Earticle

Transmission losses of various reactive silencers are predicted, using a time accurate finite difference method. The numerical scheme is the 3rd order upwind scheme for axisymmetric Euler equations. Main advantage of the present method is that it can simulate linear and non-linear wave propagation phenomena in a flow field directly with minimum numerical oscillation errors. The special treatments of incident wave condition, i.e. multiple harmonics of the transparent acoustic condition are applied to the transmission loss prediction for calculation efficiency. For the validation of the present approach, circular expansion chamber silencers without mean flow and an exponential pipe with mean flow are simulated in case of linear incident wave. The computed transmission losses have quite good agreements with those of the others. The nonlinear incident wave case is also investigated to check the usefulness of this method. The periodic N wave is clearly captured without numerical oscillation errors, and the insertion losses of two different incident frequencies are compared.

The lattice Boltzman method (LBM) and the finite difference-based lattice Boltzmann method (FDLBM) are quite recent approaches for simulating fluid flow, which have been proven as valid and efficient tools in a variety of complex flow problems. They are considered attractive alternatives to conventional finite-difference schemes because they recover the Navier-Stokes equations and are computationally more stable, and easily parallelizable. However, most models of the LBM or FDLBM are for incompressible fluids because of the simplicity of the structure of the model. Although some models for compressible thermal fluids have been introduced, these models are for monatomic gases, and suffer from the instability in calculations. A lattice BGK model based on a finite difference scheme with an internal degree of freedom is employed and it is shown that a diatomic gas such as air is successfully simulated. In this research we present a 2-dimensional edge tone to predict the frequency characteristics of discrete oscillations of a jet-edge feedback cycle by the FDLBM in which any specific heat ratio $\gamma$ can be chosen freely. The jet is chosen long enough in order to guarantee the parabolic velocity profile of a jet at the outlet, and the edge is of an angle of $\alpha$=23$^{o}$. At a stand-off distance w, the edge is inserted along the centerline of the jet, and a sinuous instability wave with real frequency is assumed to be created in the vicinity of the nozzle exit and to propagate towards the downstream. We have succeeded in capturing very small pressure fluctuations resulting from periodic oscillation of the jet around the edge.

본 연구에서는 실린더 형 쉘 구조물의 구조적 안정성에 대하여 해석 하였다. 임계하중은 하중을 점차적으로 증가하여 구조물이 파괴가 발생 할 때의 상태에서 가장 작은 하중을 의미한다. 셸 구조의 안정성을 임계하중의 크기로 기초를 두고 해석 하였다. 실린더 형 쉘의 차분해석은 일차적 원통형 판구조와 같으므로 최근에 많은 연구의 대상이 되어왔다. 차분법은 복잡한 구조물에서도 물론, 다양한 경계조건을 포함하는 문제에 이르기까지 효과적인 수치방법이다. 본 연구에서는 기본 쉘의 지배방정식을 유도하고 차분화 하여 직접적으로 접근하였다. 등분포 하중의 내압을 받고 있는 갇힌 실린더 형 쉘의 처짐 및 응력을 해석 하였다. 수치해석 결과를 해석해와 비교 검토하였으며 안정성에 대하여 임계 하중강도의 범위를 산출하였다

본 연구는 Navier-Stokes 식의 모형방정식으로 이류 및 확산거동을 갖는 선형화된 Burgers 방정식과 비선형 형태의 Burgers 방정식을 선택하여, 이에 대한 유한차분법과 유한해석법의 수치해를 해석해와 비교하여 봄으로써, 유한해석법의 응용성에 대해 고찰한 것이다. 본 연구를 통하여 얻어진 성과를 요약하면 다음과 같다. Burgers 방정식 및 선형화된 Burgers 방정식의 정상상태의 해석해를 사용하여 두 수치기법에 따른 수치해를 비교해 본 결과, 해석해와의 근사정도를 동일 기준 하에서 살펴볼 때, 유한해석법이 유한차분법보다 우수한 것으로 나타났다. Burgers 방정식의 비정상상태의 해석해에 대한 정확성 또한 유한해석법이 보다 잘 일치하는 것으로 나타났다. 특히 유한해석법은 유한차분법의 사용시 격자 크기의 선택에 따라 해의 수렴과정에서 발생할 수 있는 위상오차에 기인한 진동현상이 전혀 발생하지 않는다는 것을 확인할 수 있었으며, 따라서 유한해석법은 수치기법상 위상오차로부터 자유로운 안정된 해석기법이라고 판단된다.

