• Journal for History of Mathematics
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• v.15 no.3
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• pp.59-64
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• 2002

This paper investigates a history of Fourier Series for the heat equation and how deeply it is related to modern famous three equations, Navier-Stokes equations in fluid dynamics, drift-diffusion equations in semiconductor, and Black-Scholes equation in finance. We also propose improved models for the heat equation with finite propagation speeds.

• Geophysics and Geophysical Exploration
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• v.15 no.2
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• pp.57-65
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• 2012

Various numerical methods in simulation of seismic wave propagation have been developed. Recently an innovative numerical method called as the Spectral Element Method (SEM) has been developed and used in wave propagation in 3-D elastic media. The SEM that easily implements the free surface of topography combines the flexibility of a finite element method with the accuracy of a spectral method. It is generally used a weak formulation of the equation of motion which are solved on a mesh of hexahedral elements based on the Gauss-Lobatto-Legendre integration rule. Variational formulations of velocity-stress motion are newly modified in order to implement ADE-PML (Auxiliary Differential Equation of Perfectly Matched Layer) in wave propagation in 3-D elastic media, because a general weak formulation has a difficulty in adapting CFS (Complex Frequency Shifted) PML (Perfectly Matched Layer). SEM of Velocity-Stress motion having ADE-PML that is very efficient in absorbing waves reflected from finite boundary is verified with simulation of 1-D and 3-D wave propagation.

• The Journal of the Acoustical Society of Korea
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• v.7 no.4
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• pp.110-116
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• 1988

The thermal front over the shallow coastal seas of Korea during the winter season provides very unique acoustic media such that wave equation is easily separable and the solutions turn out to be very simple and well known. In steady of using the WKB method to solve the radial equation the mode technique have been applied to obtain the solution. The radial propagation is rather weakly influenced by the presence of the thermal front that causes the horizontal variations of the sound speed. The physical description of the sound propagation is also presented in terms of ray tracing.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2010.04a
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• pp.459-464
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• 2010

The dynamic propagation of an interface crack between two dissimilar functionally graded layers under anti-plane shear is analyzed using the integral transform method. The properties of the functionally graded layers vary continuously along the thickness. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a dual integral equation, which is then expressed to a Fredholm integral equation of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented. Followings are helpful to increase of the resistance of the interface crack propagation of FGM: a) increase of the gradient of material properties; b) increase of the material properties from the interface to the upper and lower free surface; c) increase of the thickness of FGM layer. The DERR increases or decreases with increase of the crack moving velocity.

• Earthquakes and Structures
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• v.26 no.2
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• pp.163-174
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• 2024

Dynamic equilibrium equations for finite element analysis were derived for the free field one-dimensional shear wave propagation through the horizontally layered soil deposits with the elastic half-space. We expressed Rayleigh's viscous damping consisting of mass and stiffness proportional terms. We considered two cases where damping matrices are defined in the total and relative displacement fields. Two forms of equilibrium equations are presented; one in terms of total motions and the other in terms of relative motions. To evaluate the performance of new equilibrium equations, we conducted two sets of site response analyses and directly compared them with the exact closed-form frequency domain solution. Results show that the base shear force as earthquake load represents the simpler form of equilibrium equation to be used for the finite element method. Conventional finite element procedure using base acceleration as earthquake load predicts exact solution reasonably well even in soil deposits with unrealistically high damping.

• Ocean and Polar Research
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• v.29 no.1
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• pp.9-31
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• 2007

Most seismic sea waves in the East Sea originate from earthquakes occurring near the Japanese west coast. While the waves propagate in the East Sea, they are deformed by refraction, diffraction and scattering. Though the Boussinesq equation is most applicable for such wave phenomena, it was not used in numerical modelling of seismic sea waves in the East Sea. To examine characteristics of seismic sea waves in the East Sea, numerical models based on the Boussinesq equation are established and used to simulate recent tsunamis. By considering Ursell parameter and Kajiura parameter, it is proved that Boussinesq equation is a proper equation for seismic sea waves in the East Sea. Two models based on the Boussinesq equation and linear wave equation are executed with the same initial conditions and grid size ($1min{\times}1min$), and the results are compared in various respects. The Boussinesq equation model produced better results than the linear model in respect to wave propagation and concentration of wave energy. It is also certified that the Boussinesq equation model can be used for operational purpose if it is optimized. Another Boussinesq equation model whose grid size is $40sec{\times}30sec$ is set up to simulate the 1983 and 1993 tsunamis. As the result of simulation, new propagation charts of 2 seismic sea waves focused on the Korean east coast are proposed. Even though the 1983 and 1993 tsunamis started at different areas, the propagation paths near the Korean east coast are similar and they can be distinguished into 4 paths. Among these, total energy and propagating time of the waves passing over North Korea Plateau(NKP) and South Korea Plateau(SKP) determine wave height at the Korean east coast. In case of the 1993 tsunami, the wave passing over NKP has more energy than the wave over SKP. In case of the 1983 tsunami, the huge energy of the wave passing over SKP brought about great maximum wave heights at Mukho and Imwon. The Boussinesq equation model established in this study is more useful for simulation of seismic sea waves near the Korean east coast than it is the Japanese coast. To improve understanding of seismic sea waves in shallow water, a coastal area model based on the Boussinesq equation is also required.

