-Raphson iterative method together with the proposed load transfer technique. This concept is to eliminate additional buses by transferring installed loads at the new buses to existing two adjacent buses. Thus, the total number of power flow equations is not changed. The overall computational speed is expectedly shorter than that of solving the problem without applying the load transfer technique. A 15-bus test system is employed for test to evaluate the effectiveness of the proposed load transfer technique. As a result, the total number of iteration required and execution time is significantly reduced.

Keywords: Load transfer technique, Newton-Raphson power flow, ill-condition

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Authors: Jingbo Yang, Hong Li, Yang Liu, Siriguleng He

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In this paper, a nonconforming mixed finite element method is studied for semilinear pseudo-hyperbolic partial integrodifferential equations. By use of the interpolation technique instead of the generalized elliptic projection, the optimal error estimates of the corresponding unknown function are given.

Keywords: Pseudo-hyperbolic partial integro-differential equations, Nonconforming mixed element method, Semilinear, Error estimates.

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Authors: Phongsak Phakamach

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A DC servomotor position control system using a Fuzzy Logic Sliding mode Model Following Control or FLSMFC approach is presented. The FLSMFC structure consists of an integrator and variable structure system. The integral control is introduced into it in order to eliminated steady state error due to step and ramp command inputs and improve control precision, while the fuzzy control would maintain the insensitivity to parameter variation and disturbances. The FLSMFC strategy is implemented and applied to a position control of a DC servomotor drives. Experimental results indicated that FLSMFC system performance with respect to the sensitivity to parameter variations is greatly reduced. Also, excellent control effects and avoids the chattering phenomenon.

Keywords: Sliding mode model following control, fuzzy logic, DC servomotor.

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Authors: Pierre Abou-Haila, Richard Hall, Mark Dawes

Abstract:

Instead of representing individual cognition only, population cognition is represented using artificial neural networks whilst maintaining individuality. This population network trains continuously, simulating adaptation. An implementation of two coexisting populations is compared to the Lotka-Volterra model of predator-prey interaction. Applications include multi-agent systems such as artificial life or computer games.

Keywords: Collective unconsciousness, neural networks, adaptation, predator-prey simulation.

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Authors: S. Panda, S. K. Tomar, R. Prasad, C. Ardil

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Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaining the time moments and/or Markov parameters of original system. By this method the reduced order model guarantees stability if the original high order model is stable. In the second method Particle Swarm Optimization (PSO) is employed to reduce the higher order model. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical examples.

Keywords: Model Order Reduction, Markov Parameters, Routh Approximation, Particle Swarm Optimization, Integral Squared Error, Steady State Stability.

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Authors: Minghui Wang

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The real representation of the quaternionic matrix is definited and studied. The relations between the positive (semi)define quaternionic matrix and its real representation matrix are presented. By means of the real representation, the relation between the positive (semi)definite solutions of quaternionic matrix equations and those of corresponding real matrix equations is established.

Keywords: Matrix equation, Quaternionic matrix, Real representation, positive (semi)definite solutions.

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

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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: Thomas Kampke

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Equations with differentials relating to the inverse of an unknown function rather than to the unknown function itself are solved exactly for some special cases and numerically for the general case. Invertibility combined with differentiability over connected domains forces solutions always to be monotone. Numerical function inversion is key to all solution algorithms which either are of a forward type or a fixed point type considering whole approximate solution functions in each iteration. The given considerations are restricted to ordinary differential equations with inverted functions (ODEIs) of first order. Forward type computations, if applicable, admit consistency of order one and, under an additional accuracy condition, convergence of order one.

Keywords: Euler method, fixed points, golden section, multi-step procedures, Runge Kutta methods.

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Authors: Mujde Turkkan, Nurkan Yagiz

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The vibrations, caused by the irregularities of the road surface, are to be suppressed via suspension systems. In this paper, sliding mode control for a half bus model with air suspension system is presented. The bus is modelled as five degrees of freedom (DoF) system. The mathematical model of the half bus is developed using Lagrange Equations. For time domain analysis, the bus model is assumed to travel at certain speed over the bump road. The numerical results of the analysis indicate that the sliding mode controllers can be effectively used to suppress the vibrations and to improve the ride comfort of the busses.

Keywords: Sliding mode control, bus model, air suspension.

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Authors: Gowrishankar Kasilingam

Abstract:

This paper proposes a swarm intelligence method that yields optimal Proportional-Integral-Derivative (PID) Controller parameters of a power system stabilizer (PSS) in a single machine infinite bus system. The proposed method utilizes the Particle Swarm Optimization (PSO) algorithm approach to generate the optimal tuning parameters. The paper is modeled in the MATLAB Simulink Environment to analyze the performance of a synchronous machine under several load conditions. At the same operating point, the PID-PSS parameters are also tuned by Ziegler-Nichols method. The dynamic performance of proposed controller is compared with the conventional Ziegler-Nichols method of PID tuning controller to demonstrate its advantage. The analysis reveals the effectiveness of the proposed PSO based PID controller.

