On a Sharp Interface Model for Phase Transitions in Steel

Steel is one of the most widely used materials in the world with a broad spectrum of properties. The microstructure and the distribution of the different phases are of great importance, since they each possess different properties. A sharp interface model for the austenite-ferrite phase transition is presented. Mechanical effects due to eigenstrains resulting from the different densities of the phases are taken into account. The governing PDEs in each phase are a diffusion equation for the carbon concentration and the balance of momentum. Across the free interface, separating the two phases, the physical quantities may have discontinuities, which are controlled by jump conditions. Consistency of the model with the 2nd law of thermodynamics is shown. Numerical simulations for these types of free boundary problems are quite complex and involve appropriate methods to determine the interface position. One possibility to circumvent the explicit determination of the free boundary is the use of regularization techniques in form of phase field methods, where the interface is tracked implicitly. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Transitions in a Stochastic Game

Mathematical Notes -     

相变演化Suliciu模型中波的相互作用

描述相变演化的Suliciu模型，其基本波可由行波分析得到．对于任何给定分两段常值的初始状态，相应的Riemann解是某些基本波的组合．对分三段常值的初始状态，解由上述Piemann解构成，其中相邻两状态间以基本波连接．当基本波发生碰撞时，新的Riemann问题形成．通过研究这些Riemann。问题，可以在适当的参数空间中对基本波之间复杂的相互作用加以分类．     

守恒高阶交通流模型的相平面分析

在Lagrange坐标下,运用微分方程定性理论中的相平面分析方法,研究一个近期所提出的守恒高阶交通流模型的行波解,讨论系统的平衡点类型及其稳定性状态,分析相平面中的轨线全局分布结构,验证数值解与解析解的一致性.从而,能够较好地解释现实交通中的时停时走波和瓶颈处的振荡现象,表明所讨论的模型能够描述复杂的拥挤交通.     

交通流非参数回归模型

交通流宏观参数流量、速度和占有率在交通工程和管理中具有非常重要的作用 ,对这三者关系的刻划反映了道路本身特性和交通流的规律。到目前为止 ,基本上采用线性或非线性的参数模型来描述 ,此类模型在应用中具有一定的局限性 ,并交通变量时间序列进行预测 ,数据拟合表明 ,选择适当的核函数或邻近数以及窗宽 ,可以达到比较满意的效果     

Zeroth-Order Phase Transitions

In the theory of superfluidity and superconductivity, a jump of the free energy was discovered theoretically and was naturally called a zeroth-order phase transition. We present an example of an exactly solvable problem in which such a phase transition occurs.     

Irreversible Phase Transitions in Steel

We present a mathematical model for the austenite–pearlite and austenite–martensite phase transitions in eutectoid carbon steel. The austenite–pearlite phase change is described by the Additivity Rule. For the austenite–martensite phase change we propose a new rate law, which takes into account its irreversibility. We investigate questions of existence and uniqueness for the three-dimensional model and finally present numerical calculations of a continuous cooling transformation diagram for the eutectoid carbon steel C1080. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Phase Transitions in Gravitational Allocation

Given a Poisson point process of unit masses (“stars”) in dimension d ≥ 3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. In earlier work, we showed the diameters of these cells have exponential tails. Here we analyze the quantitative geometry of the cells and show that their large deviations occur at the stretched-exponential scale. More precisely, the probability that mass exp(−R ^γ ) in a cell travels distance R decays like exp(-R^f_d(g)){\left(-R^{f_d(\gamma)}\right)} where we identify the functions f _d (·) exactly. These functions are piecewise smooth and the discontinuities of f^￠_d{f^{\prime}_d} represent phase transitions. In dimension d = 3, the large deviation is due to a “distant attracting galaxy” but a phase transition occurs when f ₃(γ) = 1 (at that point, the fluctuations due to individual stars dominate). When d ≥ 5, the large deviation is due to a thin tube (a “wormhole”) along which the star density increases monotonically, until the point f _d (γ) = 1 (where again fluctuations due to individual stars dominate). In dimension 4 we find a double phase transition, where the transition between low-dimensional behavior (attracting galaxy) and highdimensional behavior (wormhole) occurs at γ = 4/3.     

离散交通流模型的反馈线性化与拥堵控制

在自动化高速公路环境下,提出一种改进的宏观离散交通流模型密度控制方法.利用反馈线性化方法,将宏观离散交通流模型转换为一般容易处理的线性系统模型,简化了密度控制器的设计.利用线性系统中具有输入变换的跟踪反馈控制方法,对线性化后的系统模型设计控制律.通过控制该线性系统的状态变量,间接稳定离散交通流模型中的交通流密度,达到对道路交通流拥堵的控制.同时给出设计方法和步骤,仿真实例说明了方法的实用性.     

