Application of Block-centered Finite Difference Formulation for Non-linear Finite Strain Consolidation

이동섭,성치훈,이철호,최항석 대한토목학회 2014 KSCE JOURNAL OF CIVIL ENGINEERING Vol.18 No.7

A one-dimensional block-centered finite-difference model has been developed to estimate the rate of non-linear finite strain consolidation. The governing equations including the hydrodynamic and constitutive equations are presented. The hypothesis of the uniqueness of the End-of-Primary (EOP) void ratio – effective stress relationship is adopted to calculate the primary consolidation settlement. The explicit block-centered finite difference formulations and boundary conditions are presented and discussed. The developed model was compared with a point-centered finite-difference program, ILLICON to show the efficiency of the block- centered model. The block-center model provides an efficient tool to deal with interface boundaries and has advantageous ability to take into consideration the time-dependent loading, layered soil systems, and variable soil properties.

제어 알고리즘 구현을 위한 새로운 미분값 유도 방법

김태엽 한국전기전자학회 2019 전기전자학회논문지 Vol.23 No.3

마이크로프로세서를 이용한 제어알고리즘 구현에서 차분방정식이 매우 유용하게 사용된다. 샘플링 데이터로부터 미분 값을 추정하기 위해 전향, 후향 및 중심 차분 방식이 사용되어왔다. 차분 값을 계산하기 위해서는 차분계수가 매우 중요하다. 본 논문에서는 유한 차분 계수를 계산하기 위한 새로운 방식을 제시하고자 한다. 제안된 방식의 유효성을 입증하기 위해RLS 알고리즘을 적용한 파라미터 추정에 대하여 적용하였다. Difference equation is useful for control algorithm in the microprocessor. To approximate a derivative values fromsampled data, it is used the methods of forward, backward and central differences. The key of computing discretederivative values is the finite difference coefficient. The focus of this paper is a new approach method of finite differenceformula. And we apply the proposed method to the recursive least squares(RLS) algorithm.

A FINITE DIFFERENCE/FINITE VOLUME METHOD FOR SOLVING THE FRACTIONAL DIFFUSION WAVE EQUATION

Sun, Yinan,Zhang, Tie Korean Mathematical Society 2021 대한수학회지 Vol.58 No.3

In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: ∂^β_tu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂^β_tu by using an interpolation of derivative type. The truncation error of this formula is of O(△t^2+δ-β)-order if function u(t) ∈ C^2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t^3-β) if u(t) ∈ C³ [0, T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H¹-norm and L₂-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.

Time-varying meshing stiffness calculation of an internal gear pair with small tooth number difference by considering the multi-tooth contact problem

Guangjian Wang,Qing Luo,Shuaidong Zou 대한기계학회 2021 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.35 No.9

Due to the multiple tooth contact problem involving internal gear pair with small tooth number difference (IGPSTND), the existing analytical methods applied for standard spur or helix gear pairs to calculate the time-varying meshing stiffness (TVMS) are not suitable. In this paper, two methods are proposed for calculating the time-varying meshing stiffness in internal gear pairs with small tooth difference. In the first method, an analytical model is established by using the potential energy method, considering the clearance of initial contact tooth and the external load. The second method proposes the application of a hybrid finite elementanalytical method. The proposed two methods are validated by the application of the finite element method. By taking the results of finite element analysis as a comparative reference, the results show that the finite element - analytical method is closer to the reference results than the results obtained by the analytical method, and both methods are less computationally expensive than finite element analysis.

Calculation of Ion-Flow Field of HVdc Transmission Lines in the Presence of Wind Using Finite Element-Finite Difference Combined Method With Domain Decomposition

Ji Qiao,Jun Zou,Jiansheng Yuan,Lee, J. B.,Munno Ju IEEE 2016 IEEE transactions on magnetics Vol.52 No.3

<P>A combined method adopting the domain decomposition is proposed for analyzing the ion-flow field of high-voltage direct current transmission lines, including the effect of the transverse wind. The upstream finite-element method is used with a dense triangle mesh in the vicinity of bundle conductors to guarantee the accuracy. The upstream finite-difference method is proposed and the larger uniform quadrilateral grid is applied to simulate the ion-flow field in the rest of the region with a satisfactory precision. The calculation process of the Poisson equation is iterated with the Dirichlet-Neumann algorithm to coordinate the solution between adjacent subdomains. The proposed approach improves the efficiency and remains stable with high wind speed. Finally, the influence of wind on the ion-flow field is analyzed and calculations are in good agreement with experimental data and results in the previous literature.</P>