Keywords: Particle Swarm Optimization, PID Controller, Power System Stabilizer.

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Authors: Saeed Sarabadan, Kamal Rashedi

Abstract:

This article presents a numerical method to find the heat flux in an inhomogeneous inverse heat conduction problem with linear boundary conditions and an extra specification at the terminal. The method is based upon applying the satisfier function along with the Ritz-Galerkin technique to reduce the approximate solution of the inverse problem to the solution of a system of algebraic equations. The instability of the problem is resolved by taking advantage of the Landweber’s iterations as an admissible regularization strategy. In computations, we find the stable and low-cost results which demonstrate the efficiency of the technique.

Keywords: Inverse problem, parabolic equations, heat equation, Ritz-Galerkin method, Landweber iterations.

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Authors: A. Javed, K. Djidjeli, J. T. Xing

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The concept of adaptive shape parameters (ASP) has been presented for solution of incompressible Navier Strokes equations using mesh-free local Radial Basis Functions (RBF). The aim is to avoid ill-conditioning of coefficient matrices of RBF weights and inaccuracies in RBF interpolation resulting from non-optimized shape of basis functions for the cases where data points (or nodes) are not distributed uniformly throughout the domain. Unlike conventional approaches which assume globally similar values of RBF shape parameters, the presented ASP technique suggests that shape parameter be calculated exclusively for each data point (or node) based on the distribution of data points within its own influence domain. This will ensure interpolation accuracy while still maintaining well conditioned system of equations for RBF weights. Performance and accuracy of ASP technique has been tested by evaluating derivatives and laplacian of a known function using RBF in Finite difference mode (RBFFD), with and without the use of adaptivity in shape parameters. Application of adaptive shape parameters (ASP) for solution of incompressible Navier Strokes equations has been presented by solving lid driven cavity flow problem on mesh-free domain using RBF-FD. The results have been compared for fixed and adaptive shape parameters. Improved accuracy has been achieved with the use of ASP in RBF-FD especially at regions where larger gradients of field variables exist.

Keywords: CFD, Meshless Particle Method, Radial Basis Functions, Shape Parameters

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Authors: G.V.K Murthy, S. Sivanagaraju, S. Satyanarayana, B. Hanumantha Rao

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Distributed Generation (DG) systems are considered an integral part in future distribution system planning. Appropriate size and location of distributed generation plays a significant role in minimizing power losses in distribution systems. Among the benefits of distributed generation is the reduction in active power losses, which can improve the system performance, reliability and power quality. In this paper, Artificial Bee Colony (ABC) algorithm is proposed to determine the optimal DG-unit size and location by loss sensitivity index in order to minimize the real power loss, total harmonic distortion (THD) and voltage sag index improvement. Simulation study is conducted on 69-bus radial test system to verify the efficacy of the proposed method.

Keywords: Distributed generation, artificial bee colony method, loss reduction, radial distribution network.

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Authors: Taher Lotfi , Parisa Bakhtiari , Katayoun Mahdiani , Mehdi Salimi

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In this paper, verified extension of the Ostrowski method which calculates the enclosure solutions of a given nonlinear equation is introduced. Also, error analysis and convergence will be discussed. Some implemented examples with INTLAB are also included to illustrate the validity and applicability of the scheme.

Keywords: Iinterval analysis, nonlinear equations, Ostrowski method.

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Authors: Shield B. Lin, Ziraguen O. Williams

Abstract:

In a task to assist NASA in analyzing the dynamic forces caused by operational countermeasures of an astronaut’s exercise platform impacting the spacecraft, an active proportional-integral-derivative controller commanding a linear actuator is proposed in a vibration isolation system to regulate the movement of the exercise platform. Computer simulation shows promising results that most exciter forces can be reduced or even eliminated. This paper emphasizes on parameter uncertainties, variations and exciter force variations. Drift and variations of system parameters in the vibration isolation system for astronaut’s exercise platform are analyzed. An active controlled scheme is applied with the goals to reduce the platform displacement and to minimize the force being transmitted to the spacecraft structure. The controller must be robust enough to accommodate the wide variations of system parameters and exciter forces. Computer simulation for the vibration isolation system was performed via MATLAB/Simulink and Trick. The simulation results demonstrate the achievement of force reduction with small platform displacement under wide ranges of variations in system parameters. 

Keywords: control, counterweight, isolation, vibration

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Authors: Kourosh Parand, Zahra Delafkar, Fatemeh Baharifard

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The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy-s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.

Keywords: Tau method, semi-infinite, nonlinear ODE, rational Chebyshev, porous media.

