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 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 Dynamic Modelling of Shape Memory Alloys Based on Isogeometric Analysis

2012 ◽  
Vol 78 ◽  
pp. 63-68 ◽  
Author(s):  
Rakesh Dhote ◽  
Hector Gomez ◽  
Roderick Melnik ◽  
Jean Zu
Keyword(s):  
Shape Memory Alloys ◽  
Numerical Simulations ◽  
Shape Memory ◽  
Microstructure Evolution ◽  
Phase Field ◽  
Isogeometric Analysis ◽  
Field Model ◽  
Phase Field Model ◽  
Higher Order ◽  
Martensitic Phase Transformations

Shape Memory Alloys (SMAs) exhibit complex behaviors as a result of their constituent phases and microstructure evolution. In this paper, we focus on the numerical simulations of microstructure evolution in SMAs using a phase-field model for the two dimensional square-to-rectangular martensitic phase transformations. The phase-field model, based on the Ginzburg-Landau theory, has strong non-linearity, thermo-mechanical coupling, and higher-order differential terms and presents substantial challenges for numerical simulations. The isogeometric analysis, developed in this paper using the rich NURBS basis functions, offers several advantages in solving such complex problems with higher-order partial differential equations as the problem at hand. To our best knowledge, we report here for the first time the use of the new method in the study of microstructure evolution in SMAs. The numerical experiments of microstructure evolution have been carried out on the FePd SMA specimen. The results are in good agreement with those previously reported in the literature.

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

2007 ◽  
Author(s):  
Hector Gomez ◽  
Victor M. Calo ◽  
Yuri Bazilevs ◽  
Thomas J. Hughes
Keyword(s):  
Phase Field ◽  
Isogeometric Analysis ◽  
Field Model ◽  
Phase Field Model

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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