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 .jp-TerminalIcon{background-image:var(--jp-icon-terminal)}.jp-TextEditorIcon{background-image:var(--jp-icon-text-editor)}.jp-TocIcon{background-image:var(--jp-icon-toc)}.jp-TreeViewIcon{background-image:var(--jp-icon-tree-view)}.jp-TrustedIcon{background-image:var(--jp-icon-trusted)}.jp-UndoIcon{background-image:var(--jp-icon-undo)}.jp-UserIcon{background-image:var(--jp-icon-user)}.jp-UsersIcon{background-image:var(--jp-icon-users)}.jp-VegaIcon{background-image:var(--jp-icon-vega)}.jp-WordIcon{background-image:var(--jp-icon-word)}.jp-YamlIcon{background-image:var(--jp-icon-yaml)}.jp-Icon,.jp-MaterialIcon{background-position:center;background-repeat:no-repeat;background-size:16px;min-width:16px;min-height:16px}.jp-Icon-cover{background-position:center;background-repeat:no-repeat;background-size:cover}.jp-Icon-16{background-size:16px;min-width:16px;min-height:16px}.jp-Icon-18{background-size:18px;min-width:18px;min-height:18px}.jp-Icon-20{background-size:20px;min-width:20px;min-height:20px}.lm-TabBar .lm-TabBar-addButton{align-items:center;display:flex;padding:4px;padding-bottom:5px;margin-right:1px;background-color:var(--jp-layout-color2)}.lm-TabBar .lm-TabBar-addButton:hover{background-color:var(--jp-layout-color1)}.lm-DockPanel-tabBar .lm-TabBar-tab{width:var(--jp-private-horizontal-tab-width)}.lm-DockPanel-tabBar .lm-TabBar-content{flex:unset}.lm-DockPanel-tabBar[data-orientation='horizontal']{flex:1 1 auto}.jp-icon0[fill]{fill:var(--jp-inverse-layout-color0)}.jp-icon1[fill]{fill:var(--jp-inverse-layout-color1)}.jp-icon2[fill]{fill:var(--jp-inverse-layout-color2)}.jp-icon3[fill]{fill:var(--jp-inverse-layout-color3)}.jp-icon4[fill]{fill:var(--jp-inverse-layout-color4)}.jp-icon0[stroke]{stroke:var(--jp-inverse-layout-color0)}.jp-icon1[stroke]{stroke:var(--jp-inverse-layout-color1)}.jp-icon2[stroke]{stroke:var(--jp-inverse-layout-color2)}.jp-icon3[stroke]{stroke:var(--jp-inverse-layout-color3)}.jp-icon4[stroke]{stroke:var(--jp-inverse-layout-color4)}.jp-icon-accent0[fill]{fill:var(--jp-layout-color0)}.jp-icon-accent1[fill]{fill:var(--jp-layout-color1)}.jp-icon-accent2[fill]{fill:var(--jp-layout-color2)}.jp-icon-accent3[fill]{fill:var(--jp-layout-color3)}.jp-icon-accent4[fill]{fill:var(--jp-layout-color4)}.jp-icon-accent0[stroke]{stroke:var(--jp-layout-color0)}.jp-icon-accent1[stroke]{stroke:var(--jp-layout-color1)}.jp-icon-accent2[stroke]{stroke:var(--jp-layout-color2)}.jp-icon-accent3[stroke]{stroke:var(--jp-layout-color3)}.jp-icon-accent4[stroke]{stroke:var(--jp-layout-color4)}.jp-icon-none[fill]{fill:none}.jp-icon-none[stroke]{stroke:none}.jp-icon-brand0[fill]{fill:var(--jp-brand-color0)}.jp-icon-brand1[fill]{fill:var(--jp-brand-color1)}.jp-icon-brand2[fill]{fill:var(--jp-brand-color2)}.jp-icon-brand3[fill]{fill:var(--jp-brand-color3)}.jp-icon-brand4[fill]{fill:var(--jp-brand-color4)}.jp-icon-brand0[stroke]{stroke:var(--jp-brand-color0)}.jp-icon-brand1[stroke]{stroke:var(--jp-brand-color1)}.jp-icon-brand2[stroke]{stroke:var(--jp-brand-color2)}.jp-icon-brand3[stroke]{stroke:var(--jp-brand-color3)}.jp-icon-brand4[stroke]{stroke:var(--jp-brand-color4)}.jp-icon-warn0[fill]{fill:var(--jp-warn-color0)}.jp-icon-warn1[fill]{fill:var(--jp-warn-color1)}.jp-icon-warn2[fill]{fill:var(--jp-warn-color2)}.jp-icon-warn3[fill]{fill:var(--jp-warn-color3)}.jp-icon-warn0[stroke]{stroke:var(--jp-warn-color0)}.jp-icon-warn1[stroke]{stroke:var(--jp-warn-color1)}.jp-icon-warn2[stroke]{stroke:var(--jp-warn-color2)}.jp-icon-warn3[stroke]{stroke:var(--jp-warn-color3)}.jp-icon-contrast0[fill]{fill:var(--jp-icon-contrast-color0)}.jp-icon-contrast1[fill]{fill:var(--jp-icon-contrast-color1)}.jp-icon-contrast2[fill]{fill:var(--jp-icon-contrast-color2)}.jp-icon-contrast3[fill]{fill:var(--jp-icon-contrast-color3)}.jp-icon-contrast0[stroke]{stroke:var(--jp-icon-contrast-color0)}.jp-icon-contrast1[stroke]{stroke:var(--jp-icon-contrast-color1)}.jp-icon-contrast2[stroke]{stroke:var(--jp-icon-contrast-color2)}.jp-icon-contrast3[stroke]{stroke:var(--jp-icon-contrast-color3)}.jp-icon-dot[fill]{fill:var(--jp-warn-color0)}.jp-jupyter-icon-color[fill]{fill:var(--jp-jupyter-icon-color,var(--jp-warn-color0))}.jp-notebook-icon-color[fill]{fill:var(--jp-notebook-icon-color,var(--jp-warn-color0))}.jp-json-icon-color[fill]{fill:var(--jp-json-icon-color,var(--jp-warn-color1))}.jp-console-icon-color[fill]{fill:var(--jp-console-icon-color,white)}.jp-console-icon-background-color[fill]{fill:var(--jp-console-icon-background-color,var(--jp-brand-color1))}.jp-terminal-icon-color[fill]{fill:var(--jp-terminal-icon-color,var(--jp-layout-color2))}.jp-terminal-icon-background-color[fill]{fill:var(--jp-terminal-icon-background-color,var(--jp-inverse-layout-color2))}.jp-text-editor-icon-color[fill]{fill:var(--jp-text-editor-icon-color,var(--jp-inverse-layout-color3))}.jp-inspector-icon-color[fill]{fill:var(--jp-inspector-icon-color,var(--jp-inverse-layout-color3))}.jp-DirListing-item.jp-mod-selected .jp-icon-selectable[fill]{fill:#fff}.jp-DirListing-item.jp-mod-selected .jp-icon-selectable-inverse[fill]{fill:var(--jp-brand-color1)}
diff --git a/feed.xml b/feed.xml
index 6dd48691..e1933c6f 100644
--- a/feed.xml
+++ b/feed.xml
@@ -1 +1 @@
-<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en"><generator uri="https://jekyllrb.com/" version="4.3.3">Jekyll</generator><link href="https://mdeluci.github.io/feed.xml" rel="self" type="application/atom+xml"/><link href="https://mdeluci.github.io/" rel="alternate" type="text/html" hreflang="en"/><updated>2024-08-20T22:50:54+00:00</updated><id>https://mdeluci.github.io/feed.xml</id><title type="html">Mario de Lucio’s website</title><subtitle>Mario de Lucio&apos;s website </subtitle><entry><title type="html">Phase-field Modeling of Dendritic Solidification</title><link href="https://mdeluci.github.io/blog/2024/Solidification/" rel="alternate" type="text/html" title="Phase-field Modeling of Dendritic Solidification"/><published>2024-06-20T06:52:00+00:00</published><updated>2024-06-20T06:52:00+00:00</updated><id>https://mdeluci.github.io/blog/2024/Solidification</id><content type="html" xml:base="https://mdeluci.github.io/blog/2024/Solidification/"><![CDATA[<h2 id="phase-field-equation-for-solidification-processes">Phase-field equation for solidification processes</h2> <p>Liquid-solid phase transformations usually refer to solidification and melting. The formation of complex microstructures during the solidification from a liquid phase, such as formation of snow flakes or metallic alloys, and the accompanying diffusion and convection processes in the liquid and solid have been widely studied in the literature <a class="citation" href="#boettinger2002phase">(Boettinger et al., 2002; Flemings, 1974; Kühn et al., 1986)</a>. In particular, the evolution of microstructural scale of dendrites during the solidification process determines many physical and mechanical properties of metals, since almost every metallic system originates from the liquid state.</p> <div class="container"> <div class="row justify-content-center mt-3"> <div class="col mt-3 mt-md-0 d-flex justify-content-center"> <div style="width: 90%;"> <figure> <video src="/assets/video/solidification.mp4" class="img-fluid rounded z-depth-1" width="auto" height="auto" autoplay="" controls=""/> </figure> </div> </div> </div> <div class="caption text-center"> Crystal growth from a square-shaped seed located in an undercooled region. Phase-field (top row). Temperature (bottom row) </div> </div> <p>A phase-field formulation that describes the solidification process is as follows: \begin{equation} \frac{\partial \phi}{\partial t} = \Delta \phi - \frac{W’(\phi)}{\epsilon^2} + \frac{\rho H}{\sqrt{2\sigma}} \frac{\theta - \theta_m}{\theta_m} G’(\phi) \label{1} \end{equation} \begin{equation} \rho C_v \frac{\partial \theta}{\partial t} + L \phi’ \frac{\partial \phi}{\partial t} = \nabla \cdot (k(\phi) \nabla \theta) \label{2} \end{equation}</p> <p>where \(\phi \in [-1,1]\) represents the phase-field variable, which describes the location of different components (e.g., \(\phi = -1\) in the solid and \(\phi = 1\) in the liquid), \(\theta\) is the temperature, \(\theta_m\) is the melting temperature, \(H\) represents the interfacial enthalpy per unit mass, \(\omega\) is the kinetic undercooling coefficient, \(C_v\) is the heat capacity per unit mass, \(\rho\) is the density, \(l\) is specific latent heat (energy per unit mass), \(k\) is the thermal conductivity, \(\sigma\) is the surface tension and \(h\) is an interpolatory function that verifies \(h(+1) = 1, h(-1) = 0\), e.g., \(h = \frac{1}{2} (1 + \phi)\). The thermal conductivity is considered to be a function of the phase-field to account for different conductivity in the solid and liquid phases. We take \(k(\phi) = (1 + \phi) k_s + (1 - \phi) k_l\), which satisfies that \(k(+1) = k_s\), \(k(-1) = k_l\). The function \(W(\phi)\), also referred to as the double-well potential, is defined such that it has two local minima, which makes possible the coexistence of the different phases. Some important examples of double well functions can be found in <a class="citation" href="#gomez2017computational">(Gomez &amp; van der Zee, 2017)</a>. In this work we will take the classical quartic potential \(W(\phi) = \frac{1}{4} (1 - \phi^2)^2\). The function \(G(\phi)\) vanishes in the pure phases. Depending on the form of the functional \(G(\phi)\), the phase-field will converge faster or slower to the generalized Stefan problem. Here we will use the expression \(G(\phi) = (1 - \phi^2)^2\). One key aspect to achieve good agreement with the reality of interest behind dendritic solidification is surface tension anisotropy. Herein, we introduce anisotropy by assuming that \(\sigma\) depends on the unit normal to the liquid-solid interface, this is,</p> <p>\begin{equation} \sigma = \sigma_0(1 + \delta \cos(\alpha - \alpha_0)), \end{equation}</p> <p>where \(\bar{\sigma}\) is the mean value of \(\sigma\), \(\delta\) is the strenght of the anisotropy, \(q\) is the mode number of anisotropy, and \(\alpha\) is the initial offset angle. The angle of the normal to the surface, \(\alpha\), is defined from the phase-field as</p> <p>\begin{equation} \alpha = \arctan\left(\frac{\partial \phi / \partial y}{\partial \phi/ \partial x}\right). \end{equation}</p> <h2 id="weak-form">Weak form</h2> <p>The strong form of our model defined by Eqs. \eqref{1} and Eqn.