scholarly journals Numerical approximations for a three-component Cahn–Hilliard phase-field model based on the invariant energy quadratization method

2017 ◽  
Vol 27 (11) ◽  
pp. 1993-2030 ◽  
Author(s):  
Xiaofeng Yang ◽  
Jia Zhao ◽  
Qi Wang ◽  
Jie Shen
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Approximation Schemes ◽  
Numerical Schemes ◽  
Time Step ◽  
Invariant Energy Quadratization ◽  
The Stability ◽  
2D And 3D ◽  
Well Posed

How to develop efficient numerical schemes while preserving energy stability at the discrete level is challenging for the three-component Cahn–Hilliard phase-field model. In this paper, we develop a set of first- and second-order temporal approximation schemes based on a novel “Invariant Energy Quadratization” approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to well-posed linear systems with a linear symmetric, positive definite at each time step. We prove that the developed schemes are unconditionally energy stable and present various 2D and 3D numerical simulations to demonstrate the stability and the accuracy of the schemes.

Phase-field model: Boundary layer, velocity of propagation, and the stability spectrum

1992 ◽  
Vol 46 (24) ◽  
pp. 16045-16057 ◽  
Author(s):  
Raz Kupferman ◽  
Ofer Shochet ◽  
Eshel Ben-Jacob ◽  
Zeev Schuss
Keyword(s):  
Boundary Layer ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Stability Spectrum ◽  
Layer Velocity ◽  
Velocity Of Propagation ◽  
The Stability

Triangulation Based Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

2018 ◽  
Author(s):  
Ruochun Zhang ◽  
Xiaoping Qian
Keyword(s):  
Convergence Analysis ◽  
Phase Field ◽  
Isogeometric Analysis ◽  
Field Model ◽  
Complex Geometry ◽  
Phase Field Model ◽  
Initial Condition ◽  
Isoperimetric Problems ◽  
Time Step ◽  
System Evolution

This paper presents the triangulation based isogeometric analysis of the Cahn–Hilliard phase-field model. We validate our method by convergence analysis, show detailed system evolution from a randomly perturbed initial condition and then discuss related isoperimetric problems. Lastly an example highlighting its efficacy in complex geometry is provided. Triangulation based isogeometric analysis shows time step stability and complex geometry adaptability in our experiments.

scholarly journals Comparative analysis of different numerical schemes in solute trapping simulations by using the phase-field model with finite interface dissipation

2016 ◽  
Vol 52 (1) ◽  
pp. 77-85 ◽  
Author(s):  
X. Yang ◽  
Y. Tang ◽  
D. Cai ◽  
L. Zhang ◽  
Y. Du ◽  
...  
Keyword(s):  
Comparative Analysis ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Explicit Scheme ◽  
Numerical Schemes ◽  
Phase Field Simulation ◽  
Solute Trapping ◽  
Field Simulation ◽  
Relaxation Scheme

Two different numerical schemes, the standard explicit scheme and the time-elimination relaxation one, in the framework of phase-field model with finite interface dissipation were employed to investigate the solute trapping effect in a Si-4.5 at.% As alloy during rapid solidification. With the equivalent input, a unique solute distribution under the steady state can be obtained by using the two schemes without restriction to numerical length scale and interface velocity. By adjusting interface width and interface permeability, the experimental solute segregation coefficients can be well reproduced. The comparative analysis of advantages and disadvantages in the two numerical schemes indicates that the time-elimination relaxation scheme is preferable in one-dimensional phase-field simulation, while the standard explicit scheme seems to be the only choice for two- or three dimensional phase-field simulation. Furthermore, the kinetic phase diagrams in the Si-As system were predicted by using the phase-field simulation with the time-elimination relaxation scheme.

scholarly journals Hierarchy of consistent n-component Cahn–Hilliard systems

2014 ◽  
Vol 24 (14) ◽  
pp. 2885-2928 ◽  
Author(s):  
Franck Boyer ◽  
Sebastian Minjeaud
Keyword(s):  
Phase Field ◽  
Multiphase Flows ◽  
Field Model ◽  
Phase Field Model ◽  
Numerical Results ◽  
Phase Model ◽  
Navier Stokes ◽  
Two Phase ◽  
Two Phases ◽  
Well Posed

In this paper, we propose a new generalization of the well-known Cahn–Hilliard two-phase model for the modeling of n-phase mixtures. The model is derived using the consistency principle: we require that our n-phase model exactly coincides with the classical two-phase model when only two phases are present in the system. We give conditions for the model to be well-posed. We also present numerical results (including simulations obtained when coupling the Cahn–Hilliard system with the Navier–Stokes so as to obtain a phase-field model for multiphase flows) to illustrate the capability of such modeling.

Numerical Simulation of Homogeneous Polycrystalline Grain Formation Using Multi-Phase-Field Model

2012 ◽  
Vol 197 ◽  
pp. 628-632 ◽  
Author(s):  
Takuya Uehara ◽  
Hideyuki Suzuki
Keyword(s):  
Grain Size ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Homogeneous Microstructure ◽  
Grain Formation ◽  
Polycrystalline Microstructure ◽  
The Stability ◽  
Multi Phase ◽  
Original Formula

A modified multi-phase-field model for regenerating a homogeneous polycrystalline microstructure was presented. An extra term was introduced to the original formula by Steinbach et al. by assuming that the stability of every grain constituting the microstructure depends on the grain size distribution. The effect of the term on the obtained microstructure was then verified by numerical simulations, and it was found that a homogeneous microstructure having nearly the same shape and size was generated. The influence of the parameter was also investigated, and it revealed that the parameter was dominative on the grain size at the steady state.

The Phase-Field Simulation of the Grain Growth under External Field

2011 ◽  
Vol 299-300 ◽  
pp. 69-72
Author(s):  
Li Li Zhang ◽  
Ming Gao ◽  
Shao Nan Tang
Keyword(s):  
External Field ◽  
Grain Growth ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Grain Number ◽  
Grain Distribution ◽  
2D And 3D ◽  
The Relationship ◽  
New Phase

A new phase field model coupled with external field was used to simulate 2D and 3D grain growth processing. The simulation results showed that 2D grain size distribution under external field was inhomogeneous. The evolutions of 2D grain under different external field strength were roughly the same, but the grain number was obviously different with the changes of external fields. Under the external field, the relationship between 2D grain area and time was linear. 3D grain distribution was inhomogeneous in the direction of the external field gradient. The nucleation gradually decreased in the direction of change.

On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

2012 ◽  
Vol 4 (06) ◽  
pp. 665-684 ◽  
Author(s):  
Christopher E. Goodyer ◽  
Peter K. Jimack ◽  
Andrew M. Mullis ◽  
Hongbiao Dong ◽  
Yu Xie
Keyword(s):  
Binary Alloy ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Second Order ◽  
Three Dimensions ◽  
Computational Techniques ◽  
Time Step ◽  
Alloy Solidification ◽  
Fully Implicit

AbstractA fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

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