• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1035-1054
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• 2010

We consider a mixed type singularly perturbed one dimensional elliptic problem with discontinuous source term. The domain under consideration is partitioned into two subdomains. A convection-diffusion and a reaction-diffusion type equations are posed on the first and second subdomains respectively. Two hybrid difference schemes on Shishkin mesh are constructed and we prove that the schemes are almost second order convergence in the maximum norm independent of the diffusion parameter. Error bounds for the numerical solution and its numerical derivative are established. Numerical results are presented which support the theoretical results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.3
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• pp.201-215
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• 2018

In this paper we consider a class of singularly perturbed system of delay differential equations of convection diffusion type with integral boundary conditions. A finite difference scheme on an appropriate piecewise Shishkin type mesh is suggested to solve the problem. We prove that the method is of almost first order convergent. An error estimate is derived in the discrete maximum norm. Numerical experiments support our theoretical results.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.31 no.12
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• pp.1002-1008
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• 2007

Natural convection flows in a doubly-inclined cubical air-filled cavity are numerically simulated by a solution code(PowerCFD) using unstructured cell-centered method. For a physical realizability, the cavity has one pair of opposing isothermal faces at different temperatures, $T_h\;and\;T_c$, respectively, the remaining four faces having a linear variation from $T_c\;to\;T_h$. The paper redefines a new doubly-inclined orientation for the cubical-cavity benchmark problem. Special attention is paid to three-dimensional thermal characteristics in natural convection according to the new orientation at $Ra=4\times10^4$. Comparisons of the average Nusselt number at the cold face are made with benchmark solutions and experimental results found in the literature. It is found that the average Nusselt number at the cold face has a maximum value at the doubly-inclined angle ranging from $40^{\circ}\;to\; 45^{\circ}$ We also report the effect of new orientation on the type of temperature structure in a doubly-inclined cubical-cavity.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.8 no.2
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• pp.267-278
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• 1996

Steady, laminar, forced convection flow and heat transfer in the entrance region of an internally finned circular duct with a finite thermal conductivity has been analyzed numerically. The problem under investigation is a three-dimensional boundary layer problem, and is solved by employing a marching-type procedure which involves solution of a series of 2-dimensional elliptic problems in the cross-stream plane. Two types of inlet hydrodynamic conditions are considered : (a) uniform velocity flow and (b) fully developed flow. From the above inlet conditions, the effects of the fin height(h), fin number(N) and conductivity ratio($k_r$) on the flow and thermal characteristics are investigated. The numerical results show that the height and number of fins, and ratio of the solid to fluid thermal conductivity have pronounced effect on the solution. Considering pressure drop, optimized dimensionless fin height is 0.4.

• 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.

• Proceedings of the Korean Society for Technology of Plasticity Conference
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• 2006.05a
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• pp.99-104
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• 2006

A semi-Lagrangian finite element scheme with objective time stepping algorithm for solving viscoelastic flow problem is presented. The convection terms in the momentum and constitutive equations are treated using a quasi-monotone semi-Lagrangian scheme, in which characteristic feet on a regular grid are traced backwards over a single time-step. Concerned with the generalized midpoint rule type of algorithms formulated to exactly preserve objectivity, we use the geometric transformation such as pull-back, push-forward operation. The method is applied to the 4:1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions.

• Proceedings of the KSME Conference
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• 2004.04a
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• pp.531-536
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• 2004

In-arm type hydropeumatic suspension unit(ISU) is an equipment of armed tracked vehicle to absorb impact load and vibration from the irregular ground. During the operation of ISU, main piston moves forward and backward and oil flowing through damper transmits the external impact load to floating piston. Heat is generated in ISU by the oil pressure drop through the damper orifice and the friction between cylinder wall and two pistons. On the other hand, internal heat dissipatis outside via heat convection. Occurrence of high temperature can deteriorate durability of major components and basic function of ISU. And, it can cause fatal problem in the ISU life time and the sealing performance of piston rings. As well, the spring constant change of nitrogen gas that is caused by the temperature rise exerts the negative effect to the vehicle stability. Therefore, in this paper, we analyze the heat transfer analysis of the entire ISU unit, by finite element method, with the outside flow velocities 8m/s and 10m/s.

• Journal of Mechanical Science and Technology
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• v.16 no.10
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• pp.1264-1275
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• 2002

This paper is concerned with the investigation on the stability and convergence characteristics of the Crank-Nicolson-Galerkin scheme that is widely being employed for the numerical approximation of parabolic-type partial differential equations. Here, we present the theoretical analysis on its consistency and convergence, and we carry out the numerical experiments to examine the effect of the time-step size △t on the h- and P-convergence rates for various mesh sizes h and approximation orders P. We observed that the optimal convergence rates are achieved only when △t, h and P are chosen such that the total error is not affected by the oscillation behavior. In such case, △t is in linear relation with DOF, and furthermore its size depends on the singularity intensity of problems.

• 한국전산유체공학회:학술대회논문집
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• 2011.05a
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• pp.86-87
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• 2011

In this paper, a conjugate heat transfer around cylinder with heat generation was investigated. Both forced convection and conduction was considered in the present finite element simulation. A finite element formulation based on SIMPLE type algorithm was adopted for the solution of the incompressible Navier-Stokes equations. We compared the finite element solution with that of Ansys fluent 12.0, in which finite volume method was employed for spatial discretization. It was found that the finite element method gave more accurate solution than Ansys fluent 12.0. Further, it was found that the maximum temperature inside cylinder is positioned at the rear side due to the flow separation.