• Journal for History of Mathematics
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• v.31 no.6
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• pp.273-289
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• 2018

The KaiFangFa開方法 of traditional mathematics was completed in ${\ll}$JiuZhang SuanShu九章算術${\gg}$ originally, and further organized in Song宋 $Yu{\acute{a}}n$元 dinasities. The former is the ShiSuoKaiFangFa釋鎖開方法 using the coefficients of the polynomial expansion, and the latter is the ZengChengKaiFangFa增乘開方法 obtaining the solution only by some mechanical numerical manipulations. ${\ll}$SanHak JeongEui算學正義${\gg}$ basically used the latter and improved the estimate-value array by referring to the written-calculation in ${\ll}$ShuLi JingYun數理精蘊${\gg}$. As a result, ZengChengKaiFangFa was more refined so that the KaiFangFa algorithm is more consistent.

• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3494-3515
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• 2022

This paper presents the development of a nodal diffusion code, RAST-V, and its verification and validation for VVER (vodo-vodyanoi energetichesky reactor) analysis. A VVER analytic solver has been implemented in an in-house nodal diffusion code, RAST-K. The new RAST-K version, RAST-V, uses the triangle-based polynomial expansion nodal method. The RAST-K code provides stand-alone and two-step computation modes for steady-state and transient calculations. An in-house lattice code (STREAM) with updated features for VVER analysis is also utilized in the two-step method for cross-section generation. To assess the calculation capability of the formulated analysis module, various verification and validation studies have been performed with Rostov-II, and X2 multicycles, Novovoronezh-4, and the Atomic Energy Research benchmarks. In comparing the multicycle operation, rod worth, and integrated temperature coefficients, RAST-V is found to agree with measurements with high accuracy which RMS differences of each cycle are within ±47 ppm in multicycle operations, and ±81 pcm of the rod worth of the X2 reactor. Transient calculations were also performed considering two different rod ejection scenarios. The accuracy of RAST-V was observed to be comparable to that of conventional nodal diffusion codes (DYN3D, BIPR8, and PARCS).

• Geophysics and Geophysical Exploration
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• v.13 no.1
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• pp.118-127
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• 2010

Resistivity logging instruments are designed to measure the electrical resistivity of a formation, and this can be directly interpreted to provide a water-saturation profile. However, resistivity logs are sensitive to borehole and shoulder-bed effects, which often result in misinterpretation of the results. These effects are emphasised more in the presence of tool eccentricity. For precise interpretation of short- and long-normal logging measurements in the presence of tool eccentricity, we simulate and analyse eccentricity effects by combining the use of a Fourier series expansion in a new system of coordinates with a 2D goal-oriented high-order self-adaptive hp finite-element refinement strategy, where h denotes the element size and p the polynomial order of approximation within each element. The algorithm automatically performs local mesh refinement to construct an optimal grid for the problem under consideration. In addition, the proper combination of h and p refinements produces highly accurate simulations even in the presence of high electrical resistivity contrasts. Numerical results demonstrate that our algorithm provides highly accurate and reliable simulation results. Eccentricity effects are more noticeable when the borehole is large or resistive, or when the formation is highly conductive.

• Journal of the Korea Concrete Institute
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• v.16 no.2 s.80
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• pp.155-162
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• 2004

An analytical method to predict the time dependent flexural behavior of composite girder is presented based on sectional analysis. The time dependent constitutive relation accounting for the early-age concrete properties including maturing of elastic modulus, creep and shrinkage is derived in an incremental format by the first order Taylor series expansion. The sectional analysis calculates the axial and curvature strains based on the force and moment equilibriums. The deflection curve of the girder approximated by the quadratic polynomial function is calculated by applying to the proper boundary conditions in the consecutive segments. Numerical applications are made for the 3-span double composite steel box girder which is a composite bridge girder filled with concrete at the bottom of the steel box in the negative moment region. The calculated results are compared with those by finite element analysis results. Close agreement is observed between the two approaches.