\eqref{2} is now cast in weak form and discretized using the Galerkin approach. Let us define the functional space \(V \subset H^1\), where \(H^1\) is the Sobolev space of square-integrable functions with square-integrable first derivative in the domain \(\Omega\). To perform space discretization we introduce the finite-dimensional space \(V_h \subset V\), where \(V_h = \text{span} \{ N_A \}_{A=1}^{n_f}\), and \(n_f\) is the number of functions on the basis. The space of weighting functions will also be \(V_h\), giving rise to a Galerkin formulation. We define discrete approximations to the problem’s solution denoted by \(\phi^h\) and \(\theta^h\). Their corresponding weighting functions are \(w_{\phi}^h\) and \(w_{\theta}^h\). Then, the variational formulation of Eqs. \eqref{1} and Eqn.\eqref{2} over the finite-dimensional space \(V_h\) can be stated as follows: find \(U_h = \{\phi^h, \theta^h\} \in V_h \subset V\)</p> <p>such that \(\forall W_h = \{w_{\phi}^h, w_{\theta}^h\} \in V_h \subset V\), \begin{equation} \int_{\Omega} w_{\phi}^h \frac{\partial \phi^h}{\partial t} \, d\Omega + \int_{\Omega} w_{\phi}^h \frac{W’(\phi^h)}{\epsilon^2} \, d\Omega + \int_{\Omega} \nabla w_{\phi}^h \cdot \nabla \phi^h \, d\Omega + \int_{\Omega} w_{\phi}^h \frac{\rho H}{\sqrt{2 \sigma}} \left( \frac{G’(\phi^h) (\theta^h - \theta_m)}{\theta_m} \right) \, d\Omega =0, \end{equation}</p> <p>\begin{equation} \int_{\Omega} w_{\theta}^h \rho C_v \frac{\partial \theta^h}{\partial t} \, d\Omega - \int_{\Omega} w_{\theta}^h L \phi’(\phi^h) \frac{\partial \phi^h}{\partial t} \, d\Omega + \int_{\Omega} \nabla w_{\theta}^h \cdot k(\phi^h) \nabla \theta^h \, d\Omega=0. \end{equation}</p> <p>The solutions \(\phi^h\) and \(\theta^h\) are defined as \begin{equation} \phi^h(x,t) = \sum_{A=1}^{n_f} \phi_A(t) N_A(x), \end{equation} \begin{equation} \theta^h(x,t) = \sum_{A=1}^{n_f} \theta_A(t) N_A(x). \end{equation}</p> <h2 id="results">Results</h2> <p>Our computational domain is the square \(\Omega= \left[0,0.1 \right] \times \left[0,0.1 \right]\), where the units are in cm. We employ a computational mesh comprised of $512^2$ \(\mathcal{C}^1\)-quadratic elements. We impose periodic boundary conditions for \(\phi\) and \(\theta\). We choose a suitable time step of \(\Delta t = 10e-6\) s to ensure good numerical stability, specially at the beginning of the simulation. As an initial condition, we set a square-shaped seed located at the center of the domain. The reason for using a square-shaped crystal, instead of one with a smoother shape like a circle, is because the corners of the square will accelerate the formation of dendritic structures. The temperatures inside and outside the seed are 1700 K and 1500 K, respectively. The parameters used in all the simulations are summarized in <a class="citation" href="#gomez2019review">(Gomez et al., 2019)</a>. These parameters correspond to an Nickel-Copper alloy. In the following simulations, we will show the relations between the anisotropy modes and the shapes of crystals.</p> <p>The Figures below show the evolution of the phase-field variable and the temperature field over time for different values of \(q\). For all simulations nucleation starts at the center of the domain, and it triggers the solidification process. The phase change is achieved by undercooling the liquid phase below its melting temperature.</p> <p>In the first computation, perfect isotropic growth is considered, i.e. \(\sigma=\bar{\sigma}\) throughout the entire simulation. For this case we obtain symmetric dendritic patterns similar to those formed during viscous fingering. Tip splitting is observed as the crystal grows, producing new branches with tips that mantain a circular shape. For the anisotropic case with \(q = 4\), we see features of both the isotropic case and the dendritic structure. The dendritic structure is present in the formation of the principal branches, which are four in this case due to the anisotropy mode chosen. The isotropic features are observed in the viscous finger-like structures that emanate from the principal branches. The overall look of the pattern is symmetric, even though we are considering surface tension anisotropy. When we increase the anisotropy mode number to \(q = 6\), we get a completely asymmetric pattern. We can observe six main branches and four secondary branches. For \(q = 8\) we get the typical snowflake-like pattern with a non-symmetric side branching. In general, dendrites whose preferred direction is inclined with respect to the thermal gradient grow faster in order to maintain their relative tip positions during the splitting. These tips of the inclined dendrites lie slightly behind those of well aligned dendrites. We can say that there is a growth competition among the different dendrites, which results in a natural selection of the grains that grow with a smaller angle with respect to the temperature</p> <div class="row mt-3"> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/sol_1-480.webp 480w,/assets/img/sol_1-800.webp 800w,/assets/img/sol_1-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/sol_1.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/sol_2-480.webp 480w,/assets/img/sol_2-800.webp 800w,/assets/img/sol_2-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/sol_2.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> </div> <div class="caption"> Crystal growth from a square-shaped seed located in an undercooled region. Here we show the time evolution of dendritic solidification for different mode numbers of anisotropy. Time evolution of the temperature field from the initial square-shaped seed for different mode numbers of anisotropy. </div> <h2 id="conclusions">Conclusions</h2> <p>We presented a numerical methodology to solve a phase-field model of dendritic solidification with surface tension anisotropy on a two-dimensional domain. Our computational method is based on isogeometric analysis, which allows us to generate the \(\mathcal{C}^1\)-continuous functions required to solve this equation in a variational framework. Time discretazion is performed using the generalized \(\alpha\)-method, which provides second order accuracy and \(A\)-stability. We analyzed the evolution of the phase-field variable and the temperature field over time for different anisotropy mode numbers. Our results show that symmetric dendritic finger patterns can form in the presence of anisotropy.</p> <h2 id="references">References</h2> <ol class="bibliography"><li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="boettinger2002phase" class="col-sm-8"> <div class="title"><a href="https://doi.org/10.1146/annurev.matsci.32.101901.155803">Phase-field simulation of solidification</a></div> <div class="author"> W. J. Boettinger,&nbsp;J. A. Warren,&nbsp;C. Beckermann,&nbsp;and&nbsp;A. Karma </div> <div class="periodical"> <em>Annual Review of Materials Research</em>, 2002 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="flemings1974solidification" class="col-sm-8"> <div class="title"><a href="">Solidification processing</a></div> <div class="author"> M. C. Flemings </div> <div class="periodical"> <em>Metallurgical Transactions</em>, 1974 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="kuhn1986fundamentals" class="col-sm-8"> <div class="title"><a href="">Fundamentals of solidification</a></div> <div class="author"> G. Kühn,&nbsp;W. Kurz,&nbsp;and&nbsp;D. J. Fisher </div> <div class="periodical"> <em>Crystal Research and Technology</em>, 1986 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="gomez2017computational" class="col-sm-8"> <div class="title"><a href="">Computational Phase-Field Modeling</a></div> <div class="author"> H. Gomez,&nbsp;and&nbsp;K. G. Zee </div> <div class="periodical"> <em>In Encyclopedia of Computational Mechanics Second Edition</em> , 2017 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="gomez2019review" class="col-sm-8"> <div class="title"><a href="">A review on computational modelling of phase-transition problems</a></div> <div class="author"> H. Gomez,&nbsp;M. Bures,&nbsp;and&nbsp;A. Moure </div> <div class="periodical"> <em>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</em>, 2019 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li></ol>]]></content><author><name></name></author><category term="Projects"/><category term="Phase-field"/><category term="PDEs"/><category term="IGA"/><summary type="html"><![CDATA[we solve a phase-field model of dendritic soldification with surface tension anisotropy by means of isogeometric analysis.]]