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 .jp-AddAboveIcon{background-image:var(--jp-icon-add-above)}.jp-AddBelowIcon{background-image:var(--jp-icon-add-below)}.jp-AddIcon{background-image:var(--jp-icon-add)}.jp-BellIcon{background-image:var(--jp-icon-bell)}.jp-BugDotIcon{background-image:var(--jp-icon-bug-dot)}.jp-BugIcon{background-image:var(--jp-icon-bug)}.jp-BuildIcon{background-image:var(--jp-icon-build)}.jp-CaretDownEmptyIcon{background-image:var(--jp-icon-caret-down-empty)}.jp-CaretDownEmptyThinIcon{background-image:var(--jp-icon-caret-down-empty-thin)}.jp-CaretDownIcon{background-image:var(--jp-icon-caret-down)}.jp-CaretLeftIcon{background-image:var(--jp-icon-caret-left)}.jp-CaretRightIcon{background-image:var(--jp-icon-caret-right)}.jp-CaretUpEmptyThinIcon{background-image:var(--jp-icon-caret-up-empty-thin)}.jp-CaretUpIcon{background-image:var(--jp-icon-caret-up)}.jp-CaseSensitiveIcon{background-image:var(--jp-icon-case-sensitive)}.jp-CheckIcon{background-image:var(--jp-icon-check)}.jp-CircleEmptyIcon{background-image:var(--jp-icon-circle-empty)}.jp-CircleIcon{background-image:var(--jp-icon-circle)}.jp-ClearIcon{background-image:var(--jp-icon-clear)}.jp-CloseIcon{background-image:var(--jp-icon-close)}.jp-CodeCheckIcon{background-image:var(--jp-icon-code-check)}.jp-CodeIcon{background-image:var(--jp-icon-code)}.jp-CollapseAllIcon{background-image:var(--jp-icon-collapse-all)}.jp-ConsoleIcon{background-image:var(--jp-icon-console)}.jp-CopyIcon{background-image:var(--jp-icon-copy)}.jp-CopyrightIcon{background-image:var(--jp-icon-copyright)}.jp-CutIcon{background-image:var(--jp-icon-cut)}.jp-DeleteIcon{background-image:var(--jp-icon-delete)}.jp-DownloadIcon{background-image:var(--jp-icon-download)}.jp-DuplicateIcon{background-image:var(--jp-icon-duplicate)}.jp-EditIcon{background-image:var(--jp-icon-edit)}.jp-EllipsesIcon{background-image:var(--jp-icon-ellipses)}.jp-ErrorIcon{background-image:var(--jp-icon-error)}.jp-ExpandAllIcon{background-image:var(--jp-icon-expand-all)}.jp-ExtensionIcon{background-image:var(--jp-icon-extension)}.jp-FastForwardIcon{background-image:var(--jp-icon-fast-forward)}.jp-FileIcon{background-image:var(--jp-icon-file)}.jp-FileUploadIcon{background-image:var(--jp-icon-file-upload)}.jp-FilterDotIcon{background-image:var(--jp-icon-filter-dot)}.jp-FilterIcon{background-image:var(--jp-icon-filter)}.jp-FilterListIcon{background-image:var(--jp-icon-filter-list)}.jp-FolderFavoriteIcon{background-image:var(--jp-icon-folder-favorite)}.jp-FolderIcon{background-image:var(--jp-icon-folder)}.jp-HomeIcon{background-image:var(--jp-icon-home)}.jp-Html5Icon{background-image:var(--jp-icon-html5)}.jp-ImageIcon{background-image:var(--jp-icon-image)}.jp-InfoIcon{background-image:var(--jp-icon-info)}.jp-InspectorIcon{background-image:var(--jp-icon-inspector)}.jp-JsonIcon{background-image:var(--jp-icon-json)}.jp-JuliaIcon{background-image:var(--jp-icon-julia)}.jp-JupyterFaviconIcon{background-image:var(--jp-icon-jupyter-favicon)}.jp-JupyterIcon{background-image:var(--jp-icon-jupyter)}.jp-JupyterlabWordmarkIcon{background-image:var(--jp-icon-jupyterlab-wordmark)}.jp-KernelIcon{background-image:var(--jp-icon-kernel)}.jp-KeyboardIcon{background-image:var(--jp-icon-keyboard)}.jp-LaunchIcon{background-image:var(--jp-icon-launch)}.jp-LauncherIcon{background-image:var(--jp-icon-launcher)}.jp-LineFormIcon{background-image:var(--jp-icon-line-form)}.jp-LinkIcon{background-image:var(--jp-icon-link)}.jp-ListIcon{background-image:var(--jp-icon-list)}.jp-MarkdownIcon{background-image:var(--jp-icon-markdown)}.jp-MoveDownIcon{background-image:var(--jp-icon-move-down)}.jp-MoveUpIcon{background-image:var(--jp-icon-move-up)}.jp-NewFolderIcon{background-image:var(--jp-icon-new-folder)}.jp-NotTrustedIcon{background-image:var(--jp-icon-not-trusted)}.jp-NotebookIcon{background-image:var(--jp-icon-notebook)}.jp-NumberingIcon{background-image:var(--jp