• Journal of the Society of Naval Architects of Korea
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• v.28 no.1
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• pp.92-98
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• 1991

The structures such as railway bridges can be modelled as the multi-span beam on the elastic foundation. These structures are usually subject to the moving load, which has a great effect on dynamic stresses and can cause severe motions, especially at high velocities. In this paper, the dynamic responses of the multi-span beam on the elastic foundation were obtained by using the Galerkin's method and the numerical time integration technique. As trial functions, the same orthogonal polynomial functions obtained in part 1, were used. From the numerical results, it was found that the one term expansion of the assumed solution usually leads to the accurate solutions. However, in the case that the stiffness of the transnational spring is very high or the rotational spring is placed where the slope of the first mode is zero, the higher modes must be included to obtain the accurate solutions.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.18 no.3
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• pp.291-302
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• 2005

Due to the importance of the parameter in structural response, the uncertain elastic modulus was located at the center of stochastic analysis, where the response variability caused by the uncertain system parameters is pursued. However when we analyze the so-called stochastic systems, as many parameters as possible must be included in the analysis if we want to obtain the response variability that can reach a true one, even in an approximate sense. In this paper, a formulation to determine the statistical behavior of in-plane structures due to multiple uncertain material parameters, i.e., elastic modulus and Poisson's ratio, is suggested. To this end, the polynomial expansion on the coefficients of constitutive matrix is employed. In constructing the modified auto-and cross-correlation functions, use is made of the general equation for n-th moment. For the computational purpose, the infinite series of stochastic sub-stiffness matrices is truncated preserving required accuracy. To demons4rate the validity of the proposed formulation, an exemplary example is analyzed and the results are compared with those obtained by means of classical Monte Carlo simulation, which is based on the local averaging scheme.

• Proceedings of the Korea Water Resources Association Conference
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• 2020.06a
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• pp.381-381
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• 2020

자료동화(Data Assimilation) 기법은 실시간 수문학적 예측에 있어 정확도 향상을 위해 필수적인 과정이다. 가장 대중적으로 사용되는 기법들 중 하나가 모델 상태변수와 매개변수를 동시에 업데이트할 수 있는 이중 앙상블 칼만 필터(Dual Ensemble Kalman Filter)이다. 이 방법은 정확도 개선 및 적용의 용이성 때문에 많은 연구 분야에서 사용되어져 왔지만, 앙상블을 생성하는 과정에서 상당시간이 소요되는 단점이 존재한다. 본 연구에서는 상태변수와 매개변수를 동시에 업데이트 하면서 홍수 예측의 정확성을 보장할 뿐만 아니라, 앙상블 생성에 있어 계산 효율을 크게 향상시킬 수 있는 기법을 제안한다. Polynomial Chaos Expansion(PCE) 기법을 사용하여 앙상블 칼만 필터를 모방(mimic)할 수 있는 새로운 대체필터(Surrogate Filter)를 개발하는 것을 목표로 한다. 구체적으로 대체필터를 구성하기 위한 다양한 필터를 설계하였다. 첫째 시간에 대해서 PCE가 변화하지 않는 '불변 필터'(즉, 전체 예측기간에 대해 하나의 필터를 사용하여 자료동화할 수 있는 대체필터)와, 매 시간마다 PCE가 변화하는 '시변 필터'(즉, 예측하는 매 시간마다 새로운 필터를 생성해야 하는 대체필터)를 설계하여 적용성, 정확성, 예측성 등을 비교하였다. 또한, PCE의 하이퍼 매개변수를 최적화하기 위한 최적의 프레임 워크가 제안되어, 대체필터를 구축하는 데 효율을 높이고 PCE의 과적합(overfitting) 현상을 피할 수 있도록 하였다. 본 연구에서 제안된 기법은 기존 단일 및 이중 앙상블 칼만 필터(EnKF)의 결과와 비교 검증하였으며, 그 결과는 다음과 같다. (1) 대체필터의 대부분은 원래 EnKF와 비슷한 정도의 불확실성을 설명할 수 있음; (2) 모든 대체 필터는 선행시간이 짧은 경우의 예측에 있어 우수한 결과를 제공하며, 시변 필터가 불변 필터보다 더 정확한 예측 결과를 제공함; (3) 대체필터는 원래 앙상블 칼만필터보다 최대 500배 빠른 속도로 성능을 향상시킬 수 있음. 제안된 대체필터는 자료동화를 수행하는 기존필터와 비슷한 정도의 정확성, 매우 향상된 효율성을 보장함을 확인할 수 있었다.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.1
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• pp.199-219
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• 2011

