• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.623-640
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• 2007

The Lotka-McKendrick model which describes the evolution of a single population is developed from the well known Malthus model. In this paper, we introduce the Lotka-McKendrick model. We approximate the solution to the model using hp-discontinuous Galerkin finite element method. The numerical results show that the presented hp-discontinuous Galerkin method is very efficient in case that the solution has a sharp decay.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.10 no.4
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• pp.174-186
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• 1998

A Galerkin's finite element model incorporating infinite elements for modeling of radiation condition at infinity has been developed, which is based on an extended mild-slope equation. To illustrate the validity and applicability of the present model, the example analyses were carried out for a resonance problem in the rectangular harbor of Ippen and Goda (1963) and for wave transformations over circular shoals of Sharp (1968) and Chandrasekera and Cheung (1997). Comparisons with the results obtained by hydraulic experiments and hybrid element method showed that the present model gives very good results in spite of the rapidly varying topography. Numerical experiments were also performed for wave transformations over a circular concave well which may be an alternative to conventional wave barriers.

• The Journal of the Korea Contents Association
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• v.11 no.1
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• pp.404-410
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• 2011

For the simulation of one-dimensional unsteady flow, the Taylor-Galerkin finite element method was adopted to the discretization of the Saint Venant equation. The model was applied to the backwater problem in a single channel and the flood routing in dendritic channel networks. The numerical solutions were compared with previously published results of finite difference and finite element methods and good agreement was observed. The model solves the continuity and the momentum equations in a sequential manner and this leads to easy implementation. Since the final system of matrix is tri-diagonal with a few additional entry due to channel junctions, the tri-diagonal matrix solution algorithm can be used with minor modification. So it is fast and economical in terms of memory for storing matrices.

• Bulletin of the Society of Naval Architects of Korea
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• v.19 no.4
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• pp.19-29
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• 1982

Galerkin's finite element method is applied to a two-dimensional heat convection-diffusion problem arising in the hydrodynamic lubrication of thrust bearings used in naval vessels. A parabolized thermal energy equation for the lubricant, and thermal diffusion equations for both bearing pad and the collar are treated together, with proper juncture conditions on the interface boundaries. it has been known that a numerical instability arises when the classical Galerkin's method, which is equivalent to a centered difference approximation, is applied to a parabolic-type partial differential equation. Probably the simplest remedy for this instability is to use a one-sided finite difference formula for the first derivative term in the finite difference method. However, in the present coupled heat convection-diffusion problem in which the governing equation is parabolized in a subdomain(Lubricant), uniformly stable numerical solutions for a wide range of the Peclet number are obtained in the numerical test based on Galerkin's classical finite element method. In the present numerical convergence errors in several error norms are presented in the first model problem. Additional numerical results for a more realistic bearing lubrication problem are presented for a second numerical model.

• The Sea:JOURNAL OF THE KOREAN SOCIETY OF OCEANOGRAPHY
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• v.12 no.3
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• pp.133-141
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• 2007

A finite element model is developed for the extended Boussinesq equations that is capable of simulating the dynamics of long and short waves. Galerkin weighted residual method and the introduction of auxiliary variables for 3rd spatial derivative terms in the governing equations are used for the model development. The Adams-Bashforth-Moulton Predictor Corrector scheme is used as a time integration scheme for the extended Boussinesq finite element model so that the truncation error would not produce any non-physical dispersion or dissipation. This developed model is applied to the problems of solitary wave propagation. Predicted results is compared to available analytical solutions and laboratory measurements. A good agreement is observed.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.15 no.3
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• pp.159-166
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• 2003

In this study, we develop a finite element model to directly solve the Laplace equation while keeping the same computational efficiency as the models based on the extended mild-slope equation which has been widely used for calculation of wave transformation in shallow water. For this, the computational domain is discretized into finite elements with a single layer in the vertical direction. The velocity potential in the element is then expressed in terms of the potentials at the nodes located at water surface, and the Galerkin method is used to construct the numerical model. A common shape function is adopted in horizontal direction, and the cosine hyperbolic function in vertical direction, which describes the vertical behavior of progressive waves. The model was developed for vertical two-dimensional problems. In order to verify the developed model, it is applied to vertical two-dimensional problems of wave reflection and transmission. It is shown that the present finite element model is comparable to the models based on extended mild-slope equations in both computational efficiency and accuracy.

• Coupled systems mechanics
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• v.4 no.4
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• pp.365-383
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• 2015

Strip footing is an important type of shallow foundations and is commonly used beneath the walls. Analysis of shallow foundation involves the determination of stresses and deformations. Element free Galerkin method, one of the important mesh free methods, is used for the determination of stresses and deformations. Element free Galerkin method is an efficient and accurate method as compared to finite element method. The Element Free Galerkin method uses only a set of nodes and a description of model boundary is required to generate the discrete equation. Strip footing of width 2 m subjected to a loading intensity of 200 kPa is studied. The results obtained are agreeing with the values obtained using analytical solutions available in the literature. Parametric study is done and the effect of modulus of deformation, Poisson's ratio and scaling parameter on deformation and stresses are determined.

• Structural Engineering and Mechanics
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• v.61 no.6
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• pp.765-773
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• 2017

The quadratic B-spline finite element method yields mass and stiffness matrices which are half the size of matrices obtained by the conventional finite element method. We solve the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Rayleigh beam theory includes the rotary inertia effects in addition to the Euler-Bernoulli theory assumptions and presents a good mathematical model for rotating beams. Galerkin's approach is used to obtain the weak form which yields a system of symmetric matrices. Results obtained for the natural frequencies at different rotating speeds show an accurate match with the published results. A comparison with Euler-Bernoulli beam is done to decipher the variations in higher modes of the Rayleigh beam due to the slenderness ratio. The results are obtained for different values of non-uniform parameter ($\bar{n}$).

• Proceedings of the Korea Water Resources Association Conference
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• 2005.05b
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• pp.409-413
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• 2005

Even though the relative importance of length scale of flow system allow us to simplify three dimensional flow problem to one or two dimensional representation, many systems still require three dimensional analysis. The objective of this study is to develop an efficient and accurate finite element model for analyzing and predicting three dimensional flow features in natural rivers and to offend to model spreading of pollutants and transport of sediments in the future. Firstly, three dimensional Reynolds averaged Navier-Stokes equations with the hydrostatic pressure assumption in generalized curvilinear coordinates were combined with the kinematic free-surface condition. Secondly. to simulate realistic high Reynolds number flow, the model employed the Streamline Upwind/Petrov-Galerkin(SU/PG) scheme as a weighting function for the finite element method in conjunction with an appropriate turbulence model(Smagorinsky scheme for the horizontal plain and Mellor-Yamada scheme for the vertical direction). Several tests is performed for the purpose of validation and verification of the developed model. A simple rectangular channel, 5-shaped and U-shaped channel are used for tests and comparisons are made with RMA-10 model. Runs for each case is converged stably without a oscillation and calculated water-surface deformation, longitudinal and transversal velocities, and velocity vector fields are in good agreement with the results of RMA-10 model.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.419-431
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• 2023

In this paper, we propose an iterative method for a symmetric interior penalty Galerkin method for heterogeneous elliptic problems. The iterative method consists mainly of two parts based on a non-overlapping domain decomposition approach. One is an intermediate preconditioner constructed by understanding the properties of the discontinuous finite element functions and the other is a preconditioning related to the dual-primal finite element tearing and interconnecting (FETI-DP) methodology. Numerical results for the proposed method are presented, which demonstrate the performance of the iterative method in terms of various parameters associated with the elliptic model problem, the finite element discretization, and non-overlapping subdomain decomposition.