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Authors: Manal H. Saleh

Abstract:

A numerical study has been carried out to investigate the heat transfer by natural convection of nanofluid taking Cu as nanoparticles and the water as based fluid in a three dimensional annulus enclosure filled with porous media (silica sand) between two horizontal concentric cylinders with 12 annular fins of 2.4mm thickness attached to the inner cylinder under steady state conditions. The governing equations which used are continuity, momentum and energy equations under an assumptions used Darcy law and Boussinesq-s approximation which are transformed to dimensionless equations. The finite difference approach is used to obtain all the computational results using the MATLAB-7. The parameters affected on the system are modified Rayleigh number (10 ≤Ra*≤ 1000), fin length Hf (3, 7 and 11mm), radius ratio Rr (0.293, 0.365 and 0.435) and the volume fraction(0 ≤ ¤ò ≤ 0 .35). It was found that the average Nusselt number depends on (Ra*, Hf, Rr and φ). The results show that, increasing of fin length decreases the heat transfer rate and for low values of Ra*, decreasing Rr cause to decrease Nu while for Ra* greater than 100, decreasing Rr cause to increase Nu and adding Cu nanoparticles with 0.35 volume fraction cause 27.9% enhancement in heat transfer. A correlation for Nu in terms of Ra*, Hf and φ, has been developed for inner hot cylinder.

Keywords: Annular fins, laminar free convection, nanofluid, porous media, three dimensions horizontal annulus.

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Authors: S. S. Mishra, D. K. Yadav

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This paper focuses on cost and profit analysis of single-server Markovian queuing system with two priority classes. In this paper, functions of total expected cost, revenue and profit of the system are constructed and subjected to optimization with respect to its service rates of lower and higher priority classes. A computing algorithm has been developed on the basis of fast converging numerical method to solve the system of non linear equations formed out of the mathematical analysis. A novel performance measure of cost and profit analysis in view of its economic interpretation for the system with priority classes is attempted to discuss in this paper. On the basis of computed tables observations are also drawn to enlighten the variational-effect of the model on the parameters involved therein.

Keywords: Cost and Profit, Computing, Expected Revenue, Priority classes

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Authors: Xiguang Li

Abstract:

In this paper , by using fixed point theorem , upper and lower solution-s method and monotone iterative technique , we prove the existence of maximum and minimum solutions of differential equations with delay , which improved and generalize the result of related paper.

Keywords: Banach space, boundary value problem, differential equation, delay.

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Authors: S. Padma, Dr. R. Lakshmipathi, K. Ramash Kumar, P. Nandagopal

Abstract:

The power system network is becoming more complex nowadays and it is very difficult to maintain the stability of the system. Today-s enhancement of technology makes it possible to include new energy storage devices in the electric power system. In addition, with the aid of power electronic devices, it is possible to independently exchange active and reactive power flow with the utility grid. The main purpose of this paper proposes a Proportional – Integral (PI) control based 48 – pulse Inverter based Static Synchronous Series Compensator (SSSC) with and without Superconducting Magnetic Energy Storage (SMES) used for enhancing the transient stability and regulating power flow in automatic mode. Using a test power system through the dynamic simulation in Matlab/Simulink platform validates the performance of the proposed SSSC with and without SMES system.

Keywords: Flexible AC transmission system (FACTS), PIControl, Superconducting Magnetic Energy Storage (SMES), Static Synchronous Series Compensator (SSSC).

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Authors: Ze-Jun Hu, Ting-Zhu Huang, Ning-Bo Tan

Abstract:

To solve saddle point systems efficiently, several preconditioners have been published. There are many methods for constructing preconditioners for linear systems from saddle point problems, for instance, the relaxed dimensional factorization (RDF) preconditioner and the augmented Lagrangian (AL) preconditioner are used for both steady and unsteady Navier-Stokes equations. In this paper we compare the RDF preconditioner with the modified AL (MAL) preconditioner to show which is more effective to solve Navier-Stokes equations. Numerical experiments indicate that the MAL preconditioner is more efficient and robust, especially, for moderate viscosities and stretched grids in steady problems. For unsteady cases, the convergence rate of the RDF preconditioner is slightly faster than the MAL perconditioner in some circumstances, but the parameter of the RDF preconditioner is more sensitive than the MAL preconditioner. Moreover the convergence rate of the MAL preconditioner is still quite acceptable. Therefore we conclude that the MAL preconditioner is more competitive than the RDF preconditioner. These experiments are implemented with IFISS package. 

Keywords: Navier-Stokes equations, Krylov subspace method, preconditioner, dimensional splitting, augmented Lagrangian preconditioner.

