• Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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• 2006.11a
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• pp.382-385
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• 2006

A high-order spectral/boundary-element method is newly adapted as an efficient numerical tool. In this method, the velocity potential is expressed as the sum of surface potential and body potential. Then, surface potential is solved fly using the high-order spectral method and body potential is solved fly using the high-order boundary element method. Through the combination of these two methods, the wave-making problems fly a submerged sphere moving with the large amplitude oscillation are solved in time-domain. With the example calculations, nonlinear effects on free-surface profiles and hydrodynamic forces are shown and discussed.

• The Journal of the Acoustical Society of Korea
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• v.18 no.1E
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• pp.37-43
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• 1999

This paper deals with a hybrid finite element method for wave scattering problems in infinite domains. Scattering of waves involving complex geometries, in conjunction with infinite domains is modeled by introducing a mathematical boundary within which a finite element representation is employed. On the mathematical boundary, the finite element representation is matched with a known analytical solution in the infinite domain in terms of fields and their derivatives. The derivative continuity is implemented by using a slope constraint. Drilling degrees of freedom at each node of the finite element model are introduced to make the numerical model more sensitive to the transverse component of the elastodynamic field. To verify the effects of drilling degrees freedom and slope constraints individually, reflection of normally incident P and SV waves on a traction free half spaces is considered. For the P-wave incidence, the results indicate that the use of slope constraint is more effective because it suppresses artificial reflection at the mathematical boundary. For the SV-wave case, the use of drilling degrees freedom is more effective by reducing numerical error at irregular frequencies.

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.791-812
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• 2011

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

• Journal of computational fluids engineering
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• v.4 no.1
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• pp.19-26
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• 1999

This paper describes a numerical method for predicting the incompressible unsteady laminar three-dimensional flows with free-surface. The Navier-Stokes equations governing the flows have been discretized by means of finite-difference approximations, and the resulting equations have been solved via the SIMPLE-C algorithm. The free-surface is defined by the motion of a set of marker particles and the interface behaviour was investigated by means of a "Lagrangian" technique. Using the GALA concept of Spalding, the conventional mass continuity equation is modified to form a volumetric or bulk-continuity equation. The use of this bulk-continuity relation allows the hydrodynamic variables to be computed over the entire flow domain including both liquid and gas regions. Thus, the free-surface boundary conditions are imposed implicitly and the problem formulation is greatly simplified. The numerical procedure is validated by comparing the predicted results of a periodic standing waves problems with analytic solutions. The results show that this numerical method produces accurate and physically realistic predictions of three-dimensional free-surface flows.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.819-837
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• 2008

A mixed type dual to the control problem in order to unify Wolfe and Mond-Weir type dual control problem is presented in various duality results are validated and the generalized invexity assumptions. It is pointed out that our results can be extended to the control problems with free boundary conditions. The duality results for nonlinear programming problems already existing in the literature are deduced as special cases of our results.

• 한국전산유체공학회:학술대회논문집
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• 2009.04a
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• pp.165-169
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• 2009

The effect of the Dirichlet boundary condition for the redistance equation of level set method on the solutionof sloshing problem is investigated by adopting four Dirichlet boundary conditions. For the solution of the incompressible Navier-Stokes equations, P1P1 four-step fractional finite element method is employed and a least-square finite element method is used for the solutions of the two hyperbolic type equations of level set method; advection and redistance equation. ALE (Arbitrary Lagrangian Eulerian) method is used to deal with a moving computational domain. It has been shown that the free surface motion in a sloshing tank is strongly dependent on the type of the Dirichlet boundary condition and the results of broken dam and sloshing problems using various Dirichlet boundary conditions are discussed and compared with the existing experimental results.

• The Bulletin of The Korean Astronomical Society
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• v.44 no.2
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• pp.69.2-69.2
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• 2019

