• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.39-48
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• 1995

The ADI method was introduced by Peaceman and Rachford [6] in 1955, to solve the discretized boundary value problems for elliptic and parabolic PDEs. The finite difference discretization of the model elliptic problem $$ (1) -\Delta u = f, \Omega = [0, 1] \times [0, 1] $$ $$ u = 0 on \delta \Omega $$ with 5-point centered finite difference discretization, with n +2 mesh-points in the x - direction and m + 2 points in the y direction, leads to the solution of a linear system of equations of the form $$ (2) Au = b $$ where A is a matrix of dimension $N = n \times m$. Without loss of generality and for the sake of simplicity, we will assume for the remainder of this paper that m = n, so that $N = n^2$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.139-156
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• 2014

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

• Journal of the Society of Naval Architects of Korea
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• v.28 no.1
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• pp.19-26
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• 1991

The complete, fully-elliptic Reynolds-averaged Navier-Stokes equations have been solved using a two-layer model, in the $\kappa-\varepsilon$ turbulence model, for the axisymmetric body. Numerically generated boundary-fitted coordinate system and the finite analytic methods are used to solve the governing equations. Calculations are started after the middle body with given inlet conditions. The velocities and the turbulent quantities at the inlet section are specified by solving the boundary layer equations or by standard flat-plate boundary profiles. The effects of the inlet conditions on the solution are investigated.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.7 no.2
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• pp.95-111
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• 2003

BDM mixed methods are obtained for a good approximation of velocity for flow equations. In this paper, we study an implementation issue of solving the algebraic system arising from the BDM mixed finite elements. First we discuss post-processing based on the use of Lagrange multipliers to enforce interelement continuity. Furthermore, we establish an equivalence between given mixed methods and projection finite element methods developed by Chen. Finally, we present the implementation of the first order BDM on rectangular grids and show it is as simple as solving the pressure equation.

• Transactions of the Korean Society of Mechanical Engineers
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• v.14 no.2
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• pp.470-483
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• 1990

Turbulent jet in a crossflow, issuing from a row of holes on a convex surface of 90 .deg. bend duct, is predicted by a 3-dimensional numerical method. The Cartesian coordinate system in adopted in upstream and downstream tangents and the cylindrical polar coordinate system in curved region. The Reynolds stresses and heat fluxes are obtained from a standard k-e model in the core region and van Driest model in the vicinity of the wall. The governing equations are discretized by a finite volume method and solutions are obtained by a locally elliptic calculation procedure. Pressure and convective terms are treated by SIMPLE algorithm and hybrid scheme respectively. A vortex initially induced by the injected jet has been built up due to the interaction with the secondary flow caused by pressure gradient and centrifugal force. The vortex structure has a strong influence on the wall cooling effectiveness. Another vortex like horseshoe is formed in the vicinity of the injection hole and its strength is getting weak as it moves downward.

• Journal of the Korean Institute of Navigation
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• v.19 no.2
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• pp.13-22
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• 1995

This paper presents the numerical analysis results of the viscous flow for a full ship model. The mass and momentum conservation equations are used for governing equations, and the flow field is discretized by the Finite-Volume Method for the numerical calculation. An algebraic grid and elliptic grid generation techniques are adopted for generation of the body-fitted coordinates system, which is suitable to ship's hull forms. Time-marching procedure is used to solve the three-dimensional unsteady problem, where the convection terms are approximated by the QUICK scheme and the 2nd-order central differencing scheme is used for other spatial derivatives. A Sub-Grid Scale turbulence model is used to approximate the turbulence, and the wall function is used at the body surface. Pressure and velocity fields are calculated by the simultaneous iteration method. Numerical calculations were accomplished for the Crude Oil Tanker(DWT 95,000tons, Cb=0.805) model. Calculation results are compared to the experimental results and show good agreements.

• Transactions of the Korean Society of Automotive Engineers
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• v.2 no.1
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• pp.9-25
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• 1994

In the present study, the Reynolds-averaged Navier-Stokes equations, together with the equations of the $k-{\varepsilon}$ model of turbulence, were solved numerically in a general body-fitted coordinate system for three-dimensional turbulent flows around the six basic shapes of the magnetically levitated train(MAGLEV). The numerical computations were conducted on the MAGLEV model configurations to provide information on shapes of this type very near the elevated track at a constant Reynolds number of $1.48{\times}10^{6}$ based on the body length. The coordinate system was generated by numerically solving a set of Poisson equations. The convective transport equations were discretized using the finite-analytic scheme which employed analytic solutions of the locally-linearized equations. A time marching algorithm was employed to enable future extensions to be made to handle unsteady and fully-elliptic problems. The pressure-velocity coupling was treated with the SIMPLER-algorithm. Of particular interests were wall effect by the elevated track on the aerodynamic forces and flow characteristics of the six models calculated. The results indicated that the half-circle configuration with extended sides and with smooth curvature of sides was desirable because of the low aerodynamic forces and pitching moment. And it was found that the separation bubble was occured at wake region in near the elevated track.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.377-385
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• 2002

In this paper, we concentrate on the uniquencess of the positive solution for the general elliptic system $\Delta$u+u($g_1$(u)-$g_2$(v))=0 $\Delta$u+u($h_1$(u)-$h_2$(v))=0 in$R_{+}$ $\times$ $\Omega$, $u\mid\partial\Omega = u\mid\partial\Omega = 0$. This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

• Journal of Theoretical and Applied Mechanics
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• v.2 no.1
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• pp.51-70
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• 1996

This study is generalization of the study of Miles[Physica 11D, 1984, pp.309-323]on the resonant motion of a spherical pendulum, which is equivalent to a particle on a spherical container subject to a linear, horizontal excitation. This study covers an arbitrary shape of container and a more general excitation (horizontal but elliptic motion). The averaging method is applied to reduce the governing equations to an autonomous system with cubic nonlinear terms, under the assumption of small amplitude of the container motion. It is shown that both the container shape and the excitation pattern affect the particle dynamics. Under the linear excitation, the anharmonic motion of the particle is possible only for a certain finite range of the parameter a controling the container shape. Stability of the particle's harmonic motion is also influenced by the excitation pattern; as the excitation trajectory becomes closer to a circle, the particle's motion has a stronger tendency to become stable and to follow the rotational direction of the excitation. Under a circular excitation, the motion is always stable and circular with the same rotational direction as the excitation. Analogy between the present model and that of the surface wave inside a circular is studied quantitatively.

• Journal of KSNVE
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• v.3 no.3
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• pp.271-278
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• 1993

Acoustic characteristics of silencer system are affected by various geometric parameters such as cross sectional geometry, size of chamber, and location of inlet-outlet port. It is impossible to obtain exact solutions of the equations of acoustic wave propagation except few simple cases. So, we resort to numerical techniques to analyze performance of acoustic system. In this work, finite element formulation has been obtained to predict transmission loss of an arbitrary 3-dimensional muffler in the presence of mean flow of low mach number. The effect of the degree of the ellipticity of expansion chambers on the transmission loss has been studied using the resulting finite element equation.