></summary></entry><entry><title type="html">Solving the Cahn-Hilliard equation on complex geometries</title><link href="https://mdeluci.github.io/blog/2024/CahnHilliard/" rel="alternate" type="text/html" title="Solving the Cahn-Hilliard equation on complex geometries"/><published>2024-06-19T06:52:00+00:00</published><updated>2024-06-19T06:52:00+00:00</updated><id>https://mdeluci.github.io/blog/2024/CahnHilliard</id><content type="html" xml:base="https://mdeluci.github.io/blog/2024/CahnHilliard/"><![CDATA[<h2 id="the-cahn-hilliard-equation">The Cahn-Hilliard equation</h2> <p>The Cahn–Hilliard equation describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. It was originally proposed in 1958 to model phase separation in binary alloys <a class="citation" href="#Cahn-Hilliard">(Cahn &amp; Hilliard, 1958; Cahn, 1959)</a>. Since then it has been used to describe various phenomena, such as spinodal decomposition <a class="citation" href="#MARALDI201231">(Maraldi et al., 2012)</a>, diblock copolymer <a class="citation" href="#Choksi">(Choksi et al., 2009)</a>, image inpainting, tumor growth simulation <a class="citation" href="#tumor">(Hilhorst et al., 2015)</a> or multiphase fluid flows <a class="citation" href="#BADALASSI2003371">(Badalassi et al., 2003)</a>.</p> <div class="container"> <div class="row justify-content-center mt-3"> <div class="col mt-3 mt-md-0 d-flex justify-content-center"> <div style="width: 90%;"> <figure> <video src="/assets/video/ch.mp4" class="img-fluid rounded z-depth-1" width="auto" height="auto" autoplay="" controls=""/> </figure> </div> </div> </div> <div class="caption text-center"> Solution of the Cahn-Hilliard equation on a quarter of anulus from a randomly perturbed initial condition. </div> </div> <p>There are two main approaches to describing phase transition phenomena: sharp-interface models and phase-field models. Sharp-interface models involve the resolution of a moving boundary problem, meaning that partial differential equations have to be solved for each phase. This can lead to physical <a class="citation" href="#Anderson">(Anderson et al., 1998)</a> and computational complications (missing reference), such as jump discontinuities across the interface. Phase-field models replace sharp-interfaces by thin transitions regions where the interfacial forces are smoothly distributed (diffuse interfaces). For this reason, phase-field models are also referred to as \emph{diffuse interface} models, and they are emerging as a promising tool to treat problems with interfaces.</p> <p>From the mathematical point of view, the Cahn-Hilliard equation is a stiff, fourth-order, nonlinear parabolic partial differential equation. The elementary form of the Cahn-Hilliard equation is given by</p> <p>\begin{equation} \frac{\partial \phi}{\partial t}=\nabla \cdot \left(m\left(\phi\right) \nabla \left(W’\left(\phi\right) -\epsilon ^2 \Delta \phi \right) \right) \label{element} \end{equation}</p> <p>where \(\phi\left(\mathbf{x},t\right) \in [-1,1]\) represents the phase-field variable, which describes the location of different components (e.g., \(\phi=-1\) in oil and \(\phi=1\) in water), \(m\left(\phi\right)\) is the mobility, \(W\left(\phi\right)\) is the so-called double-well potential, and \(\epsilon\) is a length scale related to the thickness of the diffuse interface.</p> <p>Two different approaches have been used to solve the Cahn-Hilliard equation: solving the original form given by Eqn.\eqref{element}, which involves fourth-order operators, or by splitting it into two second order partial differential equations by introducing an auxiliary variable — the chemical potential \(\mu\). The split version of the equation is given by</p> <p>\begin{equation} \frac{\partial \phi}{\partial t}=\nabla \cdot \left(m\left(\phi\right) \nabla \mu \right) \label{split1} \end{equation} \begin{equation} \mu=W’\left(\phi \right) - \epsilon^2 \Delta \phi. \label{chempot} \end{equation}</p> <p>The splitting strategy was proposed by Elliot <a class="citation" href="#Elliott1989ASO">(Elliott et al., 1989)</a> to avoid constraints related to the continuity of the basis functions when using the finite element method with a primal formulation. It also facilitates the imposition of the boundary conditions on a circular geometry, which is the case we are interested in. Unfortunately, this comes at a price of an additional degree of freedom introduced into the solver. Nevertheless, this reformulation reduces the complexity of the problem from the numerical point of view (see <a class="citation" href="#Zee">(van der Zee et al., 2011)</a>).</p> <p>The spatial discretization of the governing equations is performed using isogeometric analysis (IGA). IGA is a generalization of the finite-element method that uses non-uniform rational B-splines (NURBS) to define the discrete spaces. A significant advantage of IGA with respect to finite element analysis is that the basis functions can be constructed with controllable continuity across the element boundaries, even on mapped geometries. The higher-order global continuity of the basis functions has been shown to produce higher accuracy than classical \(\mathcal{C}^0\)-continuous finite elements for the same number of degrees of freedom; and has been widely used to solve partial differential equations with high-order spatial derivatives <a class="citation" href="#GOMEZ20084333">(Gomez et al., 2008; Gomez et al., 2010; Gomez &amp; Nogueira, 2012)</a>. Here, we will use \(\mathcal{C}^1\)-continuous quadratic splines for \(\phi\) and \(\mu\).</p> <h2 id="strong-form">Strong form</h2> <p>The strong form of the problem is formulated as follows: find \(\phi:\Omega \times \left[0,T\right]\rightarrow \mathbb{R}\) and \(\mu:\Omega \times \left[0,T\right]\rightarrow \mathbb{R}\) (being \(\mu\) an auxiliary variable) such that:</p> <p>\begin{equation} \frac{\partial \phi}{\partial t}=\nabla \cdot \left(m\left(\phi\right) \nabla \mu\right) \quad \text{in} \ \Omega \times \left[0,T\right], \label{st1} \end{equation} <br/> \begin{equation} \mu=W’\left(\phi\right)-\epsilon^2 \Delta \phi \quad \text{in} \ \Omega \times \left[0,T\right], \label{st2} \end{equation} \begin{equation} \phi=g \quad \text{on} \ \Gamma_g \times \left[0,T\right], \label{st3} \end{equation} \begin{equation} <br/> m\left(\phi\right)\epsilon^2\Delta\phi=q \quad \text{on} \ \Gamma_q \times \left[0,T\right], \label{st4} \end{equation} \begin{equation} \nabla\left(m\left(\phi\right)\mu\right)\cdot \mathbf{n}=s \quad \text{on} \ \Gamma_s \times \left[0,T\right], \label{st5} \end{equation} \begin{equation} \nabla\left(m\left(\phi\right)\phi\right)\cdot \mathbf{n}=h \quad \text{on} \ \Gamma_h \times \left[0,T\right], \label{st6}<br/> \end{equation} \begin{equation} \phi\left(\mathbf{x},0\right)=\phi_0\left(\mathbf{x}\right) \quad \text{in} \ \Omega. \label{st7}<br/> \end{equation}</p> <p>The double well function \(W\left(\phi\right)\) is defined such that it has two local minima, which makes possible the coexistence of the different phases. Some important examples of double well functions can be found in <a class="citation" href="#Gomez2">(Gomez &amp; van der Zee, 2017)</a>. In this work we will take the classical quartic potential</p> <p>\begin{equation} W\left(\phi\right)=\frac{1}{4}\left(1-\phi^2\right)^2, \end{equation}</p> <p>which is a qualitative approximation of the logarithmic double well.</p> <p>The mobility \(m\left(\phi\right)\), which can be linked with the diffusivity, controls the time scale of the problem. The most common approach is to set it as a constant, \(m\left(\phi\right)=m\). Such an approach is extensively applied in both analytical and numerical studies. We will take \(m\) constant and equal to one as recommended in <a class="citation" href="#GOMEZ20084333">(Gomez et al., 2008)</a>.</p> <p>The parameter \(\epsilon\) is a positive constant that defines the length scale for the problem. According to <a class="citation" href="#GOMEZ20084333">(Gomez et al., 2008)</a>, the solution to the Cahn-Hilliard equations depends only on the dimensionless number</p> <p>\begin{equation} \alpha=\frac{L_0^2}{3\epsilon^2}, \end{equation}</p> <p>also referred to as sharpness parameter, where \(L_0\) is a length scale of the problem. Here, we will assume that \(L_0\) is a measure of the computational domain size. Without loss of generality, \(L_0=R_2=1.0\) will be the outer radius of the quarter of annulus. The sharpness parameter \(\alpha\) usually takes values in the range \(10^3-10^4\). We will study the evolution of the phase-field variable for different values of \(\alpha\).</p> <h2 id="weak-form">Weak form</h2> <p>The strong form of our model defined by Eqns. \eqref{st1}–\eqref{st7} is now cast in weak form and discretized using the Galerkin approach. We will only work with free-flux boundary conditions. Thus, all boundary integrals will vanish in the weak form. Let us define the functional space \(\mathcal{V} \subset \mathcal{H}^1\), where \(\mathcal{H}^1\) is the Sobolev space of square-integrable functions with square-integrable first derivative in the domain \(\Omega\). To perform space discretization we introduce the finite-dimensional space \(\mathcal{V}^h\subset \mathcal{V}\), where \(\mathcal{V}^h ={\rm span}\left \{ N_A \right \}_{A=1,\dots,n_f}\) and \(n_f\) is the number of functions on the basis. The space of weighting functions will also be \(\mathcal{V}^h\), giving rise to a Galerkin formulation. We define discrete approximations to the problem’s solution denoted by \(\phi^h\) and \(\mu^h\). Their corresponding weighting functions are \(w_{\phi}^h\) and \(w_{\mu}^h\). Then, the variational formulation of Eqns. \eqref{st1}–\eqref{st7} over the finite-dimensional space \(\mathcal{V}^h\) can be stated as follows: find \(\phi^h\), \(\mu^h\) \(\in \mathcal{V}^h\subset \mathcal{V}\) such that \(\forall\) \(w_{\phi}^h\), \(w_{\mu}^h\) \(\in \mathcal{V}^h \subset \mathcal{V}\)</p> <p>\begin{equation} \int_{\Omega}w_{\phi}^h\frac{\partial\phi}{\partial t}d\Omega +\int_{\Omega}m\nabla w_{\phi}^h \cdot \nabla \mu^h d\Omega =0, \label{wf1} \end{equation} \begin{equation} \int_{\Omega}w_{\mu}^h \mu^h d\Omega - \int_{\Omega}w_{\mu}^h \phi^h \left( \left(\phi^h \right)^2-1 \right)d\Omega- \int_{\Omega} \epsilon^2 \nabla \phi^h \cdot \nabla w_{\mu}^h d\Omega= 0. \end{equation}</p> <p>The solutions \(\phi^h\) and \(\mu^h\) are defined as \begin{equation} \phi^h(\mathbf{x},t)=\sum_{A=1}^{n_f}\phi_A\left ( t \right )N_A\left ( \mathbf{x} \right ), \end{equation} \begin{equation} \mu^h(\mathbf{x},t)=\sum_{A=1}^{n_f}\mu_A\left ( t \right )N_A\left ( \mathbf{x} \right ). \end{equation}</p> <h2 id="results">Results</h2> <p>We consider the split form of the Cahn-Hilliard equation on an open quarter-annulus domain \begin{equation} \Omega = { (x,y) \in \mathbb{R}^2 : x &gt; 0, y &gt; 0, R_1^2 &lt; x^2 + y^2 &lt; R_2^2 }, \end{equation} with inner radius \(R_1=0.5\) and outer radius \(R_2=1\). We employ a computational mesh comprised of \(128^2\) \(\mathcal{C}^1\)-quadratic elements; see the figure below.