• Journal of applied mathematics & informatics
• /
• 제27권3_4호
• /
• pp.517-529
• /
• 2009

In this paper, a singularly perturbed reaction-convection-diffusion problem with two parameters is considered. A parameter-uniform error bound for the numerical derivative is derived. The numerical method considered here is a standard finite difference scheme on piecewise-uniform Shishkin mesh, which is fitted to both boundary and initial layers. Numerical results are provided to illustrate the theoretical results.

• 대한수학회논문집
• /
• 제34권3호
• /
• pp.1015-1027
• /
• 2019

A class of systems of singularly perturbed convection diffusion type equations with integral boundary conditions is considered. A numerical method based on a finite difference scheme on a Shishkin mesh is presented. The suggested method is of almost first order convergence. An error estimate is derived in the discrete maximum norm. Numerical examples are presented to validate the theoretical estimates.

• 정보처리학회논문지B
• /
• 제9B권3호
• /
• pp.277-286
• /
• 2002

양자역학 섭동이론과 유전자프로그래밍(GP) 기법을 접목시킴으로써 실세계(Real-world)에서 발생하는 카오스 시계열에 대하여 수학모델을 구축, 예측하기 위한 새로운 알고리즘을 개발하였다. 시계열 분석과 양자역학 파동방정식의 해를 구하는 섭동이론과의 절차적 유사성을 논하고, 이것을 GP로 구현하는 전형적 접근방안을 제시한다. 함수집합(Function Set)으로서 직교함수(Orthogonal Functions)를 이용하고 병렬 집단을 사용하는 GP를 이용하여 원 시계열에 대한 초기 수학모델을 구하고, 원 시계열 데이터로부터 모델의 평가값을 뺀 나머지로 구성되는 잔여 시계열에 대하여 다시 GP를 적용하는 과정을 일정한 종료조건이 충족될 때가지 반복함으로써 실세계 카오스 시계열에 대한 정확성 높은 수학모델을 구축하는데 성공하였다. 타 방법론과의 비교와 향후 해결과제에 대하여도 소개한다.

• Geomechanics and Engineering
• /
• 제19권5호
• /
• pp.463-472
• /
• 2019

The mode perturbation method (MPM) is suitable and efficient for solving the eigenvalue problem of a nonuniform soil deposit whose property varies with depth. However, results of the MPM do not always converge to the exact solution, when the variation of soil deposit property is discontinuous. This discontinuity is typical because soil is usually made up of sedimentary layers of different geologic materials. Based on the energy integral of the variational principle, a new mode perturbation method, the energy-based mode perturbation method (EMPM), is proposed to address the convergence of the perturbation solution on the natural frequencies and the corresponding mode shapes and is able to find solution whether the soil properties are continuous or not. First, the variational principle is used to transform the variable coefficient differential equation into an equivalent energy integral equation. Then, the natural mode shapes of the uniform shear beam with same height and boundary conditions are used as Ritz function. The EMPM transforms the energy integral equation into a set of nonlinear algebraic equations which significantly simplifies the eigenvalue solution of the soil layer with variable properties. Finally, the accuracy and convergence of this new method are illustrated with two case study examples. Numerical results show that the EMPM is more accurate and convergent than the MPM. As for the mode shapes of the uniform shear beam included in the EMPM, the additional 8 modes of vibration are sufficient in engineering applications.

• 대한기계학회논문집A
• /
• 제33권3호
• /
• pp.243-250
• /
• 2009

The adjoint variable method can reduce computation time and save computer resources because it can selectively provide the sensitivity information for the positions that designers wish to measure. However, the adjoint variable method commonly employs exact analytical differentiation with respect to the design variables. It can be cumbersome to precisely differentiate every given type of finite element. This trouble can be overcome only if the numerical differentiation scheme can replace this exact manner of differentiation. But, the numerical differentiation scheme causes of severe inaccuracy due to the perturbation size dilemma. For assuring the accurate sensitivity without any dependency of perturbation size, this paper employs a complex variable that has been mainly used for computational fluid dynamics problems. The adjoint variable method combined with complex variables is applied to obtain the shape and size sensitivity for structural optimization. Numerical examples demonstrate that the proposed method can predict stable sensitivity results and that its accuracy is remarkably superior to traditional sensitivity evaluation methods.

• Journal of applied mathematics & informatics
• /
• 제39권3_4호
• /
• pp.419-435
• /
• 2021

This paper deals with numerical treatment of singularly perturbed differential equations involving small delays. The highest order derivative in the equation is multiplied by a perturbation parameter 𝜀 taking arbitrary values in the interval (0, 1]. For small 𝜀, the problem involves a boundary layer of width O(𝜀), where the solution changes by a finite value, while its derivative grows unboundedly as 𝜀 tends to zero. The considered problem contains delay on the convection and reaction terms. The terms with the delays are approximated using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. Inducing exponential fitting factor for the term containing the singular perturbation parameter and using central finite difference for the derivative terms, numerical scheme is developed. The stability and uniform convergence of difference schemes are studied. Using a priori estimates we show the convergence of the scheme in maximum norm. The scheme converges with second order of convergence for the case 𝜀 = O(N^-1) and for the case 𝜀 ≪ N^-1, the scheme converge uniformly with first order of convergence, where N is number of mesh intervals in the domain discretization. We compare the accuracy of the developed scheme with the results in the literature. It is found that the proposed scheme gives accurate result than the one in the literatures.

