A nonconforming finite element method for the Cahn–Hilliard equation

Zhang, Shuo; Wang, Ming

doi:10.1016/j.jcp.2010.06.020

Cited by 50 publications

(35 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [108], a convexity-splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable was used. And the scheme preserved the mass conservation and energy dissipation properties of the original problem.…”

Section: Cahn-hilliard Solvermentioning

confidence: 99%

Phase-Field Models for Multi-Component Fluid Flows

Kim

2012

Commun. comput. phys.

471

282

View full text Add to dashboard Cite

Abstract. In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.

show abstract

Section: Cahn-hilliard Solvermentioning

confidence: 99%

Phase-Field Models for Multi-Component Fluid Flows

Kim

2012

Commun. comput. phys.

471

282

View full text Add to dashboard Cite

show abstract

“…However, there are only a limited number of finite elements that fulfill the above continuity condition, especially in two and three dimensions. This continuity requirement was a primary motivation for the development and deployment of methods like the natural element method [22], nonconforming finite element method [17], NURBS-based variational formulation [19], and discontinuous Galerkin method [23] to solve the Cahn-Hilliard equation in its original form (with impressive results). However, this continuity requirement can be avoided by a simple splitting strategy, following which standard C 0 -continuous finite elements can be utilized.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Some terms are treated implicitly while others are treated explicitly. One such approach splits the free energy term into expansive and contractive parts, and treats the former explicitly and the later implicitly [28,17]. Then, the contractive part of the free energy can be linearized, similar to the biharmonic term of the Cahn-Hilliard equation.…”

Section: Temporal Discretizationmentioning

confidence: 99%

“…Work by Langer et al [10] is probably the first numerical study. Since then multiple models based on finite difference [11], finite volume [12,13], finite element [14][15][16][17] and spectral methods [18], all with their advantages and drawbacks, have been proposed. It has to be noted that the majority of effort was concentrated on two dimensional problems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

Wodo

Ganapathysubramanian

2011

Journal of Computational Physics

134

106

View full text Add to dashboard Cite

a b s t r a c tWe present an efficient numerical framework for analyzing spinodal decomposition described by the Cahn-Hilliard equation. We focus on the analysis of various implicit time schemes for two and three dimensional problems. We demonstrate that significant computational gains can be obtained by applying embedded, higher order Runge-Kutta methods in a time adaptive setting. This allows accessing time-scales that vary by five orders of magnitude. In addition, we also formulate a set of test problems that isolate each of the sub-processes involved in spinodal decomposition: interface creation and bulky phase coarsening. We analyze the error fluctuations using these test problems on the split form of the Cahn-Hilliard equation solved using the finite element method with basis functions of different orders. Any scheme that ensures at least four elements per interface satisfactorily captures both sub-processes. Our findings show that linear basis functions have superior error-to-cost properties.This strategy -coupled with a domain decomposition based parallel implementationlet us notably augment the efficiency of a numerical Cahn-Hillard solver, and open new venues for its practical applications, especially when three dimensional problems are considered. We use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions. The framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3D -sphere, cylinder, lamella, doubly periodic surface with genus two (Lawson surface) and triply periodic minimal surface (P Schwarz surface).

show abstract

“…In applied sciences, many model problems take the formulation of fourth order elliptic perturbation problems, such as, e.g., the linearized Cahn-Hilliard equation [9,31,39,43,[51][52][53], and the strain gradient problem [1,10,11,28,33,49]. In order for the robust discretisation of such problems, numerical schemes that work for both fourth and second order problems are needed.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

2016

Self Cite

View full text Add to dashboard Cite

In this paper, we present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h 2 ) order in L 2 norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h 2 ) order in energy norm, and the convergence rate in L 2 norm is still of O(h 2 ) order, which can not be improved. Numerical examples are presented to demonstrate our theoretical results.

show abstract

A nonconforming finite element method for the Cahn–Hilliard equation

Cited by 50 publications

References 45 publications

Phase-Field Models for Multi-Component Fluid Flows

Phase-Field Models for Multi-Component Fluid Flows

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

Contact Info

Product

Resources

About