• Korean Journal of Mathematics
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• 제12권1호
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• pp.67-75
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• 2004

Paul Turan observed that the Legendre polynomials satisfy the inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)$ > 0, $-1{\leq}x{\leq}1$. And G. Gasper(ref. [6], ref. [7]) proved such an inequality for Jacobi polynomials and J. Bustoz and N. Savage (ref. [2]) proved $P^{\alpha}_n(x)P^{\beta}_{n+1}(x)-P^{\alpha}_{n+1}(x)P{\beta}_n(x)$ > 0, $\frac{1}{2}{\leq}{\alpha}$ < ${\beta}{\leq}{\alpha}+2.0$ < $x$ < 1, for the ultraspherical polynomials (respectively, Laguerre ploynomials). The Bustoz-Savage inequalities hold for Laguerre and ultraspherical polynomials which are symmetric. In this paper, we prove some similar inequalities for non-symmetric Jacobi polynomials.

• 대한전자공학회논문지SD
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• 제39권2호
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• pp.27-38
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• 2002

양자 우물 반도체 소자 모델링에 있어서 양자 우물의 다중 에너지 부준위 각각에 대한 Boltzmann 방정식의 해를 직접적으로 구하는 self-consistent한 방법을 개발하였다 양자 우물의 특성을 고려하여 Schrodinger 방정식과 Poisson 방정식 및 Boltzmann 방정식으로 구성된 양자 우물 소자 모델을 설정하였으며 이들의 직접적인 해를 유한 차분법과 Gummel-type iteration scheme에 의해 구하였다. Si MOSFET의 inversion 영역에 형성되는 양자 우물에 적용하여 그 시뮬레이션 결과로부터 본 방법의 타당성 및 효율성을 보여 주었다.

• 대한토목학회논문집
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• 제12권1호
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• pp.43-50
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• 1992

본(本) 연구의 목적(目的)은 평면응력/변형과 축대칭 및 쉘문제를 포함하는 다양한 응용문제에서 계층적(階層的) 성질을 갖는 적분형 르장드르 형상함수에 의한 P-version 모델의 통용성(通用性)을 확인하는 것이다. 현대 유한요소 해석에서 정확도를 확보하지 못하는 가장 큰 이유는 비(非)압축성 재료와 망목(網目)설계시 요소의 형상비(形狀比), 사다리꼴 요소에서 변(邊)의 감소비(減少比)와 평행사변형 요소의 왜곡도(歪曲度) 등을 갖는 불규칙 형상에서 나타나는 가상메카니즘과 Locking 현상이다. 조건수(條件數)와 에너지 노름이 계산오차, 수렴성 및 알고리즘의 효율성을 검증하는데 사용되었으며 해석결과는 NASTRAN과 SAP90 및 Cheung이 제안한 Hybrid 요소와 비교되었다. NASTRAN을 제외한 SAP90 및 P-version 프로그램은 16 Bit 소형컴퓨터에 의해 실행되었다.

• 대한조선학회논문집
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• 제29권2호
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• pp.66-72
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• 1992

유한수심에 대한 Michell적분의 계산을 위해 선각함수를 종방향 및 수직방향에 대해 Legendre다항식으로 전개하여 조파저항계수를 형상계수와 유체동력학적계수의 곱에 대한 4중급수로 구할 수 있는 식을 얻었다. 여기서 형상계수는 선각의 기하학적 형상만의 함수이고, 유체동력학적계수는 수심에 근거한 Fn와 수심과 홀수의 배의 길이에 대한 비들만의 함수이다. Wigley의 포물선형 선각과 Series 60의 $C_B$ 0.6에 대한 계산을 수행하고 그 결과를 기존의 실험결과(무한수심) 및 다른 이론결과(유한수심)와 비교하였다.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1055-1071
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• 2010

Recently, a number of algorithms have been proposed for numerical evaluation of Hankel transforms as these transforms arise naturally in many areas of science and technology. All these algorithms depend on separating the integrand $rf(r)J_{\upsilon}(pr)$ into two components; the slowly varying component rf(r) and the rapidly oscillating component $J_{\upsilon}(pr)$. Then the slowly varying component rf(r) is expanded either into a Fourier Bessel series or various wavelet series using different orthonormal bases like Haar wavelets, rationalized Haar wavelets, linear Legendre multiwavelets, Legendre wavelets and truncating the series at an optimal level; or approximating rf(r) by a quadratic over the subinterval using the Filon quadrature philosophy. The purpose of this communication is to take a different approach and replace rapidly oscillating component $J_{\upsilon}(pr)$ in the integrand by its Bernstein series approximation, thus avoiding the complexity of evaluating integrals involving Bessel functions. This leads to a very simple efficient and stable algorithm for numerical evaluation of Hankel transform.