In this study, the applicability of finite analytic method (FAM) is studied by selecting linearized-Burgers equation and Burgers equation which have convective and diffusive behaviors as the model equation of Navier-Stokes equations and by comparing numerical solution of finite difference method (FDM) and finite analytic method. The results are as follows. It is shown that the convergence of FAM for steady-state analytic solution of linearized-Burgers equation and Burgers equation is better than that of FDM under the same criteria. Also the accuracy of FAM for transient solution of Burgers equation is excellent. Especially, it is shown that oscillation phenomenon due to dispersion errors which occur according to the choice of grid size in FDM does not occur in FAM at all. So, it can be thought that FAM is numerically very stable scheme, which is free from dispersion errors.

본 연구는 균열선단에서 응력특이성을 갖는 탄성균열문제를 해석하기 위한 이동최소제곱 유한차분법을 제시한다. 응력특이성을 유발하는 균열선단 주변장을 모형화하기 위해 근사식에 선단주변함수를 내재적으로 도입하여 이동최소제곱 근사의 틀을 그대로 유지하면서 실제 미분계산을 거의 하지 않고 미분근사를 할 수 있는 이동최소제곱 Taylor 다항식 근사의 장점을 살렸다. 균열문제 정식화시 시간소모적인 적분과정이 필요한 약정식화 대신 해석영역에 배치된 절점에서 지배 미분방정식에 대한 차분식을 직접 구성하는 강정식화를 적용하여 계산 효율성을 향상시켰다. 균열문제 해석을 통해 내적확장된 이동최소제곱 유한차분법이 응력 특이성을 내포한 선단주변 변위장을 정확히 묘사할 수 있을 뿐만 아니라 응력확대계수를 정확히 계산 할 수 있음을 보였다.

This study presents a moving least squares (MLS) finite difference method for solving elastic crack problems with stress singularity at the crack tip. Near-tip functions are intrinsically employed in the MLS approximation to model near-tip field inducing singularity in stress field. employment of the functions does not lose the merit of the MLS Taylor polynomial approximation which approximates the derivatives of a function without actual differentiating process. In the formulation of crack problem, computational efficiency is considerably improved by taking the strong formulation instead of weak formulation involving time consuming numerical quadrature Difference equations are constructed on the nodes distributed in computational domain. Numerical experiments for crack problems show that the intrinsically enriched MLS finite difference method can sharply capture the singular behavior of near-tip stress and accurately evaluate stress intensity factors.

시간영역에서 변위만을 이용하여 수행되는 3차원 변위근사 유한차분 탄성파 모델링에서 자유면 경계조건을 정확히 묘사하기 위한 방법으로 육면체 내부에 매질의 물성을 정의할 것을 제안한다. 매질의 물성과 변위를 모두 격자점에 정의하는 기존의 방법과는 달리, 매질의 물성을 육면체 내부에 정의할 경우 추가적인 자유면 경계조건을 사용하지 않고도 매질의 물성의 변화만으로 자유면에서 응력이 사라진다는 자유면 경계조건을 만족시킬 수 있다. 무한 균질 매질과 반무한 균질 매질에 대하여 수치적인 해와 해석적인 해를 비교함으로써 육면체 내부에 매질의 물성을 정의하는 3차원 유한차분 탄성파 모델링법의 정확성을 확인하였다. 송신원과 수진기의 위치가 서로 바뀐 경우에 대하여 수치적으로 구한 해들을 비교함으로써 육면체 내부에 매질의 물성을 정의하는 3차원 유한차분 탄성파 모델링 알고리듬이 상반원리 (reciprocity theorem)를 만족시킨다는 것을 알 수 있었다. 또한, 수평층 구조 및 단층구조에 대하여 합성탄성파 단면도 및 스냅단면도를 성공적으로 작성할 수 있었다.

We describe a 3-D time-domain, displacement-based, finite-difference elastic wave modeling algorithm that is constructed by defining the material properties within cubes. In the conventional displacement-based finite-difference method, both displacements and material properties are defined at the nodal points, whereas in our finite-difference algorithm, displacements are still assigned to the nodal points but material properties are defined within cubes. In this case, free-surface boundary conditions, which describe stress-free at the free surface, are naturally satisfied by the changes of material properties. Through numerical examples for infinite homogeneous and semi-infinite homogeneous models, we could examine the accuracy of the 3-D finite-difference elastic wave modeling algorithm. Some numerical examples showed that the 3-D finite-differenc elastic wave modeling algorithm satisfies the reciprocity theorem and successfully generates synthetic seismograms and snapshots.

