Relevant bibliographies by topics / Allen-Cahn equation

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Allen-Cahn equation'

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Published: 4 June 2021
Last updated: 12 June 2025

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Journal articles on the topic "Allen-Cahn equation"

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Kim, Yongho, Gilnam Ryu, and Yongho Choi. "Fast and Accurate Numerical Solution of Allen–Cahn Equation." Mathematical Problems in Engineering 2021 (December 6, 2021): 1–12. http://dx.doi.org/10.1155/2021/5263989.

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Simulation speed depends on code structures. Hence, it is crucial how to build a fast algorithm. We solve the Allen–Cahn equation by an explicit finite difference method, so it requires grid calculations implemented by many for-loops in the simulation code. In terms of programming, many for-loops make the simulation speed slow. We propose a model architecture containing a pad and a convolution operation on the Allen–Cahn equation for fast computation while maintaining accuracy. Also, the GPU operation is used to boost up the speed more. In this way, the simulation of other differential equations can be improved. In this paper, various numerical simulations are conducted to confirm that the Allen–Cahn equation follows motion by mean curvature and phase separation in two-dimensional and three-dimensional spaces. Finally, we demonstrate that our algorithm is much faster than an unoptimized code and the CPU operation.
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Beneš, Michal, Shigetoshi Yazaki, and Masato Kimura. "Computational studies of non-local anisotropic Allen-Cahn equation." Mathematica Bohemica 136, no. 4 (2011): 429–37. http://dx.doi.org/10.21136/mb.2011.141702.

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Colli, Pierluigi, and Takeshi Fukao. "Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type." Advances in Nonlinear Analysis 9, no. 1 (2018): 16–38. http://dx.doi.org/10.1515/anona-2018-0055.

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Abstract The well-posedness of a system of partial differential equations with dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk Ω and on the boundary Γ. The Poisson equation for the chemical potential and the Allen–Cahn equation for the order parameter in the bulk Ω are considered as auxiliary conditions for solving the Cahn–Hilliard equation on the boundary Γ. Recently, the well-posedness of this equation with a dynamic boundary condition, both of Cahn–Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved.
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Lee, Dongsun, and Seunggyu Lee. "Image Segmentation Based on Modified Fractional Allen–Cahn Equation." Mathematical Problems in Engineering 2019 (January 30, 2019): 1–6. http://dx.doi.org/10.1155/2019/3980181.

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We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.
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Bhatt, Harish, Janak Joshi, and Ioannis Argyros. "Fourier Spectral High-Order Time-Stepping Method for Numerical Simulation of the Multi-Dimensional Allen–Cahn Equations." Symmetry 13, no. 2 (2021): 245. http://dx.doi.org/10.3390/sym13020245.

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This paper introduces the Fourier spectral method combined with the strongly stable exponential time difference method as an attractive and easy-to-implement alternative for the integration of the multi-dimensional Allen–Cahn equation with no-flux boundary conditions. The main advantages of the proposed method are that it utilizes the discrete fast Fourier transform, which ensures efficiency, allows an extension to two and three spatial dimensions in a similar fashion as one-dimensional problems, and deals with various boundary conditions. Several numerical experiments are carried out on multi-dimensional Allen–Cahn equations including a two-dimensional Allen–Cahn equation with a radially symmetric circular interface initial condition to demonstrate the fourth-order temporal accuracy and stability of the method. The numerical results show that the proposed method is fourth-order accurate in the time direction and is able to satisfy the discrete energy law.
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Cui, Runqing. "On time fractional Cahn-Allen equation." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 3 (2017): 7272–74. http://dx.doi.org/10.24297/jam.v13i3.6219.

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In [1], Ozkan Guner et al. obtained some exact solutions of the time fractional Cahn-Allen equation. By using the method proposed in [10], we have tested these solutions and have found that they are not the solutions of this equation.
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AKAGI, GORO, and MESSOUD EFENDIEV. "Allen–Cahn equation with strong irreversibility." European Journal of Applied Mathematics 30, no. 04 (2018): 707–55. http://dx.doi.org/10.1017/s0956792518000384.

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This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by derivingpartialenergy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solutionu(x,t) converges to a solution of an elliptic obstacle problem ast→ +∞.
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Pisante, Adriano, and Fabio Punzo. "Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows." Communications in Contemporary Mathematics 17, no. 05 (2015): 1450041. http://dx.doi.org/10.1142/s0219199714500412.

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We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.
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Zafar, Asim, Hadi Rezazadeh, and Khalid K. Ali. "On finite series solutions of conformable time-fractional Cahn-Allen equation." Nonlinear Engineering 9, no. 1 (2020): 194–200. http://dx.doi.org/10.1515/nleng-2020-0008.