• Proceedings of the Korean Society of Propulsion Engineers Conference
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• 2005.11a
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• pp.422-425
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• 2005

In an transversely isotropic composite laminates, the velocities, the particle directions and the amplitudes of reflected and transmitted waves were obtained using the equation of motion, the constitutive equation, and the displacement equation expressed by wave number and frequency Eigenvalue problem involving a velocity was solved by Snell's law. Finally, the results were confirmed by T300 Carbon fiber/5208 Epoxy materials. This approach could be applied to the detection of flaws in a transversely isotropic composite laminates by the water immersion C-scan procedure.

• Journal of the Korean Society of Propulsion Engineers
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• v.10 no.2
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• pp.62-69
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• 2006

In transversely isotropic composite laminates, the velocities, the particle directions and the amplitudes of reflected and transmitted waves were obtained using the equation of motion, the constitutive equation, and the displacement equation expressed by wave number and frequency. Eigenvalue problem involving a velocity was solved by Snell's law. Finally, the results were confirmed by 7300 Carbon fiber/5208 Epoxy materials. This approach could be applied to the detection of flaws in transversely isotropic composite laminates by the water immersion C-scan procedure.

• Proceedings of the Earthquake Engineering Society of Korea Conference
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• 2000.10a
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• pp.81-88
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• 2000

The elastic wave equation is solved using the finite-difference method in 3D space to simulate the seismic wave propagation. It is based on the velocity-stress formulation of the equation of motion on a staggered grid. The nonreflecting boundary conditions are used to attenuate the wave field close to the numerical boundary. To satisfy the stress-free conditions at the free-surface boundary, a new formulation combining the zero-stress formalism with the vacuum one is applied. The effective media parameters are employed to satisfy the traction continuity condition across the media interface. With use of the moment-tensor components, the wide range of source mechanism parameters can be specified. The numerical experiments are carried out in order to test the applicability and accuracy of this scheme and to understand the fundamental features of the wave propagation under the generalized elastic media structure. Computational results show that the scheme is sufficiently accurate for modeling wave propagation in 3D elastic media and generates all the possible phases appropriately in under the given heterogeneous velocity structure. Also the characteristics of the ground motion in an sedimentary basin such as the amplification, trapping, and focusing of the elastic wave energy are well represented. These results demonstrate the use of this simulation method will be helpful for modeling the ground motion of seismological and engineering purpose like earthquake hazard assessment, seismic design, city planning, and etc..

• Structural Engineering and Mechanics
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• v.66 no.4
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• pp.495-504
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• 2018

The current study is dedicated to study the thermal effects of wave propagation in beams, reinforced by carbon nanotubes (CNT). Beams, made up of carbon nanotube reinforced composite (CNTRC) are the future materials in various high tech industries. Herein a Winkler elastic foundation is assumed in order to make the model more realistic. Mostly, CNTs are pervaded in cross section of beam, in various models. So, it is tried to use four of the most profitable reconstructions. The homogenization of elastic and thermal properties such as density, Yong's module, Poisson's ratio and shear module of CNTRC beam, had been done by the demotic rule of mixture to homogenize, which gives appropriate traits in such settlements. To make this investigation, a perfect one, various shear deformation theories had been utilized to show the applicability of this theories, in contrast to their theoretical face. The reigning equation had been derived by extended Hamilton principle and the culminant equation solved analytically by scattering relations for propagation of wave in solid bodies. Results had been verified by preceding studies. It is anticipated that current results can be applicable in future studies.