The disaster from functional gastrointestinal disorders (FGID) has detrimental impact on the quality of life of the affected population. There are, however, rare diagnostic methods for FGID. Our research group identified recently that the gastrointestinal tract well of the patients with FGID became more rigid than that of healthy people when palpating the abdominal regions overlaying the gastrointestinal tract. The objective of the current study is, therefore, to identify feasibility of a diagnostic system for FGID based on ultrasound technique, which can quantify the characteristics above. Two-dimensional finite difference (FD) models (one normal and two rigid models) were developed to analyze the reflective characteristic (displacement) on each soft-tissue layer responded after application of ultrasound signals. Based on the results from FD analysis, the ultrasound system for diagnosis of the FGID was developed and clinically tested via application of it to 40 human subjects with/without FGID who were assigned to Normal and Patient Groups. The results from FD analysis showed that the maximum displacement amplitude in the rigid models (0.12 and 0.16) at the interface between the fat and muscle layers was explicitly less than that in the normal model (0.29). The results from actual specimens showed that the maximum amplitude of the ultrasound reflective signal in the rigid models $(0.2{\pm}0.1Vp-p)$ at the interface between the fat and muscle layers was explicitly higher than that in the normal model $(0.1{\pm}0.0Vp-p)$. Clinical tests using our customized ultrasound system showed that the maximum amplitudes of the ultrasound reflective signals near to the gastrointestinal tract well for the patient group $(2.6{\pm}0.3Vp-p)$ were generally higher than those in normal group $(0.1{\pm}0.2Vp-p)$. These findings suggest that our customized ultrasound system using the ultrasound reflective signal may be helpful to the diagnosis of the FGID.

Since seismic anisotropy often appears in geological media, which may result from various causes, we need to describe anisotropic features in seismic modeling and inversion. Although a number of modeling algorithms were developed to describe seismic anisotropy, we still need to develop a simple but accurate modeling algorithm to simulate geological scale models. For such a modeling algorithm for anisotropic media, we extend a time-domain cell-based finite-difference method to anisotropic media. Since the cell-based finite-difference scheme only employs displacements, we can expect that our anisotropic modeling algorithm is computationally more efficient than the staggered-grid finite-difference method. Because our algorithm does not require any interpolations, it’s possible to simulate a model whose material properties abruptly change. In order to suppress artificial reflections originating from the outer boundaries of a given model, we apply Higdon’s absorbing boundary conditions. Numerical examples show that our modeling algorithm can properly describe anisotropic features.

본 연구에서는 연약지반의 압밀해석 시에 물성이 다른 두 층간의 경계면 해석을 보다 정확히 수행할 수 있는 유한차분 해석기법을 제안하였다. 제안된 해석기법을 이용하여 다층지반 및 스미어가 발생된 지반에 대하여 정확한 압밀해석이 가능한 유한차분 해석프로그램을 개발하였으며, 다양한 지반조건에서 해석을 수행하여 제안된 해석 기법을 근간으로 개발된 해석프로그램의 적용성을 검증하였다. 해석결과 다층지반의 압밀속도는 배수면 근처의 투수계수에 지배적인 영향을 받는 것으로 나타났으며, 스미어 발생지반의 압밀속도는 스미어 영역에서의 투수계수 분포가 선형인 경우보다 일정한 경우 압밀이 더 지체되는 것으로 나타났다. 또한 스미어 영향을 고려한 연약지반의 압밀해석에서 스미어로 인한 주변지반의 투수계수 감소가 상대적으로 작은 경우에는 스미어 영역 전체의 투수계수 감소율이 일정하다고 가정한 단순 해석으로도 정확한 압밀거동 예측이 가능하지만, 스미어 발생으로 배수재 주변지반의 투수계수의 감소가 큰 경우에는 배수재로부터의 거리에 따른 투수계수 분포형태를 정확히 가정하고 해석을 수행해야 압밀거동을 정확히 예측할 수 있는 것으로 나타났다. 마지막으로 부산지역의 다층지반 조건을 고려한 해석결과, 압밀완료 시간을 정확히 예측하기 위해서는 다층지반에 대한 해석이 필요한 것으로 나타났다.

In this research, finite difference (FD) scheme for the interface of the layer between different soil characteristics was suggested. Based on the suggested scheme, FD analysis program for the consolidation analysis of the multi-layered and smeared soil was developed. And the applicability of the program was investigated by the FD analysis conducted for the various soil conditions. Analysis results showed that the permeability near the drainage boundary had a dominant effect on the consolidation rate. And the consolidation rate of the soil with the constant permeability in smeared area was retarded more than the soil with the linear variation of permeability in smeared area. Simple assumption of the constant variation of permeability in smeared area could be used when the decreasing rate of permeability in smeared area was relatively low. But exact assumption of the permeability variation in smeared area should be considered when the decreasing rate of permeability in smeared area was relatively high. Finally, based on the analysis result on Busan area, the analysis considering multi-layered soil should be needed to exactly evaluate time for the completion of consolidation.