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AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.
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Yokus, Asıf, and Hasan Bulut. "On the numerical investigations to the Cahn-Allen equation by using finite difference method." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 9, no. 1 (2018): 18–23. http://dx.doi.org/10.11121/ijocta.01.2019.00561.

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In this study, by using the finite difference method (FDM for short) and operators, the discretized Cahn-Allen equation is obtained. New initial condition for the Cahn-Allen equation is introduced, considering the analytical solution given in Application of the modified exponential function method to the Cahn-Allen equation, AIP Conference Proceedings 1798, 020033 [1]. It is shown that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in and Furthermore, the FDM is treated in order to obtain the numerical results and to construct a table including numerical and exact solutions as well as absolute measuring error. A comparison between the numerical and the exact solutions is supported with two and three dimensional graphics via Wolfram Mathematica 11.
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Dissertations / Theses on the topic "Allen-Cahn equation"

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Weber, Simon. "The sharp interface limit of the stochastic Allen-Cahn equation." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/66775/.

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We study the Allen-Cahn equation ut = ϵ2uxx + f(u) + ϵγW with an additive noise term ϵγW for small ϵ > 0, and in particular the limit ϵ → 0. This is a reaction-diffusion equation, where f (∙) is the negative derivative of a symmetric double-well potential. We study this equation in the interval (0, 1) with symmetric boundary conditions, and relatively general initial conditions, for W we take space-time white noise. Brassesco et al., Funaki and other authors showed (with different boundary conditions) that if we can project the solution of the equation to an energy-optimal deterministic solution with just one zero, then in the sharp interface limit ϵ → 0 of the solution appropriately rescaled in time is a standard Brownian motion. In this work, we extend these results to a much more general case: We start with fairly general initial conditions, show that after some time we are able to project the solution onto energy-optimal deterministic solutions with finitely many zeroes, after which we derive a semimartingale representation for the interfaces; this representation holds until two interfaces get close to each other and annihilate. In the sharp interface limit ϵ → 0, the appropriately time-rescaled interface position of the solution converges weakly to annihilating independent standard Brownian motions. We also derive an analogous result for smooth noise with trace-class covariance operator, in this case the phenomenon happens on a different timescale than for space-time white noise.
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Saoud, Wafa. "Etude d'un modèle d'équations couplées Cahn-Hilliard/Allen-Cahn en séparation de phase." Thesis, Poitiers, 2018. http://www.theses.fr/2018POIT2285/document.

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Cette thèse est une étude théorique d’un système d’équations de Cahn-Hilliard/Allen-Cahn couplées qui représente un mélange binaire en séparation de phase. Le but principal de l’étude est le comportement asymptotique des solutions en termes d’attracteurs exponentiels/globaux. Pour cette raison, l’existence et l’unicité de la solution sont étudiées tout d’abord. Une des principales applications de ce modèle d’équations est la cristallographie.Dans la première partie de la thèse, on examine le modèle proposé avec des conditions de type Dirichlet sur le bord et une non linéarité régulière de type polynomial : on réussit à trouver un attracteur exponentiel et par conséquence un attracteur global de dimension finie. Une non linéarité singulière de type logarithmique est ensuite prise dans la deuxième partie, cette fonction étant approchée par une suite de fonctions régulières et l’existence d’un attracteur global est démontrée sous des conditions au bord de type Dirichlet.Enfin, dans la dernière partie, le système est couplé avec une équation pour la température: suivant la loi de Fourrier premièrement, puis la loi de type III de la thermo-élasticité. Dans les deux cas, la dynamique de l’équation est étudiée et un attracteur exponentiel est trouvé malgré la difficulté créée par l’équation hyperbolique dans le deuxième cas<br>This thesis is a theoretical study of a coupled system of equations of Cahn-Hilliard and Allen-Cahn that represents phase separation of binary alloys. The main goal of this study is to investigate the asymptotic behavior of the solution in terms of exponential/global attractors. For this reason, the existence and unicity of the solution are first studied. One of the most important applications of this proposed model of equations is crystallography. In the first part of the thesis, the system is studied with boundary conditions of Dirichlet type and a regular nonlinearity (a polynomial). There, we prove the existence of an exponential attractor that leads to the existence of a global attractor of finite dimension. Then, a singular nonlinearity (a logarithmic potential) is considered in the second part. This function is approximated by a sequence of regular ones and a global attractor is found.At the end, the system of equations is coupled with temperature: with the Fourrier law in the first case, then with the type III law (in the context of thermoelasticity) in the second case. The dynamics of the equations are studied and the existence of an exponential attractor is obtained
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Schindler, Alexander [Verfasser], and Dirk [Akademischer Betreuer] Blömker. "Interface motion for the stochastic Allen-Cahn and Cahn-Hilliard equation / Alexander Schindler ; Betreuer: Dirk Blömker." Augsburg : Universität Augsburg, 2021. http://d-nb.info/1235326780/34.

