• The Pure and Applied Mathematics
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• v.26 no.3
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• pp.157-175
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• 2019

In this paper we define higher-order Stieltjes derivatives, and using Schaefer's fixed point theorem we investigate the existence of solutions for a class of differential equations involving second-order Stieltjes derivatives with two-point boundary conditions. The equations include ordinary and impulsive differential equations, and difference equations.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.21 no.4
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• pp.391-399
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• 2008

In this paper, we further generalize the work of Lin and Abel to the case of the first and the second order derivatives of energy release rates for two-dimensional, multiply cracked systems. The direct integral expressions are presented for the energy release rates and their first and second order derivatives. The salient feature of this numerical method is that the energy release rates and their first and second order derivatives can be computed in a single analysis. It is demonstrated through a set of examples that the proposed method gives expectedly decreasing, but acceptably accurate results for the energy release rates and their first and second order derivatives. The computed errors were approximately 0.5% for the energy release rates, $3\sim5%$ for their first order derivatives and $10\sim20%$ for their second order derivatives for the mesh densities used in the examples. Potential applications of the present method include a universal size effect model and a probabilistic fracture analysis of cracked structures.

• The Pure and Applied Mathematics
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• v.24 no.1
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• pp.1-19
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• 2017

In this paper we obtain some integral inequalities by using Stieltjes derivatives, and we apply our results to the study of asymptotic behavior of a certain second-order integro-differential equation.

• Transactions of the Korean Society of Machine Tool Engineers
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• v.15 no.3
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• pp.8-16
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• 2006

This paper discusses design sensitivity analysis and its application to a structural dynamics modification. Eigenvalue derivatives are determined with respect to the element parameters, which include intrinsic property parameters such as Young's modulus, density of the material, diameter of a beam element, thickness of a plate element, and shape parameters. Derivatives of stiffness and mass matrices are directly calculated by derivatives of element matrices. The first and the second order derivatives of the eigenvalues are then mathematically derived from a dynamic equation of motion of FEM model. The calculation of the second order eigenvalue derivative requires the sensitivity of its corresponding eigenvector, which are developed by Nelson's direct approach. The modified eigenvalue of the structure is then evaluated by the Taylor series expansion with the first and the second derivatives of eigenvalue. Numerical examples for simple beam and plate are presented. First, eigenvalues of the structural system are numerically calculated. Second, the sensitivities of eigenvalues are then evaluated with respect to the element intrinsic parameters. The most effective parameter is determined by comparing sensitivities. Finally, we predict the modified eigenvalue by Taylor series expansion with the derivatives of eigenvalue for single parameter or multi parameters. The examples illustrate the effectiveness of the eigenvalue sensitivity analysis for the optimization of the structures.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.11a
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• pp.721-726
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• 2004

A simplified method for the computation of first second and higher order derivatives of eigenvalues and eigenvectors derivatives associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m is the number of multiplicity of the repeated eigenvalues. The algebraic equation developed can be used to compute derivatives of both eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space it is numerically stable and very efficient compared to previous methods. This method can be consistently applied to structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam is considered.

• The Mathematical Education
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• v.53 no.1
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• pp.25-40
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• 2014

Mathematical concept exists in the structural form, not in the independent form. The purpose of this study is to consider the network which students actually have for the mathematical concept structure related to the properties of derivatives. First, we analyzed the properties of derivatives in 'Mathematics II' and showed the mathematical concept structure of the relations among derivatives, functions, and primitive functions as a network. Also, we investigated the understanding of high school students for the mathematical concept structure between derivatives and functions, and the structure between functions and second order derivatives when the functional formula is not given, and only the graph is given. The results showed that students mainly focus on the relation of 'function-derivatives', the thinking process for direction of derivative and the thinking style for algebra. On this basis, we suggest the educational implication that is necessary for students to build the network properly.

• Journal of Hydrospace Technology
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• v.1 no.1
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• pp.66-74
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• 1995

Unknown parameters can be determined by system identification techniques. Extended Kalman filter method was introduced as a real time estimator of hydrodynamic derivatives but it has the problem named the coefficient drift. In this study, 2nd order filter estimates hydrodynamic derivatives in Abkowitz model In order to reduce the coefficient drift, parallel processing is used. The measured state and ship trajectory are compared with the estimated values. Parallel processing of 2nd order filter gives very similar results to parallel processing of extended Kalman filter. Parallel processing cannot not remove the coefficient drift perfectly, but it reduces the estimation error.

• 한국전산유체공학회:학술대회논문집
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• 1996.05a
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• pp.46-51
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• 1996

The flow field around a three dimensional minivan-like body has been simulated. This study solves 3-D unsteady incompressible Navier-Stokes equations on a non-orthogonal curvilinear coordinate system using second-order accurate schemes for the time derivatives, and third/second-order scheme for the spatial derivatives. The Marker-and-Cell concept is applied to efficiently solve continuity equation. The fourth -order artificial damping is added to the continuity equation for numerical stability. A H-H type multi-block grid system is generated around a three dimensional minivan-like body. Turbulent flows have been modeled by the Baldwin-Lomax turbulent model. The simulation shows three dimensional vortex-pair just behind body. And the flow separation is also observed the rear of the body. It has concluded that the results of present study properly agree with physical flow phenomena.

• 한국전산유체공학회:학술대회논문집
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• 1997.10a
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• pp.86-91
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• 1997

This paper describes analysis of complex flow around Car-like body using Chimera grid technique. As a computational algorithm, Pullboat and Chaussee's Diagonal algorithm is selected to reduce computational time. Introducing hole points flag to this Diagonal algorithm, an algorithm for Chimera grid is generated easily. This study solves 3-D unsteady incompressible Navier-Stokes equations on a non-orthogonal curvilinear coordinate system using second-order accurate schemes for the time derivatives, and third/second-order scheme for the spatial derivatives. The Marker-and-Cell concept is applied to efficiently solve continuity equation. The fourth-order artificial damping is added to the continuity equation for numerical stability, It has concluded that the results of present study properly agree with physical flow phenomena.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.1
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• pp.133-143
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• 2014

A new three-step quadraparametric family of eighth-order iterative methods free from second derivatives are proposed in this paper to find a simple root of a nonlinear equation. Convergence analysis as well as numerical experiments confirms the eighth-order convergence and asymptotic error constants.