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Authors: G.Ashwini, A.T.Eswara

Abstract:

This paper examines the forced convection flow of incompressible, electrically conducting viscous fluid past a sharp wedge in the presence of heat generation or absorption with an applied magnetic field. The system of partial differential equations governing Falkner - Skan wedge flow and heat transfer is first transformed into a system of ordinary differential equations using similarity transformations which is later solved using an implicit finite - difference scheme, along with quasilinearization technique. Numerical computations are performed for air (Pr = 0.7) and displayed graphically to illustrate the influence of pertinent physical parameters on local skin friction and heat transfer coefficients and, also on, velocity and temperature fields. It is observed that the magnetic field increases both the coefficients of skin friction and heat transfer. The effect of heat generation or absorption is found to be very significant on heat transfer, but its effect on the skin friction is negligible. Indeed, the occurrence of overshoot is noticed in the temperature profiles during heat generation process, causing the reversal in the direction of heat transfer.

Keywords: Heat generation / absorption, MHD Falkner- Skan flow, skin friction and heat transfer

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Authors: Mohammed A. Hajeeh

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This paper examines the behavior of a system, which upon failure is either replaced with certain probability p or imperfectly repaired with probability q. The system is analyzed using Kolmogorov's forward equations method; the analytical expression for the steady state availability is derived as an indicator of the system’s performance. It is found that the analysis becomes more complex as the number of imperfect repairs increases. It is also observed that the availability increases as the number of states and replacement probability increases. Using such an approach in more complex configurations and in dynamic systems is cumbersome; therefore, it is advisable to resort to simulation or heuristics. In this paper, an example is provided for demonstration.

Keywords: Repairable models, imperfect, availability, exponential distribution.

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Authors: A. Elsayed Ahmed, A. Hafez, A. N. Ouda, H. Eldin Hussein Ahmed, H. Mohamed Abd-Elkader

Abstract:

Unmanned aircraft systems (UAS) are playing increasingly prominent roles in defense programs and defense strategies around the world. Technology advancements have enabled the development of it to do many excellent jobs as reconnaissance, surveillance, battle fighters, and communications relays. Simulating a small unmanned aerial vehicle (SUAV) dynamics and analyzing its behavior at the preflight stage is too important and more efficient. The first step in the UAV design is the mathematical modeling of the nonlinear equations of motion. . In this paper, a survey with a standard method to obtain the full non-linear equations of motion is utilized, and then the linearization of the equations according to a steady state flight condition (trimming) is derived. This modeling technique is applied to an Ultrastick-25e fixed wing UAV to obtain the valued linear longitudinal and lateral models. At the end the model is checked by matching between the behavior of the states of the nonlinear UAV and the resulted linear model with doublet at the control surfaces.

Keywords: Equations of motion, linearization, modeling, nonlinear model, UAV.

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Authors: Ali Essam El Shazly

Abstract:

The social logic of 'Sequina' slum area in Alexandria details the integral measure of space syntax at the room-level of twenty-building samples. The essence of spatial structure integrates the central 'visitor' domain with the 'living' frontage of the 'children' zone against the segregated privacy of the opposite 'parent' depth. Meanwhile, the multifunctioning of shallow rooms optimizes the integral 'visitor' structure through graph and visibility dimensions in contrast to the 'inhabitant' structure of graph-tails out of sight. Common theme of the layout integrity increases in compensation to the decrease of room visibility. Despite the 'pheno-type' of collective integration, the individual layouts observe 'geno-type' structure of spatial diversity per room adjoins. In this regard, the layout integrity alternates the cross-correlation of the 'kitchen & living' rooms with the 'inhabitant & visitor' domains of 'motherhood' dynamic structure. Moreover, the added 'grandparent' restructures the integral measure to become the deepest space, but opens to the 'living' of 'household' integrity. Some isomorphic layouts change the integral structure just through the 'balcony' extension of access, visual or ignored 'ringiness' of space syntax. However, the most integrated or segregated layouts invert the 'geno-type' into a shallow 'inhabitant' centrality versus the remote 'visitor' structure. Overview of the multivariate social logic of spatial integrity could never clarify without the micro-data analysis.

Keywords: Alexandria, Sequina slum, spatial integration, space syntax.

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Authors: O. Miraliyari, M.M. Najafizadeh, A.R. Rahmani, A. Momeni Hezaveh

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This paper presents the buckling analysis of short and long functionally graded cylindrical shells under thermal and mechanical loads. The shell properties are assumed to vary continuously from the inner surface to the outer surface of the shell. The equilibrium and stability equations are derived using the total potential energy equations, Euler equations and first order shear deformation theory assumptions. The resulting equations are solved for simply supported boundary conditions. The critical temperature and pressure loads are calculated for both short and long cylindrical shells. Comparison studies show the effects of functionally graded index, loading type and shell geometry on critical buckling loads of short and long functionally graded cylindrical shells.

Keywords: Buckling, Functionally graded materials, Short and long cylindrical shell, Thermal and mechanical loads.

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