In this thesis, we designed a experimental learning-teaching plan of 'decimal fraction concept' at the 4-th grade level. We rest our plan on two basic premises. One is the fact that a essential concept of decimal fraction is 'polynomial of which indeterminate is 10', and another is the fact that the origin of decimal fraction is successive measurement activities which improving accuracy through decimal partition of measuring unit. The main features of our experimental learning-teaching plan is as follows. Firstly, students can experience a operation which generate decimal unit system through decimal partitioning of measuring unit. Secondly, the decimal fraction expansion will be initially introduced and the complete representation of decimal fraction according to positional notation will follow. Thirdly, such various interpretations of decimal fraction as 3.751m, 3m+7dm+5cm+1mm, $(3+\frac{7}{10}+\frac{5}{100}+\frac{1}{1000})m$ and $\frac{3751}{1000}m$ will be handled. Fourthly, decimal fraction will not be introduced with 'unit decimal fraction' such as 0.1, 0.01, 0.001, ${\cdots}$ but with 'natural number+decimal fraction' such as 2.345. Fifthly, we arranged a numeration activity ruled by random unit system previous to formal representation ruled by decimal positional notation. A experimental learning-teaching plan which presented in this thesis must be examined through teaching experiment. It is necessary to successive research for this task.

• Korean Journal of Optics and Photonics
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• v.29 no.4
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• pp.166-172
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• 2018

In this paper, we propose a new method to predict a mirror's surface deformation due to the stress of moisture release by a coating in the environment of outer space. We measured the surface deformation of circular samples 50 mm in diameter and 1.03 mm thick, using an interferometer. The results were analyzed using Zernike fringe polynomials. The coating stress caused by moisture release was calculated to be 152.7 MPa. This value was applied to an analytic model of a 1.25 mm thickness sample mirror, confirming that the change of surface deformation could be predicted within the standard deviation of the measurement result ($78.9{\pm}5.9nm$). Using this methodology, we predicted the surface deformation of 600 mm hyperbolic mirror for the Compact Advanced Satellite, which will be launched in 2019. The result is only $2.005{\mu}m$ of focal shift, leading to 2.3% degradation of modulation transfer function (MTF) at the Nyquist frequency, which satisfies the requirement.

• KIPS Transactions on Software and Data Engineering
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• v.12 no.3
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• pp.141-148
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• 2023

Accurate overlay metrology is essential to achieve high yields of semiconductor products. Overlay metrology performance is greatly affected by overlay target design and measurement method. Therefore, in order to improve the performance of the overlay target, measurement methods applicable to various targets are required. In this study, we propose a new algorithm that can measure image-based overlay. The proposed measurement algorithm can estimate the sub-pixel position by using a motion vector. The motion vector may estimate the position of the sub-pixel unit by applying a quadratic equation model through polynomial expansion using pixels in the selected region. The measurement method using the motion vector can calculate the stacking error in all directions at once, unlike the existing correlation coefficient-based measurement method that calculates the stacking error on the X-axis and the Y-axis, respectively. Therefore, more accurate overlay measurement is possible by reflecting the relationship between the X-axis and the Y-axis. However, since the amount of computation is increased compared to the existing correlation coefficient-based algorithm, more computation time may be required. The purpose of this study is not to present an algorithm improved over the existing method, but to suggest a direction for a new measurement method. Through the experimental results, it was confirmed that measurement results similar to those of the existing method could be obtained.