• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.277-302
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• 2017

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter ${\varepsilon}$ multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.

• Geomechanics and Engineering
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• v.14 no.3
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• pp.293-299
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• 2018

This paper has the aim of estimating the applicability of a numerical approach to the Hyperstatic Reaction Method (HRM) for the analysis of segmental tunnel linings. For this purpose, a simplified three-dimensional (3D) numerical model, using the $FLAC^{3D}$ finite difference software, has been developed, which allows analysing in a rigorous way the effect of the lining segmentation on the overall behaviour of the lining. Comparisons between the results obtained with the HRM and those determined by means of the simplified 3D numerical model show that the proposed HRM method can be used to investigate the behaviour of a segmental tunnel lining.

• 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 Ship and Ocean Technology
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• v.8 no.3
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• pp.8-17
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• 2004

A numerical method is developed for computing the free surface flows around a transom stern of a ship at a high Froude number. At high speed, the flow may be detached from the flat transom stern. In the limit of the high Froude number, the problem becomes a planning problem. In the present study, we make the finite-element computations for a transom stern flows around a wedge-shaped floating ship. The numerical method is based on the Hamilton's principle. The problem is formulated as an initial value problem with nonlinear free surface conditions. In the numerical procedures, the domain was discretized into a set of finite elements and the numerical quadrature was used for the functional equation. The time integrations of the nonlinear free surface condition are made iteratively at each time step. A set of large algebraic equations is solved by GMRES(Generalized Minimal RESidual, Saad and Schultz 1986) method which is proven very efficient. The computed results are compared with previous numerical results obtained by others.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.603-618
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• 2022

In this paper, a numerical solution of the generalized diffusion equation with a delay has been obtained by a numerical technique based on the Galerkin finite element method by applying the cubic B-spline basis functions. The time discretization process is carried out using the forward Euler method. The numerical scheme is required to preserve the delay-independent asymptotic stability with an additional restriction on time and spatial step sizes. Both the theoretical and computational rates of convergence of the numerical method have been examined and found to be in agreement. As it can be observed from the numerical results given in tables and graphs, the proposed method approximates the exact solution very well. The accuracy of the numerical scheme is confirmed by computing L₂ and L_∞ error norms.

• Journal of Mechanical Science and Technology
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• v.14 no.1
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• pp.103-112
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• 2000

Numerical simulations of unsteady opposed-flow flames are performed using an adaptive time integration method designed for differential-algebraic systems. The compressibility effect is considered in deriving the system of equations, such that the numerical difficulties associated with a high-index system are alleviated. The numerical method is implemented for systems with detailed chemical mechanisms and transport properties by utilizing the Chemkin software. Two test simulations are performeds hydrogen/air diffusion flames with an oscillatory strain rate and transient ignition of methane against heated air. Both results show that the rapid transient behavior is successfully captured by the numerical method.

• International Journal of Control, Automation, and Systems
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• v.1 no.1
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• pp.127-133
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• 2003

This paper proposes a new numerical method to check the stability of a general class of time delay systems. The proposed method checks whether there are characteristic roots whose real values are nonnegative through two steps. Firstly, rectangular bounds of characteristic roots whose real values are nonnegative are computed. Secondly, the existence of roots inside the bounds are checked using the numerical computation of argument principles. An adaptive discretization is proposed for the numerical computation of argument principles.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.51-67
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• 1999

The numerical solution of the Navier-Stokes equations in a constricted stepped channel problem has been obtained using a fourth order numerical method. Transformations are made to have a fine grid near the sharp corner and a long channel downstream. The derivatives in the Navier-Stokes equations are replaced by fourth order central differences which result a 29-point computational stencil. A procedure is used to avoid extra numerical boundary conditions near the solid walls. Results have been obtained for Reynolds numbers up to 1000.

• Journal of KSNVE
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• v.7 no.5
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• pp.773-780
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• 1997

Using Generalized Inverse Method presented by Udwadia and Kalaba in 1992, we can obtain equations to exactly describe the motion of constrained systems. When the differential equations are numerically integrated by any numerical integration scheme, the numerical results are generally found to veer away from satisfying constraint equations. Thus, this paper deals with the numerical integration of the differential equations describing constrained systems. Based on Baumgarte method, we propose numerical methods for reducing the errors in the satisfaction of the constraints.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.28 no.10
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• pp.1590-1597
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• 2004

Three methods of design sensitivity analysis for structures such as numerical method, analytical method and semi-analytical method have been developed for the last three decades. Although analytical design sensitivity analysis can provide very exact result, it is difficult to implement into practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable fur most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate in nonlinear design sensitivity analysis because its computational cost depends on the number of design variables and large numerical errors can be included. Thus the semi-analytical method is more suitable for complicated design problems. Moreover, semi-analytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure fur the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and the computational technique is proposed for evaluating the partial differentiation of internal nodal force, so called pseudo-load. Numerical examples coupled with commercial finite element package are shown to verify usefulness of proposed semi-analytical sensitivity analysis procedure and computational technique for pseudo-load.