Development of Reliability Evaluation System for Numerical Solutions by Introducing Stochastic Approach and Application to Complicated Fluid Dynamics Simulation

引入随机方法的数值解可靠性评估系统开发及其在复杂流体动力学模拟中的应用

基本信息

批准号：
18540118
负责人：
HATAUE Itaru
金额：
$ 2.44万
依托单位：
Kanazawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540118/
关键词：
Randomness Stochastic difference equation Simulation Stability Traffic flow Macroscopic model Flux-free

项目摘要

For the purpose of development of reliability evaluation system for numerical solutions, we tried to discuss how structure of numerical solutions of a stochastic difference equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Errors in solving the nonlinear systems are inserted randomly and structure of solutions becomes very complicated We try to investigate the dependence of the structure of numerical solutions on insertion of random errors As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation and the Lorenz equations are considered. A new approach, sample mean dynamical system (SMDS), is proposed in order to analyze the dependence of the structure of numerical solutions of discretized dynamical system on insertion of random errors and the relation between the size of noise and characteristics of obtained numerical solutions is discussed.In addition, we appli … More ed them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations. Dependence of the averaged structure of numerical solutions of fluid simulations on forcibly added random errors are discussed. Next, we give some theoretical considerations on the flux-free finite-element method for the generalized Stokes interface problem arising from the immiscible two-fluid flow problems. In the flux-free finite-element method, the flux constraint is posed as another Lagrange multiplier to keep the zero-flux on the interface. As a result, the mass of each fluid is expected to be preserved at every time step. We fast study the effect of discontinuous coefficients(viscosity and density)on the error of the standard finite element approximations very carefully. Then, the analysis is extended to the flux-free finite element method. As for the problem of traffic jam, the formation of the traffic congestion in two-lane traffic flow is studied. The two-lane macroscopic model using the optimal velocity model which has been introduced in the microscopic model is constructed on the basis of the one-lane model. We adopt different optimal velocity function to each lane and new rules of changing lanes are introduced. Numerical simulations are performed in order to investigate the characteristic phenomena of two-lane traffic flow In particular, we concentrate the discussion about the property of "Synchronized flow', one of the most characteristic phenomena of two-lane traffic flow Furthermore, the fundamental diagrams from the simulations are compared with those observed in a highway. Less

为了开发数值解的可靠性评估系统，我们试图从求解非线性系统的误差的角度来讨论随机差分方程数值解的结构如何通过随机式的误差插入而改变。插入，解的结构变得非常复杂我们尝试研究数值解的结构对随机误差插入的依赖性作为基础研究，考虑基于确定性 Logistic 微分方程和洛伦兹方程 A 的随机微分方程。为了分析离散动力系统数值解结构对随机误差插入的依赖性，提出了样本平均动力系统（SMDS）新方法，并讨论了噪声大小与所获得数值解特征之间的关系。此外，我们将它们应用于实际流体计算问题和交通拥堵问题，以分析控制非线性现象的几个因素。首先，我们尝试讨论不可压缩数值解的结构的依赖性。流体方程求解联立方程中随机误差的插入。接下来，我们讨论了流体模拟数值解的平均结构对强制添加随机误差的依赖性。在无通量有限元方法中，通量约束被视为另一个拉格朗日乘子，以保持界面上的零通量。我们希望在每个时间步都保留每种流体的质量，然后非常仔细地研究不连续系数（粘度和密度）对标准有限元近似误差的影响。针对交通拥堵问题，在微观模型中引入最优速度模型，研究了两车道交通流中交通拥堵的形成。的基础我们对每条车道采用最优的不同速度函数，并引入了新的换道规则，以研究双车道交通流的特征现象，特别是对其性质进行了讨论。 “同步流”是双车道交通流最典型的现象之一。此外，还将模拟的基本图与在高速公路中观察到的图进行了比较。更少