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Zúñiga, Munizaga Andrés Jahir. "Entire solutions to the inhomogeneous allen-cahn equation in R^2, with a transition on a noncompact curve." Tesis, Universidad de Chile, 2012. http://www.repositorio.uchile.cl/handle/2250/111186.

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Ingeniero Civil Matemático<br>Este trabajo de memoria de título presenta un estudio de la ecuación de perturbación singular de Allen-Cahn con inhomogeneidad: \begin{equation}\ep^2\div\left(a(x)\cdot\nabla_{x}u(x)\right)+a(x)f(u(x))=0,\quad\text{ en }\quad\R^2 \label{AllenCahnEq}\end{equation} donde $\varepsilon>0$ es un parámetro pequeño, $a(x)$ es un potencial uniformemente positivo y suave, que induce una forma de medir distancias para puntos en $\R^2$, y $f$ es la nolinealidad dada por $f(u)=u-u^3$. El estudio aborda la construcción de soluciones enteras de~\eqref{AllenCahnEq}, bajo la condición que $u$ se anule cerca de una curva $\Gamma\subset \R^2$. El enfoque propuesto asume que $\Gamma$ es una curva no acotada, geodésica no-degenerada relativa al funcional de longitud de arco $\int_{\Gamma}a(\vec{x})$, con curvatura $k_{\Gamma}$ suave que decae a una tasa polinomial. Es de interés el estudio de la ecuación de Allen-Cahn con presencia de un término de inhomogeneidad $a(x)\not\equiv 1$, ya que esto conlleva el estudio de curvas geodésicas para una métrica no trivial de $\R^2$. Además, es relevante considerar que el conjunto nodal de $u$ yace cerca de una curva no acotada, pues esto se refleja en el estudio de ecuaciones diferenciales en contextos no compactos. El resultado principal asegura la existencia de una solución de~\eqref{AllenCahnEq}, la cual converge exponencialmente a $\pm 1$ cuando $x$ se aleja de $\Gamma$. Un segundo resultado entrega ejemplos de potenciales $a(x)$ y curvas $\Gamma$, para los cuales es posible construir una solución $u$ con el comportamiento antes descrito. La demostración de este resultado está basada en una técnica conocida como reducción infinito dimensional de Lyapunov-Schmidt, la cual motiva a la elección de un candidato a solución del tipo $u = w + \phi$, donde $w$ en coordenadas adecuadas resuelve $w''+f(w)=0$, y determina el perfil de $u$ a orden principal. Además $\phi$ es una función de corrección, con el fin de convertir a $u$ en solución exacta de~\eqref{AllenCahnEq}, lo que obliga a $\phi$ a resolver una ecuación diferencial no lineal. De ahí en más, el problema consiste en estudiar la existencia y unicidad de la última ecuación en un espacio funcional adecuado. Esto se realizó analizando el operador linealizado asociado a la ecuación de Allen-Cahn, y luego el problema no-lineal que es resuelto mediante un esquema de punto fijo. Para el ultimo análisis, fue necesario ajustar $\Gamma$ en un parámetro de perturbación $h$, lo que equivale a una EDO no lineal en $h$ donde participa la segunda variación del funcional de largo $l_{a,\Gamma}$ asociado a $\int_{\Gamma}a(\vec{x})$. Finalmente, el método utilizado no sólo provee la existencia de una solución $u$ de~\eqref{AllenCahnEq}, sino que además entrega una caracterizacón completa de ésta, tanto en tamaño como en comportamiento cualitativo en coordenadas relacionadas a la curva $\Gamma$.
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Kusche, Tobias. "Spectral analysis for linearizations of the Allen-Cahn equation around rescaled stationary solutions with triple-junction." [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=979448786.

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Li, Yichen. "Phase-field Modeling of Phase Change Phenomena." Thesis, Virginia Tech, 2020. http://hdl.handle.net/10919/99148.

