Phase transition in binary systems and the binary alloy problem

In this paper we consider a phase transition in binary systems describing the solidification of a dilute alloy in one space dimension, in which the equations of heat and mass transfer are coupled through the conditions at the free boundary. A kinetic condition is introduced to suggest a regularization of the model, which admits a unique solution, and to obtain the generalized solution of the dilute-alloy problem under an additional assumption.     

Inverse problem for the model of evaporation of a binary alloy

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 10–13, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Modeling binary alloy solidification by a random projection method

This paper addresses the numerical modeling of the solidification of a binary alloy that obeys a liquidus–solidus phase diagram. In order to capture the moving melting front, we introduce a Lagrange projection scheme based on a random sampling projection. Using a finite volume formulation, we define accurate numerical fluxes for the temperature and concentration fields which guarantee the sharp treatment of the boundary conditions at the moving front, especially the jump of the concentration according to the liquidus–solidus diagram. We provide some numerical illustrations which assess the good behavior of the method: maximum principle, stability under CFL condition, numerical convergence toward self‐similar solutions, ability to handle two melting fronts.     

Static conductivity of a binary alloy in the traveling-cluster approximation

A study is made of the problem of finding the static electrical conductivity of a binary alloy in the tight-binding model with offdiagonal disorder in the presence of short-range order in the system. A general scheme is proposed for constructing selfconsistent approximations for averaged products of two Green's functions in order to determine the kinetic properties of disordered alloys. The approach corresponds to the traveling-cluster approximation used to find the electron spectra of binary alloys. The augmented-space formalism provides the framework for the treatment.Physicotechnical Institute, Ukrainian Academy of Sciences, Kharkov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 128–137, January, 1992.     

Global solutions to a degenerate solutal phase-field model for the solidification of a binary alloy

We consider a degenerate solutal phase-field model for the solidification of a binary alloy. This model is devoted to the evolution of the phase-field variable together with the relative concentration of the alloy for which the equation may degenerate. The existence of global weak solutions is proved for the degenerate case with a loss of regularity for the concentration in comparison with the non-degenerate case.     

Study of a system for the isothermal separation of components in a binary alloy with change of phase

An isothermal model describing the separation of the componentsof a binary metallic alloy is considered. A process of phasetransition is also assumed to occur in the solder; hence, thestate of the material is described by two order parameters,i.e. the concentration c of the first component and the phasefield . A physical derivation is provided starting from energybalance considerations. The resulting system of PDEs consistsof a rather regular second-order parabolic equation for coupledwith a fourth-order relation of Cahn–Hilliard type forc with constraint and solution-dependent mobility. Global existenceof solutions is proved and several regularity properties arediscussed under more restrictive assumptions on the physicalparameters. Continuous dependence on data is shown in a specialcase. An asymptotic analysis of the model is also performed,yielding at the limit step a coupling of the original phasefield equation with a Stefan-like system for c.     

双水平集逼近油藏模型特征的数值模拟方法

本文使用双水平集函数逼近油藏模型特征, 构造出Uzawas 算法进行数值模拟. 对于两相流渗透率的数值求解问题, 可以通过测量油井数据和地震波数据来实现. 将构造出来的带限制的最优化问题使用变异的Lagrange 方法求解. 如果使用双水平集函数逼近渗透率函数, 则需要对Lagrange 函数进行修正, 从而将带限制的最优化问题转化成无限制的最优化问题. 由于双水平集函数的优越性, 进一步构造出最速梯度下降Uzawas 算法和算子分裂格式Uzawas 算法进行求解对应的最优化子问题. 数值算例表明设计的算法是高效的、稳定的.     

Existence of solutions to a phase‐field model for the isothermal solidification process of a binary alloy

We investigate the well‐posedness of a phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger [12]. Existence of weak solution as well as regularity and uniqueness results are established under Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. Copyright © 2000 John Wiley & Sons, Ltd.     

The morphological stability of dendritic growth from the binary alloy melt with an external flow

The morphological stability of dendritic growth from the binary alloy melt with an external flow is studied by means of the matched asymptotic expansion method and multiple variable expansion method. The uniformly valid asymptotic solution is obtained for the case of the large Schmidt number. The analytical result reveals that the stability of dendritic growth depends on a critical stability number above which dendritic growth is stable. The selection condition of dendritic growth determines the Peclet number, tip growth velocity, tip radius and oscillation frequency, which is significantly affected by the external flow. The stability mechanism of dendritic growth in the binary alloy melt with the external flow remains the same as that in pure melt. In the binary alloy melt with the external flow the solute concentration destabilizes the dendritic growth system. The numerical computation for various growth conditions demonstrates the variations of the critical stability number, tip growth velocity, tip radius, and oscillatory frequency with the undercooling, external flow and morphological number.     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

A priori error estimates of a finite-element method for an isothermal phase-field model related to the solidification process of a binary alloy

We introduce a piecewise linear finite-element scheme with semi-implicittime discretization for an evolutionary phase field system modellingthe isothermal solidification process of a binary alloy. Thissystem can be written in a vectorial form as a nonlinear parabolicsystem. The convergence of the scheme with error estimate isthen proved by introducing a generalized vectorial ellipticprojector.     

BDD-based heuristics for binary optimization

In this paper we introduce a new method for generating heuristic solutions to binary optimization problems. We develop a technique based on binary decision diagrams. We use these structures to provide an under-approximation to the set of feasible solutions. We show that the proposed algorithm delivers comparable solutions to a state-of-the-art general-purpose optimization solver on randomly generated set covering and set packing problems.     

Multi-resolution method for binary tomography

Multi-resolution and region-growing strategies have been successfully used in several fields of image processing. In this paper we investigate how these two strategies can be applied for binary tomography. We included these strategies into a reconstruction method using simulated annealing and tested these new methods on different images.     

Extractors for binary elliptic curves

We propose a simple and efficient deterministic extractor for an ordinary elliptic curve E, defined over $\mathbb{F}_{2^n}$ , where n = 2? and ? is a positive integer. Our extractor, for a given point P on E, outputs the first ${\mathbb{F}}_{2^\ell}$ -coefficient of the abscissa of the point P. We also propose a deterministic extractor for the main subgroup G of E, where E has minimal 2-torsion. We show that if a point P is chosen uniformly at random in G, the bits extracted from the point P are indistinguishable from a uniformly random bit-string of length ?.     

Swan's theorem for binary tetranomials

Swan's theorem [Pacific J. Math. 12 (1962) 1099–1106] determines the parity of the number of irreducible factors of a binary trinomial. This paper does the same for a binary tetranomial. When phrased in terms of the periodic portion of the factor-parity sequence, the result in several cases is comparable in simplicity to Swan's result for square-free trinomials.     

On binary codes for identification

A code C ⊆ 𝔽₂ⁿ is called t‐identifying if the sets B_t(x) ∩ C are all nonempty and different. Constructions of t‐identifying codes are given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 151–156, 2000