• 대한화학회지
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• 제22권3호
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• pp.117-127
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• 1978

두점 A와 B에 위치한 Slater원자궤도함수의 spherical harmonics부와 지름부를 기준점 A를 중심으로 전개하여 공통좌표상에 기술하였다. 이 Slater 원자궤도함수의 전개식을 사용하여 two center overlap integral의 기본식을 유도하였으며 이 기본식을 이용하여 $CH_4,\;H_2O,\;NH_3,\;C_2H_6$ 및 $PH_3$ 분자의 two center overlap integral을 계산하였을 때 이 값이 Mulliken의 값과 일치하였다.

• Nuclear Engineering and Technology
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• 제53권12호
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• pp.3861-3878
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• 2021

The steady-state simplified spherical harmonics equations (SP_N equations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SP_N equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full system.

• Nuclear Engineering and Technology
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• 제54권9호
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• pp.3478-3487
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• 2022

In this study, we incorporate an anisotropic scattering scheme involving spherical harmonics into the method of characteristics (MOC). The neutron transport solution in a light water reactor can be significantly improved because of the impact of an anisotropic scattering source with the MOC flat source approximation. Several problems are selected to verify the proposed scheme and investigate its effects and accuracy. The MOC anisotropic scattering source is based on the expansion of spherical harmonics with Legendre polynomial functions. The angular flux, scattering source, and cross section are expanded in terms of the surface spherical harmonics. Later, the polynomial is expanded to achieve the odd and even parity of the source components. Ultimately, the MOC angular and scalar fluxes are calculated from a combination of two sources. This paper presents various numerical examples that represent the hot and cold conditions of a reactor core with boron concentration, burnable absorbers, and control rod materials, with and without a reflector or baffle. Moreover, a small critical core problem is considered which involves significant neutron leakage at room temperature. We demonstrate that an anisotropic scattering source significantly improves solution accuracy for the small core high-leakage problem, as well as for practical large core analyses.

• 대한화학회지
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• 제23권3호
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• pp.125-131
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• 1979

한 쌍의 slater type orbitals에 대한 two overlap integrals을 계산하는 방법이 Mulliken등에 의하여 발전되었다. 이 방법으로 two center integrals을 계산하기 위해서는 한쌍의 Slater type orbital,에 대한 극좌표들 타원좌표로 변환해야한다. 두 점에 위치한 Slater type orbital을 공통좌표상에 전개시키는 새로운 방법 즉 spherical harmonics의 전개방법이 two center overlap integrals, 을 계산하는데 응용되었다. 이 새로운 방법에서는 Slater type orbitals, 을 기준점에 대해 전개시키는 것이 필요하다. 본 연구에서는 two center overlap integral을 계산하기 위한 spherical harmonics 전개방법을 $|3s{\g}$, $|5s{\g}$및 $|5s{\g}$에 까지 확장시켰다. 이들 원자궤도함수의 전개식을 사용하여 two center overlap integrals의 기본식을 유도하였으며, 이 기본식을 사용하여 가상적인 NO 분자에 대한 two cunter overlap integrals의 계산값이 이미 보고된 값과 일치하였다.

• Bulletin of the Korean Chemical Society
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• 제7권4호
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• pp.299-304
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• 1986

Master formulas for overlap integrals and the dipole moments involving |4p > atomic orbitals have been derived by the expansion method for spherical harmonics. The radial integrals for the hyperfine interaction have also been derived for |4p > atomic orbitals. The calculated values of the overlap integrals and dipole moment matrix elements by the expansion method for spherical harmonics for a hypothetical NO molecule are exactly in agreement with those of Mulliken's method. The radial integrals for the hyperfine interaction may be used to calculate the chemical shift for |4p > atomic orbitals.

• 대한화학회지
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• 제22권6호
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• pp.357-364
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• 1978

Spherical harmonics의 전개방법에 의하여 쌍극자모멘트의 행렬요소를 계산하는 방법을 사용하여 $NH_3$, HF, CO, HCHO, HCN, PO, $PO^-\;및\;H_2O$분자의 쌍극자모멘트를 계산하였다. 이 방법에 의하여 계산한 쌍극자모멘트의 값이 다른 방법의 값보다 실험치에 가까웠다.

• Journal of the Korean Statistical Society
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• 제33권3호
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• pp.255-269
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• 2004

This paper develops log-density estimation for directional data. The methodology is to use expansions with respect to spherical harmonics followed by estimating the unknown parameters by maximum likelihood. Minimax rates of convergence in terms of the Kullback-Leibler information divergence are obtained.

• 대한기계학회논문집
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• 제15권3호
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• pp.1021-1029
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• 1991

본 연구에서는 2차원 정사각형 밀폐공간내에서 흡수 및 방사하는 회기체에 대 한 자연대류-복사 열전달을 P-1 및 P-3 근사법을 이용하고 수치해석을 통하여 유동 및 열전달 특성을 연구하였고 Plank 수, 광학두께 및 벽방사율의 영향을 조사하였다. 또한 P-3 근사해와 비교함으로써 P-1 근사해의 적용범위를 고찰하였다.

• 한국전자파학회논문지
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• 제26권1호
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• pp.88-93
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• 2015

대규모 물체의 RCS(Radar Cross Section)값을 임의의 각도에서 예측하기 위해, 미리 계산된 샘플링 지점 외 각도에서는 보간법을 사용한다. 대규모 물체의 경우, RCS 데이터베이스를 구성하기 위해 많은 입사각도에서 RCS값을 계산해야 한다. 이는 수치적으로 시간이 많이 걸려, 실질적으로 필요한 모든 입사각에서 RCS값을 미리 계산하기는 어렵다. 그러므로 가능한 적은 샘플을 이용하여 데이터베이스를 구축하고, 이를 보간하여 RCS값을 예측하는 방법이 필요하다. 본 논문에서는 계산된 RCS를 임의의 각도에서 예측하기 위한 보간법으로 Sinc 함수 및 VSH(Vector Spherical Harmonics) 함수를 이용한 방법을 고려하고, 그 정확성을 시뮬레이션을 통하여 검증한다.

• 한국지구과학회지
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• 제34권5호
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• pp.393-401
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• 2013

Spherical harmonics power spectrum of the geopotential field of Gaussian-bell type on the sphere was investigated using integral formula that is associated with Legendre polynomials. The geopotential field of Gaussian-bell type is defined as a function of sine of angular distance from the bell's center in order to guarantee the continuity on the global domain. Since the integral-formula associated with the Legendre polynomials was represented with infinite series of polynomial, an estimation method was developed to make the procedure computationally efficient while preserving the accuracy. The spherical harmonics power spectrum was shown to vary significantly depending on the scale parameter of the Gaussian bell. Due to the accurate procedure of the new method, the power (degree variance) spanning over orders that were far higher than machine roundoff was well explored. When the scale parameter (or width) of the Gaussian bell is large, the spectrum drops sharply with the total wavenumber. On the other hand, in case of small scale parameter the spectrum tends to be flat, showing very slow decaying with the total wavenumber. The accuracy of the new method was compared with theoretical values for various scale parameters. The new method was found advantageous over discrete numerical methods, such as Gaussian quadrature and Fourier method, in that it can produce the power spectrum with accuracy and computational efficiency for all range of total wavenumber. The results of present study help to determine the allowable maximum scale parameter of the geopotential field when a Gaussian-bell type is adopted as a localized function.