</p> <div class="container"> <div class="row justify-content-center mt-3"> <div class="col mt-3 mt-md-0 d-flex justify-content-center"> <div style="width: 50%;"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/mesh-480.webp 480w,/assets/img/mesh-800.webp 800w,/assets/img/mesh-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/mesh.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> </div> </div> <div class="caption text-center"> Geometry definition and mesh used for the calculations. The circular inset shows a detail of the mesh. </div> </div> <p>The usual setup for numerical simulations of the Cahn-Hilliard equation is as follows: a randomly perturbed homogeneous solution is allowed to evolve in a sufficiently large computational domain with periodic boundary conditions. Periodic boundary conditions would make no sense in our case, since we are not considering a square domain. Instead, we prescribe free-flux boundary conditions on the entire domain for \(\phi\) and \(\mu\). The initial condition is given by \begin{equation} \phi_0\left(\mathbf{x}\right)=\bar{\phi}+r, \end{equation} where \(\bar{\phi}\) is a constant (referred to as the volume fraction) and \(r\) is a random variable with uniform distribution in \([-0.05,0.05]\).</p> <p>With this numerical setup we present two test cases which are defined by the sharpness parameter \(\alpha\) and the volume fraction \(\bar{\phi}\). We take \(\alpha=1500\), \(\bar{\phi}=0.3\) for the first case, and \(\alpha=3000\), \(\bar{\phi}=0.1\) for the second case.</p> <p>The left figure below shows snapshots of the solution at different times for \(\alpha=1500\) and \(\bar{\phi}=0.3\). We observe two main processes that take place at very different time scales: phase separation and coarsening. The Cahn-Hilliard equation is unstable under random perturbations within the spinodal region, and, thus, two phases will develop. Phase separation takes place at \(t\approx 0.01\) s. The concentration is driven to the binodal points in a very fast process. After the phase separation, the coarsening process starts. During the coarsening process, bubbles merge with each other to reduce the surface energy, acquiring a circular shape. This process finishes when there are only two regions occupied by the two components of the mixture and they achieve the equilibrium topology. This is occurs at the steady state (data not shown).</p> <blockquote> <p>Note: Spinodal decomposition occurs when one thermodynamic phase spontaneusly (i.e., without nucleation) separates into two phases. The spinodal region is the region of the phase diagram where this occurs.</p> </blockquote> <p>The right figure below shows snapshots of the solution at different times for \(\alpha=1500\) and \(\bar{\phi}=0.3\). This test case is more challenging than the previous one, since the parameter \(\alpha\) is larger, and, thus, the interfaces are thinner. The parameter \(\alpha\) not only affects the thickness of the interface but also modifies the time scale of the problem. The separation process is faster for \(\alpha=3000\) than for \(\alpha=1500\). Phase separation in this case takes place at the beginning of the simulation. On the other hand, the coarsening process is slower. The bubbles start acquiring the equilibrium topology, that is, a circular shape, later in time compared to the previous example. The initial volume fraction \(\bar{\phi}\) plays an important role in the topology of the solution. The topology of the solution for \(\bar{\phi}=0.1\) shows a more connected pattern than that of the solution for \(\bar{\phi}=0.3\). In contrast to the case defined by \(\bar{\phi}=0.3\), for \(\bar{\phi}=0.1\) there is no nucleation, and the coarsening process is more continuous over the simulation.</p> <div class="row mt-3"> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/fig2_ch-480.webp 480w,/assets/img/fig2_ch-800.webp 800w,/assets/img/fig2_ch-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/fig2_ch.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/fig3_ch-480.webp 480w,/assets/img/fig3_ch-800.webp 800w,/assets/img/fig3_ch-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/fig3_ch.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> </div> <div class="caption"> Evolution of the phase-field from a randomly perturbed initial condition. Left and right show the snapshots over time for different initial conditions. </div> <h2 id="conclusions">Conclusions</h2> <p>We presented a numerical methodology to solve the Cahn-Hilliard equation on a quarter-annulus domain. Our computational method is based on isogeometric analysis, which allows us to generate the \(\mathcal{C}^1\)-continuous functions required to solve this equation in a variational framework. We adopt the split form of the equation to reduce its order, and facilitate the imposition of the boundary conditions on the circular geometry. Time discretazion is performed using the generalized \(\alpha\)-methods, which provides second order accuracy and \(A\)-stability. We analyzed the solution over time for different values of \(\alpha\) and \(\bar{\phi}\). The parameter \(\alpha\) affects the thickness of the interface and the time scale of the problem. As we increased \(\alpha\), the separation process was faster, and the thickness of the interface became thinner. The initial volume fraction \(\bar{\phi}\) modifies the topology of the solution. Smaller values of \(\bar{\phi}\) produce more continuous patterns without nucleation.</p> <h2 id="references">References</h2> <ol class="bibliography"><li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Cahn-Hilliard" class="col-sm-8"> <div class="title"><a href="">Free Energy of a Nonuniform System. I. Interfacial Free Energy</a></div> <div class="author"> John W. Cahn,&nbsp;and&nbsp;John E. Hilliard </div> <div class="periodical"> <em>The Journal of Chemical Physics</em>, 1958 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Cahn-Hilliard2" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1063/1.1730145 ">Free Energy of a Nonuniform System. II. Thermodynamic Basis</a></div> <div class="author"> John W. Cahn </div> <div class="periodical"> <em>The Journal of Chemical Physics</em>, 1959 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="MARALDI201231" class="col-sm-8"> <div class="title"><a href="http://www.sciencedirect.com/science/article/pii/S0020722511001832">A unified thermodynamic framework for the modelling of diffusive and displacive phase transitions</a></div> <div class="author"> M. Maraldi,&nbsp;L. Molari,&nbsp;and&nbsp;D. Grandi </div> <div class="periodical"> <em>International Journal of Engineering Science</em>, 2012 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Choksi" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1137/080728809 ">On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn–Hilliard Functional</a></div> <div class="author"> Rustum Choksi,&nbsp;Mark A. Peletier,&nbsp;and&nbsp;J. F. Williams </div> <div class="periodical"> <em>SIAM Journal on Applied Mathematics</em>, 2009 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="tumor" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1142/S0218202515500268 ">Formal asymptotic limit of a diffuse-interface tumor-growth model</a></div> <div class="author"> Danielle Hilhorst,&nbsp;Johannes Kampmann,&nbsp;Thanh Nam Nguyen,&nbsp;and&nbsp;Kristoffer George Van Der Zee </div> <div class="periodical"> <em>Mathematical Models and Methods in Applied Sciences</em>, 2015 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="BADALASSI2003371" class="col-sm-8"> <div class="title"><a href="http://www.sciencedirect.com/science/article/pii/S0021999103002808">Computation of multiphase systems with phase field models</a></div> <div class="author"> V.E. Badalassi,&nbsp;H.D. Ceniceros,&nbsp;and&nbsp;S. Banerjee </div> <div class="periodical"> <em>Journal of Computational Physics</em>, 2003 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Anderson" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1146/annurev.fluid.30.1.139 ">DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS</a></div> <div class="author"> D. M. Anderson,&nbsp;G. B. McFadden,&nbsp;and&nbsp;A. A. Wheeler </div> <div class="periodical"> <em>Annual Review of Fluid Mechanics</em>, 1998 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Elliott1989ASO" class="col-sm-8"> <div class="title"><a href="">A second order splitting method for the Cahn-Hilliard equation</a></div> <div class="author"> C. M. Elliott,&nbsp;D. French,&nbsp;and&nbsp;F. Milner </div> <div class="periodical"> <em>Numerische Mathematik</em>, 1989 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Zee" class="col-sm-8"> <div class="title"><a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/num.20638">Goal-oriented