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Journal articles on the topic 'Cahn-Hillard equation'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Theljani, Anis, Hamdi Houichet, and Anis Mohamed. "An adaptive Cahn-Hilliard equation for enhanced edges in binary image inpainting." Journal of Algorithms & Computational Technology 14 (January 2020): 174830262094143. http://dx.doi.org/10.1177/1748302620941430.

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We consider the Cahn-Hilliard equation for solving the binary image inpainting problem with emphasis on the recovery of low-order sets (edges, corners) and enhanced edges. The model consists in solving a modified Cahn-Hilliard equation by weighting the diffusion operator with a function which will be selected locally and adaptively. The diffusivity selection is dynamically adopted at the discrete level using the residual error indicator. We combine the adaptive approach with a standard mesh adaptation technique in order to well approximate and recover the singular set of the solution. We give some numerical examples and comparisons with the classical Cahn-Hillard equation for different scenarios. The numerical results illustrate the effectiveness of the proposed model.

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2

Songmu, Zheng. "Asymptotic behavior of solution to the Cahn-Hillard equation." Applicable Analysis 23, no. 3 (December 1986): 165–84. http://dx.doi.org/10.1080/00036818608839639.

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3

Radkevich, E., and M. Zakharchenko. "The Singular Limit Problem to the Extended Cahn–Hillard Equation." Journal of Mathematical Sciences 120, no. 3 (March 2004): 1424–41. http://dx.doi.org/10.1023/b:joth.0000016059.66277.da.

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4

Львов, П. Е., and В. В. Светухин. "Влияние флуктуаций на образование выделений вторых фаз на границах зерен." Физика твердого тела 61, no. 2 (2019): 357. http://dx.doi.org/10.21883/ftt.2019.02.47138.232.

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AbstractBased on the free energy density functional method (modified Cahn–Hillard–Cook equation), the formation kinetics of secondary phases in binary alloys is considered in the presence of composition fluctuations and with inclusion of the grain boundaries influences. It is revealed that the existence of grain boundaries and the fluctuations at the initial stage of the phase transition can lead to the appearance of anomalous growth rate of the average precipitate size due to a competition of various decomposition mechanisms.

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5

Nastar, Maylise. "Beyond the Cahn-Hilliard Equation: a Vacancy-Based Kinetic Theory." Solid State Phenomena 172-174 (June 2011): 321–30. http://dx.doi.org/10.4028/www.scientific.net/ssp.172-174.321.

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A Self-Consistent Mean Field (SCMF) kinetic theory including an explicit description ofthe vacancy diffusion mechanism is developed. The present theory goes beyond the usual local equi-librium hypothesis. It is applied to the study of the early time spinodal decomposition in alloys. Theresulting analytical expression of the structure function highlights the contribution of the vacancydiffusion mechanism. Instead of the single amplification rate of the Cahn-Hillard linear theory, thelinearized SCMF kinetic equations involve three constant rates, first one describing the vacancy re-laxation kinetics, second one related to the kinetic coupling between local concentrations and paircorrelations and the third one representing the spinodal amplification rate. Starting from the same va-cancy diffusion model, we perform kineticMonte Carlo simulations of a Body Centered Cubic (BCC)demixting alloy. The resulting spherically averaged structure function is compared to the SCMF pre-dictions. Both qualitative and quantitative agreements are satisfying.

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6

王, 乙竹. "Regularity Criteria for the Navier-Stokes-Cahn-Hillard Equation in the Morrey-Campanato Space." Advances in Applied Mathematics 10, no. 06 (2021): 2095–104. http://dx.doi.org/10.12677/aam.2021.106219.

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7

Hernandez, Alvaro, and Michal Kowalczyk. "Nondegeneracy and the Jacobi fields of rotationally symmetric solutions to the Cahn-Hillard equation." Indiana University Mathematics Journal 68, no. 4 (2019): 1047–87. http://dx.doi.org/10.1512/iumj.2019.68.7718.

