Two-phase flow analysis based on a phase-field model using orthogonal basis bubble function finite element method

In this study, a finite element method based on a phase-field model for gas–liquid two-phase flow is proposed. MINI element based on a bubble function element stabilisation method is employed for the incompressible Navier–Stokes equations. The Cahn–Hilliard equation is employed to estimate the interface of gas and liquid. The orthogonal basis bubble function element is used to solve the Cahn–Hilliard equation. In particular, a detailed explanation for solving the Cahn–Hilliard equation based on a finite element method is given.     

Effect of elastic strain energy on grain growth and texture in AZ31 magnesium alloy by phase-field simulation

A phase-field model is modified to investigate the grain growth and texture evolution in AZ31 magnesium alloy during stressing at elevated temperatures. The order parameters are defined to represent a physical variable of grain orientation in terms of three angles in spatial coordinates so that the grain volume of different order parameters can be used to indicate the texture of the alloy. The stiffness tensors for different grains are different because of elastic anisotropy of the magnesium lattice. The tensor is defined by transforming the standard stiffness tensor according to the angle between the (0001) plane of a grain and the direction of applied stress. Therefore, different grains contribute to different amounts of work under applied stress. The simulation results are well-explained by using the limited experimental data available, and the texture results are in good agreement with the experimental observations. The simulation results reveal that the applied stress strongly influences AZ31 alloy grain growth and that the grain-growth rate increases with the applied stress increasing, particularly when the stress is less than 400 MPa. A parameter (△d) is introduced to characterize the degree of grain-size variation due to abnormal grain growth; the △d increases with applied stress increasing and becomes considerably large only when the stress is greater than 800 MPa. Moreover, the applied stress also results in an intensive texture of the 〈0001〉 axis parallel to the direction of compressive stress in AZ31 alloy after growing at elevated temperatures, only when the applied stress is greater than 500 MPa.     

二元合金多晶粒的枝晶生长的等温相场模型

基于Ginzburg-Landau理论和单晶粒的枝晶生长模型,发展了一个单相场控制的多个晶粒的枝晶生长模型. 采用相场和溶质场耦合的方法,以Al-2%Cu合金为例模拟了二元合金等温凝固过程中多个晶粒的生长过程. 结果表明,这个模型的计算结果展现了多个晶粒枝晶的竞争生长,能较真实的再现凝固过程中的枝晶的生长过程. 关键词： 相场法 多晶粒 等温凝固 二元合金     

Phase-field simulation of the coupled evolutions of grain and twin boundaries in nanotwinned polycrystals

Nanotwinned polycrystals exhibit an excellent strength-ductility combination due to nanoscale twins and grains. However, nanotwin-assisted grain coarsening under mechanical loading reported in recent experiments may result in strength drop based on the Hall-Petch law. In this paper, a phase-field model is developed to investigate the effect of coupled evolutions of twin and grain boundaries on nanotwin-assisted grain growth. The simulation result demonstrates that there are three pathways for coupled motions of twin and grain boundaries in a bicrystal under the applied loading, dependent on the amplitude of applied loading and misorientation of the bicrystal. It reveals that a large misorientation angle and a large applied stress promote the twinning-driven grain boundary migration. The resultant twin-assisted grain coarsening is confirmed in the simulations for the microstructural evolutions in twinned and un-twinned polycrystals under a high applied stress.     

各向异性相态场模型的Gibbs—Thompson关系

本文利用奇异摄动的内、外解匹配方法,分析了各向异性时在相态场模型的边界层上表面张菌、法向速度、平均曲率和各向异性函数的影响,得到了各向异性时的Ｇｉｂｂｓ－Ｔｈｏｍｐｓｏｎ关系,以及边界层所满足的方程。     

Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

三维枝晶生长的相场法数值模拟研究

基于薄界面限制、耦合界面能各向异性的相场模型,采用动态计算区域的加速算法,对纯物质的三维枝晶生长进行了定量模拟,真实再现了枝晶的生长过程.对枝晶尖端进行了剖切分析,表明主枝截面上各向异性没有主枝方向各向异性明显;对枝晶尖端生长速度、尖端半径、Peclet数及临界稳定性参数σ^*进行模拟计算,并与同条件下报道值进行了对比分析,两者符合良好,并得到了与结晶理论相一致的枝晶生长规律,证实了相场方法模拟三维空间枝晶生长可行有效. 关键词： 相场方法 枝晶生长 微观组织 三维数值模拟     

Phase-field lattice-Boltzmann investigation of dendritic evolution under different flow modes

ABSTRACT

Numerical modelling offers an effective method to investigate the effect of convection on dendritic growth; however, the current numerical approach to modelling convection behaviour is simplified, e.g. only simulating a shear flow by setting a constant inlet velocity and a zero-velocity-gradient outlet boundary condition. In this work, based on a phase-field lattice-Boltzmann approach, the effect of various flow modes on dendritic growth is investigated by introducing an external force term to induce flow. Numerical tests (2-D and 3-D) validate that the results according to the force-induced flow agree well with those by the velocity-imposed flow. Intricate convection effects under complex boundary conditions are discussed in detail. Furthermore, this force-induced flow mode allows additional freedom by eliminating the restriction on the initial position of the nuclei, which provides new ways to the microstructure modelling under complex convection.     

Maximal attractors for the phase-field equations of penrose-fife type

In this paper, the dynamics for the phase-field equations of Penrose-Fife type arising from the study of phase transitions is investigated. One of important features of this problem is that the metric space H we work with is incomplete.     

Phase-field study of spinodal decomposition under effect of grain boundary

Grain boundary directed spinodal decomposition has a substantial effect on the microstructure evolution and properties of polycrystalline alloys. The morphological selection mechanism of spinodal decomposition at grain boundaries is a major challenge to reveal, and remains elusive so far. In this work, the effect of grain boundaries on spinodal decomposition is investigated by using the phase-field model. The simulation results indicate that the spinodal morphology at the grain boundary is anisotropic bicontinuous microstructures different from the isotropic continuous microstructures of spinodal decomposition in the bulk phase. Moreover, at grain boundaries with higher energy, the decomposed phases are alternating α/β layers that are parallel to the grain boundary. On the contrary, alternating α/β layers are perpendicular to the grain boundary.