error estimation for Cahn–Hilliard models of binary phase transition</a></div> <div class="author"> Kristoffer G. Zee,&nbsp;J. Tinsley Oden,&nbsp;Serge Prudhomme,&nbsp;and&nbsp;Andrea Hawkins-Daarud </div> <div class="periodical"> <em>Numerical Methods for Partial Differential Equations</em>, 2011 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="GOMEZ20084333" class="col-sm-8"> <div class="title"><a href="">Isogeometric analysis of the Cahn–Hilliard phase-field model</a></div> <div class="author"> Hector Gomez,&nbsp;Victor M. Calo,&nbsp;Yuri Bazilevs,&nbsp;and&nbsp;Thomas J.R. Hughes </div> <div class="periodical"> <em>Computer Methods in Applied Mechanics and Engineering</em>, 2008 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="GOMEZ20101828" class="col-sm-8"> <div class="title"><a href="">Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations</a></div> <div class="author"> Hector Gomez,&nbsp;Thomas J.R. Hughes,&nbsp;Xesús Nogueira,&nbsp;and&nbsp;Victor M. Calo </div> <div class="periodical"> <em>Computer Methods in Applied Mechanics and Engineering</em>, 2010 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="GOMEZ201252" class="col-sm-8"> <div class="title"><a href="">An unconditionally energy-stable method for the phase field crystal equation</a></div> <div class="author"> Hector Gomez,&nbsp;and&nbsp;Xesús Nogueira </div> <div class="periodical"> <em>Computer Methods in Applied Mechanics and Engineering</em>, 2012 </div> <div class="periodical"> Higher Order Finite Element and Isogeometric Methods </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Gomez2" class="col-sm-8"> <div class="title"><a href="">Computational Phase-Field Modeling</a></div> <div class="author"> Hector Gomez,&nbsp;and&nbsp;Kristoffer G. Zee </div> <div class="periodical"> 2017 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li></ol>]]></content><author><name></name></author><category term="Projects"/><category term="Cahn-Hilliard"/><category term="PDEs"/><category term="IGA"/><summary type="html"><![CDATA[We solve the Cahn-Hilliard equation on a quarter of annulus by means of isogeometric analysis. We adopt the split form of the equation to avoid introducing fourth-order operators and facilitate the imposition of the boundary conditions on the circular geometry,]]></summary></entry></feed>
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+<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en"><generator uri="https://jekyllrb.com/" version="4.3.3">Jekyll</generator><link href="https://mdeluci.github.io/feed.xml" rel="self" type="application/atom+xml"/><link href="https://mdeluci.github.io/" rel="alternate" type="text/html" hreflang="en"/><updated>2024-08-20T22:51:46+00:00</updated><id>https://mdeluci.github.io/feed.xml</id><title type="html">Mario de Lucio’s website</title><subtitle>Mario de Lucio&apos;s website </subtitle><entry><title type="html">Phase-field Modeling of Dendritic Solidification</title><link href="https://mdeluci.github.io/blog/2024/Solidification/" rel="alternate" type="text/html" title="Phase-field Modeling of Dendritic Solidification"/><published>2024-06-20T06:52:00+00:00</published><updated>2024-06-20T06:52:00+00:00</updated><id>https://mdeluci.github.io/blog/2024/Solidification</id><content type="html" xml:base="https://mdeluci.github.io/blog/2024/Solidification/"><![CDATA[<h2 id="phase-field-equation-for-solidification-processes">Phase-field equation for solidification processes</h2> <p>Liquid-solid phase transformations usually refer to solidification and melting. The formation of complex microstructures during the solidification from a liquid phase, such as formation of snow flakes or metallic alloys, and the accompanying diffusion and convection processes in the liquid and solid have been widely studied in the literature <a class="citation" href="#boettinger2002phase">(Boettinger et al., 2002; Flemings, 1974; Kühn et al., 1986)</a>. In particular, the evolution of microstructural scale of dendrites during the solidification process determines many physical and mechanical properties of metals, since almost every metallic system originates from the liquid state.</p> <div class="container"> <div class="row justify-content-center mt-3"> <div class="col mt-3 mt-md-0 d-flex justify-content-center"> <div style="width: 90%;"> <figure> <video src="/assets/video/solidification.mp4" class="img-fluid rounded z-depth-1" width="auto" height="auto" autoplay="" controls=""/> </figure> </div> </div> </div> <div class="caption text-center"> Crystal growth from a square-shaped seed located in an undercooled region. Phase-field (top row). Temperature (bottom row) </div> </div> <p>A phase-field formulation that describes the solidification process is as follows: \begin{equation} \frac{\partial \phi}{\partial t} = \Delta \phi - \frac{W’(\phi)}{\epsilon^2} + \frac{\rho H}{\sqrt{2\sigma}} \frac{\theta - \theta_m}{\theta_m} G’(\phi) \label{1} \end{equation} \begin{equation} \rho C_v \frac{\partial \theta}{\partial t} + L \phi’ \frac{\partial \phi}{\partial t} = \nabla \cdot (k(\phi) \nabla \theta) \label{2} \end{equation}</p> <p>where \(\phi \in [-1,1]\) represents the phase-field variable, which describes the location of different components (e.g., \(\phi = -1\) in the solid and \(\phi = 1\) in the liquid), \(\theta\) is the temperature, \(\theta_m\) is the melting temperature, \(H\) represents the interfacial enthalpy per unit mass, \(\omega\) is the kinetic undercooling coefficient, \(C_v\) is the heat capacity per unit mass, \(\rho\) is the density, \(l\) is specific latent heat (energy per unit mass), \(k\) is the thermal conductivity, \(\sigma\) is the surface tension and \(h\) is an interpolatory function that verifies \(h(+1) = 1, h(-1) = 0\), e.g., \(h = \frac{1}{2} (1 + \phi)\). The thermal conductivity is considered to be a function of the phase-field to account for different conductivity in the solid and liquid phases. We take \(k(\phi) = (1 + \phi) k_s + (1 - \phi) k_l\), which satisfies that \(k(+1) = k_s\), \(k(-1) = k_l\). The function \(W(\phi)\), also referred to as the double-well potential, is defined such that it has two local minima, which makes possible the coexistence of the different phases. Some important examples of double well functions can be found in <a class="citation" href="#gomez2017computational">(Gomez &amp; van der Zee, 2017)</a>. In this work we will take the classical quartic potential \(W(\phi) = \frac{1}{4} (1 - \phi^2)^2\). The function \(G(\phi)\) vanishes in the pure phases. Depending on the form of the functional \(G(\phi)\), the phase-field will converge faster or slower to the generalized Stefan problem. Here we will use the expression \(G(\phi) = (1 - \phi^2)^2\). One key aspect to achieve good agreement with the reality of interest behind dendritic solidification is surface tension anisotropy. Herein, we introduce anisotropy by assuming that \(\sigma\) depends on the unit normal to the liquid-solid interface, this is,</p> <p>\begin{equation} \sigma = \sigma_0(1 + \delta \cos(\alpha - \alpha_0)), \end{equation}</p> <p>where \(\bar{\sigma}\) is the mean value of \(\sigma\), \(\delta\) is the strenght of the anisotropy, \(q\) is the mode number of anisotropy, and \(\alpha\) is the initial offset angle. The angle of the normal to the surface, \(\alpha\), is defined from the phase-field as</p> <p>\begin{equation} \alpha = \arctan\left(\frac{\partial \phi / \partial y}{\partial \phi/ \partial x}\right). \end{equation}</p> <h2 id="weak-form">Weak form</h2> <p>The strong form of our model defined by Eqs. \eqref{1} and Eqn.\eqref{2} is now cast in weak form and discretized using the Galerkin approach. Let us define the functional space \(V \subset H^1\), where \(H^1\) is the Sobolev space of square-integrable functions with square-integrable first derivative in the domain \(\Omega\). To perform space discretization we introduce the finite-dimensional space \(V_h \subset V\), where \(V_h = \text{span} \{ N_A \}_{A=1}^{n_f}\), and \(n_f\) is the number of functions on the basis. The space of weighting functions will also be \(V_h\), giving rise to a Galerkin formulation. We define discrete approximations to the problem’s solution denoted by \(\phi^h\) and \(\theta^h\). Their corresponding weighting functions are \(w_{\phi}^h\) and \(w_{\theta}^h\). Then, the variational formulation of Eqs. \eqref{1} and Eqn.\eqref{2} over the finite-dimensional space \(V_h\) can be stated as follows: find \(U_h = \{\phi^h, \theta^h\} \in V_h \subset V\)</p> <p>such that \(\forall W_h = \{w_{\phi}^h, w_{\theta}^h\} \in V_h \subset V\), \begin{equation} \int_{\Omega} w_{\phi}^h \frac{\partial \phi^h}{\partial t} \, d\Omega + \int_{\Omega} w_{\phi}^h \frac{W’(\phi^h)}{\epsilon^2} \, d\Omega + \int_{\Omega} \nabla w_{\phi}^h \cdot \nabla \phi^h \, d\Omega + \int_{\Omega} w_{\phi}^h \frac{\rho H}{\sqrt{2 \sigma}} \left( \frac{G’(\phi^h) (\theta^h - \theta_m)}{\theta_m} \right) \, d\Omega =0, \end{equation}</p> <p>\begin{equation} \int_{\Omega} w_{\theta}^h \rho C_v \frac{\partial \theta^h}{\partial t} \, d\Omega - \int_{\Omega} w_{\theta}^h L \phi’(\phi^h) \frac{\partial \phi^h}{\partial t} \, d\Omega + \int_{\Omega} \nabla w_{\theta}^h \cdot k(\phi^h) \nabla \theta^h \, d\Omega=0. \end{equation}</p> <p>The solutions \(\phi^h\) and \(\theta^h\) are defined as \begin{equation} \phi^h(x,t) = \sum_{A=1}^{n_f} \phi_A(t) N_A(x), \end{equation} \begin{equation} \theta^h(x,t) = \sum_{A=1}^{n_f} \theta_A(t) N_A(x). \end{equation}</p> <h2 id="results">Results</h2> <p>Our computational domain is the square \(\Omega= \left[0,0.1 \right] \times \left[0,0.1 \right]\), where the units are in cm. We employ a computational mesh comprised of $512^2$ \(\mathcal{C}^1\)-quadratic elements. We impose periodic boundary conditions for \(\phi\) and \(\theta\). We choose a suitable time step of \(\Delta t = 10e-6\) s to ensure good numerical stability, specially at the beginning of the simulation. As an initial condition, we set a square-shaped seed located at the center of the domain. The reason for using a square-shaped crystal, instead of one with a smoother shape like a circle, is because the corners of the square will accelerate the formation of dendritic structures. The temperatures inside and outside the seed are 1700 K and 1500 K, respectively. The parameters used in all the simulations are summarized in <a class="citation" href="#gomez2019review">(Gomez et al., 2019)</a>. These parameters correspond to an Nickel-Copper alloy. In the following simulations, we will show the relations between the anisotropy modes and the shapes of crystals.