• 한국전산구조공학회논문집
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• 제23권6호
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• pp.615-626
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• 2010

The discrete-layer model is proposed to analyze the edge-effect problem of laminates under extension and flexure. Based on three-dimensional elasticity theory, the displacement fields of each layer in a laminate have been treated discretely in terms of three displacement components across the thickness. The displacement fields at bottom and top surfaces within a layer are approximated by two-dimensional shape functions. Then two surfaces are connected by one-dimensional high order shape functions. Thus the p-convergent refinement on approximated one- and two-dimensional shape functions can be implemented independently of each other. The quality of present model is mostly determined by polynomial degrees of shape functions for given displacement fields. For nodal modes with physical meaning, the linear Lagrangian polynomials are considered. Additional modes without physical meaning, which are created by increasing nodeless degrees of shape functions, are derived from integrals of Legendre polynomials which have an orthogonality property. Also, it is assumed that mapping functions are linear in the light of shape of laminated plates. The results obtained by this proposed model are compared with those available in literatures. Especially, three-dimensional out-of-plane stresses in the interior and near the free edges are evaluated and convergence performance of the present model is established with the stress results.

• Kyungpook Mathematical Journal
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• 제50권2호
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• pp.177-193
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• 2010

Working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the type $F_n(x)=\sum\limits_{k=0}^{n}A^n_kG_k(x)$ and $ G_n(x)=\sum\limits_{k=0}^{n}B_k^nF_k(x)$, where 0, 1, 2,$\cdots$. Here $F_n(x)$, $G_n(x)$ denote special polynomial functions, and $A_k^n$, $B_k^n$ denote coefficients found by use of the orthogonal properties of $F_n(x)$ and $G_n(x)$, or by skillful series manipulations. Typically $G_n(x)=x^n$ and $F_n(x)=P_n(x)$, the n-th Legendre polynomial. We give a collection of inverse series pairs of the type $f(n)=\sum\limits_{k=0}^{n}A_k^ng(k)$ if and only if $g(n)=\sum\limits_{k=0}^{n}B_k^nf(k)$, each pair being based on some reasonably well-known special function. We also state and prove an interesting generalization of a theorem of Rainville in this form.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2007년도 정기 학술대회 논문집
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• pp.481-486
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• 2007

The Zienkiewicz-Zhu(Z/Z) error estimate is slightly modified for the hierarchical p-refinement, and is then applied to L-shaped plates subjected to bending to demonstrate its effectiveness. An adaptive procedure in finite element analysis is presented by p-refinement of meshes in conjunction with a posteriori error estimator that is based on the superconvergent patch recovery(SPR) technique. The modified Z/Z error estimate p-refinement is different from the conventional approach because the high order shape functions based on integrals of Legendre polynomials are used to interpolate displacements within an element, on the other hand, the same order of basis function based on Pascal's triangle tree is also used to interpolate recovered stresses. The least-square method is used to fit a polynomial to the stresses computed at the sampling points. The strategy of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed such that a particular element has to be refined automatically to obtain an acceptable level of accuracy by increasing p-levels non-uniformly or selectively. It is noted that the error decreases rapidly with an increase in the number of degrees of freedom and the sequences of p-distributions obtained by the proposed error indicator closely follow the optimal trajectory.

• 항공우주기술
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• 제13권2호
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• pp.60-65
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• 2014

본 논문에서는 적분 행렬을 이용한 수치적분 방법을 연구하였다. 비선형 운동방정식에 대한 빠른 수치적분을 수행하기 위해, 적분 행렬을 이용한 개선된 fixed point iteration 방법을 소개한다. 예제로는 궤도 운동에 대한 수치적분 예를 고려한다. 수치 예제를 통하여, 본 논문에서 연구되는 알고리듬이 적분의 정밀도는 크게 저하시키지 않음과 동시에, 계산시간 측면에서 효과적이라는 것을 보인다.

• Structural Engineering and Mechanics
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• 제2권3호
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• pp.245-256
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• 1994

A new axisymmetric crack model is proposed on the basis of p-version of the finite element method limited to theory of small scale yielding. To this end, axisymmetric stress element is formulated by integrals of Legendre polynomial which has hierarchical nature and orthogonality relationship. The virtual crack extension method has been adopted to calculate the stress intensity factors for 3-D axisymmetric cracked bodies where the potential energy change as a function of position along the crack front is calculated. The sensitivity with respect to the aspect ratio and Poisson locking has been tested to ascertain the robustness of p-version axisymmetric element. Also, the limit value that is an exact solution obtained by FEM when degree of freedom is infinite can be estimated using the extrapolation equation based on error prediction in energy norm. Numerical examples of thick-walled cylinder, axisymmetric crack in a round bar and internal part-thorough cracked pipes are tested with high precision.