Merge pull request #1522 from wd15/fix-bm5-bm6-links

fix bm5 and bm6 link to benchmark paper

benchmarks/benchmark5-hackathon.ipynb

@@ -149,10 +149,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "See the Overleaf document entitled [\"Phase Field Benchmark Problems for Dendritic Growth and Linear Elasticity\"][overleaf] for more details about the benchmark problems. Furthermore, read [the extended essay][benchmarks] for a discussion about the need for benchmark problems.\n",
+    "See the publicaton entitled [\"Phase field benchmark problems targeting fluid flow and electrochemistry\"][benchmark-paper] for more details about this benchmark problem. Furthermore, read [the extended essay][benchmarks] for a discussion about the need for benchmark problems.\n",
     "\n",
     "[benchmarks]: ../ \n",
-    "[overleaf]: https://www.overleaf.com/read/nqjkdwyybvdz"
+    "[benchmark-paper]: https://doi.org/10.1016/j.commatsci.2020.109548"
    ]
   },
   {

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@@ -168,9 +173,11 @@ <h1 id="Benchmark-Problem-5:-Stokes-Flow">Benchmark Problem 5: Stokes Flow<a cla
 
 
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@@ -181,38 +188,37 @@ <h1 id="Benchmark-Problem-5:-Stokes-Flow">Benchmark Problem 5: Stokes Flow<a cla
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-<p>See the Overleaf document entitled <a href="https://www.overleaf.com/read/nqjkdwyybvdz">"Phase Field Benchmark Problems for Dendritic Growth and Linear Elasticity"</a> for more details about the benchmark problems. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
+<p>See the publicaton entitled <a href="https://doi.org/10.1016/j.commatsci.2020.109548">"Phase field benchmark problems targeting fluid flow and electrochemistry"</a> for more details about this benchmark problem. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
 
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-<h1 id="Overview">Overview<a class="anchor-link" href="#Overview">&#182;</a></h1><p>Flow of a liquid can be incorporated into phase field models, so we present this benchmark problem for incompressible fluid flow through a channel (the flow of many liquids can be modeled as incompressible). The flow of a fluid can generally be modeled via the Navier-Stokes equations.  When length scales are small, fluid velocities are low, and/or viscosity is large, such that the Reynolds number $Re<<1$, inertial forces are small compared with viscous forces, resulting in a simplification of Navier-Stokes flow to Stokes flow.</p>
+<h1 id="Overview">Overview<a class="anchor-link" href="#Overview">&#182;</a></h1><p>Flow of a liquid can be incorporated into phase field models, so we present this benchmark problem for incompressible fluid flow through a channel (the flow of many liquids can be modeled as incompressible). The flow of a fluid can generally be modeled via the Navier-Stokes equations.  When length scales are small, fluid velocities are low, and/or viscosity is large, such that the Reynolds number $Re&lt;&lt;1$, inertial forces are small compared with viscous forces, resulting in a simplification of Navier-Stokes flow to Stokes flow.</p>
 