-icon-numbering)}.jp-OfflineBoltIcon{background-image:var(--jp-icon-offline-bolt)}.jp-PaletteIcon{background-image:var(--jp-icon-palette)}.jp-PasteIcon{background-image:var(--jp-icon-paste)}.jp-PdfIcon{background-image:var(--jp-icon-pdf)}.jp-PythonIcon{background-image:var(--jp-icon-python)}.jp-RKernelIcon{background-image:var(--jp-icon-r-kernel)}.jp-ReactIcon{background-image:var(--jp-icon-react)}.jp-RedoIcon{background-image:var(--jp-icon-redo)}.jp-RefreshIcon{background-image:var(--jp-icon-refresh)}.jp-RegexIcon{background-image:var(--jp-icon-regex)}.jp-RunIcon{background-image:var(--jp-icon-run)}.jp-RunningIcon{background-image:var(--jp-icon-running)}.jp-SaveIcon{background-image:var(--jp-icon-save)}.jp-SearchIcon{background-image:var(--jp-icon-search)}.jp-SettingsIcon{background-image:var(--jp-icon-settings)}.jp-ShareIcon{background-image:var(--jp-icon-share)}.jp-SpreadsheetIcon{background-image:var(--jp-icon-spreadsheet)}.jp-StopIcon{background-image:var(--jp-icon-stop)}.jp-TabIcon{background-image:var(--jp-icon-tab)}.jp-TableRowsIcon{background-image:var(--jp-icon-table-rows)}.jp-TagIcon{background-image:var(--j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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 4536511f..1f637b93 100644
--- a/feed.xml
+++ b/feed.xml
@@ -1,4 +1,4 @@
-<?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>2024-06-19T18:55:56+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">Solving the Cahn-Hilliard equation on complex geometries</title><link href="https://mdeluci.github.io/blog/2024/math/" 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/math</id><content type="html" xml:base="https://mdeluci.github.io/blog/2024/math/"><![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 \cite{Cahn-Hilliard, Cahn-Hilliard2}. Since then it has been used to describe various phenomena, such as spinodal decomposition \cite{MARALDI201231}, diblock copolymer \cite{Choksi,JEONG20141263}, image inpainting \cite{4032803}, tumor growth simulation \cite{tumor,WISE2008524} or multiphase fluid flows \cite{BADALASSI2003371, Kotschote}.</p> <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 \cite{Anderson} and computational complications \cite{Barrett}, 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 \cite{Elliott1989ASO} 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 \cite{M2AN_2009__43_5_1003_0,Zee}).</p> <p>The spatial discretization of the governing equations is performed using isogeometric analysis (IGA) \cite{HUGHES20054135}. 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 \cite{akkerman2008role,evans2009n}; and has been widely used to solve partial differential equations with high-order spatial derivatives \cite{GOMEZ20084333, GOMEZ20101828, GOMEZ201252}. 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 \cite{Gomez2}. 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 \cite{Rajagopal,polymer,CuetoFelgueroso2008ATF}. We will take \(m\) constant and equal to one as recommended in \cite{GOMEZ20084333}.</p> <p>The parameter \(\epsilon\) is a positive constant that defines the length scale for the problem. According to \cite{GOMEZ20084333}, 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 \(10^3-10^4\) \cite{GOMEZ20084333,Gomez2}. We will study the evolution of the phase-field variable for different values of \(\alpha\).</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.png" 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.png" 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 for $$\alpha=1500$$ and $$\bar{\phi}=0.3$$ (left). Evolution of the phase-field from a randomly perturbed initial condition for $$\alpha=3000$$ and $$\bar{\phi}=0.1$$ (right). </div>]]></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><entry><title type="html">displaying beautiful tables with Bootstrap Tables</title><link href="https://mdeluci.github.io/blog/2023/tables/" rel="alternate" type="text/html" title="displaying beautiful tables with Bootstrap Tables"/><published>2023-03-20T18:37:00+00:00</published><updated>2023-03-20T18:37:00+00:00</updated><id>https://mdeluci.github.io/blog/2023/tables</id><content type="html" xml:base="https://mdeluci.github.io/blog/2023/tables/"><![CDATA[<p>Using markdown to display tables is easy. Just use the following syntax:</p> <div class="language-markdown highlighter-rouge"><div class="highlight"><pre class="highlight"><code>| Left aligned | Center aligned | Right aligned |
