• 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.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.219-244
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• 2020

In this paper, we approximate the solution of the coupled nonlinear Schrödinger equations by using a fully discrete finite element scheme based on the standard Galerkin method in space and implicit midpoint discretization in time. The proposed scheme guarantees the conservation of the total mass and the energy. First, a priori error estimates for the fully discrete Galerkin method is derived. Second, the existence of the approximated solution is proved by virtue of the Brouwer fixed point theorem. Moreover, the uniqueness of the solution is shown. Finally, convergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.10 no.4
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• pp.141-150
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• 1990

Numerical experiments of sballow water equations are performed under various boundary conditions by finite element method to simulate the circulation in estuaries and coastal areas. Galerkin method is employed to discretize spatial domain, and for time integration, finite difference method (Crank-Nicolson scheme) is used. This method is tested in five problems, in which first four cases have analytic solutions. The computed values are well in agreement with the analytic solutions in four experiments and the result of the last 2-dimensional ease is resonable. Implicit and two step Lax-Wendroff schemes in time domain are compared, and the results when using four node bilinear and triangular elements are presented. Consequently it takes very long time for complex problems requiring many elements to integrate all the time steps using the implicit schemes. And the explicit scheme requires careful consideration in selecting the time step and the grid size to obtain the desired accuracy.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.8 s.179
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• pp.2042-2049
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• 2000

This paper is concerned with the comparison of forging characteristics between forward and backward processes, through the three-dimensional finite element simulation, for the aluminum powder forging of engine pistons. Starting from the theoretical formulation of velocity and temperature fields in the sintered preform during the process, we examine the comparative distributions of relative density, effective stress and temperature as well as the variations of total forging load and total volume reduction. Through the comparative results, we find out that the forward method provides better forging characteristics than the backward method.

• Proceedings of the Korea Water Resources Association Conference
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• 2005.05b
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• pp.414-418
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• 2005

A finite element model is developed for the numerical simulation of bed elevation change in a curved channel. The SU/PG (Streamline-Upwind/Petrov-Galerkin) method is used to solve 2D shallow water equations and the BG (Bubnov-Galerkin) method is used for the Exner equation. For the time derivative terms, the Crank-Nicolson scheme is used. The developed model is a decoupled model in a sense that the bed elevation does not change simultaneously with the flow during the computational time step. The total load formula with is used for the sediment transport model. The slip conditions are described along the lateral boundaries. The effects of gravity force due to geometry change and the secondary flows in a curved channel are considered in the model. For the verification, the model is applied to two laboratory experiments. The first is $140^{\circ}$ bended channel data at Delft Hydraulics Laboratory and the second is $140^{\circ}$ bended channel data at Laboratory of Fluid Mechanics of the Delft University of Technology. The finite element grid is constructed with linear quadrilateral elements. It is found that the computed results are in good agreement with measured data, showing a point bar at the inner bank and a pool at the outer bank.

• Proceedings of the KSME Conference
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• 2001.06e
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• pp.386-391
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• 2001

A finite element code based on P2P1 tetra element has been developed for the large eddy simulation (LES) of turbulent flows around a complex geometry. Fractional 4-step algorithm is employed to obtain time accurate solution since it is less expensive than the integrated formulation, in which the velocity and pressure fields are solved at the same time. Crank-Nicolson method is used for second order temporal discretization and Galerkin method is adopted for spatial discretization. For very high Reynolds number flows, which would require a formidable number of nodes to resolve the flow field, SUPG (Streamline Upwind Petrov-Galerkin) method is applied to the quadratic interpolation function for velocity variables, Noting that the calculation of intrinsic time scale is very complicated when using SUPG for quadratic tetra element of velocity variables, the present study uses a unique intrinsic time scale proposed by Codina et al. since it makes the present three-dimensional unstructured code much simpler in terms of implementing SUPG. In order to see the effect of numerical diffusion caused by using an upwind scheme (SUPG), those obtained from P2P1 Galerkin method and P2P1 Petrov-Galerkin approach are compared for the flow around a sphere at some Reynolds number. Smagorinsky model is adopted as subgrid scale models in the context of P2P1 finite element method. As a benchmark problem for code validation, turbulent flows around a sphere and a MIRA model have been studied at various Reynolds numbers.