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8

Sigala-García, Darío A., Víctor M. López-Hirata, Maribel L. Saucedo-Muñoz, Héctor J. Dorantes-Rosales, and José D. Villegas-Cárdenas. "Phase-Field Simulation of Spinodal Decomposition in Mn-Cu Alloys." Metals 12, no. 7 (July 19, 2022): 1220. http://dx.doi.org/10.3390/met12071220.

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The spinodal decomposition was studied in the aged Mn-40 at. %Cu, Mn-30 at. %Cu, Mn-20 at. %Cu alloys using a phase-field model based on the Cahn–Hillard equation, considering a subregular solution model and the energy contribution of the magnetic behavior. The simulations were performed at aging temperatures of 300, 400, and 500 °C for times from 1 to 240 min. The growth kinetics of the Mn concentration profiles with time indicated clearly that the phase decomposition of the supersaturated solid solution γ into a mixture of Mn-rich γ′ and Cu-rich γ phases occurred by the spinodal decomposition mechanism. Moreover, the phase decomposition at the early stages of aging exhibited the characteristic morphology of spinodal decomposition, an interconnected and percolated microstructure of the decomposed phases. The most rapid growth kinetics of spinodal decomposition occurred for the aging of Mn-20 and 30 at. %Cu alloys because of the higher driving force. The presence of the phase decomposition is responsible for the increase in hardness, as well as the improvement of the damping capacity of Mn-Cu alloys.

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9

李, 亚楠. "Finite Element Method for the Modified Cahn-Hilliard Equation with the Concentration Mobility and the Logarithmic Potential." Advances in Applied Mathematics 09, no. 09 (2020): 1383–93. http://dx.doi.org/10.12677/aam.2020.99164.

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10

Polat, M., A. O. Çelebı, and N. Çali⋅kan. "Global attractors for the 3D viscous Cahn–Hillard equations in an unbounded domain." Applicable Analysis 88, no. 8 (August 2009): 1157–71. http://dx.doi.org/10.1080/00036810903156172.

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11

Vorobev, Anatoliy, and Tatyana Lyubimova. "Vibrational convection in a heterogeneous binary mixture. Part 1. Time-averaged equations." Journal of Fluid Mechanics 870 (May 14, 2019): 543–62. http://dx.doi.org/10.1017/jfm.2019.282.

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High-frequency vibrations of a container filled with a fluid generate pulsation flows that however are barely visible with the naked eye, and induce the slow but large-amplitude averaged flows that are important for various practical applications. In this work we derive a theoretical model that gives the averaged description of the influence of uniform high-frequency vibrations on an isothermal mixture of two slowly miscible liquids. The miscible multiphase system is described within the framework of the phase-field approach. The full Cahn–Hillard–Navier–Stokes equations are split into the separate systems for the quasi-acoustic, pulsating and averaged flow fields, eliminating the need for the resolution of the short time scale pulsation motion and thus making the analysis of the long-term evolution much more efficient. The resultant averaged model includes the effects of concentration diffusion and barodiffusion, the dynamic interfacial stresses and the generation of the hydrodynamic flows by non-homogeneities of the concentration field (when they are combined with the effects of gravity and vibrations). The resultant model for the vibrational convection in a heterogeneous mixture of two fluids separated by diffusive boundaries could be used for the description of processes of mixing/de-mixing, solidification/melting, polymerisation, etc. in the presence of vibrations.

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12

Akinyemi, Lanre, Olaniyi S. Iyiola, and Udoh Akpan. "Iterative methods for solving fourth‐ and sixth‐order time‐fractional Cahn‐Hillard equation." Mathematical Methods in the Applied Sciences, January 20, 2020. http://dx.doi.org/10.1002/mma.6173.

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13

Promislow, Keith, and Qiliang Wu. "Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation." Discrete & Continuous Dynamical Systems - S, 2022, 0. http://dx.doi.org/10.3934/dcdss.2022035.