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 <h1 id="Model-Formulation">Model Formulation<a class="anchor-link" href="#Model-Formulation">&#182;</a></h1><h2 id="Governing-Equations">Governing Equations<a class="anchor-link" href="#Governing-Equations">&#182;</a></h2><p>In this problem, two variables are used: the fluid velocity, $\textbf{u}$, which is a vector field, and the fluid pressure, $p$, which is a scalar field.  The Stokes momentum equation is given as</p>
-\begin{equation}
+<p>\begin{equation}
 -\mu \nabla^{2} \textbf{u} + \nabla p - \rho \textbf{g} = 0,
-\end{equation}<p>where $\rho$ is the density, assumed constant in this problem, $\mu$ is the dynamic viscosity, and $\textbf{g}$ is the acceleration due to gravity.  To fully describe fluid flow, the momentum balance equation is supplemented with the continuity equation for mass flow,</p>
-\begin{equation}
+\end{equation}</p>
+<p>where $\rho$ is the density, assumed constant in this problem, $\mu$ is the dynamic viscosity, and $\textbf{g}$ is the acceleration due to gravity.  To fully describe fluid flow, the momentum balance equation is supplemented with the continuity equation for mass flow,</p>
+<p>\begin{equation}
 \frac{d\rho}{dt}+\nabla\cdot\left(\rho{\mathbf u}\right)=0;
-\end{equation}<p>this simplifies to 
+\end{equation}</p>
+<p>this simplifies to 
 \begin{equation}
 \nabla \cdot \textbf{u} = 0
 \end{equation}</p>
@@ -223,21 +229,20 @@ <h3 id="Figure-1:-Domain-for-variation-(b)">Figure 1: Domain for variation (b)<a
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 <h2 id="Boundary-Conditions">Boundary Conditions<a class="anchor-link" href="#Boundary-Conditions">&#182;</a></h2><p>All solid surfaces, including the boundary for the obstruction, have no-slip boundary conditions, that is, $u_x=u_y=0$.  The inlet velocity, on the left boundary, follows a parabolic profile described by</p>
-\begin{equation}
+<p>\begin{equation}
 u_x(0,y) = -0.001(y-3)^2+0.009.
-\end{equation}<p>The outlet velocity (on the right boundary) is left to the solver to determine, as is the pressure over the entire domain.  However, we specify that the pressure at point (30, 6) is zero.  Finally, the obstruction is described by an ellipse centered at (7, 2.5).  The semi-major axis (in the <em>y</em>-direction) is $a=1.5$, and the semi-minor axis (in the <em>x</em>-direction) is $b=1$.</p>
+\end{equation}</p>
+<p>The outlet velocity (on the right boundary) is left to the solver to determine, as is the pressure over the entire domain.  However, we specify that the pressure at point (30, 6) is zero.  Finally, the obstruction is described by an ellipse centered at (7, 2.5).  The semi-major axis (in the <em>y</em>-direction) is $a=1.5$, and the semi-minor axis (in the <em>x</em>-direction) is $b=1$.</p>
 
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benchmarks/benchmark6-hackathon.ipynb

@@ -149,10 +149,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "See the Overleaf document entitled [\"Phase Field Benchmark Problems for Dendritic Growth and Linear Elasticity\"][overleaf] for more details about the benchmark problems. Furthermore, read [the extended essay][benchmarks] for a discussion about the need for benchmark problems.\n",
+    "See the publicaton entitled [\"Phase field benchmark problems targeting fluid flow and electrochemistry\"][benchmark-paper] for more details about this benchmark problem. Furthermore, read [the extended essay][benchmarks] for a discussion about the need for benchmark problems.\n",
     "\n",
     "[benchmarks]: ../ \n",
-    "[overleaf]: https://www.overleaf.com/read/nqjkdwyybvdz"
+    "[benchmark-paper]: https://doi.org/10.1016/j.commatsci.2020.109548"
    ]
   },
   {

benchmarks/benchmark6-hackathon.ipynb.raw.html

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@@ -181,52 +188,57 @@ <h1 id="Benchmark-Problem-6:-Electrostatics">Benchmark Problem 6: Electrostatics
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-<p>See the Overleaf document entitled <a href="https://www.overleaf.com/read/nqjkdwyybvdz">"Phase Field Benchmark Problems for Dendritic Growth and Linear Elasticity"</a> for more details about the benchmark problems. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
+<p>See the publicaton entitled <a href="https://doi.org/10.1016/j.commatsci.2020.109548">"Phase field benchmark problems targeting fluid flow and electrochemistry"</a> for more details about this benchmark problem. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
 
