• Computational Structural Engineering
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• v.9 no.3
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• pp.187-197
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• 1996

An explicit direct time integration method based solution algorithm is presented to predict dynamic buckling response of Euler column. On the basis of large deflection beam theory, a plane frame finite element is formulated and implemented into the solution algorithm. The element formulation takes into account geometrical nonlinearity and overall buckling of steel structural frames. The solution algorithm employs the central difference method. Using the computer program developed by the author, dynamic instability behavior of Euler column under impact loading is investigated by considering the time variation of load, load magnitude, and load duration. The free vibration of Euler column caused by a short duration impact load is also studied. The validity and efficiency of the present formulation and solution algorithm are verified through illustrative numerical examples.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.31-40
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• 1992

We introduce the generalized Runge-Kutta methods with the exponentially dominant order .omega. in [3], and the convergence theorems of the generalized explicit Euler method are derived in [4]. In this paper we will study the convergence of the generalized implicit Euler method.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.13-36
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• 2018

In this paper, we obtain some formulae for harmonic sums, alternating harmonic sums and Stirling number sums by using the method of integral representations of series. As applications of these formulae, we give explicit formula of several quadratic and cubic Euler sums through zeta values and linear sums. Furthermore, some relationships between harmonic numbers and Stirling numbers of the first kind are established.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.3
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• pp.197-207
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• 2013

The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn-Hillard equation, many numerical methods have been proposed such as the explicit Euler's, the implicit Euler's, the Crank-Nicolson, the semi-implicit Euler's, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler's method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank-Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.

• Journal of computational fluids engineering
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• v.12 no.1
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• pp.43-52
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• 2007

A comparative study on flux functions for the 2-dimensional Euler equations has been conducted. Explicit 4-stage Runge-Kutta method is used to integrate the equations. Flux functions used in the study are Steger-Warming's, van Leer's, Godunov's, Osher's(physical order and natural order), Roe's, HLLE, AUSM, AUSM+, AUSMPW+ and M-AUSMPW+. The performance of MUSCL limiters and MLP limiters in conjunction with flux functions are compared extensively for steady and unsteady problems.

• 한국전산유체공학회:학술대회논문집
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• 2006.10a
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• pp.36-40
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• 2006

A comparative study on flux functions for the 2-dimensional Euler equations has been conducted. Explicit 4-stage Runge-Kutta method is used to integrate the equations. Flux functions used in the study are Steger-Warming's, van Leer's. Godunov's, Osher's(physical order and natural order), Roe's, HILE, AUSM, AUSM+ and AUSMPW+. The performance of MUSCL limiters and MLP limiters in conjunction with flux functions are compared extensively for steady and unsteady problems.

• Journal of the Korea Computer Graphics Society
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• v.5 no.1
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• pp.27-33
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• 1999

In this paper, we propose an efficient technique for the animation of flexible thin objects. Mass-spring model was employed to represent the flexible objects. Till now, many techniques have used the mass-spring model to generate plausible animation of soft objects. A straight-forward approach to the animation with mass-spring model is explicit Euler method, but the explicit Euler method has serious disadvantage that it suffers from 'instability problem'. The implicit integration method is a possible solution to overcome the instability problem. However, the most critical flaw of the implicit method is that it involves a large linear system. This paper presents a fast animation technique for mass-spring model with approximated implicit method. The proposed technique stably updates the state of n mass-points in O(n) time when the number of total springs are O(n). We also consider the interaction of the flexible object and air in order to generate plausible results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.1
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• pp.49-66
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• 2022

The G-Euler process has been proposed to overcome the difficulties of the calculation of the exponential function of the Jacobian. It is an explicit method that uses the exponential function of the scalar skew-symmetric matrix. We define the moving shapes of true solutions and the moving shapes of numerical solutions. It is discussed whether the moving shape of the numerical solution matches the moving shape of the true solution. The match rates of these two kinds of moving shapes are sequentially calculated by the G-Euler process without using the true solution. It is shown that the closer the minimum match rate is to 100%, the more closely the numerical solutions follow the true solutions to the end. The minimum match rate indicates the reliability of the numerical solution calculated by the G-Euler process. The graphs of the Lorenz system in Perko [1] are different from those drawn by the G-Euler process. By the way, there is no basis for claiming that the Perko's graphs are reliable.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.9 no.2
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• pp.286-290
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• 2008

This paper summarizes the components of an explicit aeroelastic solver developed especially for the simulation of dynamic fracture events occurring during the flight of solid propellant rockets. The numerical method combines an explicit Arbitrary Lagrangian Eulerian (ALE) version of the Cohesive Volumetric Finite Element (CVFE) scheme, used to simulate the spontaneous motion of one or more cracks propagating dynamically through a domain with regressing boundaries, and an explicit unstructured finite volume Euler code to follow the flow field during the failure event. A key feature of the algorithm is the ability to adaptively repair and expand the fluid mesh to handle the large geometrical changes associated with grain deformation and crack motion.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.985-1001
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• 2023

Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums S⁺⁺_p,q (a, b) and S^+-_p,q (a, b) defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.