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The phase-field method has become a popular numerical tool for moving boundary problems in recent years. In this method, the interface is intrinsically diffuse and stores a mixing energy that is equivalent to surface tension. The major advantage of this method is its energy formulation which makes it easy to incorporate different physics. Meanwhile, the energy decay property can be used to guide the design of energy stable numerical schemes. In this dissertation, we investigate the application of the Allen-Cahn model, a member of the phase-field family, in the simulation of phase change problems. Because phase change is usually accompanied with latent heat, heat transfer also needs to be considered. Firstly, we go through different theoretical aspects of the Allen-Cahn model for nonconserved interfacial dynamics. We derive the equilibrium interface profile and the connection between surface tension and mixing energy. We also discuss the well-known convex splitting algorithm, which is linear and unconditionally energy stable. Secondly, by modifying the free energy functional, we give the Allen-Cahn model for isothermal phase transformation. In particular, we explain how the Gibbs-Thomson effect and the kinetic effect are recovered. Thirdly, we couple the Allen-Chan and heat transfer equations in a way that the whole system has the energy decay property. We also propose a convex-splitting-based numerical scheme that satisfies a similar discrete energy law. The equations are solved by a finite-element method using the deal.ii library. Finally, we present numerical results on the evolution of a liquid drop in isothermal and non-isothermal settings. The numerical results agree well with theoretical analysis.<br>Master of Science<br>Phase change phenomena, such as freezing and melting, are ubiquitous in our everyday life. Mathematically, this is a moving boundary problem where the phase front evolves based on the local temperature. The phase change is usually accompanied with the release or absorption of latent heat, which in turn affects the temperature. In this work, we develop a phase-field model, where the phase front is treated as a diffuse interface, to simulate the liquid-solid transition. This model is consistent with the second law of thermodynamics. Our finite-element simulations successfully capture the solidification and melting processes including the interesting phenomenon of recalescence.
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Makki, Ahmad. "Étude de modèles en séparation de phase tenant compte d'effets d'anisotropie." Thesis, Poitiers, 2016. http://www.theses.fr/2016POIT2288/document.

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Cette thèse se situe dans le cadre de l'analyse théorique et numérique de modèles en séparation de phase qui tiennent compte d'effets d'anisotropie. Ceci est pertinent, par exemple, pour l'évolution de cristaux dans leur matrice liquide pour lesquels ces effets d'anisotropie sont très forts. On étudie l'existence, l'unicité et la régularité de la solution des équations de Cahn-Hilliard et d'Allen-Cahn ainsi que son comportement asymptotique en terme d'existence d'un attracteur global de dimension fractale finie. La première partie de la thèse concerne certains modèles de séparation de phase qui, en particulier, décrivent la formation de motifs dendritiques. D'abord, on étudie les équations de Cahn-Hilliard et d'Allen-Cahn qui prennent en compte les effets d'anisotropie forts en dimension un avec des conditions de type Neumann sur le bord et une non linéarité régulière de type polynomial. En particulier, ces modèles contiennent un terme supplémentaire appelé régularisation de Willmore. Ensuite, on étudie ces modèles avec des conditions de type périodique (respectivement, Dirichlet) sur le bord pour l'équation de Cahn-Hilliard (respectivement, d'Allen-Cahn) mais en dimension spatiales plus élevées. Finalement, on étudie la dynamique des équations de Cahn-Hilliard et d'Allen-Cahn visqueux avec des conditions de type Neumann et Dirichlet respectivement sur le bord et une non linéarité régulière et en plus, la présence de simulations numériques qui montrent les effets du terme de viscosité sur l'anisotropie et l'isotropie dans l'équation de Cahn-Hilliard. Dans le dernier chapitre, on étudie le comportement en temps long en termes d'attracteurs de dimension finie, d'une classe d'équations doublement non linéaires de type Allen-Cahn avec des conditions de type Dirichlet sur le bord et une non linéarité singulière<br>This thesis is situated in the context of the theoretical and numerical analysis of models in phase separation which take into account the anisotropic effects. This is relevant, for example, for the development of crystals in their liquid matrix for which the effects of anisotropy are very strong. We study the existence, uniqueness and the regularity of the solution of Cahn-Hilliard and Alen-Cahn equations and the asymptotic behavior in terms of the existence of a global attractor with finite fractal dimension. The first part of the thesis concerns some models in phase separation which, in particular, describe the formation of dendritic patterns. We start by study- ing the anisotropic Cahn-Hilliard and Allen-Cahn equations in one space dimension both associated with Neumann boundary conditions and a regular nonlinearity. In particular, these two models contain an additional term called Willmore regularization. Furthermore, we study these two models with Periodic (respectively, Dirichlet) boundary conditions for the Cahn-Hilliard (respectively, Allen-Cahn) equation but in higher space dimensions. Finally, we study the dynamics of the viscous Cahn-Hilliard and Allen-Cahn equations with Neumann and Dirichlet boundary conditions respectively and a regular nonlinearity in the presence of the Willmore regularization term and we also give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equations. In the last chapter, we study the long time behavior, in terms of finite dimensional attractors, of a class of doubly nonlinear Allen-Cahn equations with Dirichlet boundary conditions and singular potentials
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Pagliardini, Dayana. "Fractional minimal surfaces and Allen-Cahn equations." Doctoral thesis, Scuola Normale Superiore, 2018. http://hdl.handle.net/11384/85738.