A LINEARIZED FINITE-DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF THE NONLINEAR CUBIC SCHRODINGER EQUATION

Bratsos, A.G. 한국전산응용수학회 2001 The Korean journal of computational & applied math Vol.8 No.3

A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrodinger equation into a linear algebraic system. This method is developed by replacing the time and the nonlinear term by an appropriate parametric linearized scheme based on Taylor’s expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given. AMS Mathematics Subject Classification : 65J15, 47H17, 49D15

엇격자 유한차분법을 이용한 극지해역 지진파 모델링

오주원,민동주,이호용,박민규 한국지구과학회 2011 한국지구과학회지 Vol.32 No.6

We simulate the propagation of earthquake waves in the continental margin of Antarctica using the elastic wave modeling algorithm, which is modified to be suitable for acoustic-elastic coupled media and earthquake source. To simulate the various types of earthquake source, the staggered-grid finite-difference method, which is composed of velocity-stress formulae, can be more appropriate to use than the conventional, displacement-based, finite-difference method. We simulate the elastic wave propagation generated by earthquakes combining 3D staggered-grid finite-difference algorithm composed of displacement-velocity-stress formulae with double couple mechanisms for earthquake source. Through numerical tests for left-lateral strike-slip fault, normal fault and reverse fault, we could confirm that the first arrival of P waves at the surface is in a good agreement with the theoretically-predicted results based on the focal mechanism of an earthquake. Numerical results for a model made after the subduction zone in the continental margin of Antarctica showed that earthquake waves, generated by the reverse fault and propagating through the continental crust, the oceanic crust and the ocean, are accurately described. 이 연구에서는 기존의 탄성파 모델링 알고리즘에 지진 송신원을 적용하고, 음향-탄성파 결합 매질을 구현하여남극대륙 주변과 같은 극지해역에서 발생할 수 있는 지진파의 거동을 모사한다. 기존의 변위근사 유한차분법과 달리 속도-응력 식으로 구성되는 엇격자 유한차분법의 경우 다양한 송신원을 구현하는데 적합하므로 변위-속도-응력 식에 기초하여 개발된 3차원 엇격자 유한차분법 알고리즘과 이중 우력(Double Couple Forces)을 이용하여 구현한 지진 송신원을접목시켜 지진파의 거동을 모사한다. 좌수향 주향이동단층, 정단층, 역단층 형태의 지진 송신원에 대해서 개발된 알고리즘을 검증한 결과 이론적으로 예측되는 P파의 초동을 정확히 모사할 수 있었고, 섭입대 모델에 대한 수치모형실험 결과 섭입대에서 역단층에 의해 발생된 후 대륙지각, 해양지각 및 해양에서 전파되는 지진파의 거동양상이 정확하게 모사되는 것을 확인할 수 있었다.

고차 셀 기반 유한 차분법을 이용한 음향파 파동 전파 모델링

조준현,하완수 한국자원공학회 2021 한국자원공학회지 Vol.58 No.4

We propose high-order finite-difference methods for acoustic wave propagation modeling considering the P-wave velocity and density. We can simulate wave propagation without directly differentiating discontinuous medium parameters using the cell-based finite-difference method. However, a limitation of the conventional cell-based finite-difference method using the second-order scheme is that it requires a large number of grids per wavelength to obtain an accurate result. We improve the conventional cell-based method to exploit arbitrary high-order schemes. In a numerical example, we compare results of the proposed methods with that of the analytic solution, and show that we can obtain accurate results with small number of grids per wavelength using the high-order cell-based methods. 이 연구에서는 P파 속도와 밀도를 고려한 음향파 파동 전파 모델링을 위한 고차 셀 기반 유한 차분법을 제안하였다. 셀 기반 유한 차분법을 사용하면 불연속적인 매질 특성을 직접 미분하지 않고도파동 전파를 모델링할 수 있다. 그러나 기존의 2차 차분식을 사용하는 셀 기반 유한 차분법에서는정확한 결과를 얻기 위해 파장당 격자수가 커야 한다는 한계가 있었다. 이 연구에서는 임의의 고차차분식을 사용할 수 있도록 기존 기법을 개선하였다. 수치 예제에서 해석해와의 비교를 통해 고차셀 기반 유한 차분법을 이용하면 파장당 격자수가 작아도 정확한 결과를 얻을 수 있음을 보였다.

