• International Journal of Aeronautical and Space Sciences
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• 제18권4호
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• pp.816-826
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• 2017

In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2007년도 정기 학술대회 논문집
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• pp.501-506
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• 2007

A generalized finite difference method for solving solid mechanics problems such as elasticity and crack problems is presented. The method is constructed in framework of Taylor polynomial based on the Moving Least Squares method and collocation scheme based on the diffuse derivative approximation. The governing equations are discretized into the difference equations and the nodal solutions are obtained by solving the system of equations. Numerical examples successfully demonstrate the robustness and efficiency of the proposed method.

• Journal of applied mathematics & informatics
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• 제8권2호
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• pp.473-489
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• 2001

The notion of difference for two convex compact sets in Rⁿ, proposed by Rubinov et al, is generalized to R/sub mxn/. A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of function, is presented.

• 대한수학회지
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• 제34권3호
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• pp.515-531
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• 1997

In this paper, finite difference method is applied to approximate the generalized solutions of Sobolev equations. Using the Steklov mollifier and Bramble-Hilbert Lemma, a priori error estimates in discrete $L^2$ as well as in discrete $H^1$ norms are derived frist for the semidiscrete methods. For the fully discrete schemes, both backward Euler and Crank-Nicolson methods are discussed and related error analyses are also presented.

• 대한수학회보
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• 제38권1호
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• pp.17-27
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• 2001

We discuss a finite difference preconditioner for the$C^1$ Lagrance quadratic spline collocation method for a uniformly elliptic operator with homogeneous Dirichlet boundary conditions. Using the generalized field of values argument, we analyzed eigenvalues of the matrix preconditioned by the matrix corresponding to a finite difference operator with zero boundary condition.

• Structural Engineering and Mechanics
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• 제49권1호
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• pp.41-64
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• 2014

The main objective of this article is the exploitation of a stochastic hybrid mesh-free method based on stochastic generalized finite difference (SGFD), Newmark finite difference (NFD) methods and Monte Carlo simulation for thermoelastic wave propagation and coupled thermoelasticity analysis based on GN theory (without energy dissipation). A thick hollow cylinder with Gaussian uncertainty in mechanical properties is considered as an analyzed domain for the problem. The effects of uncertainty in mechanical properties with various coefficients of variations on thermo-elastic wave propagation are studied in details. Also, the time histories and distribution on thickness of cylinder of maximum, mean and variance values of temperature and radial displacement are studied for various coefficients of variations (COVs).

• Steel and Composite Structures
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• 제27권3호
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• pp.255-271
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• 2018

This paper deals with the transient dynamic analysis and elastic wave propagation in a functionally graded graphene platelets (FGGPLs)-reinforced composite thick hollow cylinder, which is subjected to shock loading. A micromechanical model based on the Halpin-Tsai model and rule of mixture is modified for nonlinear functionally graded distributions of graphene platelets (GPLs) in polymer matrix of composites. The governing equations are derived for an axisymmetric FGGPLs-reinforced composite cylinder with a finite length and then solved using a hybrid meshless method based on the generalized finite difference (GFD) and Newmark finite difference methods. A numerical time discretization is performed for the dynamic problem using the Newmark method. The dynamic behaviors of the displacements and stresses are obtained and discussed in detail using the modified micromechanical model and meshless GFD method. The effects of the reinforcement of the composite cylinder by GPLs on the elastic wave propagations in both displacement and stress fields are obtained for various parameters. It is concluded that the proposed micromechanical model and also the meshless GFD method have a high capability to simulate the composite structures under shock loadings, which are reinforced by FGGPLs. It is shown that the modified micromechanical model and solution technique based on the meshless GFD method are accurate. Also, the time histories of the field variables are shown for various parameters.

• 한국수산과학회지
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• 제27권6호
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• pp.782-786
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• 1994

Dey and Morrison (1979)이 육상의 전기탐사문제의 해결에 성공적으로 적용한 적분차분법(integral finite different method)의 해양에서의 응용성을 연구해보기 위해, 해석해가 존재하는 연안역의 해수고유진동 문제를 도출하여 기존의 고유진동문제에 적용하여 보았다. 그 응용성의 평가는 기존 해양에 널리 적용되는 기존차분법(conventional finite different method)으로 구한 수치결과와 적분차분법으로 구한 결과를 해석해와의 비교검증을 통하여 실시되었다. 그 결과 적분차분법으로 구한 고유치와 고유함수값이 기존차분법으로 구한 결과보다 좋은것으로 나타났다. 이러한 결과는 적분차분법의 경우 원래의 기본방정식에 Green's theorem을 적용함으로써, 기본방정식에 존재하는 2계 미분연산자가 1계 미분연산자로 해석적으로 처리되었기 때문으로 사료된다. 따라서 적분차분법을 이용하여 해수고유진동문제를 비롯한 다른 유사문제를 풀 경우 기본차분법보다 좋은 결과가 나을 것으로 사료된다. 또한 미분방정식의 수치해를 구할 경우 적분법이 차분법보다 좋은해를 줄 수 있다는 것을 증명한 것으로 사료된다.

• 대한기계학회논문집A
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• 제24권6호
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• pp.1520-1528
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• 2000

This study presents a method for determining bearing stiffness and damping coefficients of air-lubricated slider bearing, and shows influences of air-bearing surface geometry(recess depth, crown an d pivot location) on flying attitude and dynamic characteristics. To derive the dynamic lubrication equation, the perturbation method is applied to the generalized lubrication equation which based on linearized Boltzmann equation. The generalized lubrication equation and the dynamic lubrication equation are converted to a control volume formulation, and then, the static and dynamic pressure distributions are calculated by finite difference method. The recess depth and crown of the slider show significantly influence on flying attitude and dynamic characteristics comparing with those of pivot location.

• 한국해안해양공학회지
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• 제10권2호
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• pp.93-99
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• 1998

타원형 유한차분모형에서 개방 경계조건으로 포물선 근사식과 스폰지층 경계를 사용하여 모형의 개량을 도모하였다. 수치기법은 GCG(Generalized conjugate gradient)기법을 사용하였고 구형해저실험에서 포물형 근사식을 사용하여 부적절한 반사파를 상당 부분 제거할 수 있었다. 스폰지층 경계의 경우 2파장 이상의 스폰지층을 사용할 때 포물형 근사식과 유사한 결과를 얻을 수있었다. 직사각형 항만에 대한 실험을 통하여 임의 형상의 대상 해역에도 쉽게 모형을 적용할 수 있음을 확인하였다.