</p> <p>The Figures below show the evolution of the phase-field variable and the temperature field over time for different values of \(q\). For all simulations nucleation starts at the center of the domain, and it triggers the solidification process. The phase change is achieved by undercooling the liquid phase below its melting temperature.</p> <p>In the first computation, perfect isotropic growth is considered, i.e. \(\sigma=\bar{\sigma}\) throughout the entire simulation. For this case we obtain symmetric dendritic patterns similar to those formed during viscous fingering. Tip splitting is observed as the crystal grows, producing new branches with tips that mantain a circular shape. For the anisotropic case with \(q = 4\), we see features of both the isotropic case and the dendritic structure. The dendritic structure is present in the formation of the principal branches, which are four in this case due to the anisotropy mode chosen. The isotropic features are observed in the viscous finger-like structures that emanate from the principal branches. The overall look of the pattern is symmetric, even though we are considering surface tension anisotropy. When we increase the anisotropy mode number to \(q = 6\), we get a completely asymmetric pattern. We can observe six main branches and four secondary branches. For \(q = 8\) we get the typical snowflake-like pattern with a non-symmetric side branching. In general, dendrites whose preferred direction is inclined with respect to the thermal gradient grow faster in order to maintain their relative tip positions during the splitting. These tips of the inclined dendrites lie slightly behind those of well aligned dendrites. We can say that there is a growth competition among the different dendrites, which results in a natural selection of the grains that grow with a smaller angle with respect to the temperature</p> <div class="row mt-3"> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/sol_1-480.webp 480w,/assets/img/sol_1-800.webp 800w,/assets/img/sol_1-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/sol_1.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/sol_2-480.webp 480w,/assets/img/sol_2-800.webp 800w,/assets/img/sol_2-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/sol_2.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> </div> <div class="caption"> Crystal growth from a square-shaped seed located in an undercooled region. Here we show the time evolution of dendritic solidification for different mode numbers of anisotropy. Time evolution of the temperature field from the initial square-shaped seed for different mode numbers of anisotropy. </div> <h2 id="conclusions">Conclusions</h2> <p>We presented a numerical methodology to solve a phase-field model of dendritic solidification with surface tension anisotropy on a two-dimensional domain. Our computational method is based on isogeometric analysis, which allows us to generate the \(\mathcal{C}^1\)-continuous functions required to solve this equation in a variational framework. Time discretazion is performed using the generalized \(\alpha\)-method, which provides second order accuracy and \(A\)-stability. We analyzed the evolution of the phase-field variable and the temperature field over time for different anisotropy mode numbers. Our results show that symmetric dendritic finger patterns can form in the presence of anisotropy.</p> <h2 id="references">References</h2> <ol class="bibliography"><li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="boettinger2002phase" class="col-sm-8"> <div class="title"><a href="https://doi.org/10.1146/annurev.matsci.32.101901.155803">Phase-field simulation of solidification</a></div> <div class="author"> W. J. Boettinger,&nbsp;J. A. Warren,&nbsp;C. Beckermann,&nbsp;and&nbsp;A. Karma </div> <div class="periodical"> <em>Annual Review of Materials Research</em>, 2002 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="flemings1974solidification" class="col-sm-8"> <div class="title"><a href="">Solidification processing</a></div> <div class="author"> M. C. Flemings </div> <div class="periodical"> <em>Metallurgical Transactions</em>, 1974 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="kuhn1986fundamentals" class="col-sm-8"> <div class="title"><a href="">Fundamentals of solidification</a></div> <div class="author"> G. Kühn,&nbsp;W. Kurz,&nbsp;and&nbsp;D. J. Fisher </div> <div class="periodical"> <em>Crystal Research and Technology</em>, 1986 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="gomez2017computational" class="col-sm-8"> <div class="title"><a href="">Computational Phase-Field Modeling</a></div> <div class="author"> H. Gomez,&nbsp;and&nbsp;K. G. Zee </div> <div class="periodical"> <em>In Encyclopedia of Computational Mechanics Second Edition</em> , 2017 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="gomez2019review" class="col-sm-8"> <div class="title"><a href="">A review on computational modelling of phase-transition problems</a></div> <div class="author"> H. Gomez,&nbsp;M. Bures,&nbsp;and&nbsp;A. Moure </div> <div class="periodical"> <em>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</em>, 2019 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li></ol>]]></content><author><name></name></author><category term="Projects"/><category term="Phase-field"/><category term="PDEs"/><category term="IGA"/><summary type="html"><![CDATA[we solve a phase-field model of dendritic soldification with surface tension anisotropy by means of isogeometric analysis.]]></summary></entry><entry><title type="html">Solving the Cahn-Hilliard equation on complex geometries</title><link href="https://mdeluci.github.io/blog/2024/CahnHilliard/" rel="alternate" type="text/html" title="Solving the Cahn-Hilliard equation on complex geometries"/><published>2024-06-19T06:52:00+00:00</published><updated>2024-06-19T06:52:00+00:00</updated><id>https://mdeluci.github.io/blog/2024/CahnHilliard</id><content type="html" xml:base="https://mdeluci.github.io/blog/2024/CahnHilliard/"><![CDATA[<h2 id="the-cahn-hilliard-equation">The Cahn-Hilliard equation</h2> <p>The Cahn–Hilliard equation describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. It was originally proposed in 1958 to model phase separation in binary alloys <a class="citation" href="#Cahn-Hilliard">(Cahn &amp; Hilliard, 1958; Cahn, 1959)</a>. Since then it has been used to describe various phenomena, such as spinodal decomposition <a class="citation" href="#MARALDI201231">(Maraldi et al., 2012)</a>, diblock copolymer <a class="citation" href="#Choksi">(Choksi et al., 2009)</a>, image inpainting, tumor growth simulation <a class="citation" href="#tumor">(Hilhorst et al., 2015)</a> or multiphase fluid flows <a class="citation" href="#BADALASSI2003371">(Badalassi et al., 2003)</a>.</p> <div class="container"> <div class="row justify-content-center mt-3"> <div class="col mt-3 mt-md-0 d-flex justify-content-center"> <div style="width: 90%;"> <figure> <video src="/assets/video/ch.mp4" class="img-fluid rounded z-depth-1" width="auto" height="auto" autoplay="" controls=""/> </figure> </div> </div> </div> <div class="caption text-center"> Solution of the Cahn-Hilliard equation on a quarter of anulus from a randomly perturbed initial condition. </div> </div> <p>There are two main approaches to describing phase transition phenomena: sharp-interface models and phase-field models. Sharp-interface models involve the resolution of a moving boundary problem, meaning that partial differential equations have to be solved for each phase. This can lead to physical <a class="citation" href="#Anderson">(Anderson et al., 1998)</a> and computational complications (missing reference), such as jump discontinuities across the interface. Phase-field models replace sharp-interfaces by thin transitions regions where the interfacial forces are smoothly distributed (diffuse interfaces). For this reason, phase-field models are also referred to as \emph{diffuse interface} models, and they are emerging as a promising tool to treat problems with interfaces.</p> <p>From the mathematical point of view, the Cahn-Hilliard equation is a stiff, fourth-order, nonlinear parabolic partial differential equation. The elementary form of the Cahn-Hilliard equation is given by</p> <p>\begin{equation} \frac{\partial \phi}{\partial t}=\nabla \cdot \left(m\left(\phi\right) \nabla \left(W’\left(\phi\right) -\epsilon ^2 \Delta \phi \right) \right) \label{element} \end{equation}</p> <p>where \(\phi\left(\mathbf{x},t\right) \in [-1,1]\) represents the phase-field variable, which describes the location of different components (e.g., \(\phi=-1\) in oil and \(\phi=1\) in water), \(m\left(\phi\right)\) is the mobility, \(W\left(\phi\right)\) is the so-called double-well potential, and \(\epsilon\) is a length scale related to the thickness of the diffuse interface.