• 터널과지하공간
• /
• 제13권5호
• /
• pp.389-396
• /
• 2003

본 연구는 정상류 Navier-Stokes 방정식에 섭동(perturbation) 이론을 적용하여 주기함수 간극에 대한 삼승법칙의 수정에 대해 논하였다. 이를 위해, 주기함수를 진폭과 파장에 대한 무차원 함수로 전환한 뒤 미소 계수에 대한 무차원 유동함수와 연속 방정식을 적용하였다. 이러한 과정을 통해 정상류 Navier-Stokes 방정식의 섭동 근사해를 구하였으며 이를 유한 차분법에 적용하였다. 단일 절리 모델에 대한유한 차분 수치해석을 통해, 수정된 삼승 법칙이 주기함수 간극의 유체 유동에 대한 정상류 Navier-Stokes 방정식의 섭동 근사해와 잘 일치하는 것으로 나타났다. 이를 통해 본 연구에서 제시된 삼승 법칙이 간극 분포에 따른 유체 유동의 평가에 있어 유용하게 적용될 수 있는 것으로 나타났다.

• 한국항공우주학회지
• /
• 제31권3호
• /
• pp.31-38
• /
• 2003

본 연구는 최적설계에 사용되는 역섭동법을 구조물의 손상탐지에 적용하였다. 이 방법은 손상의 위치를 정확하게 탐지하기 위하여 미지수보다 많은 수의 구속조건이 필요하므로, 최적설계와는 달리 비선형 회소자승법을 수치기법에 사용한다. 한편 손상탐지의 경우, 모든 자유도의 응답이 측정가능한 것은 아니며 제한된 수의 센서에서 부분적인 진동모드만 측정할 수 있다. 이처럼 부분적인 정보를 사용하여, 손상진단을 성공적으로 수행하기 위해서는 사용될 센서의 수와 위치를 결정하는 연구가 매우 중요한다. 본 논문에서는 센서의 개수가 결정되었을 때, 손상탐지에 적합한 센서 위치의 선정방법에 관하여 연구하였다. 이러한 연구의 결과로 순차적 소거법이 역섭동법을 이용한 손상탐지에 가장 적합한 센서위치의 선정방법임을 수치 예에서 확인하였다.

• Smart Structures and Systems
• /
• 제23권6호
• /
• pp.641-651
• /
• 2019

The stochastic stability control of the parameter-excited vibration of an inclined stay cable with multiple modes coupling under random and periodic combined support disturbances is studied by using the direct eigenvalue analysis approach based on the response moment stability, Floquet theorem, Fourier series and matrix eigenvalue analysis. The differential equation with time-varying parameters for the transverse vibration of the inclined cable with control under random and deterministic support disturbances is derived and converted into the randomly and deterministically parameter-excited multi-degree-of-freedom vibration equations. As the stochastic stability of the parameter-excited vibration is mainly determined by the characteristics of perturbation moment, the differential equation with only deterministic parameters for the perturbation second moment is derived based on the $It{\hat{o}}$ stochastic differential rule. The stochastically and deterministically parameter-excited vibration stability is then determined by the deterministic parameter-varying response moment stability. Based on the Floquet theorem, expanding the periodic parameters of the perturbation moment equation and the periodic component of the characteristic perturbation moment expression into the Fourier series yields the eigenvalue equation which determines the perturbation moment behavior. Thus the stochastic stability of the parameter-excited cable vibration under the random and periodic combined support disturbances is determined directly by the matrix eigenvalues. The direct eigenvalue analysis approach is applicable to the stochastic stability of the control cable with multiple modes coupling under various periodic and/or random support disturbances. Numerical results illustrate that the multiple cable modes need to be considered for the stochastic stability of the parameter-excited cable vibration under the random and periodic support disturbances, and the increase of the control damping rather than control stiffness can greatly enhance the stochastic stability of the parameter-excited cable vibration including the frequency width increase of the periodic disturbance and the critical value increase of the random disturbance amplitude.

• 한국전산유체공학회:학술대회논문집
• /
• 한국전산유체공학회 2009년 춘계학술대회논문집
• /
• pp.155-158
• /
• 2009

We developed a numerical method to analyze the contact-line problems, incorporating a dynamic contact angle model. We used level set method to capture free surface. The method is applied to the analysis of dynamic behavior of a droplet in DC electrowetting. The result is compared with an experimental data and result of perturbation method.