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In recent years fractional operators have received considerable attention both in pure and applied mathematics. They appear in biological observations, finance, crystal dislocation, digital image reconstruction and minimal surfaces. In this thesis we study nonlocal minimal surfaces which are boundaries of sets minimizing certain integral norms and can be interpreted as a non-infinitesimal version of classical minimal surfaces. In particular, we consider critical points, with or withouth constraints, of suitable functionals, or approximations through diffuse models as the Allen-Cahn’s. In the first part of the thesis we prove an existence and multiplicity result for critical points of the fractional analogue of the Allen-Cahn equation in bounded domains. We bound the functional using a standard nonlocal tool: we split the domain in two regions and we analyze the three significative interactions. Then, the proof becomes an application of a classical Krasnoselskii’s genus result. Then, we consider a fractional mesoscopic model of phase transition i.e. the fractional Allen-Cahn equation with the addition of a mesoscopic term changing the ‘pure phases’ ±1 in periodic functions. We investigate geometric properties of the interface of the associated minimal solutions. Then we construct minimal interfaces lying to a strip of prescribed direction and universal width. We provide a geometric and variational technique adapted to deal with nonlocal interactions. In the last part of the thesis, we study functionals involving the fractional perimeter. In particular, first we study the localization of sets with constant nonlocal mean curvature and small prescribed volume in an open bounded domain, proving that these sets are ‘sufficiently close’ to critical points of a suitable potential. The proof is an application of the Lyupanov-Schmidt reduction to the fractional perimeter. Finally, we consider the fractional perimeter in a half-space. We prove the existence of a minimal set with fixed volume and some of its properties as intersection with the hyperplane {xN = 0}, symmetry, to be a graph in the xN-direction and smoothness.
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Willoughby, Mark Ryerson. "High-order time-adaptive numerical methods for the Allen-Cahn and Cahn-Hilliard equations." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/39533.

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In some nonlinear reaction-diffusion equations of interest in applications, there are transition layers in solutions that separate two or more materials or phases in a medium when the reaction term is very large. Two well known equations that are of this type: The Allen-Cahn equation and the Cahn-Hillard equation. The transition layers between phases evolve over time and can move very slowly. The models have an order parameter epsilon. Fully developed transition layers have a width that scales linearly with epsilon. As epsilon goes to 0, the time scale of evolution can also change and the problem becomes numerically challenging. We consider several numerical methods to obtain solutions to these equations, in order to build a robust, efficient and accurate numerical strategy. Explicit time stepping methods have severe time step constraints, so we direct our attention to implicit schemes. Second and third order time-adaptive methods are presented using spectral discretization in space. The implicit problem is solved using the conjugate gradient method with a novel preconditioner. The behaviour of the preconditioner is investigated, and the dependence on epsilon and time step size is identified. The Allen-Cahn and Cahn-Hilliard equations have been used extensively to model phenomena in materials science. We strongly believe that our high order adaptive approach is also easily extensible to higher order models with application to pore formation in functionalized polymers and to cancerous tumor growth simulation. This is the subject of ongoing research.
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Nguyen, Thanh Nam. "Equations d'évolution non locales et problèmes de transition de phase." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00919784.

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L'objet de cette thèse est d'étudier le comportement en temps long de solutions d'équations d'évolution non locales ainsi que la limite singulière d'équations et de systèmes d'équations aux dérivées partielles, où intervient un petit paramètre epsilon. Au Chapitre 1, nous considérons une équation de réaction-diffusion non locale avec conservation au cours du temps de l'intégrale en espace de la solution; cette équation a été initialement proposée par Rubinstein et Sternberg pour modéliser la séparation de phase dans un mélange binaire. Le problème de Neumann associé possède une fonctionnelle de Lyapunov, c'est-à-dire une fonctionnelle qui décroit selon les orbites. Après avoir prouvé que la solution est confinée dans une région invariante, nous étudions son comportement en temps long. Nous nous appuyons sur une inégalité de Lojasiewicz pour montrer qu'elle converge vers une solution stationnaire quand t tend vers l'infini. Nous évaluons également le taux de la convergence et calculons précisément la solution stationnaire limite en dimension un d'espace. Le Chapitre 2 est consacré à l'étude de l'équation différentielle non locale que l'on obtient en négligeant le terme de diffusion dans l'équation d'Allen-Cahn non locale étudiée au Chapitre 1. Sans le terme de diffusion, la solution ne peut pas être plus régulière que la fonction initiale. C'est la raison pour laquelle on ne peut pas appliquer la méthode du Chapitre 1 pour l'étude du comportement en temps long de la solution. Nous présentons une nouvelle méthode basée sur la théorie des réarrangements et sur l'étude du profil de la solution. Nous montrons que la solution est stable pour les temps grands et présentons une caractérisation détaillée de sa limite asymptotique quand t tend vers l'infini. Plus précisément, la fonction limite est une fonction en escalier, qui prend au plus deux valeurs, qui coïncident avec les points stables d'une équation différentielle associée. Nous montrons aussi par un contre-exemple non trivial que, quand une hypothèse sur la fonction initiale n'est pas satisfaite, la fonction limite peut prendre trois valeurs, qui correspondent aux points instable et stables de l'équation différentielle associée. Nous étudions au Chapitre 3 une équation différentielle ordinaire non locale qui a éte proposée par M. Nagayama. Une difficulté essentielle est que le dénominateur dans le terme de réaction non local peut s'annuler. Nous appliquons un théorème de point fixe lié a une application contractante pour démontrer que le problème à valeur initiale correspondant possède une solution unique qui reste connée dans un ensemble invariant. Ce problème possède une fonctionnelle de Lyapunov, qui est un ingrédient essentiel pour démontrer que la solution converge vers une solution stationnaire constante par morceaux quand t tend vers l'infini. Au Chapitre 4, nous considérons un modèle d'interface diffuse pour la croissance de tumeurs, où intervient une équation d'ordre quatre de type Cahn Hilliard. Après avoir introduit un modèle de champ de phase associé, on étudie formellement la limite singulière de la solution quand le coefficient du terme de réaction tend vers l'infini. Plus précisément, nous montrons que la solution converge vers la solution d'un problème à frontière libre. AMS subject classifications. 35K57, 35K50, 35K20, 35R35, 35R37, 35B40, 35B25.
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Book chapters on the topic "Allen-Cahn equation"