</p> <p>Two different approaches have been used to solve the Cahn-Hilliard equation: solving the original form given by Eqn.\eqref{element}, which involves fourth-order operators, or by splitting it into two second order partial differential equations by introducing an auxiliary variable — the chemical potential \(\mu\). The split version of the equation is given by</p> <p>\begin{equation} \frac{\partial \phi}{\partial t}=\nabla \cdot \left(m\left(\phi\right) \nabla \mu \right) \label{split1} \end{equation} \begin{equation} \mu=W’\left(\phi \right) - \epsilon^2 \Delta \phi. \label{chempot} \end{equation}</p> <p>The splitting strategy was proposed by Elliot <a class="citation" href="#Elliott1989ASO">(Elliott et al., 1989)</a> to avoid constraints related to the continuity of the basis functions when using the finite element method with a primal formulation. It also facilitates the imposition of the boundary conditions on a circular geometry, which is the case we are interested in. Unfortunately, this comes at a price of an additional degree of freedom introduced into the solver. Nevertheless, this reformulation reduces the complexity of the problem from the numerical point of view (see <a class="citation" href="#Zee">(van der Zee et al., 2011)</a>).</p> <p>The spatial discretization of the governing equations is performed using isogeometric analysis (IGA). IGA is a generalization of the finite-element method that uses non-uniform rational B-splines (NURBS) to define the discrete spaces. A significant advantage of IGA with respect to finite element analysis is that the basis functions can be constructed with controllable continuity across the element boundaries, even on mapped geometries. The higher-order global continuity of the basis functions has been shown to produce higher accuracy than classical \(\mathcal{C}^0\)-continuous finite elements for the same number of degrees of freedom; and has been widely used to solve partial differential equations with high-order spatial derivatives <a class="citation" href="#GOMEZ20084333">(Gomez et al., 2008; Gomez et al., 2010; Gomez &amp; Nogueira, 2012)</a>. Here, we will use \(\mathcal{C}^1\)-continuous quadratic splines for \(\phi\) and \(\mu\).</p> <h2 id="strong-form">Strong form</h2> <p>The strong form of the problem is formulated as follows: find \(\phi:\Omega \times \left[0,T\right]\rightarrow \mathbb{R}\) and \(\mu:\Omega \times \left[0,T\right]\rightarrow \mathbb{R}\) (being \(\mu\) an auxiliary variable) such that:</p> <p>\begin{equation} \frac{\partial \phi}{\partial t}=\nabla \cdot \left(m\left(\phi\right) \nabla \mu\right) \quad \text{in} \ \Omega \times \left[0,T\right], \label{st1} \end{equation} <br/> \begin{equation} \mu=W’\left(\phi\right)-\epsilon^2 \Delta \phi \quad \text{in} \ \Omega \times \left[0,T\right], \label{st2} \end{equation} \begin{equation} \phi=g \quad \text{on} \ \Gamma_g \times \left[0,T\right], \label{st3} \end{equation} \begin{equation} <br/> m\left(\phi\right)\epsilon^2\Delta\phi=q \quad \text{on} \ \Gamma_q \times \left[0,T\right], \label{st4} \end{equation} \begin{equation} \nabla\left(m\left(\phi\right)\mu\right)\cdot \mathbf{n}=s \quad \text{on} \ \Gamma_s \times \left[0,T\right], \label{st5} \end{equation} \begin{equation} \nabla\left(m\left(\phi\right)\phi\right)\cdot \mathbf{n}=h \quad \text{on} \ \Gamma_h \times \left[0,T\right], \label{st6}<br/> \end{equation} \begin{equation} \phi\left(\mathbf{x},0\right)=\phi_0\left(\mathbf{x}\right) \quad \text{in} \ \Omega. \label{st7}<br/> \end{equation}</p> <p>The double well function \(W\left(\phi\right)\) is defined such that it has two local minima, which makes possible the coexistence of the different phases. Some important examples of double well functions can be found in <a class="citation" href="#Gomez2">(Gomez &amp; van der Zee, 2017)</a>. In this work we will take the classical quartic potential</p> <p>\begin{equation} W\left(\phi\right)=\frac{1}{4}\left(1-\phi^2\right)^2, \end{equation}</p> <p>which is a qualitative approximation of the logarithmic double well.</p> <p>The mobility \(m\left(\phi\right)\), which can be linked with the diffusivity, controls the time scale of the problem. The most common approach is to set it as a constant, \(m\left(\phi\right)=m\). Such an approach is extensively applied in both analytical and numerical studies. We will take \(m\) constant and equal to one as recommended in <a class="citation" href="#GOMEZ20084333">(Gomez et al., 2008)</a>.</p> <p>The parameter \(\epsilon\) is a positive constant that defines the length scale for the problem. According to <a class="citation" href="#GOMEZ20084333">(Gomez et al., 2008)</a>, the solution to the Cahn-Hilliard equations depends only on the dimensionless number</p> <p>\begin{equation} \alpha=\frac{L_0^2}{3\epsilon^2}, \end{equation}</p> <p>also referred to as sharpness parameter, where \(L_0\) is a length scale of the problem. Here, we will assume that \(L_0\) is a measure of the computational domain size. Without loss of generality, \(L_0=R_2=1.0\) will be the outer radius of the quarter of annulus. The sharpness parameter \(\alpha\) usually takes values in the range \(10^3-10^4\). We will study the evolution of the phase-field variable for different values of \(\alpha\).</p> <h2 id="weak-form">Weak form</h2> <p>The strong form of our model defined by Eqns. \eqref{st1}–\eqref{st7} is now cast in weak form and discretized using the Galerkin approach. We will only work with free-flux boundary conditions. Thus, all boundary integrals will vanish in the weak form. Let us define the functional space \(\mathcal{V} \subset \mathcal{H}^1\), where \(\mathcal{H}^1\) is the Sobolev space of square-integrable functions with square-integrable first derivative in the domain \(\Omega\). To perform space discretization we introduce the finite-dimensional space \(\mathcal{V}^h\subset \mathcal{V}\), where \(\mathcal{V}^h ={\rm span}\left \{ N_A \right \}_{A=1,\dots,n_f}\) and \(n_f\) is the number of functions on the basis. The space of weighting functions will also be \(\mathcal{V}^h\), giving rise to a Galerkin formulation. We define discrete approximations to the problem’s solution denoted by \(\phi^h\) and \(\mu^h\). Their corresponding weighting functions are \(w_{\phi}^h\) and \(w_{\mu}^h\). Then, the variational formulation of Eqns. \eqref{st1}–\eqref{st7} over the finite-dimensional space \(\mathcal{V}^h\) can be stated as follows: find \(\phi^h\), \(\mu^h\) \(\in \mathcal{V}^h\subset \mathcal{V}\) such that \(\forall\) \(w_{\phi}^h\), \(w_{\mu}^h\) \(\in \mathcal{V}^h \subset \mathcal{V}\)</p> <p>\begin{equation} \int_{\Omega}w_{\phi}^h\frac{\partial\phi}{\partial t}d\Omega +\int_{\Omega}m\nabla w_{\phi}^h \cdot \nabla \mu^h d\Omega =0, \label{wf1} \end{equation} \begin{equation} \int_{\Omega}w_{\mu}^h \mu^h d\Omega - \int_{\Omega}w_{\mu}^h \phi^h \left( \left(\phi^h \right)^2-1 \right)d\Omega- \int_{\Omega} \epsilon^2 \nabla \phi^h \cdot \nabla w_{\mu}^h d\Omega= 0. \end{equation}</p> <p>The solutions \(\phi^h\) and \(\mu^h\) are defined as \begin{equation} \phi^h(\mathbf{x},t)=\sum_{A=1}^{n_f}\phi_A\left ( t \right )N_A\left ( \mathbf{x} \right ), \end{equation} \begin{equation} \mu^h(\mathbf{x},t)=\sum_{A=1}^{n_f}\mu_A\left ( t \right )N_A\left ( \mathbf{x} \right ). \end{equation}</p> <h2 id="results">Results</h2> <p>We consider the split form of the Cahn-Hilliard equation on an open quarter-annulus domain \begin{equation} \Omega = { (x,y) \in \mathbb{R}^2 : x &gt; 0, y &gt; 0, R_1^2 &lt; x^2 + y^2 &lt; R_2^2 }, \end{equation} with inner radius \(R_1=0.5\) and outer radius \(R_2=1\). We employ a computational mesh comprised of \(128^2\) \(\mathcal{C}^1\)-quadratic elements; see the figure below.</p> <div class="container"> <div class="row justify-content-center mt-3"> <div class="col mt-3 mt-md-0 d-flex justify-content-center"> <div style="width: 50%;"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/mesh-480.webp 480w,/assets/img/mesh-800.webp 800w,/assets/img/mesh-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/mesh.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> </div> </div> <div class="caption text-center"> Geometry definition and mesh used for the calculations. The circular inset shows a detail of the mesh. </div> </div> <p>The usual setup for numerical simulations of the Cahn-Hilliard equation is as follows: a randomly perturbed homogeneous solution is allowed to evolve in a sufficiently large computational domain with periodic boundary conditions. Periodic boundary conditions would make no sense in our case, since we are not considering a square domain. Instead, we prescribe free-flux boundary conditions on the entire domain for \(\phi\) and \(\mu\). The initial condition is given by \begin{equation} \phi_0\left(\mathbf{x}\right)=\bar{\phi}+r, \end{equation} where \(\bar{\phi}\) is a constant (referred to as the volume fraction) and \(r\) is a random variable with uniform distribution in \([-0.05,0.05]\).</p> <p>With this numerical setup we present two test cases which are defined by the sharpness parameter \(\alpha\) and the volume fraction \(\bar{\phi}\). We take \(\alpha=1500\), \(\bar{\phi}=0.3\) for the first case, and \(\alpha=3000\), \(\bar{\phi}=0.1\) for the second case.