• Nuclear Engineering and Technology
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• v.53 no.5
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• pp.1391-1402
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• 2021

The aim of this work is to prepare a neutron noise calculator based on the second order of average current nodal expansion method (ACNEM). Generally, nodal methods have the ability to fulfill the neutronic analysis with adequate precision using coarse meshes as large as a fuel assembly size. But, for the zeroth order of ACNEM, the accuracy of neutronic simulations may not be sufficient when coarse meshes are employed in the reactor core modeling. In this work, the capability of second order ACNEM is extended for solving the neutron diffusion equation in the frequency domain using coarse meshes. For this purpose, two problems are modeled and checked including a slab reactor and 2D BIBLIS PWR. For validating of results, a semi-analytical solution is utilized for 1D test case, and for 2D problem, the results of both forward and adjoint neutron noise calculations are exploited. Numerical results indicate that by increasing the order of method, the errors of frequency dependent coarse mesh solutions are considerably decreased in comparison to the reference. Accordingly, the accuracy of second order ACNEM can be acceptable for the neutron noise calculations by using coarse meshes in the nuclear reactor core.

• Structural Engineering and Mechanics
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• v.58 no.6
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• pp.949-966
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• 2016

The second order analysis taking place due to non-linear behavior of the structures under the mechanical and geometric factors through implementing exact and approximate methods is an indispensible issue in the analysis of such structures. Among the exact methods is the slope-deflection method that due to its simplicity and efficiency of its relationships has always been in consideration. By solving the differential equations of the modified slope-deflection method in which the effect of axial compressive force is considered, the stiffness matrix including trigonometric entries would be obtained. The complexity of computations with trigonometric functions causes replacement with their Maclaurin expansion. In most cases only the first two terms of this expansion are used but to obtain more accurate results, more elements are needed. In this paper, the effect of utilizing higher order terms of Maclaurin expansion on reducing the number of required elements and attaining more rapid convergence with less error is investigated for the Bernoulli beam with various boundary conditions. The results indicate that when using only one element along the beam length, utilizing higher order terms in Maclaurin expansion would reduce the relative error in determining the critical buckling load and kinematic parameters in the second order analysis.

• Journal of the Korean Statistical Society
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• v.22 no.2
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• pp.191-200
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• 1993

An asymptotic estimation problem of the unknown mean is studied under a general loss function. The proof of this result is based on the asymptotic expansion of the risk function. Also conditions for second order admissibility and minimaxity of a class of estimators depending only on the sample mean are established.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.889-900
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• 2005

We consider the second order mock theta functions defined by McIntosh and define generalized functions. We give integral representation and multibasic expansion of these functions. We also show that they are $F_q$-functions.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.141-152
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• 2007

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.

• Transactions of the Korean Society of Machine Tool Engineers
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• v.15 no.3
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• pp.8-16
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• 2006

This paper discusses design sensitivity analysis and its application to a structural dynamics modification. Eigenvalue derivatives are determined with respect to the element parameters, which include intrinsic property parameters such as Young's modulus, density of the material, diameter of a beam element, thickness of a plate element, and shape parameters. Derivatives of stiffness and mass matrices are directly calculated by derivatives of element matrices. The first and the second order derivatives of the eigenvalues are then mathematically derived from a dynamic equation of motion of FEM model. The calculation of the second order eigenvalue derivative requires the sensitivity of its corresponding eigenvector, which are developed by Nelson's direct approach. The modified eigenvalue of the structure is then evaluated by the Taylor series expansion with the first and the second derivatives of eigenvalue. Numerical examples for simple beam and plate are presented. First, eigenvalues of the structural system are numerically calculated. Second, the sensitivities of eigenvalues are then evaluated with respect to the element intrinsic parameters. The most effective parameter is determined by comparing sensitivities. Finally, we predict the modified eigenvalue by Taylor series expansion with the derivatives of eigenvalue for single parameter or multi parameters. The examples illustrate the effectiveness of the eigenvalue sensitivity analysis for the optimization of the structures.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.337-346
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• 2001

A linearization owing to Oseen originally is performed to study the recirculating Navier-Stokes flows at high Reynolds numbers. The procedure is generalized to produce higher order asymptotic expansion for the flow velocity. We call this the Oseen-type expansion of the given flow. As a concrete example, the velocity of a steady Navier-Stockes flow due to a swimming flexible sheet in two-dimensional infinite strip domain is calculated by an asymptotic expansion technic with two-parameters, the Reynolds number R and the perturbation parameter $\varepsilon$ first and then R secondly. The asymptotic result is up to second order in $\varepsilon$.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.277-302
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• 2017

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter ${\varepsilon}$ multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.

• Structural Engineering and Mechanics
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• v.44 no.4
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• pp.521-538
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• 2012

In this study, the transverse vibrations of an axially moving flexible beams resting on multiple supports are investigated. The time-dependent velocity is assumed to vary harmonically about a constant mean velocity. Simple-simple, fixed-fixed, simple-simple-simple and fixed-simple-fixed boundary conditions are considered. The equation of motion becomes independent from geometry and material properties and boundary conditions, since equation is expressed in terms of dimensionless quantities. Then the equation is obtained by assuming small flexural rigidity. For this case, the fourth order spatial derivative multiplies a small parameter; the mathematical model converts to a boundary layer type of problem. Perturbation techniques (The Method of Multiple Scales and The Method of Matched Asymptotic Expansions) are applied to the equation of motion to obtain approximate analytical solutions. Outer expansion solution is obtained by using MMS (The Method of Multiple Scales) and it is observed that this solution does not satisfy the boundary conditions for moment and incline. In order to eliminate this problem, inner solutions are obtained by employing a second expansion near the both ends of the flexible beam. Then the outer and the inner expansion solutions are combined to obtain composite solution which approximately satisfying all the boundary conditions. Effects of axial speed and flexural rigidity on first and second natural frequency of system are investigated. And obtained results are compared with older studies.

• Honam Mathematical Journal
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• v.18 no.1
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• pp.59-77
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• 1996

In this paper, we study how the second coefficient in the power series expansion of functions in $P_n$(a; b, M), $F_n$(a; b, M) and $G_n$(a; b, M) influences certain properties like distortion, ${\gamma}-spiral$ radius and radius of starlikeness of these classes.