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Bartels, Sören. "The Allen–Cahn Equation." In Springer Series in Computational Mathematics. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-13797-1_6.

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Rabinowitz, Paul H., and Ed Stredulinsky. "Solutions of an Allen-Cahn Model Equation." In Nonlinear Equations: Methods, Models and Applications. Birkhäuser Basel, 2003. http://dx.doi.org/10.1007/978-3-0348-8087-9_19.

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Folino, Raffaele. "Metastability for Hyperbolic Variations of Allen–Cahn Equation." In Theory, Numerics and Applications of Hyperbolic Problems I. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91545-6_42.

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Zhou, Xiaolan, Mejdi Azaiez, and Chuanju Xu. "SAV Method Applied to Fractional Allen-Cahn Equation." In Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-39647-3_39.

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Xu, Huiying, Jie Chen, and Fei Ma. "Adaptive Deep Learning Approximation for Allen-Cahn Equation." In Computational Science – ICCS 2022. Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-08760-8_23.

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Funaki, Tadahisa. "Sharp Interface Limits for a Stochastic Allen-Cahn Equation." In Lectures on Random Interfaces. Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-0849-8_4.

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Uzunca, Murat, and Bülent Karasözen. "Energy Stable Model Order Reduction for the Allen-Cahn Equation." In Model Reduction of Parametrized Systems. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-58786-8_25.

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Brkić, Antun Lovro, and Andrej Novak. "A Nonlocal Image Inpainting Problem Using the Linear Allen–Cahn Equation." In Lecture Notes in Electrical Engineering. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-17344-9_17.

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Alikakos, Nicholas D., Giorgio Fusco, and Panayotis Smyrnelis. "Symmetry and the Vector Allen–Cahn Equation: The Point Group in ℝn." In Progress in Nonlinear Differential Equations and Their Applications. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-90572-3_6.

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Alikakos, Nicholas D., Giorgio Fusco, and Panayotis Smyrnelis. "Symmetry and the Vector Allen–Cahn Equation: Crystalline and Other Complex Structures." In Progress in Nonlinear Differential Equations and Their Applications. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-90572-3_7.

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Conference papers on the topic "Allen-Cahn equation"

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Beneš, Michal, Danielle Hilhorst, and Rémi Weidenfeld. "Interface dynamics for an anisotropic Allen-Cahn equation." In Nonlocal Elliptic and Parabolic Problems. Institute of Mathematics Polish Academy of Sciences, 2004. http://dx.doi.org/10.4064/bc66-0-3.

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Yamazaki, Noriaki, Takeshi Fukao, and Mohammad Hassan Farshbaf-Shaker. "Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier." In The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Madrid, Spain). American Institute of Mathematical Sciences, 2015. http://dx.doi.org/10.3934/proc.2015.0418.

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Bulut, Hasan. "Application of the modified exponential function method to the Cahn-Allen equation." In ICNPAA 2016 WORLD CONGRESS: 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences. Author(s), 2017. http://dx.doi.org/10.1063/1.4972625.

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Ramsay, Travis. "Uncertainty Quantification of Allen-Cahn Phase Field Parameters in Multiphysics Simulation of Oil Shale Radio Frequency Heating." In SPE Annual Technical Conference and Exhibition. SPE, 2021. http://dx.doi.org/10.2118/205866-ms.

