Fast and accurate coarsening simulation with an unconditionally stable time step

Vollmayr-Lee, Benjamin; Rutenberg, Andrew D.

doi:10.1103/physreve.68.066703

Cited by 125 publications

(105 citation statements)

References 6 publications

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“…The equation can be solved efficiently through fast Fourier transform methods. Similar ideas were also used to simulate coarsening in the Cahn-Hilliard equation (Vollmayr- Lee & Rutenberg 2003) and surface diffusion (Smereka 2003). The diffuseinterface model imposes a constraint on the spatial resolution in order to resolve the transition layer, x 6 C .…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

A diffuse-interface model for electrowetting drops in a Hele-Shaw cell

Glasner

Bertozzi

et al. 2007

J. Fluid Mech.

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Electrowetting has recently been explored as a mechanism for moving small amounts of fluids in confined spaces. We propose a diffuse-interface model for drop motion, due to electrowetting, in a Hele-Shaw geometry. In the limit of small interface thickness, asymptotic analysis shows that the model is equivalent to Hele-Shaw flow with a voltage-modified Young-Laplace boundary condition on the free surface. We show that details of the contact angle significantly affect the time scale of motion in the model. We measure receding and advancing contact angles in the experiments and derive their influence through a reduced-order model. These measurements suggest a range of time scales in the Hele-Shaw model which include those observed in the experiment. The shape dynamics and topology changes in the model agree well with the experiment, down to the length scale of the diffuse-interface thickness.

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Section: Numerical Simulations and Discussionmentioning

confidence: 99%

A diffuse-interface model for electrowetting drops in a Hele-Shaw cell

Glasner

Bertozzi

et al. 2007

J. Fluid Mech.

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“…This way to examine the stability of a central difference scheme by following the energy was already proposed to nonlinear problems a long time ago by Park [28]. We should mention that a similar analysis, based also on the Eyre's approach, performing numerical tests of stability and with a very complete classification scheme for the stable values of ∆t for the CH and Allen-Cahn equation in 2D was recently developed [22].…”

Section: The Properties Of the Cahn-hilliard Equationmentioning

confidence: 98%

Numerical study of the Cahn–Hilliard equation in one, two and three dimensions

Mello¹,

Filho²

2005

Physica A: Statistical Mechanics and its Applications

102

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The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it numerically increases with the dimensionality and therefore, there are several published numerical studies in one dimension (1D), dealing with different approaches, and much fewer in two dimensions (2D). In three dimensions (3D) there are very few publications, usually concentrate in some specific result without the details of the used numerical scheme. We present here a stable and fast conservative finite difference scheme to solve the Cahn-Hilliard with two improvements: a splitting potential into a implicit and explicit in time part and a the use of free boundary conditions. We show that gradient stability is achieved in one, two and three dimensions with large time marching steps than normal methods.

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“…The usual approach presented in the literature for simplified versions of the Cahn-Hilliard equation has been to use a few (2 or 3) different timestep sizes during the simulation [14]. These time steps are not selected by means of accuracy criteria, but by using approximate theories of the latetime behavior of the Cahn-Hilliard equation [73]. In this paper we propose an adaptive-in-time method where the time step is selected by using an accuracy criterion.…”

Section: Time Discretizationmentioning

confidence: 99%

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

Gómez¹,

Calo²,

Bazilevs³

et al. 2007

135

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The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions are not common because primal variational formulations of fourthorder operators are well defined and integrable only if the finite element basis functions are piecewise smooth and globally C 1 -continuous. There are a very limited number of two-dimensional finite elements possessing C 1 -continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two-and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of C 1 and higher-order continuity. A NURBSbased variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology. There are a very limited number of two-dimensional finite elements possessing C1-continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two-and three-dimensional geometric exibility, compact support, and, most importantly, the possibility of C1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

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Fast and accurate coarsening simulation with an unconditionally stable time step

Cited by 125 publications

References 6 publications

A diffuse-interface model for electrowetting drops in a Hele-Shaw cell

A diffuse-interface model for electrowetting drops in a Hele-Shaw cell

Numerical study of the Cahn–Hilliard equation in one, two and three dimensions

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

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