연직드레인의 유효반경내에서 시간과 압밀도 관계를 계산하기 위해 음적차분해법이 적용되었으며 계산시 과잉간극수압의 소산은 두 방향으로 수행된다. 다단계 성토에 의한 지중응력 증분을 계산하기 위해, 지표면하의 연약지반은 등방균질 탄성체로서 간주하였고 각 단계 순간성토시 초기 과잉간극수압은 포화된 점성토내에서 평면변형률 조건과 탄성단계의 간극수압 응답 상태에 대한 Skempton의 간극수압계수를 이용해 계산하였다. 침하에 대해서는, 즉시 및 1 차 압밀침하량만 계산하였고, 2차 압밀침하는 고려하지 않았다. 계산된 과잉간극수압과 지표침하량이 경과시간에 대한 현성측성치와 유사한 것으로 판정되었으며, 본 연구에 적용된 계산기법(압밀특성이 다른 다층지반으로 구성된 연약지반내에 연직배수공법을 적용하고 성토가 다단계 순간성토로 이루어질 경우 과잉간극수압 소산과정을 음적 차분해법으로 근사계산)은 각 연약층의 시간-압밀도 관계를 예측하는데 이용가능할 것으로 생각된다.

The implicit finite difference program was developed to evaluate the relationship between time and consolidation ratio within the zone of vertical drain effective radius. In the evaluation, the excess pore water pressure was considered to dissipate in two directions, namely, vertical and radial flow direction. To calculate subsoil stress increments in the soil due to multi-step embanking, the foundation soil was assumed to be an isotropic and homogeneous elastic medium and the initial excess pore water pressure was estimated by using Skempton's parameters whose condition is plane strain and elastic phase of pore pressure response within the soft ground. Regarding to the settlement estimation, immediate and primary consolidation settlements were calculated. The secondary or delayed consolidation settlement was not considered. Numerically calculated excess pore water pressure and settlements were similar to the measured data in situ. Thus, this method can be used to predict the time-consolidation ratio of each layer treated by vertical drain method.

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.

The straight rectangular fin is analyzed using the one-dimensional analytic method and the finite difference method. For the finite difference method, the numbers of nodes vary from 20 to 100. The relative errors of heat loss and temperature between the analytic method and the finite difference method are represented as a function of Biot Number and dimensionless fin length. One of the results shows that the relative error between the analytic method and the finite difference method decreases as the numbers of nodes for finite difference method increase.

In this paper, we develop an accurate explicit finite difference method for the two-dimensional Black-Scholes equation with a hybrid boundary condition. In general, the correlation term in multi-asset options is problematic in numerical treatments partially due to cross derivatives and numerical boundary conditions at the far field domain corners. In the proposed hybrid boundary condition, we use a linear boundary condition at the boundaries where at least one asset is zero. After updating the numerical solution by one time step, we reduce the computational domain so that we do not need boundary conditions. To demonstrate the accuracy and efficiency of the proposed algorithm, we calculate option prices and their Greeks for the two-asset European call and cash-or-nothing options. Computational results show that the proposed method is accurate and is very useful for nonlinear boundary conditions.

We study the pricing of reverse step-down equity-linked securities by means of an implicit finite difference method applied to the Black-Scholes partial differential equation. In the classical step-down structure, early redemption is triggered when the underlying asset price rises above a fixed level. Unlike this structure, the reverse variant redeems early when the price falls below a prescribed threshold. This inversion of the redemption condition places the capital-loss zone on the upside. The structure is therefore attractive to investors who anticipate a decline in the underlying asset. We formulate the governing equation and boundary conditions for both structures. We then discretise them on a uniform spatial grid and solve the resulting tridiagonal systems by the Thomas algorithm at each backward time step. The method is applied to three market instruments (the KOSPI200 index, SK Hynix stock, and Tesla stock) with volatilities obtained from reported market data. We also carry out a systematic spatial grid convergence study for the KOSPI200 reverse product. The study shows that the computed price converges monotonically to a stable reference value as the grid spacing decreases. The empirical convergence rates are presented at each refinement level. The grid spacing is chosen to make all barrier and strike levels coincide with grid nodes at every refinement level. For the grid sequence considered, this alignment yields a monotone error sequence and avoids the oscillations that misaligned grids can produce. We further report a temporal convergence study, a sensitivity analysis of the price to the truncation boundary and to the reverse knock-in barrier, and an independent Monte Carlo validation of the computed prices.

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.

In this paper, we aim to construct a nonstandard finite difference (NSFD) scheme to solve numerically a mathematical model for cholera epidemic dynamics. We first show that if the basic reproduction number is less than unity, the disease-free equilibrium (DFE) is locally asymptotically stable. Moreover, we mainly establish the global stability analysis of the DFE and endemic equilibrium by using suitable Lyapunov functionals regardless of the time step size. Finally, numerical simulations with different time step sizes and initial conditions are carried out and comparisons are made with other well-known methods to illustrate the main theoretical results.

We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method (FChFD). The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. We tested this technique to solve numerically fractional BVPs. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The fractional derivatives are presented in terms of Caputo sense. The application of the method to fractional BVPs leads to algebraic systems which can be solved by an appropriate method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

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.