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Abstract Radio frequency (RF) heating represents a dielectric heating technique for converting kerogen-rich oil shale into liquid oil through in-situ pyrolysis. This process can be modeled using a multiphysics finite element based coupled thermal, phase field, mechanical and electromagnetic (TPME) numerical framework. This work focuses on the combination of a two-dimensional (2D) TPME multiphysics simulation with uncertainty quantification (UQ) that incorporates the Allen-Cahn phase field parameters, specifically those which describe the associated reaction-diffusion process as electromagnetic energy being converted to thermal energy in the RF heating process. The breadth of UQ performed in this study includes not only the Allen-Cahn parameters but also selected thermal, statistical rock-type distribution in the geological model, as well as electromagnetic parameters of the applied quasi-static Maxwell equation. A Non-Intrusive Polynomial Chaos (NIPC) is used for: considering the affect of Allen-Cahn phase field parameters on the evaluation of plausible conversion timelines of TPME simulation and the evaluation of summary statistics to predict the order of Polynomial Chaos Expansion (PCE) that is representative of full kerogen-rich zonal conversion response in a geologically descriptive finite element model. A sparse representation of polynomial chaos coefficients is highlighted in the process of computing summary statistics for the complex stochastically-driven TPME simulation results. Additionally, Monte Carlo (MC) simulations were performed in order to validate the results of the sparse NIPC representation. This is done considering MC is a widely recognized stochastic simulation process. Additionally, NIPC was used to illustrate the potential performance improvement that are possible, with a sparse polynomial chaos expansion enhanced by the incorporation of Least Angle Regression (LAR), as compared to MC simulation. Although the parametic uncertainty of the reaction-diffusion parameters of the Allen-Cahn was comprehensive, they did not accelerate the conversion timelines associated with the full zonal conversion of the kerogen-rich rock type in the statistical simulation results. By executing the stochastic simulations for a greater length of time the extent of full zonal conversion is examined in the RF modeling.
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Chin, Pius W. M. "Stabilized and reliable numerical scheme for a fully discrete Allen-Cahn equation on a smooth domain." In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2014 (ICNAAM-2014). AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4912366.

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He, Zhi Zhu, and Jing Liu. "Investigation of Tumor Growth Based on Phase Field Model." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-65744.

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This paper presents and investigates the tumor growth based on a phase model. The tumor core is necrotic and inhibitor chemical species are considered. The interface of tumor and health tissue is tracked using a phase field equation. The reformulation of a classical model, accounting for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion, is derived. The advantages of the finite difference methodology employed are generality and relative simplicity implication. We present simulations of the nonlinear evolution of growing tumors morphology and discuss the effects of tumor microenvironment. Mechanisms reflecting the tumor growth and development behavior was preliminarily interpreted. Recently numerous mathematical have been developed to investigate the growth dynamics of tumor [1–8]. One of most significant model developed by Wise [8] is based on Cahn-Hilliard equation, which is conservation phase field method. Allen-Chan nonconservation phase field has been developed to track the moving interface for multiphase simulation by Sun [9]. Allen-Chan equation is second order, while Cahn-Hilliard equation is fourth order in space. Thus, we introduce the Allen-Chan phase method [9–10] to simulate the tumor growth, which is very simple for numerical simulation The computation domain is illustrated in Fig. 1, where ΩH denotes host tissue, the tumor domains is comprised of viable tumor cell ΩV and dead tumor cell ΩD. The numerical results are presented at Fig. (2–4). One can find that the growth of tumor strongly depend on the nutrients and nonlinear unstable growth may lead to finger shaped pattern, which is in agreement with recent experimental observations [7] of in vivo tumor. In summary, a phase method has been developed to study diffusion and consumption of the nutrients and tumor cell proliferation, necrosis and migration, which discloses the evolution of complex shape of tumor.
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Hameed, Raad A., Maan A. Rasheed, Ogada E. Achieng, and Wafaa M. Taha. "On the existence of periodic solutions to a p-Laplacian Allen-Cahn equation with Neumann boundary condition." In PROCEEDING OF THE 1ST INTERNATIONAL CONFERENCE ON ADVANCED RESEARCH IN PURE AND APPLIED SCIENCE (ICARPAS2021): Third Annual Conference of Al-Muthanna University/College of Science. AIP Publishing, 2022. http://dx.doi.org/10.1063/5.0093631.

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Mao, X., V. Joshi, T. P. Miyanawala, and Rajeev K. Jaiman. "Data-Driven Computing With Convolutional Neural Networks for Two-Phase Flows: Application to Wave-Structure Interaction." In ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/omae2018-78425.

