• 대한수학회논문집
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• 제20권3호
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• pp.563-578
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• 2005

In [Z. Cai and B. C. Shin, SIAM J. Numer. Anal. 40 (2002), 307-318], we developed the discrete first-order system least squares method for the second-order elliptic boundary value problem by directly approximating $H(div){\cap}H(curl)-type$ space based on the Helmholtz decomposition. Under general assumptions, error estimates were established in the $L^2\;and\;H^1$ norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. In this paper, we study solution methods for solving the system of linear equations arising from the discretization of variational formulation which possesses discrete biharmonic term and focus on numerical results including the performances of multigrid preconditioners and the finite element accuracy.

• 대한수학회논문집
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• 제15권1호
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• pp.143-153
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• 2000

In [8], Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we propose the block diagonal preconditioners. The preconditioned conjugate residual method is robust in that the convergence is uniform as the parameter, v, goes to $\sfrac{1}{2}$. Computational experiments are included.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 1999년도 춘계 학술대회논문집
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• pp.23-28
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• 1999

A mutigrid convergence acceleration technique is presented for computing hypersonic inviscid and viscous flows in equilibrium state. The governing equations are solved using an explicit Runge-Kutta method. Curve fitting data in NASA Reference Publication 1181, 1260 are used to calculate equilibrium properties. In order to ensure stability, damped prolongation and modified implicit residual smoothing are proposed. Blunt body test cases are presented to demonstrate the robustness and the efficiency in performance of the proposed methods

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 2004년도 추계 학술대회논문집
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• pp.7-14
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• 2004

It is well known that preconditioning methods are efficient for convergence acceleration at compressible low Mach number flows. In this study, the original Euler equations and three preconditioners nondimensionalized differently are implemented in two dimensional inviscid bump flows using the 3rd order MUSCL and DADI schemes as flux discretization and time integration respectively. The multigrid and local time stepping methods are also used to accelerate the convergence. The test case indicates that a properly modified local preconditioning technique involving concepts of a global preconditioning one produces Mach number independent convergence. Besides, an asymptotic analysis for properties of preconditioning methods is added.

• Nuclear Engineering and Technology
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• 제49권6호
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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.

• 지구물리와물리탐사
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• 제11권1호
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• pp.60-67
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• 2008

이 논문에서 우리는 해저용 수신기들과 이동하는 전기적 양극송신기의 한 조로 이루어진 전형적인 해양 인공송신 전자탐사 (MCSEM) 방법에 의해 얻어진 자료 해석에 전자탐사 구조보정법의 적용을 다룬다. 이 연구에서와 같이 다중 송신기와 다중 수신기를 이용해 획득된 자료는 방대한 컴퓨터 계산을 요하기 때문에 MCSEM자료의 3차원적 해석은 매우 도전적인 문제이다. 이와 동시에, 우리는 조밀하게 송신 및 수신기를 위치 시켜야 하는 이 MCSEM시스템은 구조보정법의 적용에 아주 적합하다는 것을 보여줄 것이다. 구조보정장 계산의 속도를 증가시키기 위해 우리는 직접 개발한 다중격자 준선형 (MGQL) 근사법에 기초한 적분방정식 해의 빠른 형태를 적용시켰다. 이 논문에서 공식화된 구조보정 영상 원리는 전형적인 해저 석유 저류층 모델에 적용되어 시험 되었다.

• 한국항공우주학회지
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• 제33권8호
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• pp.8-17
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• 2005

저속 압축성 유동에서 사용하는 예조건화 기법은 수렴성 증진에 효과적이다. 본 연구에서는 일반적인 오일러 지배 방정식과 각각 다르게 무차원화한 세 가지 종류의 예조건화 기법을 3차 공간 정확도의 MUSCL, DADI, 다중 격자, 국소 시간 전진 기법을 이용하여 2차원 비점성 bump 유동에 적용하였다. 결과적으로 국소 예조건화 기법에 전역 예조건화 기법의 압력 항 무차원화 방법을 적용하면, 마하수에 무관한 수렴 특성을 얻을 수 있다. 또한, 점근해석을 이용하여 각 예조건화 기법의 특성에 대해 언급하였다.

• 한국전산유체공학회지
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• 제9권1호
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• pp.34-40
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• 2004

Recently with growth in the capability of super computers and Parallel computers, shape design optimization is becoming easible for real problems. Also, Computational Fluid Dynamics(CFD) techniques have been improved for higher reliability and higher accuracy. In the shape design optimization, analysis solvers and optimization schemes are essential. In this work, the Roe's 2nd-order Upwind TVD scheme and DADI time march with multigrid were used for the flow solution with the Euler equation and FDM(Finite Differenciation Method), GA(Genetic Algorithm) and Kriging were used for the design optimization. Kriging were applied to 2-D airfoil design optimization and compared with FDM and GA's results. When Kriging is applied to the nonlinear problems, satisfactory results were obtained. From the result design optimization by Kriging method appeared as good as other methods.

• 한국정보통신학회논문지
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• 제26권2호
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• pp.181-186
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• 2022

포아즌 볼츠만 방정식 (Poisson-Boltzmann equation, PBE)은 생물물리, 콜로이드 화학 등에서 등장하는 문제들을 모델링하는데 사용되는 방정식이다. 따라서 PBE의 수치해를 효율적으로 예측하는 것은 중요한 이슈이다. 저자들은 기존의 연구에서 PBE를 풀기위한 딥러닝 방법을 제안하였으나, 딥러닝을 훈련하기 위한 샘플을 생성하는 시간이 컸다는 어려움이 있었다. 본 논문에서는 FEM 수치해를 생성하는데 걸리는 시간을 줄이는 두가지 방안을 마련하였다. 첫째로 대수 방정식을 만들 때 bilinar form에 포함되는 penalty 파라메터를 실험적으로 조정하였다. 두 번째로, 대수적멀티그리드 기법을 활용하여 대수 방정식의 컨디션 넘버를 meshsize와 무관하게 만들었다. 따라서 PBE 방정식의 대수 방정식을 풀 때 계산 시간을 효과적으로 줄였다. 이러한 대수적 멀티그리드를 사용한 방법은 다양한 분야에서 딥러닝의 샘플을 생성하는데 효과적으로 활용될 수 있을 것으로 기대된다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제17권3호
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• pp.197-207
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• 2013

The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn-Hillard equation, many numerical methods have been proposed such as the explicit Euler's, the implicit Euler's, the Crank-Nicolson, the semi-implicit Euler's, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler's method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank-Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.