</p> <p>The left figure below shows snapshots of the solution at different times for \(\alpha=1500\) and \(\bar{\phi}=0.3\). We observe two main processes that take place at very different time scales: phase separation and coarsening. The Cahn-Hilliard equation is unstable under random perturbations within the spinodal region, and, thus, two phases will develop. Phase separation takes place at \(t\approx 0.01\) s. The concentration is driven to the binodal points in a very fast process. After the phase separation, the coarsening process starts. During the coarsening process, bubbles merge with each other to reduce the surface energy, acquiring a circular shape. This process finishes when there are only two regions occupied by the two components of the mixture and they achieve the equilibrium topology. This is occurs at the steady state (data not shown).</p> <blockquote> <p>Note: Spinodal decomposition occurs when one thermodynamic phase spontaneusly (i.e., without nucleation) separates into two phases. The spinodal region is the region of the phase diagram where this occurs.</p> </blockquote> <p>The right figure below shows snapshots of the solution at different times for \(\alpha=1500\) and \(\bar{\phi}=0.3\). This test case is more challenging than the previous one, since the parameter \(\alpha\) is larger, and, thus, the interfaces are thinner. The parameter \(\alpha\) not only affects the thickness of the interface but also modifies the time scale of the problem. The separation process is faster for \(\alpha=3000\) than for \(\alpha=1500\). Phase separation in this case takes place at the beginning of the simulation. On the other hand, the coarsening process is slower. The bubbles start acquiring the equilibrium topology, that is, a circular shape, later in time compared to the previous example. The initial volume fraction \(\bar{\phi}\) plays an important role in the topology of the solution. The topology of the solution for \(\bar{\phi}=0.1\) shows a more connected pattern than that of the solution for \(\bar{\phi}=0.3\). In contrast to the case defined by \(\bar{\phi}=0.3\), for \(\bar{\phi}=0.1\) there is no nucleation, and the coarsening process is more continuous over the simulation.</p> <div class="row mt-3"> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/fig2_ch-480.webp 480w,/assets/img/fig2_ch-800.webp 800w,/assets/img/fig2_ch-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/fig2_ch.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> <div class="col-sm mt-3 mt-md-0"> <figure> <picture> <source class="responsive-img-srcset" srcset="/assets/img/fig3_ch-480.webp 480w,/assets/img/fig3_ch-800.webp 800w,/assets/img/fig3_ch-1400.webp 1400w," sizes="95vw" type="image/webp"/> <img src="/assets/img/fig3_ch.jpg" class="img-fluid rounded z-depth-1" width="100%" height="auto" loading="eager" onerror="this.onerror=null; $('.responsive-img-srcset').remove();"/> </picture> </figure> </div> </div> <div class="caption"> Evolution of the phase-field from a randomly perturbed initial condition. Left and right show the snapshots over time for different initial conditions. </div> <h2 id="conclusions">Conclusions</h2> <p>We presented a numerical methodology to solve the Cahn-Hilliard equation on a quarter-annulus domain. Our computational method is based on isogeometric analysis, which allows us to generate the \(\mathcal{C}^1\)-continuous functions required to solve this equation in a variational framework. We adopt the split form of the equation to reduce its order, and facilitate the imposition of the boundary conditions on the circular geometry. Time discretazion is performed using the generalized \(\alpha\)-methods, which provides second order accuracy and \(A\)-stability. We analyzed the solution over time for different values of \(\alpha\) and \(\bar{\phi}\). The parameter \(\alpha\) affects the thickness of the interface and the time scale of the problem. As we increased \(\alpha\), the separation process was faster, and the thickness of the interface became thinner. The initial volume fraction \(\bar{\phi}\) modifies the topology of the solution. Smaller values of \(\bar{\phi}\) produce more continuous patterns without nucleation.</p> <h2 id="references">References</h2> <ol class="bibliography"><li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Cahn-Hilliard" class="col-sm-8"> <div class="title"><a href="">Free Energy of a Nonuniform System. I. Interfacial Free Energy</a></div> <div class="author"> John W. Cahn,&nbsp;and&nbsp;John E. Hilliard </div> <div class="periodical"> <em>The Journal of Chemical Physics</em>, 1958 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Cahn-Hilliard2" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1063/1.1730145 ">Free Energy of a Nonuniform System. II. Thermodynamic Basis</a></div> <div class="author"> John W. Cahn </div> <div class="periodical"> <em>The Journal of Chemical Physics</em>, 1959 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="MARALDI201231" class="col-sm-8"> <div class="title"><a href="http://www.sciencedirect.com/science/article/pii/S0020722511001832">A unified thermodynamic framework for the modelling of diffusive and displacive phase transitions</a></div> <div class="author"> M. Maraldi,&nbsp;L. Molari,&nbsp;and&nbsp;D. Grandi </div> <div class="periodical"> <em>International Journal of Engineering Science</em>, 2012 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Choksi" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1137/080728809 ">On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn–Hilliard Functional</a></div> <div class="author"> Rustum Choksi,&nbsp;Mark A. Peletier,&nbsp;and&nbsp;J. F. Williams </div> <div class="periodical"> <em>SIAM Journal on Applied Mathematics</em>, 2009 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="tumor" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1142/S0218202515500268 ">Formal asymptotic limit of a diffuse-interface tumor-growth model</a></div> <div class="author"> Danielle Hilhorst,&nbsp;Johannes Kampmann,&nbsp;Thanh Nam Nguyen,&nbsp;and&nbsp;Kristoffer George Van Der Zee </div> <div class="periodical"> <em>Mathematical Models and Methods in Applied Sciences</em>, 2015 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="BADALASSI2003371" class="col-sm-8"> <div class="title"><a href="http://www.sciencedirect.com/science/article/pii/S0021999103002808">Computation of multiphase systems with phase field models</a></div> <div class="author"> V.E. Badalassi,&nbsp;H.D. Ceniceros,&nbsp;and&nbsp;S. Banerjee </div> <div class="periodical"> <em>Journal of Computational Physics</em>, 2003 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Anderson" class="col-sm-8"> <div class="title"><a href=" https://doi.org/10.1146/annurev.fluid.30.1.139 ">DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS</a></div> <div class="author"> D. M. Anderson,&nbsp;G. B. McFadden,&nbsp;and&nbsp;A. A. Wheeler </div> <div class="periodical"> <em>Annual Review of Fluid Mechanics</em>, 1998 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Elliott1989ASO" class="col-sm-8"> <div class="title"><a href="">A second order splitting method for the Cahn-Hilliard equation</a></div> <div class="author"> C. M. Elliott,&nbsp;D. French,&nbsp;and&nbsp;F. Milner </div> <div class="periodical"> <em>Numerische Mathematik</em>, 1989 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Zee" class="col-sm-8"> <div class="title"><a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/num.20638">Goal-oriented error estimation for Cahn–Hilliard models of binary phase transition</a></div> <div class="author"> Kristoffer G. Zee,&nbsp;J. Tinsley Oden,&nbsp;Serge Prudhomme,&nbsp;and&nbsp;Andrea Hawkins-Daarud </div> <div class="periodical"> <em>Numerical Methods for Partial Differential Equations</em>, 2011 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="GOMEZ20084333" class="col-sm-8"> <div class="title"><a href="">Isogeometric analysis of the Cahn–Hilliard phase-field model</a></div> <div class="author"> Hector Gomez,&nbsp;Victor M. Calo,&nbsp;Yuri Bazilevs,&nbsp;and&nbsp;Thomas J.R. Hughes </div> <div class="periodical"> <em>Computer Methods in Applied Mechanics and Engineering</em>, 2008 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="GOMEZ20101828" class="col-sm-8"> <div class="title"><a href="">Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations</a></div> <div class="author"> Hector Gomez,&nbsp;Thomas J.R. Hughes,&nbsp;Xesús Nogueira,&nbsp;and&nbsp;Victor M. Calo </div> <div class="periodical"> <em>Computer Methods in Applied Mechanics and Engineering</em>, 2010 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="GOMEZ201252" class="col-sm-8"> <div class="title"><a href="">An unconditionally energy-stable method for the phase field crystal equation</a></div> <div class="author"> Hector Gomez,&nbsp;and&nbsp;Xesús Nogueira </div> <div class="periodical"> <em>Computer Methods in Applied Mechanics and Engineering</em>, 2012 </div> <div class="periodical"> Higher Order Finite Element and Isogeometric Methods </div> <div class="links"> </div> </div> </div> </li> <li><div class="row"> <div class="col col-sm-2 abbr"> </div> <div id="Gomez2" class="col-sm-8"> <div class="title"><a href="">Computational Phase-Field Modeling</a></div> <div class="author"> Hector Gomez,&nbsp;and&nbsp;Kristoffer G. Zee </div> <div class="periodical"> 2017 </div> <div class="periodical"> </div> <div class="links"> </div> </div> </div> </li></ol>]]></content><author><name></name></author><category term="Projects"/><category term="Cahn-Hilliard"/><category term="PDEs"/><category term="IGA"/><summary type="html"><![CDATA[We solve the Cahn-Hilliard equation on a quarter of annulus by means of isogeometric analysis. We adopt the split form of the equation to avoid introducing fourth-order operators and facilitate the imposition of the boundary conditions on the circular geometry,]]></summary></entry></feed>
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