+<?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>2024-06-19T18:56:03+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">Solving the Cahn-Hilliard equation on complex geometries</title><link href="https://mdeluci.github.io/blog/2024/math/" 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/math</id><content type="html" xml:base="https://mdeluci.github.io/blog/2024/math/"><![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 \cite{Cahn-Hilliard, Cahn-Hilliard2}. Since then it has been used to describe various phenomena, such as spinodal decomposition \cite{MARALDI201231}, diblock copolymer \cite{Choksi,JEONG20141263}, image inpainting \cite{4032803}, tumor growth simulation \cite{tumor,WISE2008524} or multiphase fluid flows \cite{BADALASSI2003371, Kotschote}.</p> <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 \cite{Anderson} and computational complications \cite{Barrett}, 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 \cite{Elliott1989ASO} 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 \cite{M2AN_2009__43_5_1003_0,Zee}).</p> <p>The spatial discretization of the governing equations is performed using isogeometric analysis (IGA) \cite{HUGHES20054135}. 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 \cite{akkerman2008role,evans2009n}; and has been widely used to solve partial differential equations with high-order spatial derivatives \cite{GOMEZ20084333, GOMEZ20101828, GOMEZ201252}. 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 \cite{Gomez2}. 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 \cite{Rajagopal,polymer,CuetoFelgueroso2008ATF}. We will take \(m\) constant and equal to one as recommended in \cite{GOMEZ20084333}.</p> <p>The parameter \(\epsilon\) is a positive constant that defines the length scale for the problem. According to \cite{GOMEZ20084333}, 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 \(10^3-10^4\) \cite{GOMEZ20084333,Gomez2}. We will study the evolution of the phase-field variable for different values of \(\alpha\).</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.png" 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.png" 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 for $$\alpha=1500$$ and $$\bar{\phi}=0.3$$ (left). Evolution of the phase-field from a randomly perturbed initial condition for $$\alpha=3000$$ and $$\bar{\phi}=0.1$$ (right). </div>]]></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><entry><title type="html">displaying beautiful tables with Bootstrap Tables</title><link href="https://mdeluci.github.io/blog/2023/tables/" rel="alternate" type="text/html" title="displaying beautiful tables with Bootstrap Tables"/><published>2023-03-20T18:37:00+00:00</published><updated>2023-03-20T18:37:00+00:00</updated><id>https://mdeluci.github.io/blog/2023/tables</id><content type="html" xml:base="https://mdeluci.github.io/blog/2023/tables/"><![CDATA[<p>Using markdown to display tables is easy. Just use the following syntax:</p> <div class="language-markdown highlighter-rouge"><div class="highlight"><pre class="highlight"><code>| Left aligned | Center aligned | Right aligned |
 | :----------- | :------------: | ------------: |
 | Left 1       |    center 1    |       right 1 |
 | Left 2       |    center 2    |       right 2 |
diff --git a/sitemap.xml b/sitemap.xml
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