• Structural Engineering and Mechanics
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• v.39 no.6
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• pp.751-765
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• 2011

The present paper deals with the identification of a concentrated damage in an elastic parabolic arch through the minimization of an objective function which measures the differences between numerical and experimental values of static displacements. The damage consists in a notch that reduces the height of the cross section at a given abscissa and therefore causes a variation in the flexural stiffness of the structure. The analytical values of static displacements due to applied loads are calculated by means of the principle of virtual work for both the undamaged and damaged arch. First, pseudo-experimental data are used to study the inverse problem and investigate whether a unique solution can occur or not. Various damage intensities are considered to assess the reliability of the identification procedure. Then, the identification procedure is applied to an experimental case, where displacements are measured on a prototype arch. The identified values of damage parameters, i.e., location and intensity, are compared to those obtained by means of a dynamic identification technique performed on the same structure.

• Journal of Mechanical Science and Technology
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• v.16 no.10
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• pp.1264-1275
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• 2002

This paper is concerned with the investigation on the stability and convergence characteristics of the Crank-Nicolson-Galerkin scheme that is widely being employed for the numerical approximation of parabolic-type partial differential equations. Here, we present the theoretical analysis on its consistency and convergence, and we carry out the numerical experiments to examine the effect of the time-step size △t on the h- and P-convergence rates for various mesh sizes h and approximation orders P. We observed that the optimal convergence rates are achieved only when △t, h and P are chosen such that the total error is not affected by the oscillation behavior. In such case, △t is in linear relation with DOF, and furthermore its size depends on the singularity intensity of problems.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.655-676
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• 2021

A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε-uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.

• 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.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.389-399
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• 1995

We consider the reaction-diffusion system of two-component model in one-dimensional space described by $$ (1) u_s = d_1 u_{xx} + f(u, \upsilon) \upsilon_t = d_2\upsilon_{xx} + \gammag(u, \upsilon) $$ where $d_1$ and $d_2$ are the diffusion rates of u and $\upsilon$, and $\gamma$ is the ration of reaction rates. It is interesting the case of that there are differences in the diffusion and reaction rates of u and $\upsilon$.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.631-638
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• 1996

The parabolic free boundary problem with Puschino dynamics is given by (see in [3]) $$ (1) { \upsilon_t = D\upsilon_{xx} - (c_1 + b)\upsilon + c_1 H(x - s(t)) for (x,t) \in \Omega^- \cup \Omega^+, { \upsilon_x(0,t) = 0 = \upsilon_x(1,t) for t > 0, { \upsilon(x,0) = \upsilon_0(x) for 0 \leq x \leq 1, { \tau\frac{dt}{ds} = C)\upsilon(s(t),t)) for t > 0, { s(0) = s_0, 0 < s_0 < 1, $$ where $\upsilon(x,t)$ and $\upsilon_x(x,t)$ are assumed continuous in $\Omega = (0,1) \times (0, \infty)$.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.713-723
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• 2003

We study a limit problem of reflected solutions of parabolic stochastic partial differential equations driven by martingale measures. The existence of solutions is found in an extension of the work with respect to white noise by Donati-Martin and Pardoux [4]. We show that if a certain sequence of driving martingale measures converges, the corresponding solutions also converge in the Skorohod topology.

• Structural Engineering and Mechanics
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• v.75 no.2
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• pp.157-174
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• 2020

In the present paper, a simple analytical model is developed based on a new refined parabolic shear deformation theory (RPSDT) for free vibration and buckling analysis of orthotropic rectangular plates with simply supported boundary conditions. The displacement field is simpler than those of other higher-order theories since it is modeled with only two unknowns and accounts for a parabolic distribution of the transverse shear stress through the plate thickness. The governing differential equations related to the present theory are obtained from the principle of virtual work, while the solution of the eigenvalue problem is achieved by assuming a Navier technique in the form of a double trigonometric series that satisfy the edge boundary conditions of the plate. Numerical results are presented and compared with previously published results for orthotropic rectangular plates in order to verify the precision of the proposed analytical model and to assess the impacts of several parameters such as the modulus ratio, the side-to-thickness ratio and the geometric ratio on natural frequencies and critical buckling loads. From these results, it can be concluded that the present computations are in excellent agreement with the other higher-order theories.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.185-201
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• 2014

In this paper, we construct a numerical method to solve singularly perturbed one-dimensional parabolic convection-diffusion problems. We use Euler method with uniform step size for temporal discretization and exponential-spline scheme on spatial uniform mesh of Shishkin type for full discretization. We show that the resulting method is uniformly convergent with respect to diffusion parameter. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. The obtained numerical results show that the method is efficient, stable and reliable for solving convection-diffusion problem accurately even involving diffusion parameter.

• 제어로봇시스템학회:학술대회논문집
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• 1994.10a
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• pp.319-323
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• 1994

In this paper, we discuss a design method of iterative learning control systems for parabolic linear distributed parameter systems(DPSs). First, we discuss some aspects of boundary control of the DPS, and then propose to employ the Karhunen-Loeve procedure to reduce the infinite dimensional problem to a low-order finite dimensional problem. An iterative learning control(ILC) for non-square transfer function matrix is introduced finally for the reduced order system.