• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-276
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• 2019

For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn-Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn-Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.21 no.1
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• pp.1-16
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• 2017

The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. The purpose of this paper is to characterize higher order operator splitting schemes and propose several higher order methods. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.

• Journal of Ocean Engineering and Technology
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• v.10 no.3
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• pp.96-104
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• 1996

Hamilton's principle is used to derive Euler-Lagrange equations for free surface flow problems of incompressible ideal fluid. The velocity field is chosen to satisfy the continuity equation a priori. This approach results in a hierarchial set of governing equations consist of two evolution equations with respect to two canonical variables and corresponding boundary value problems. The free surface elevation and the Lagrange's multiplier are the canonical variables in Hamilton's sense. This Lagrange's multiplier is a velocity potential defined on the free surface. Energy is conserved as a consequence of the Hamiltonian structure. These equations can be applied to waves in water of finite depth including generalization of Hamilton's equations given by Miles and Salmon.

• Proceedings of the Korea Concrete Institute Conference
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• 1994.04a
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• pp.91-96
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• 1994

The growth and propagation of microcracks existed in concrete cause failure of concrete. This is called "damage". The concepts of two principles, equivalent strain principle and equivalent energy principle, are reviewed and compared in the case of uniaxial compressior to concrete. The damage evolution law and constitutive equation are derived by using the Helmholz free energy and the dissipation potential by means of the thermodynamic principles.rinciples.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.14 no.4
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• pp.751-759
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• 1994

Concrete contains numerous microcracks initially. The growth and propagation of microcracks cause failure of concrete. These processings are termed as "damage". The concepts of the continuum damage mechanics are presented and the damage evolution law and constitutive equation are derived by using the Helmholz free energy and the dissipation potential by means of the thermodynamic principles. The constitutive equation includes the effects of elasticity, damage and plasticity of concrete. The proposed model successfully predicts the nonlinear behavior of concrete subject to monotonic uniaxial and biaxial loadings.

• Journal of the Korean Society for Marine Environment & Energy
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• v.12 no.3
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• pp.197-201
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• 2009

As one of the promising model on the multiphase fluid mixtures, the Lattice-Boltzmann Method(LBM) is being developed to simulate flows containing two immisible components which are different mass values. The equilibrium function in the LBM can have a nonideal gas model for the equation of state and use the interfacial energy for the phase separation effect. An example on the phase separation has been carried out through the time evolution. The LBM based on the statistic mechanics is appropriate to solve very complicated flow problems and this model gives comparative merits rather than the continuum mechanics model.

• Computational Structural Engineering
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• v.9 no.1
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• pp.65-76
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• 1996

The initiation and growth of microcracks or microvoids inside concrete results in the progressive degradation of concrete. This damage processing along processing along with plastic deformation is main cause of nonlinear behavior of concrete. In this study, a continuum damage model of concrete is developed for the analysis of the nonlinear behavior of concrete due to damage and elasto-plastic deformation. Anisotropic damage tensor is used to describe the anisotropy of concrete and hypothesis of equivalent elastic energy is used to define the effective elastic tensor. The damage model including the damage evolution law and constitutive equation is derived with damage variable and damage surface which is defined by damage energy release rate by using the Helmholtz free energy and dissipation potential based on the thermodynamic principles. By adopting a typical plasticity model of concrete, plasticity of concrete is included to this model. Afinite element analysis program implemented with this model was developed and finite element analysis was performed for the analyses of concrete subjected to uniaxial and biaxial loadings. Comparison of the results of analysis with those of experiments and other models shows that the model successfully predicts the nonlinear behavior of concrete.