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 <h1 id="Overview">Overview<a class="anchor-link" href="#Overview">&#182;</a></h1><p>Diffusion of a charged species is often modeled with the phase field approach, such as for  batteries, electrodeposition, and electromigration.  This benchmark problem incorporates the first benchmark problem for spinodal decomposition <a href="https://doi.org/10.1016/j.commatsci.2016.09.022">Jokisaari 2017</a> and extends it to incorporate coupling with electrostatics.</p>
 <h2 id="Governing-Equations">Governing Equations<a class="anchor-link" href="#Governing-Equations">&#182;</a></h2><p>In this problem, two variables are used: $c$, the concentration field of our charged species, and $\Phi$, the electric potential field.  The free energy of the system is given as</p>
-\begin{equation}
+<p>\begin{equation}
 F=\int_{V} \left[\frac{\kappa}{2} |\nabla c|^2+f_{chem}(c)+f_{elec}(c,\Phi)\right]\,dV,
-\end{equation}<p>where $\kappa$ is the gradient energy coefficient, $f_{chem}$ is the chemical free energy, and $f_{elec}$ is the electrostatic coupling energy.  Here, $f_{chem}$ is a symmetric double-well function with minima between $0<c<1$,</p>
-\begin{equation}
+\end{equation}</p>
+<p>where $\kappa$ is the gradient energy coefficient, $f_{chem}$ is the chemical free energy, and $f_{elec}$ is the electrostatic coupling energy.  Here, $f_{chem}$ is a symmetric double-well function with minima between $0&lt;c&lt;1$,</p>
+<p>\begin{equation}
 f_{chem} = \rho (c-c^{\alpha})^2(c^{\beta}-c)^2, 
-\end{equation}<p>where $\rho$ controls the height of the double-well and $c^{\alpha}$ and $c^{\beta}$ are the compositions at which the energy is minimum.  In addition, the electrostatic coupling energy is given as</p>
-\begin{equation}
+\end{equation}</p>
+<p>where $\rho$ controls the height of the double-well and $c^{\alpha}$ and $c^{\beta}$ are the compositions at which the energy is minimum.  In addition, the electrostatic coupling energy is given as</p>
+<p>\begin{equation}
 f_{elec}=\frac{k\, c \, \Phi}{2},
-\end{equation}<p>where $k$ is a constant.</p>
+\end{equation}</p>
+<p>where $k$ is a constant.</p>
 <p>The time evolution of the system is described by the Cahn-Hilliard equation with the additional constraint that the Poisson equation must be satisfied at every point in the system at each time step (<a href="https://doi.org/10.1103/PhysRevE.69.021603">Guyer 2004a</a>, <a href="https://doi.org/10.1103/PhysRevE.69.021604">Guyer 2004b</a>).  To avoid a lengthy derivation, we simply present the equations below and suggest the readers study the given references.  The Cahn-Hillard equation is given as</p>
-\begin{equation}
+<p>\begin{equation}
 \frac{\partial c}{\partial t} = \nabla \cdot \left\lbrace M \nabla \left( 2 \rho(c-c^{\alpha})(c^{\beta}-c)(c^{\alpha}+c^{\beta}-2c) -\kappa \nabla^2 c + k \, \Phi \right) \right\rbrace, 
-\end{equation}<p>where $M$ is the mobility. The dynamics of electric relaxation occur at a much faster time scale than the diffusion of the charged species.  In this case, the Poisson equation is solved at each time step,</p>
-\begin{equation}
+\end{equation}</p>
+<p>where $M$ is the mobility. The dynamics of electric relaxation occur at a much faster time scale than the diffusion of the charged species.  In this case, the Poisson equation is solved at each time step,</p>
+<p>\begin{equation}
 \nabla^2 \Phi = \frac{-k\, c}{\epsilon}, 
-\end{equation}<p>where $\epsilon$ is the permittivity.</p>
+\end{equation}</p>
+<p>where $\epsilon$ is the permittivity.</p>
