• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.311-327
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• 2016

In this paper, we propose an error embedded Runge-Kutta method to improve the traditional embedded Runge-Kutta method. The proposed scheme can be applied into most explicit embedded Runge-Kutta methods. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. These solution and error are obtained by solving an initial value problem whose solution has the information of the error at each integration step. The constructed algorithm controls both the error and the time step size simultaneously and possesses a good performance in the computational cost compared to the original method. For the assessment of the effectiveness, the van der Pol equation and another one having a difficulty for the global error control are numerically solved. Finally, a two-body Kepler problem is also used to assess the efficiency of the proposed algorithm.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.238-266
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• 2013

The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schr$\ddot{o}$dinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve approximation such as the one resulting from the use of a splitting formula. In its standard form the IP method such as the SS method is used in conjunction with the classical 4th order Runge-Kutta (RK) scheme. However it appears to be relevant to look for RK scheme of higher order so as to improve the accuracy of the IP method. In this paper we investigate 5th order Embedded Runge-Kutta schemes suited to be used in conjunction with the IP method and designed to deliver a local error estimation for adaptive step size control.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.43 no.2
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• pp.187-193
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• 2023

This paper presents the buckling behavior of a pile considering its self-weight. The differential equation and boundary conditions governing the buckling of partially embedded piles in nonhomogeneous soils are derived. The buckling load and mode shape of the pile are numerically computed by the Runge-Kutta method combined with the Regula-Falsi algorithm. The obtained numerical solutions for bucking loads agree well with the results available from the literature. Numerical examples are given to analyze the buckling load and mode shape of the piles as affected by the self-weight, embedment ratio, slenderness ratio and boundary condition of the pile as well as the aspect ratio and rigidity ratio of the subgrade reaction. It is found that the self-weight of the pile leads to the reduction of the buckling load, indicating that neglecting the effect of self-weight may overestimate the buckling load of partially embedded piles.

• Journal of the Korean Geosynthetics Society
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• v.18 no.3
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• pp.69-77
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• 2019

This paper presents a new analytical model for estimating the free vibration of tapered friction piles. The governing differential equation for the free vibration of statically axially-loaded piles embedded in non-homogeneous soil is derived. The equation is numerically integrated by the Runge-Kutta method, and then the eigenvalue of natural frequency is determined by the Regula-Falsi scheme. For a cylindrical non-tapered pile, the computed natural frequencies compare well with the available data from literature. Numerical examples are presented to investigate the effects of the tapering, the skin friction resistance, the end condition of the pile, the vertical compressive loading, and the soil non-homogeneity on the natural frequency and mode shape of tapered friction piles.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.61 no.8
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• pp.1123-1129
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• 2012

This paper presents the fast steady state analysis using time-stepping finite element method for a high power three-phase transformer. The high power transformer spends huge computational cost of the time-stepping finite element method. It is because that the high power transformer requires a lot of time to reach steady state by its large inductance component. In order to reduce computational cost, in this paper, the adaptive time-step control algorithm combined with the embedded 2nd 4th singly diagonally implicit Runge-Kutta method and the analysis strategy using variation of the winding resistance are studied, and their numerical results are compared with those from the typical time-stepping finite element method.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.41 no.1
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• pp.39-48
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• 2021

In this paper, an analytical model is proposed for assessing the natural frequency of barrettes subjected to vertical loading. The differential equation governing the free vibration of rectangular friction piles embedded in inhomogeneous soil is derived. The governing equation is numerically integrated by Runge-Kutta technique and the eigenvalue of natural frequency is computed by Regula-Falsi method. The numerical solutions for the natural frequency of barrettes compare well with those obtained from finite element analysis. Illustrated examples show that the natural frequencies increase with an increase of the cross-sectional aspect ratio, the friction resistance ratio and the soil stiffness ratio, and decrease with an increase of the friction aspect ratio, the slenderness ratio and the load factor, respectively.

• Structural Engineering and Mechanics
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• v.89 no.1
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• pp.33-46
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• 2024

The present research delves into the analysis of nonlinear free and forced vibrations of porous functionally graded (FG) shallow shells reinforced with oblique stiffeners, which are embedded in a nonlinear elastic foundation (NEF) subjected to external excitation. Two distinct types of PFG shallow shells, characterized by even and uneven porosity distribution along the thickness direction, are considered in the research. In order to model the stiffeners, Lekhnitskii's smeared stiffeners technique is implemented. With the stress function and first-order shear deformation theory (FSDT), the nonlinear model of the oblique stiffened shallow shells is established. The strain-displacement relationships for the system are derived via the FSDT and utilization of the von-Kármán's geometric assumptions. To discretize the nonlinear governing equations, the Galerkin method is employed. The model such developed allows analysis of the effects of the stiffeners with various angles as desired, in addition to the quantitative investigation on the influence of the surrounding nonlinear elastic foundations. To numerically solve the problem of vibrations, the 4th-order P-T method is used, as this method, known for its enhanced accuracy and reliability, proves to be an effective choice. The validation of the present research findings includes a comprehensive comparison with outcomes documented in existing literature. Additionally, a comparative analysis of the numerical results against those obtained using the 4th Runge-Kutta method is performed. The impact of stiffeners with varying angles and material parameters on the vibration characteristics of the present system is also explored. The researchers and engineers working in this field may use the results of this study as benchmarks in their design and research for the considered shell systems.