To construct a coronal force-free magnetic field, we must impose the boundary normal current density (or three components of magnetic field) as well as the boundary normal field at the photosphere as boundary conditions. The only method that is known to implement these boundary conditions exactly is the method devised by Grad and Rubin (1958). However, the Grad-Rubin method and all its variations (including the fluxon method) suffer from convergence problems. The magnetofrictional method and its variations are more robust than the Grad-Rubin method in that they at least produce a certain solution irrespective of whether the global solution is compatible with the imposed boundary conditions. More than often, the influence of the boundary conditions does not reach beyond one or two grid planes next to the boundary. We have found that the 2D solenoidal gauge condition for vector potentials allows us to implement the required boundary conditions easily and effectively. The 2D solenoidal condition is translated into one scalar function. Thus, we need two scalar functions to describe the magnetic field. This description is quite similar to the Chandrasekhar-Kendall representation, but there is a significant difference between them. In the latter, the toroidal field has both Laplacian and divergence terms while in ours, it has only a 2D Laplacian term. The toroidal current density is also expressed by a 2D Laplacian. Thus, the implementation of boundary normal field and current are straightforward and their effect can permeate through the whole computational domain. In this paper, we will give detailed math involved in this formulation and discuss possible lateral and top boundary conditions and their meanings.

• Journal of Mechanical Science and Technology
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• v.17 no.3
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• pp.370-379
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• 2003

A study of free vibration of rectangular Mindlin plates is presented. The analysis is based on the Chebyshev pseudospectral method, which uses test functions that satisfy the boundary conditions as basis functions. The result shows that rapid convergence and accuracy as well as the conceptual simplicity are achieved when the pseudospectral method is applied to the solution of eigenvalue problems. Numerical examples of rectangular Mindlin plates with clamped and simply supported boundary conditions are provided for various aspect ratios and thickness-to length ratios.

• Journal of the Earthquake Engineering Society of Korea
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• v.9 no.2 s.42
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• pp.71-80
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• 2005

The solution to a liquid sloshing problem is challenge to the field of engineering. This is not only because the dynamic boundary condition at the free surface is nonlinear, but also because the position of the free surface varies with time in a manner not known a priori. Therefore, this nonlinear phenomenon, which is characterized by the oscillation of the unrestrained free surface of the fluid, is a difficult mathematical problem to solve numerically and analytically. In this study, three-dimensional boundary element method(BEM), which is based on the so-called an arbitrary Lagrangian-Eulerian(ALE) approach for the fluid flow problems with a free surface, was formulated to solve the behavior of the nonlinear free surface motion. An ALE-BEM has the advantage to track the free surface along any prescribed paths by using only one displacement variable, even for a three-dimensional problem. Also, some numerical examples were presented to demonstrate the validity and the applicability of the developed procedure.

• Ocean Systems Engineering
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• v.6 no.3
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• pp.245-264
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• 2016

The iterative boundary element method (IBEM) developed originally before for cavitating two-dimensional (2-D) and three-dimensional (3-D) hydrofoils moving under free surface is modified and applied to the case of 2-D (two-dimensional) airfoils and 3-D (three-dimensional) wings over water. The calculation of the steady-state flow characteristics of an inviscid, incompressible fluid past 2-D airfoils and 3-D wings above free water surface is of practical importance for air-assisted marine vehicles such as some racing boats including catamarans with hydrofoils and WIG (Wing-In-Ground) effect crafts. In the present paper, the effects of free surface both on 2-D airfoils and 3-D wings moving steadily over free water surface are investigated in detail. The iterative numerical method (IBEM) based on the Green's theorem allows separating the airfoil or wing problems and the free surface problem. Both the 2-D airfoil surface (or 3-D wing surface) and the free surface are modeled with constant strength dipole and constant strength source panels. While the kinematic boundary condition is applied on the airfoil surface or on the wing surface, the linearized kinematic-dynamic combined condition is applied on the free surface. The source strengths on the free surface are expressed in terms of perturbation potential by applying the linearized free surface conditions. No radiation condition is enforced for downstream boundary in 2-D airfoil and 3-D wing cases and transverse boundaries in only 3-D wing case. The method is first applied to 2-D NACA0004 airfoil with angle of attack of four degrees to validate the method. The effects of height of 2-D airfoil from free surface and Froude number on lift and drag coefficients are investigated. The method is also applied to NACA0015 airfoil for another validation with experiments in case of ground effect. The lift coefficient with different clearance values are compared with those of experiments. The numerical method is then applied to NACA0012 airfoil with the angle of attack of five degrees and the effects of Froude number and clearance on the lift and drag coefficients are discussed. The method is lastly applied to a rectangular 3-D wing and the effects of Froude number on wing performance have been investigated. The numerical results for wing moving under free surface have also been compared with those of the same wing moving above free surface. It has been found that the free surface can affect the wing performance significantly.