• Journal of the Society of Naval Architects of Korea
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• v.29 no.4
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• pp.114-131
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• 1992

In the present work, a two-dimensional boundary-value problem for a large amplitude motion is treated as an initial-value problem by satisfying the exact body-boundary and nonlinear free-surface boundary conditions. The present nonlinear numerical scheme is similar to that described by Vinje and Brevig(1981) who utilized the Cauchy's theorem and assumed the periodicity in the horizontal coordinate. In the present thesis, however, the periodicity in the horizontal coordinate is not assumed. Thus the present method can treat more realistic problems, which allow radiating waves to infinities. In the present method of solution, the original infinite fluid domain, is divided into two subdomains ; ie the inner and outer subdomains which are a local nonlinear subdomain and the truncated infinite linear subdomain, respectively. By imposing an appropriate matching condition, the computation is carried out only in the inner domain which includes the body. Here we adopt the nonlinear scheme of Vinje & Brevig only in the inner domain and respresent the solution in the truncated infinite subdomains by distributing the time-dependent Green function on the matching boundaries. The matching condition is that the velocity potential and stream function are required to be continuous across the matching boundary. In the computations we used, if necessary, a regriding algorithm on the free surface which could give converged stable solutions successfully even for the breaking waves. In harmonic oscillation problem, each harmonic component and time-mean force are obtained by the Fourier transform of the computed forces in the time domain. The numerical calculations are made for the following problems. $\cdot$ Forced harmonic large-amplitude oscillation(${\omega}{\neq}0,\;U=0$) $\cdot$ Translation with a uniform speed(${\omega}=0,\;U{\neq}0$) The computed results are compared with available experimental data and other analytical results.

• Journal of the Society of Naval Architects of Korea
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• v.35 no.1
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• pp.32-39
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• 1998

A numerical analysis for large amplitude motions of submerged circular cylinder is presented. The method is based on potential theory and two-dimensional motions in regular harmonic waves are tented as an initial value problem. The fully nonlinear free surface boundary condition is assumed in an inner domain and this solution is matched along an assumed an assumed common boundary to a linear solution in outer domain. Calculations of the large amplitude motion of a submerged circular cylinder are directly simulated in time domain. It is shown that relative motion between the body and fluid particle gives a significant effect on the lift and drift motions.

• Bulletin of the Society of Naval Architects of Korea
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• v.24 no.2
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• pp.47-54
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• 1987

Linear and nonlinear hydrodynamic force, which acts on submerged circular and eilliptic cylinders in oscillations as well as in advancing motion, are investigated as an initial-boundary value problem using a numerical method, which makes use of the source distribution on the body surface and the spectral method for treating the free surface waves. In the numerical code developed here, the boundary condition at the body surface is linearized. Using the numerical code so attained, nonlinear effects for different forward speeds and of the large-amplitude motion are computed. One of the major findings is that, when the forward speed is large, the added mass has its minimum and the damping force change rapidly around the frequency corresponding to the speed-frequency parameter, $\tau$=0.25, Compared to the result of Grue's [10], who used linear theory to get abrupt changes in values of the added mass and the damping force at the frequency corresponding to $\tau$=0.25, the present study, which takes nonlinear effects into account, shows much smoother variations near the frequency.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2006.05a
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• pp.73-80
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• 2006

In order to examine the validity of an asymptotic solution for nonlinear interaction in asymmetric vibration modes of a perfect circular plate, we obtain the numerical solution. The motion of the plate is governed by nonlinear partial differential equation. The initial and boundary value problem is solved by using the finite difference method. The numerical solution is compared with the asymptotic solution. It is found that traveling waves relating clockwise and counterclockwise as well as standing wave are depicted by the numerical solution.

• Smart Structures and Systems
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• v.30 no.3
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• pp.239-243
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• 2022

We investigate the influence of nonlinear viscoelastic damping on the response of a cantilever sensor covered by piezoelectric layers in a symmetric or asymmetric configuration. We formulate an initial-boundary-value problem which consistently incorporates both geometric and material nonlinearities including the effect of viscoelastic damping which cannot be ignored for micro- and nano-mechanical sensor operation in a vacuum environment. We employ an asymptotic multiple-scales methodology to yield the system nonlinear frequency response near its primary resonance and employ a model-based estimation procedure to deduce the system damping backone curve from controlled experiments in vacuum. We discuss the effect of nonlinear damping on sensor applications for scanning probe microscopy.

• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.679-689
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• 2022

In this paper, we establish sufficient conditions for the existence and uniqueness of solution for a class of initial boundary value problems with Dirichlet condition in regard to a category of fractional-order partial differential equations. The results are established by a method based on the theorem of Lax Milligram.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1003-1031
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• 2020

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$\frac{{\partial}u}{{\partial}t}-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)={\lambda}u^k{\displaystyle\smashmargin{2}{\int\nolimits_{\Omega}}}u^sdx-{\mu}u^l{\mid}{\nabla}u{\mid}^q$$ in a bounded domain Ω ⊂ ℝ^N, where p > 1, the parameters k, s, l, q, λ > 0 and µ ≥ 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on |∇u|. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.161-193
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• 2024

In this paper, we consider the asymptotic behavior of solutions for the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with hereditary memory and a very large class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. We first prove the existence and uniqueness of weak solutions to the initial boundary value problem for the above-mentioned model. Next, we investigate the existence of a uniform attractor of this problem, where the time-dependent forcing term h ∈ L²_b(ℝ; H^-1(ℝ^N)) is the only translation bounded instead of translation compact. Finally, we prove the regularity of the uniform attractor A, i.e., A is a bounded subset of H²(ℝ^N) × H¹(ℝ^N) × L²_µ(ℝ⁺, H²(ℝ^N)). The results in this paper will extend and improve some previously obtained results, which have not been studied before in the case of non-autonomous, exponential growth nonlinearity and contain memory kernels.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.37-56
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• 2013

This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $$\{u_t=(\mid(u^{m_1})_x{\mid}^{{p_1}^{-1}}(u^{m_1})_x)_x+u^{l_{11}}{{\int_0}^a}v^{l_{12}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T),\\{v_t=(\mid(v^{m_2})_x{\mid}^{{p_2}^{-1}}(v^{m_2})_x)_x+v^{l_{22}}{{\int_0}^a}u^{l_{21}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T)}$$ with nonlinear boundary conditions $u_x{\mid}{_{x=0}}=0$, $u_x{\mid}{_{x=a}}=u^{q_{11}}u^{q_{12}}{\mid}{_{x=a}}$, $v_x{\mid}{_{x=0}}=0$, $v_x|{_{x=a}}=u^{q21}v^{q22}|{_{x=a}}$ and the initial data ($u_0$, $v_0$), where $m_1$, $m_2{\geq}1$, $p_1$, $p_2$ > 1, $l_{11}$, $l_{12}$, $l_{21}$, $l_{22}$, $q_{11}$, $q_{12}$, $q_{21}$, $q_{22}$ > 0. Under appropriate hypotheses, the authors establish local theory of the solutions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.

• Journal of Ocean Engineering and Technology
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• v.34 no.6
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• pp.406-418
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• 2020

In this study, a nonlinear wave simulation code is developed using a higher-order spectral (HOS) method. The HOS method is very efficient because it can determine the solution of the boundary value problem using fast Fourier transform (FFT) without matrix operation. Based on the HOS order, the vertical velocity of the free surface boundary was estimated and applied to the nonlinear free surface boundary condition. Time integration was carried out using the fourth order Runge-Kutta method, which is known to be stable for nonlinear free-surface problems. Numerical stability against the aliasing effect was guaranteed by using the zero-padding method. In addition to simulating the initial wave field distribution, a nonlinear adjusted region for wave generation and a damping region for wave absorption were introduced for wave generation simulation. To validate the developed simulation code, the adjusted simulation was carried out and its results were compared to the eighth order Stokes theory. Long-time simulations were carried out on the irregular wave field distribution, and nonlinear wave propagation characteristics were observed from the results of the simulations. Nonlinear adjusted and damping regions were introduced to implement a numerical wave tank that successfully generated nonlinear regular waves. According to the variation in the mean wave steepness, irregular wave simulations were carried out in the numerical wave tank. The simulation results indicated an increase in the nonlinear interaction between the wave components, which was numerically verified as the mean wave steepness. The results of this study demonstrate that the HOS method is an accurate and efficient method for predicting the nonlinear interaction between waves, which increases with wave steepness.