• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.411-423
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• 2010

In this paper, a numerical method for singular perturbation problems arising in chemical reactor theory for general singularly perturbed two point boundary value problems with boundary layer at one end(left or right) of the underlying interval is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.20 no.2
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• pp.212-218
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• 2008

A possible source of resonant problems in a harbor is long waves generated by incident wave groups. The analytical solutions of the governing equations of second-order long waves derived using a multiple-scale perturbation method consist of the locked and free long waves. The locked long waves propagate at some group velocity, whereas the free long waves propagate at the shallow-water speed. To study the resonance of free long waves, a trapezoidally varying topography is employed. With certain combinations of incident angle, water depth, and ambient current velocity, free long waves can be trapped and resonated.

• Structural Engineering and Mechanics
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• v.20 no.3
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• pp.293-312
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• 2005

In this paper, free vibration of rotating beam with random properties is studied. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus and density are modeled as random fields and the rotational speed as a random variable. To study uncertainty, stochastic finite element method based on second order perturbation method is applied. To discretize random fields, the three methods of midpoint, interpolation and local average are applied and compared. The effects of rotational speed, setting angle, random property variances, discretization scheme, number of elements, correlation of random fields, correlation function form and correlation length on "Coefficient of Variation" (C.O.V.) of first mode eigenvalue are investigated completely. To determine the significant random properties on the variation of first mode eigenvalue the sensitivity analysis is performed. The results are studied for both Timoshenko and Bernoulli-Euler rotating beam. It is shown that the C.O.V. of first mode eigenvalue of Timoshenko and Bernoulli-Euler rotating beams are approximately identical. Also, compared to uncorrelated random fields, the correlated case has larger C.O.V. value. Another important result is, where correlation length is small, the convergence rate is lower and more number of elements are necessary for convergence of final response.

• Journal of Advanced Research in Ocean Engineering
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• v.2 no.1
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• pp.12-21
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• 2016

In this paper, the $2^{nd}$ order hydrodynamic force effect of heaving submerged circular cylinder is considered, with the linear potential theory. Boundary value problem (BVP) is expanded up to the $2^{nd}$ order by using of the perturbation method and the $2^{nd}$ order velocity potential is calculated by means of integral equation technique using the classical Green's function expressed in cylindrical coordinates. The method of solving BVP is based on eigenfunction expansions. With different cylinder heights and heaving frequencies, graphical results are presented. As a result of the study, the cause of oscillatory force pattern is analyzed with the occurrence of negative added mass when a top of the cylinder gets closer to the free surface.

• Ocean Systems Engineering
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• v.1 no.4
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• pp.317-336
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• 2011

A time domain finite element based method is employed to analyze wave radiation by multiple cylinders. The nonlinear free surface and body surface boundary conditions are satisfied based on the perturbation method up to the second order. The first- and second-order velocity potential problems at each time step are solved through a finite element method (FEM). The matrix equation of the FEM is solved through an iteration and the initial solution is obtained from the result at the previous time step. The three-dimensional (3D) mesh required is generated based on a two-dimensional (2D) hybrid mesh on a horizontal plane and its extension in the vertical direction. The hybrid mesh is generated by combining an unstructured grid away from cylinders and two structured grids near the cylinder and the artificial boundary, respectively. The fluid velocity on the free surface and the cylinder surface are calculated by using a differential method. Results for various configurations including two-cylinder and four-cylinder cases are provided to show the mutual influence due to cylinders on the first and second waves and forces.

• Bulletin of the Korean Chemical Society
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• v.19 no.9
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• pp.985-993
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• 1998

The second order effective valence shell Hamiltonian ($H^v$), which is based on quasidegencrate many-body perturbation theory, is applied to determining the potential energy surfaces and the dipole moment functions of the various valence states of $NH_2$. The $H^v$ calculated values are found to be in good agreement with those of other ab initio calculations or experiments. It signifies the fact that the $H^v$ is a good ab initio method to describe the energies and properties of various valence states with a same chemical accuracy. Furthermore, it is shown that the lowest (second order for energy and the first order for property) order $H^v$ method is very accurate for small molecules like $NH_2$ and the matrix elements of Hv which are computed only once are all we need to accurately describe all the valence states simultaneously.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.419-435
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• 2021

This paper deals with numerical treatment of singularly perturbed differential equations involving small delays. The highest order derivative in the equation is multiplied by a perturbation parameter 𝜀 taking arbitrary values in the interval (0, 1]. For small 𝜀, the problem involves a boundary layer of width O(𝜀), where the solution changes by a finite value, while its derivative grows unboundedly as 𝜀 tends to zero. The considered problem contains delay on the convection and reaction terms. The terms with the delays are approximated using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. Inducing exponential fitting factor for the term containing the singular perturbation parameter and using central finite difference for the derivative terms, numerical scheme is developed. The stability and uniform convergence of difference schemes are studied. Using a priori estimates we show the convergence of the scheme in maximum norm. The scheme converges with second order of convergence for the case 𝜀 = O(N^-1) and for the case 𝜀 ≪ N^-1, the scheme converge uniformly with first order of convergence, where N is number of mesh intervals in the domain discretization. We compare the accuracy of the developed scheme with the results in the literature. It is found that the proposed scheme gives accurate result than the one in the literatures.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.32 no.4
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• pp.432-446
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• 1996

A stochastic Hamilton variational principle(SHVP) is formulated for dynamic problems of linear continuum. The SHVP allows incorporation of probabilistic distributions into the finite element analysis. The formulation is simplified by transformation of correlated random variables to a set of uncorrelated random variables through a standard eigenproblem. A procedure based on the Fourier analysis and synthesis is presented for eliminating secularities from the perturbation approach. In addition to, a method to analyse stochastic design sensitivity for structural dynamics is present. A combination of the adjoint variable approach and the second order perturbation method is used in the finite element codes. An alternative form of the constraint functional that holds for all times is introduced to consider the time response of dynamic sensitivity. The algorithms developed can readily be adapted to existing deterministic finite element codes. The numerical results for stochastic analysis by proceeding approach of cantilever, 2D-frame and 3D-frame illustrates in this paper.

• 제어로봇시스템학회:학술대회논문집
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• 1996.10b
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• pp.225-228
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• 1996

This paper concerns a SCARA type robot with the second arm flexible. Its equations of motion are derived by the Lagrangian mechanics. For controller design, the perturbation approach is taken to separate the original equations of motion into linear equations describing small perturbed motions and nonlinear equations describing purely rigid motion of the robot. To effect the desired payload motion, open loop control inputs are first determined based on the inverse dynamics of the latter. Next, in order to reduce the positional error during maneuver, an active vibration suppression is done. To this end, a feedback control is designed for robustness against disturbance on the basis of the linear equations and the LQR theory modified with a prescribed degree of stability. The numerical simulations results show the satisfactory control performance.

• Journal of Ocean Engineering and Technology
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• v.33 no.3
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• pp.229-235
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• 2019

In this study, a two-dimensional oscillating foil with forward speed in a propagating wave flow field was considered. The time-mean power to maintain the heaving and pitching motions of the foil was analyzed using the perturbation theory in an ideal fluid. The power, which was a non-linear quantity of the second-order, was expressed in terms of the quadratic transfer functions related to the mutual product of the heaving and pitching motions and incoming vertical flow. The effects of the pivot point and phase difference among the disturbances were studied. The negative power, which indicates energy extraction from the fluid, is shown as an example calculation.