• Nuclear Engineering and Technology
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• 제52권3호
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• pp.485-498
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• 2020

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

• Structural Engineering and Mechanics
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• 제30권4호
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• pp.467-480
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• 2008

An aggregation multigrid method (AMM) is a leading iterative solver in solid mechanics. Recently, AMM is applied for solving Schur Complement system in the FE analysis of shell structures. In this work, an extended application of AMM for solving Schur Complement system in the FE analysis of continuum elements is presented. Further, the performance of the proposed AMM in multiple load cases, which is a challenging problem for an iterative solver, is studied. The proposed method is developed by combining the substructuring and the multigrid methods. The substructuring method avoids factorizing the full-size matrix of an original system and the multigrid method gives near-optimal convergence. This method is demonstrated for the FE analysis of several elastostatic problems. The numerical results show better performance by the proposed method as compared to the preconditioned conjugate gradient method. The smaller computational cost for the iterative procedure of the proposed method gives a good alternative to a direct solver in large systems with multiple load cases.

• Journal of applied mathematics & informatics
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• 제38권3_4호
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• pp.311-334
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• 2020

The solution of the elliptic partial differential equation has interface singularity at the points which are either the intersections of interfaces or the intersections of interfaces with the boundary of the domain. The singularities that arises in the elliptic interface problems are very complex. In this article we propose an exponentially accurate nonconforming spectral element method for these problems based on [7, 18]. A geometric mesh is used in the neighbourhood of the singularities and the auxiliary map of the form z = ln ξ is introduced to remove the singularities. The method is essentially a least-squares method and the solution can be obtained by solving the normal equations using the preconditioned conjugate gradient method (PCGM) without computing the mass and stiffness matrices. Numerical examples are presented to show the exponential accuracy of the method.

• 경영과학
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• 제21권1호
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• pp.77-86
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• 2004

In this paper, we present a methodology for solving the multicommodity network flow problems using interior point methods. In our method, the minimum cost network flow problem extracted from the given multicommodity network flow problem is solved by primal-dual barrier method in which normal equations are solved partially using preconditioned conjugate gradient method. Based on the solution of the minimum cost network flow problem, a warm-start point is obtained from which Castro's specialized interior point method for multicommodity network flow problem starts. In the computational experiments, the effectiveness of our methodology is shown.

• 한국해안해양공학회지
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• 제6권2호
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• pp.186-195
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• 1994

회복기법인 PCGM에 근거한 정밀하고 수렴속도가 빠른 수치기법을 개발하기 위하여 기존 기법인 Bayliss 등(1983) 혹은 Panchang 등(1991)의 기법을 포함하여 여러가지 기법을 제시하였다. 각 기법의 수치결과는 일정 수심 위를 지나는 선형 파랑의 해석해와 비교하였으며, 각 방법의 장단점을 논한 뒤 정밀도와 수렴원도를 분석하였다. 이러한 비교를 통해 본 논문의 방법이 검토한 방법중에서 완경사 파랑식을 PCGM 수치기법으로 계산하는 데 가장 적합한 방법임을 입증하였다.

• 한국원자력학회:학술대회논문집
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• 한국원자력학회 1996년도 춘계학술발표회논문집(1)
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• pp.157-162
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• 1996

A novel nodal method is developed for the two-dimensional multi-group diffusion equations based on the Spectral-Galerkin approach. In this study, the nodal diffusion equations with Robin boundary condition are reformulated in a weak (variational) form, which is then approximated spatially by choosing appropriate basis functions. For the nodal coupling relations between the neighbouring nodes, the continuity conditions of partial currents are utilized. The resulting discrete systems with sparse structured matrices are solved by the Preconditioned Conjugate Gradient Method (PCG) and sweeping technique. The method is validated on two test problems.

• 디지털콘텐츠학회 논문지
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• 제10권4호
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• pp.579-585
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• 2009

Tyrtshnikov[9]의 연구에서는 토플리츠 선형시스템에서 토플리츠 선행조건으로 일반해를 구하는 방법들을 제시하고 있다. 또한 대칭 토플리츠 행렬에서의 선행조건 행렬을 선택하는 방법도 소개 하였다. 본 연구는 토플리츠 시스템에서 새롭게 선행조건 찾는 방법을 소개하고 있으며, 선행조건행렬들의 분석을 통해 대칭 토플리츠 행렬의 고유값들과 대칭 토플리츠행렬로 부터 생성된 선행조건행렬의 고유값들이 매우 근접하다는 결과를 나타내고 있다. 즉, 선행조건시스템 $C_0^{-1}T$의 고유값들은 1에 모두 접근하게되면, 선행조건 시스템의 수렴속도는 superlinear이다. 본 연구에서 생성된 선행조건행렬 $C_0$은 선행조건시스템의 superlinear의 수렴속도로 계산하게 된다. 또한 토플리츠 행렬은 이미지 프로세싱이나 시그널 프로세싱에서 많이 응용되고 있으므로 본 연구에서 개발한 선행조건행렬로부터 다양한 응용성을 높일 수 있다. 본연구의 또 다른 특징은 토플리츠 행렬의 중요한 성질을 보존하면서 선행조건행렬을 생성하였다.

• 한국해안해양공학회지
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• 제6권2호
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• pp.164-173
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• 1994

완경사 파랑식을 PCGM기법으로 계산하기 위한 수치모형을 제시하였다. 본 논문에서는 기존 Panchang 등(1991)의 모형과 달리 정확한 경계조건을 부여하였고 보다 향상된 Preconditioner가 사용되었다. 비선형파랑에 대한 계산과정을 중점적으로 다루었고 보다 정밀한 수치모형을 개발하기 위해 발표된 문제들을 토의하였다. 수치모형의 결과를 구형 천퇴와 타원형 천퇴수리실험 자료와 비교하였다. 파랑의 진폭에 대한 수치모형 결과는 수리실험 자료와 잘 일치하였으며 본 수치모형은 복잡한 지형을 갖는 천해역의 파랑변형을 계산하는데 유용한 모형임을 입증하였다.

• 한국정밀공학회지
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• 제22권1호
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• pp.122-129
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• 2005

On the cluster system, the parallel code of rigid-plastic FEM has been developed. The cluster system, Simforge, has 15 processors and the total memory is 4.5GBytes. In the developed parallel code, the distributed data of the column-wise partitioned stiffness are stored as the compressed row storage and the diagonal preconditioned conjugate gradient solver is applied. The analysis of block upsetting is performed with the parallel code on Simforge cluster system. In this paper, the analysis results are compared and discussed.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 1998년도 가을 학술발표회 논문집
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• pp.362-369
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• 1998

The computational environment in which engineers perform their designs has been rapidly evolved from coarse serial machines to massively parallel machines. Although the recent development of high-performance computers are available for a number of years, only limited successful applications of the new computational environments in computational structural engineering field has been reported due to its limited availability and large cost associated with high-performance computing. As a new computational model for high-performance engineering computing without cost and availability problems, parallel structural analysis models for large scale structures on a network of personal computers (PCs) are presented in this paper. In structural analysis solving routine for the linear system of equations is the most time consuming part. Thus, the focus is on the development of efficient preconditioned conjugate gradient (PCG) solvers on the proposed computational model. Two parallel PCG solvers, PPCG-I and PPCG-II, are developed and applied to analysis of large scale space truss structures.