一个交通模型的行波解的渐近稳定性

主要考虑下面的交通模型的行波解的渐近稳定性.{vt-ux=0 (E)ut+p(v)x=1/ε(f(v)-u+μuxx 其中初始值为 (v,u)(x,0)=(v0(x),u0(x))→(v±,u±),v±＞0,as x→±∞.(Ⅰ)在允许流函数f不是凹函数以及初始值在无穷远处的极限不满足平衡方程的条件下,我们得到了稳定性...     

Phase Transitions on the Boundary of a Domain

We consider the minimization problem for the energy functional of a two-phase medium concentrated at the boundary of a domain. We study regularization of the functional by means of the area of the boundary of the phase interface under additional conditions on the displacement field. Bibliography: 3 titles.     

具有交通流量异质性的网络传染病模型研究

主要研究了异质集合种群网络上的移动和扩散行为对疾病传播的影响.针对现实社会中的网络所具有的异质性,分析了影响城市疾病传播的主要因素为网络拓扑结构以及城市交通流量异质性,建立了依赖于交通流量移出率的传染病动力学模型.通过分析模型的无病平衡点以及正平衡点的存在及其稳定性,发现人口流动会使交通较发达的城市拥有更多的染病者,更容易促使疾病的爆发.     

短时交通流混沌预测模型的比较研究

短时交通流预测是实现交通流诱导的关键技术之一.针对目前短时交通混沌预测模型预测结果差异较大的问题,归纳了4种基于混沌理论的短时交通流预测模型:RBF神经网络模型、最大Lyapunov指数模型、局域线性模型和Volterra滤波器自适应预测模型,并对这4种预测模型进行了比较研究.应用4种预测模型对几个典型的非线性系统进行预测,验证了算法的准确性.然后用这4种预测模型对微观实测交通流的时间序列进行实证分析.仿真结果表明,4种预测模型对典型混沌时间序列具有很好的预测效果;而对实测交通流预测,其预测精度和稳定性较差,但可以满足实时交通流预测的需要.     

Dynamics of Phase Transitions in a Piecewise Linear Diatomic Chain

We consider a diatomic chain with nearest neighbors connected by phase-transforming springs. Assuming a piecewise linear interaction force, we use the Fourier transform to construct exact traveling wave solutions representing a moving phase-transition front and examine their stability through numerical experiments. We find that the identified traveling wave solutions may be stable in some velocity intervals. We show that the kinetic relation between the driving force on the phase boundary and its velocity is significantly affected by the ratio of the two masses. When the ratio is small enough, the relation may become multivalued at some velocities, with the two solutions corresponding to the different orders in which the two springs in a dimer cell change phase. The model bears additional interesting waveforms such as the so-called twinkling phase, which is also briefly discussed and compared to its monatomic analog.     

First-Order Phase Transitions and Hysteresis

A simple phase-field model for first-order phase transitions with hysteresis is proposed. It describes both temperature- and stress-induced transitions between austenitic and (oriented) martensitic regimes in a shape memory alloy (SMA). Finally, numerical simulations of local paths of the system are performed in the (ε,σ) and (ε,θ) planes, respectively, when either stress or temperature cyclic processes are considered and phase diffusion is neglected.     

Stability of Multidimensional Phase Transitions

In this paper, the author studies the multidimensional stability of subsonic phase transitions in a steady supersonic flow of van der Waals type. The viscosity capillarity criterion （in ＂Arch. Rat. Mech. Anal., 81（4）, 1983, 301-315＂） is used to seek physical admissible planar waves. By showing the Lopatinski determinant being non-zero, it is proved that subsonic phase transitions are uniformly stable in the sense of Majda （in ＂Mem. Amer. Math. Soc., 41（275）, 1983, 1-95＂） under both one dimensional and multidimensional perturbations.     

Helbing流体力学交通流模型的守恒形式

得到了Helbing交通流流体力学模型的标准守恒形式,并证明了模型的双曲性,这对研究模型的解析性质和数值格式至关重要.基于给出的守恒形式,设计了高效求解模型方程的LDG(lo-cal discontinuous Galerkin)格式,并模拟了由不稳定平衡态到稳定的时停时走波的演化.数值模拟也表明,通过扩散系数校正确实使模型得到改进,避免了车辆碰撞和出现极端高密度.