2D time-domain acoustic-elastic coupled modeling: a cell-based finite-difference method

이호용,Seung-Chul Lim,Dong-Joo Min,Byung-Doo Kwon,Minkyu Park 한국지질과학협의회 2009 Geosciences Journal Vol.13 No.4

To describe wave propagation in a fluid-solid environment which is usually encountered during marine seismic exploration, we design a time-domain acoustic-elastic coupled modeling algorithm based on the cell-based finite-difference method. The cell-based method has proven to delineate stress-free conditions correctly at the free surface with just changes in the material properties, which indicates that it can also properly deal with subsurface interface boundaries. In the acoustic-elastic coupled modeling, we first compose finite differences individually for the acoustic and elastic media; we then combine the differences using fluid-solid interface boundary conditions. Numerical experiments show that the cell-based coupled modeling algorithm gives solutions compatible with analytic solutions and that it properly describes S- and converted waves as well as P-waves. Applying the cell-based coupled modeling algorithm to a slope model, we confirm that our coupled modeling algorithm describes irregular interfaces properly, although it employs a staircase approximation of them.

파동방정식 수치해의 일관성에 관한 연구

편석준 ( Sukjoon Pyun ),박윤희 ( Yunhui Park ) 한국지구물리·물리탐사학회 2016 지구물리와 물리탐사 Vol.19 No.3

탄성파 자료의 역산은 파동방정식에 기초하고 있으므로 파동방정식의 해를 정확하게 구하는 것이 가장 중요하다. 특히, 전파형역산은 파동장 전체를 이용하기 때문에 정문제에 해당하는 모델링이 정확하게 이루어져야 신뢰할 수 있는 결과를 얻게 된다. 파동방정식의 수치해를 구하는 대표적인 기법인 유한차분법과 유한요소법은 해의 수렴성을 보장할 수 있어야 하는데, 해의 수렴성은 이론적으로 일반화된 증명이 되어 있으나 실제 문제에 적용할 경우 일관성과 안정성을 분석해야 한다. 모델링 결과의 일관성은 송신원 함수의 구현이 매우 중요한 부분인데, 유한차분법은 디랙 델타 함수(Dirac delta function)를 나타낼 때 격자 간격으로 표준화된 싱크 함수(sinc function)를 사용해야 하는 반면 유한요소법은 격자 간격에 관계없이 기저함수 값을 사용하면 된다. 주파수 영역 파동방정식을 사용할 경우 송신 파형 함수의 스펙트럼을 정확하게 표현하기 위해 샘플링 이론으로 정의되는 시간 간격보다 더 조밀한 샘플링 간격을 사용하고 나이퀴스트(Nyquist) 주파수보다 더 높은 주파수를 최대 주파수로 사용해야 한다. 또한, 복소 각주파수를 사용하는 경우 감쇠 파동방정식을 만족하기 위해서는 송신 파형 함수를 먼저 감쇠한 후 사용해야 한다. 이러한 요건들이 모두 만족되었을 때 신뢰할 수 있는 역산 알고리즘 개발이 가능하다. Since seismic inversion is based on the wave equation, it is important to calculate the solution of wave equation exactly. In particular, full waveform inversion would produce reliable results only when the forward modeling is accurately performed because it uses full waveform. When we use finite-difference or finite-element method to solve the wave equation, the convergence of numerical scheme should be guaranteed. Although the general proof of convergence is provided theoretically, the consistency and stability of numerical schemes should be verified for practical applications. The implementation of source function is the most crucial factor for the consistency of modeling schemes. While we have to use the sinc function normalized by grid spacing to correctly describe the Dirac delta function in the finite-difference method, we can simply use the value of basis function, regardless of grid spacing, to implement the Dirac delta function in the finite-element method. If we use frequency-domain wave equation, we need to use a conservative criterion to determine both sampling interval and maximum frequency for the source wavelet generation. In addition, the source wavelet should be attenuated before applying it for modeling in order to make it obey damped wave equation in case of using complex angular frequency. With these conditions satisfied, we can develop reliable inversion algorithms.