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Fluctuating wave force on a bluff body is of great significance in many offshore and marine engineering applications. We present a Convolutional Neural Network (CNN) based data-driven computing to predict the unsteady wave forces on bluff bodies due to the free-surface wave motion. For the full-order modeling and high-fidelity data generation, the air-water interface for such wave-body problems must be captured accurately for a broad range of physical and geometric parameters. Originated from the thermodynamically consistent theories, the physically motivated Allen-Cahn phase-field method has many advantages over other interface capturing techniques such as level-set and volume-of-fluid methods. The Allen-Cahn equation is solved in the mass-conservative form by imposing a Lagrange multiplier technique. While a tremendous amount of wave-body interaction data is generated in offshore engineering via both CFD simulations and experiments, the results are generally underutilized. Design space exploration and flow control of such practical scenarios are still time-consuming and expensive. An alternative to semi-analytical modeling, CNN is a class of deep neural network for solving inverse problems which is efficient in parametric data-driven computation and can use the domain knowledge. It establishes a model with arbitrarily generated model parameters, makes predictions using the model and existing input parametric settings, and adjusts the model parameters according to the error between the predictions and existing results. The computational cost of this prediction process, compared with high-fidelity CFD simulation, is significantly reduced, which makes CNN an accessible tool in design and optimization problems. In this study, CNN-based data-driven computing is utilized to predict the wave forces on bluff bodies with different geometries and distances to the free surface. The discrete convolution process with a non-linear rectification is employed to approximate the mapping between the bluff-body shape, the distance to the free-surface and the fluid forces. The wave-induced fluid forces on bluff bodies of different shapes and submergences are predicted by the trained CNN. Finally, a convergence study is performed to identify the effective hyper-parameters of the CNN such as the convolution kernel size, the number of kernels and the learning rate. Overall, the proposed CNN-based approximation procedure has a profound impact on the parametric design of bluff bodies experiencing wave loads.
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Akhtar, M. Wasy, and Holley C. Love. "Computations of Single and Multiphase Flows Using a Lattice Boltzmann Solver." In ASME 2019 Pressure Vessels & Piping Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/pvp2019-93817.

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Abstract There is considerable interest in high fidelity simulation of both single phase incompressible flows and multiphase flows. Most commonly applied numerical methods include finite difference, finite volume, finite element and spectral methods. All of these methods attempt to capture the flow details by solving the Navier–Stokes equations. Challenges of solving the Navier–Stokes single phase incompressible flows include the non-locality of the pressure gradient, non-linearity of the advection term and handling the pressure-velocity coupling. Multiphase flow computations pose additional challenges, such as property and flow variable discontinuities at the interface, whose location and orientation is not known a priori. Further, capturing/tracking of the multiphase interface requires solution of an additional advection equation. Recently, the lattice Boltzmann method has been applied to compute fluid dynamics simulations both for single and multiphase configurations; it is considered a modern CFD approach with improved accuracy and performance. Specifically, we employ a multiple-relaxation time (MRT) technique for the collision term on a D3Q27 lattice. The multiphase interface is captured using the phase-field approach of Allen-Cahn. Test cases include lid driven cavity, vortex shedding for a double backward facing step, Rayleigh Taylor instability, Enright’s deformation test and rising bubble in an infinite domain. These test cases validate different aspects of the single and multiphase model, so that the results can be interpreted with confidence that the underlying computational framework is sufficiently accurate.
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Sahoo, Seshadev, and Kevin Chou. "Review on Phase-Field Modeling of Microstructure Evolutions: Application to Electron Beam Additive Manufacturing." In ASME 2014 International Manufacturing Science and Engineering Conference collocated with the JSME 2014 International Conference on Materials and Processing and the 42nd North American Manufacturing Research Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/msec2014-3901.

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Powder-bed electron beam additive manufacturing (EBAM) is a relatively new technology to produce metallic parts in a layer by layer fashion by melting and fusing metallic powders. EBAM is a rapid solidification process and the properties of the parts depend on the solidification behavior as well as the microstructure of the build material. Thus, the prediction of part microstructures during the process may be an important factor for process optimization. Nowadays, the increase in computational power allows for direct simulations of microstructures during materials processing for specific manufacturing conditions. Among different methods, phase-field modeling (PFM) has recently emerged as a powerful computational technique for simulating microstructure evolutions at the mesoscale during a rapid solidification process. PFM describes microstructures using a set of conserved and non-conserved field variables and the evolution of the field variables are governed by Cahn-Hilliard and Allen-Cahn equations. By using the thermodynamics and kinetic parameters as input parameters in the model, PFM is able to simulate the evolution of complex microstructures during materials processing. The objective of this study is to achieve a thorough review of PFM techniques used in various processes, attempted for an application to microstructure evolutions during EBAM. The concept of diffuse interfaces, phase field variables, thermodynamic driving forces for microstructure evolutions and the kinetic phase-field equations are described in this paper.
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