• Nuclear Engineering and Technology
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• v.49 no.6
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• pp.1114-1124
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• 2017

Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A flexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates ($S_N$) or spherical-harmonics ($P_N$) solve to accelerate convergence of a high-order $S_N$ source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving electron transport and problems with artificial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.4
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• pp.375-385
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• 2020

In this paper, a special indefinite inner product, named hyperbolic scalar product, is used and all acquired results have been raised and proved with the proviso that the space is equipped with this indefinite scalar product. The main objective is to be introduced and applied an indefinite oblique projection method, called Indefinite Lanczos J-biorthogonalizatiom process, which in addition to building a pair of J-biorthogonal bases for two used Krylov subspaces, leads to the introduction of a process for solving large non-J-symmetric linear systems, i.e., Indefinite two-sided Lanczos Algorithm for Linear systems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.1-4
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• 1999

The Helmholtz equation is very important in physics and engineering. However, solution of the Helmholtz equation is in general known as a very difficult phenomenon. For if the ${\omega}$ is negative, the FDM discretized linear system becomes indefinite, whose solution by iterative method requires a very clever preconditioner. In this paper we assume that ${\omega}$ is nonnegative, and determine the optimal ${\rho}$ parameter for the three dimensional ADI iteration for the Helmholtz equation. The ADI(Alternating Direction Implicit) method is also getting new attentions due to the fact that it is very suitable to the vector/parallel computers, for example, as a preconditioner to the Krylov subspace methods. However, classical ADI was developed for two dimensions, and for three dimensions it is known that its convergence behaviour is quite different from that in two dimensions. So far, in three dimensions the so-called Douglas-Rachford form of ADI was developed. It is known to converge for a relatively wide range of ${\rho}$ values but its convergence is very slow. In this paper we determine the necessary conditions of the ${\rho}$ parameter for the convergence and optimal ${\rho}$ for the three dimensional ADI iteration of the Peaceman-Rachford form for the real Helmholtz equation with nonnegative ${\omega}$. Also, we conducted some experiments which is in close agreement with our theory. This straightforward extension of Peaceman-rachford ADI into three dimensions will be useful as an iterative solver itself or as a preconditioner to the the Krylov subspace methods, such as CG(Conjugate Gradient) method or GMRES(m).

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.413-426
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• 2008

An asymptotic solution of a fourth order critically damped nonlinear differential system has been found by means of extended Krylov-Bogoliubov-Mitropolskii (KBM) method. The solutions obtained by this method agree with those obtained by numerical method. The method is illustrated by an example.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2006.04a
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• pp.291-298
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• 2006

A parallelized topology design optimization method is developed on a distributed memory system. The parallelization is based on a domain decomposition method and a boundary communication scheme. For the finite element analysis of structural responses and design sensitivities, the PCG method based on a Krylov iterative scheme is employed. Also a parallelized optimization method of optimality criteria is used to solve large-scale topology optimization problems. Through several numerical examples, the developed method shows efficient and acceptable topology optimization results for the large-scale problems.

• Bulletin of the Society of Naval Architects of Korea
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• v.26 no.1
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• pp.1-10
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• 1989

When a ship is travelling in following seas, the encounter frequency is reduced to be very low. n that case surf-riding, broaching-to and capsizing phenomena are most likely to occur. In the present paper, the emphasis is laid upon the transverse stability of ships in following seas, which is related to capsizing phenomenon. The authors intend to clarify the mechanism of the stability variation in following seas. In order to predict that variation, the theoretical calculation method and computer programming are developed. The theoretical calculation is based on Froude-Krylov Hypothesis and static equilibrium condition in waves. Through the application of present calculation method to cargo vessel and fishing boat, it is verified that the transverse stability in following seas with crest amidships can be reduced to a half extent in comparison with calm water stability. On the other hand, the present calculation results are compared with existing experimental data.

• International Journal of Naval Architecture and Ocean Engineering
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• v.13 no.1
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• pp.718-733
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• 2021

A prediction method, based on the Morison equation as well as Froude-Krylov formula, is presented to simulate the loads acting on the columns and caissons of the semi-submersible platform induced by Internal Solitary Wave (ISW) respectively. Combined with the experimental results, empirical formulas of the drag and inertia coefficients in Morison equation can be determined as a function of the Keulegan-Carpenter (KC) number, Reynolds number (Re) and upper layer depth h₁/h respectively. The experimental and calculated results are compared. And a good agreement is observed, which proves that the present prediction method can be used for analyzing the ISW-forces on the semi-submersible platform. Moreover, the results also demonstrate the layer thickness ratio has a significant effect upon the maximum horizontal forces on the columns and caissons, but both minimum horizontal and vertical forces are scarcely affected. In addition, the incoming wave directions may also contribute greatly to the values of horizontal forces exerted on the caissons, which can be ignored in the vertical force analysis.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.5-16
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• 1999

We introduce a new projector for accelerating convergence of a symmetric eigenvalue problem Ax = x, and devise a power/Lanczos hybrid algorithm. Acceleration can be achieved by removing the hard-to-annihilate nonsolution eigencomponents corresponding to the widespread eigenvalues with modulus close to 1, by estimating them accurately using the Lanczos method. However, the additional Lanczos results can be obtained without expensive matrix-vector multiplications but a very small amount of extra work, by utilizing simple power-Lanczos interconversion algorithms suggested. Numerical experiments are given at the end.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.145-155
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• 1997

We present a couple of tools that can be used in the solution of nonsymmetric eigenvalue problems. The first one allows us to convert power iterates into Arnoldi's results so that a few eigenpairs are easily obtainable. The other converts Arnoldi's results into power iterates to simulate the power method and improve the result. Suggestions for application are also given.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.755-765
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• 2003

Krylov-Bogoliubov-Mitropolskii method has been extended to obtain asymptotic solution of n-th order nonlinear differential system characterized by certain non-oscillatory processes. The damping force is considered in such a manner that one of the characteristic roots of the linear system becomes small and others are in integral multiple. The method is illustrated by an example. The solutions for different initial conditions show a good agreement with those obtained by numerical method.