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<p style='text-indent:20px;'>Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [<xref ref-type="bibr" rid="b8">8</xref>]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon\ll1 $\end{document}</tex-math></inline-formula>, we show that this induces undulated bilayer solutions whose width perturbations decay on an <inline-formula><tex-math id="M2">\begin{document}$ O\!\left( \varepsilon^{-1/2}\right) $\end{document}</tex-math></inline-formula> inner length scale that is long in comparison to the <inline-formula><tex-math id="M3">\begin{document}$ O(1) $\end{document}</tex-math></inline-formula> scale that characterizes the bilayer width.</p>

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14

Qiu, Rundi, Renfang Huang, Yao Xiao, Jingzhu Wang, Zhen Zhang, Jieshun Yue, Zhong Zeng, and Yiwei Wang. "Physics-informed neural networks for phase-field method in two-phase flow." Physics of Fluids, April 24, 2022. http://dx.doi.org/10.1063/5.0091063.

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The complex flow modeling based on machine learning is becoming a promising way to describe multiphase fluid systems. This work demonstrates how a physics-informed neural network promotes the combination of traditional governing equations and advanced interface evolution equations without intricate algorithms. We develop physics-informed neural networks for phase-field method (PF-PINNs) in two-dimensional immiscible incompressibletwo-phase flow. The Cahn-Hillard equation and Navier-Stokes equationsare encoded directly into the residuals of a fully connected neural network. Compared withthe traditional interface-capturing method, the phase-field model has a firm physical basis because it is based on the Ginzburg-Landau theory and conserves mass and energy. It also performs well in two-phase flow at large density ratio. However, the high-order differential nonlinear term of the Cahn-Hilliard equation poses a great challenge for obtaining numerical solutions. Thus, in this work, we adopt neural networks to tackle the challenge by solving high-order derivate terms and capture the interface adaptively. To enhance the accuracy and efficiency of PF-PINNs, we use the time-marching strategy and the forced constraint of the density and viscosity. The PF-PINNs are tested by two cases for presenting the interface-capturing ability of PINNs and evaluating the accuracy of PF-PINNs at large density ratio (up to 1000). The shape of the interface in both cases coincides well with the reference results, and the dynamic behavior of the second case is precisely captured.We also quantify the variations in the center of mass and increasing velocity over time for validation purposes. The results show that PF-PINNs exploit the automatic differentiation without sacrificing the high accuracy of the phase-field method.

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15

Qiu, Rundi, Renfang Huang, Yao Xiao, Jingzhu Wang, Zhen Zhang, Jieshun Yue, Zhong Zeng, and Yiwei Wang. "Physics-informed neural networks for phase-field method in two-phase flow." Physics of Fluids, April 24, 2022. http://dx.doi.org/10.1063/5.0091063.

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The complex flow modeling based on machine learning is becoming a promising way to describe multiphase fluid systems. This work demonstrates how a physics-informed neural network promotes the combination of traditional governing equations and advanced interface evolution equations without intricate algorithms. We develop physics-informed neural networks for phase-field method (PF-PINNs) in two-dimensional immiscible incompressibletwo-phase flow. The Cahn-Hillard equation and Navier-Stokes equationsare encoded directly into the residuals of a fully connected neural network. Compared withthe traditional interface-capturing method, the phase-field model has a firm physical basis because it is based on the Ginzburg-Landau theory and conserves mass and energy. It also performs well in two-phase flow at large density ratio. However, the high-order differential nonlinear term of the Cahn-Hilliard equation poses a great challenge for obtaining numerical solutions. Thus, in this work, we adopt neural networks to tackle the challenge by solving high-order derivate terms and capture the interface adaptively. To enhance the accuracy and efficiency of PF-PINNs, we use the time-marching strategy and the forced constraint of the density and viscosity. The PF-PINNs are tested by two cases for presenting the interface-capturing ability of PINNs and evaluating the accuracy of PF-PINNs at large density ratio (up to 1000). The shape of the interface in both cases coincides well with the reference results, and the dynamic behavior of the second case is precisely captured.We also quantify the variations in the center of mass and increasing velocity over time for validation purposes. The results show that PF-PINNs exploit the automatic differentiation without sacrificing the high accuracy of the phase-field method.

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16

Biswas, Tania, Sheetal Dharmatti, and Manil T. Mohan. "Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations." Journal of Mathematical Fluid Mechanics 22, no. 3 (June 6, 2020). http://dx.doi.org/10.1007/s00021-020-00493-8.

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