 <p>As in the <a href="{{ site.baseurl }}/benchmarks/benchmark1.ipynb">spinodal benchmark problem</a>, $c^{\alpha}=0.3$, $c^{\beta}=0.7$, $\kappa=2$, $\rho=5$, and $M=5$.  In addition, $k=0.09$ and $\epsilon=90$.</p>
 <h2 id="Domain">Domain<a class="anchor-link" href="#Domain">&#182;</a></h2><p>In this problem, the system consists of a domain which is grounded on one side and with a voltage applied on the other.  Two different 2D computational domains are given, shown below (the boundary conditions for $\Phi$ are schematically illustrated).  The first consists of a square domain, while the second consists of a half-circle with a radius of 50 units and centered at (50, 50), which is attached to a rectangle of 50 units wide by 100 units tall.  The curved boundary is the right-hand boundary, while the straight domain edges from (0, 0) to (50, 0) and (0, 100) to (50, 100) are the bottom and top boundaries, respectively. Schematic illustrations of the 2D computational domains and boundary conditions of $\Phi$ (indicated by hatching) for the electrochemical benchmark problem are shown in Figures 1 and 2.</p>
 <h3 id="Figure-1:-Domain-(a)">Figure 1: Domain (a)<a class="anchor-link" href="#Figure-1:-Domain-(a)">&#182;</a></h3><p><img src="../../images/electrochem_domain.png" alt="domaina"></p>
 <h3 id="Figure-2:-Domain-(b)">Figure 2: Domain (b)<a class="anchor-link" href="#Figure-2:-Domain-(b)">&#182;</a></h3><p><img src="../../images/electrochem_domain_curved2.png" alt="domainb"></p>
 <h2 id="Boundary-Conditions">Boundary Conditions<a class="anchor-link" href="#Boundary-Conditions">&#182;</a></h2><p>We impose no-flux boundary conditions on all boundaries, while $\Phi=0$ for the left boundary, $\nabla \Phi \cdot \hat{\textbf{n}}=0$ for the top and bottom boundaries,  and the right boundary is given by</p>
-\begin{equation}
+<p>\begin{equation}
 \Phi(x|_{boundary},y) = \sin(y/7),
-\end{equation}<p>where $x|_{boundary}=100$ for domain (a), and varies from 50 to 100 in domain (b).
+\end{equation}</p>
+<p>where $x|_{boundary}=100$ for domain (a), and varies from 50 to 100 in domain (b).
 The initial condition for $c$ is specified by</p>
-\begin{eqnarray}
-c\left(x,y\right) & = & c_{0}+c_{1}\left\{\cos\left(0.2x\right)\cos\left(0.11y\right)+\left[\cos\left(0.13x\right)\cos\left(0.087y\right)\right]^{2}\right. \\
- &  & \left.+\cos\left(0.025x-0.15y\right)\cos\left(0.07x-0.02y\right)\right\},
-\end{eqnarray}<p>where $c_{0}=0.5$ and $c_{1}=0.04$ (you may recognize this as the initial conditions for the <a href="{{ site.baseurl }}/benchmarks/benchmark1.ipynb">spinodal decomposition benchmark</a> problem with a slightly different parameterization).</p>
+<p>\begin{eqnarray}
+c\left(x,y\right) &amp; = &amp; c_{0}+c_{1}\left\{\cos\left(0.2x\right)\cos\left(0.11y\right)+\left[\cos\left(0.13x\right)\cos\left(0.087y\right)\right]^{2}\right. \\
+ &amp;  &amp; \left.+\cos\left(0.025x-0.15y\right)\cos\left(0.07x-0.02y\right)\right\},
+\end{eqnarray}</p>
+<p>where $c_{0}=0.5$ and $c_{1}=0.04$ (you may recognize this as the initial conditions for the <a href="{{ site.baseurl }}/benchmarks/benchmark1.ipynb">spinodal decomposition benchmark</a> problem with a slightly different parameterization).</p>
 <h1 id="Submission-Guidelines">Submission Guidelines<a class="anchor-link" href="#Submission-Guidelines">&#182;</a></h1><p>Both variations (a) and (b) should be run for 400 time units. The following data should be collected from each simulation,</p>
 <ul>
 <li><p>the free energy integrated across the entire domain, $F$, collected as frequently as is feasible.</p>
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