• Structural Engineering and Mechanics
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• 제76권5호
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• pp.643-651
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• 2020

Generally, it is necessary to perform transient structural analysis in order to verify and improve the seismic performance of high-rise buildings and bridges against earthquake loads. In this paper, we propose the model order reduction (MOR) method using the Krylov vectors to perform seismic analysis for linear and elastic systems in an efficient way. We then compared the proposed method with the mode superposition method (MSM) by using the limited numbers of modal vectors (or eigenvectors) calculated from the modal analysis. In the calculation, the data of the El Centro earthquake in 1940 were adopted for the seismic loading in the transient analysis. The numerical accuracy and efficiency of the two methods were compared in detail in the case of a simplified high-rise building.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제3권2호
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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 Korean Society for Industrial and Applied Mathematics
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• 제24권4호
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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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• 제22권3호
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• pp.155-162
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• 2018

We describe a parallel implementation of a relaxed Hermitian and skew-Hermitian splitting preconditioner for the numerical solution of saddle point problems arising from the steady incompressible Navier-Stokes equations. The equations are linearized by the Picard iteration and discretized with the finite element and finite difference schemes on two-dimensional and three-dimensional domains. We report strong scalability results for up to 32 cores.

• 대한수학회논문집
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• 제12권1호
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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.

• 대한기계학회논문집A
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• 제38권9호
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• pp.933-941
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• 2014

본 논문에서는 대표적 투영기반 모델차수축소법인 크리로프 부공간 모델차수축소법(KSM)과 모드중첩법(MTM)을 고려하여 주파수응답해석에 대한 수치적 정확도와 효율성을 비교하였다. 두 모델차수축소법의 수치 정확도 비교를 위하여 주파수응답해석 결과, 축소차수 및 관심주파수에 따른 상대오차를 고려하였으며 이후에 오차수렴지표를 통한 자동적인 축소차수의 결정이 가능 여부를 확인하였다. 효율성 비교를 위해서는 각 축소모델의 주파수응답 해석시간 및 축소차수에 따른 변환행렬 생성시간을 비교하였다. 자동차 현가장치에 대한 유한요소모델을 적용예제로 선정하여 수치 비교를 수행하였다.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.307-316
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• 2006

Incomplete LU factorization preconditioning techniques often have difficulty on indefinite sparse matrices. We present hybrid reordering strategies to deal with such matrices, which include new diagonal reorderings that are in conjunction with a symmetric nondecreasing degree algorithm. We first use the diagonal reorderings to efficiently search for entries of single element rows and columns and/or the maximum absolute value to be placed on the diagonal for computing a nonsymmetric permutation. To augment the effectiveness of the diagonal reorderings, a nondecreasing degree algorithm is applied to reduce the amount of fill-in during the ILU factorization. With the reordered matrices, we achieve a noticeable improvement in enhancing the stability of incomplete LU factorizations. Consequently, we reduce the convergence cost of the preconditioned Krylov subspace methods on solving the reordered indefinite matrices.

• Nuclear Engineering and Technology
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• 제49권6호
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• pp.1310-1317
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• 2017

GASFLOW-MPI is a widely used scalable computational fluid dynamics numerical tool to simulate the fluid turbulence behavior, combustion dynamics, and other related thermal-hydraulic phenomena in nuclear power plant containment. An efficient scalable linear solver for the large-scale pressure equation is one of the key issues to ensure the computational efficiency of GASFLOW-MPI. Several advanced Krylov subspace methods and scalable preconditioning methods are compared and analyzed to improve the computational performance. With the help of the powerful computational capability, the large eddy simulation turbulent model is used to resolve more detailed turbulent behaviors. A backward-facing step flow is performed to study the free shear layer, the recirculation region, and the boundary layer, which is widespread in many scientific and engineering applications. Numerical results are compared with the experimental data in the literature and the direct numerical simulation results by GASFLOW-MPI. Both time-averaged velocity profile and turbulent intensity are well consistent with the experimental data and direct numerical simulation result. Furthermore, the frequency spectrum is presented and a -5/3 energy decay is observed for a wide range of frequencies, satisfying the turbulent energy spectrum theory. Parallel scaling tests are also implemented on the KIT/IKET cluster and a linear scaling is realized for GASFLOW-MPI.

• Structural Engineering and Mechanics
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• 제57권5호
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• pp.921-936
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• 2016

In this study, a model order reduction technique is applied to solve the transient responses of submerged floating tunnel (SFT) from Mokpo to Jeju under seismic excitations. Because the SFT is a very long structure as well as a transient response analysis requires large amount of computational resources, the model order reduction is mandatory in the design stage of the SFT. Thus, we apply a model order reduction based on Krylov subspace to the simplified finite element model of the SFT. The responses of the reduced order model are compared with those of the full order model and also are verified by referring a previous work. In conclusion, the computational resources are dramatically reduced with an acceptable accuracy by using the model order reduction, which eventually is useful for designing the full-scale model of SFTs.

• 지구물리와물리탐사
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• 제7권2호
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• pp.148-154
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• 2004

자기지전류(MT)법의 3차원 모델링에 대해 소개한다. 3차원 MT 모델링은 MT 반응의 물리적 특성의 이해뿐만 아니라 지하의 3차원적 전기비저항 구조를 재구성하기 위한 역산법의 개발에도 필수적이다. 지난 20년 동안 3차원 모델링에 관한 여러 수치기법들이 개발되었으나 그 실용성에는 많은 한계가 있었다. 그러나 최근에는 컴퓨터의 급속한 발전과 대형 연립방정식에 대한 반복해법의 발전에 힘입어 이전에는 어려웠던 복잡한 3차원 구조에 대한 MT 반응을 효율적으로 모델링할 수 있게 되었다. 유한차분법에서는 자기 flux와 전류의 보존법칙을 만족하면서 전기장의 불연속을 표현할 수 있는 staggered 격자의 사용이 보편화되었다. 대형 연립방정식에 대한 수치해의 수렴성은 Krylov 부분공간법, 적당한 전처리 기술 및 정적 발산보정법을 채택함으로써 크게 향상된다. 변요소를 사용하는 벡터 유한요소법으로도 전기장의 불연속 문제를 해결할 수 있으며 이 방법이 가진 기하학적 유연성은 불규칙한 지표기복을 포함한 복잡한 구조를 모델화할 때 특히 유용하다.